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- Permanent Link:
- http://ufdc.ufl.edu/UFE0021970/00001
## Material Information- Title:
- Aeroelastic Analysis and Optimization of Membrane Micro Air Vehicle Wings
- Creator:
- Stanford, Bret Kennedy
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2008
- Language:
- english
- Physical Description:
- 1 online resource (179 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Aerospace Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- Ifju, Peter
- Committee Members:
- Lind, Richard C.
Haftka, Raphael T. Bloomquist, David G. Albertani, Roberto - Graduation Date:
- 5/1/2008
## Subjects- Subjects / Keywords:
- Aerodynamics ( jstor )
Aircraft wings ( jstor ) Carbon fibers ( jstor ) Design optimization ( jstor ) Laminates ( jstor ) Leading edges ( jstor ) Rigid wings ( jstor ) Skin ( jstor ) Topology ( jstor ) Trailing edges ( jstor ) Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF aeroelasticity, membrane - Genre:
- Electronic Thesis or Dissertation
bibliography ( marcgt ) theses ( marcgt ) Aerospace Engineering thesis, Ph.D.
## Notes- Abstract:
- Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separation erodes wing efficiency, the wings are susceptible to rolling instabilities, wind gusts can be the same size as the flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divided into two categories: load-alleviating or load-augmenting. This depends on the wing?s topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms (cambering, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical designed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2008.
- Local:
- Adviser: Ifju, Peter.
- Statement of Responsibility:
- by Bret Kennedy Stanford
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Bret Kennedy Stanford. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 7/11/2008
- Classification:
- LD1780 2008 ( lcc )
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augmentation designs are represented by a baseline PR wing and the topology optimized for maximum lift. Lift-alleviation designs are represented by a baseline BR wing and the topology optimized for minimum lift slope. CL = 0.604 CL = 0.639 CL = 0.675 CL = 0.595 CL = 0.570 w/c -0.01 0 0.01 0 0.01 0.02 0.03 ACp : : -1 -0.5 0 0.5 1 1.5 2 Figure 7-10. Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom) for baseline and optimal topology designs, a = 12, reflex wing. The differential pressure distribution over the rigid wing is largely similar to that computed with the Navier-Stokes solver in Figure 5-18 and Figure 5-19: leading edge suction due to flow stagnation, pressure recovery (and peak lift) over the camber, and negative forces over the reflex portion of the wing. As expected, the inviscid solver misses the low-pressure cells at the wingtip (from the vortex swirling system [3]), and the plateau in the pressure distribution, indicative of a separation bubble [27]. This aerodynamic loading causes a moderate wash-in of the carbon fiber wing (0.1), resulting in a computed lift coefficient of 0.604. Computed deformation of the PR wing is likewise similar to that found above (Figure 5-5), though the deformations are smaller, within the range of validity of the linear finite element solver. The sudden changes in wing geometry at the membrane/carbon fiber interfaces lead to P ~\ Similar data is given in Figure 7-21, for the trade-off between maximum lift and minimum lift slope, for a cambered wing (no reflex) at 120 angle of attack. Such a trade-off is of interest because minimizing the lift slope of a membrane MAV wing, while an effective method for delaying the onset of stall or rejecting a sudden wind gust, typically decreases the pre-stall lift in steady flight as well; a potentially unacceptable consequence. Certain aeroelastic deformations, such as a passive wing de-cambering, would provide a wing with higher lift (than the baseline carbon fiber wing, for example), but a shallower lift slope. 0. 042 0.041 0.04 0.039 0 0.038 'Ih 0.037 0.036 E 0 Sa moo i al or lo a rao baseline ie -e---- Pareto front 0.034 \ max CLa design 0.033 0.7 0.75 0.8 0.85 0.9 0.95 1 CL Figure 7-21. Trade-off between lift and lift slope, a = 12, cambered wing. Such a motion is unusual for low aspect ratio membrane structures however: none of the baseline designs have both larger lift and a smaller lift slope than the carbon fiber wing. The correlation between CL and CL, within the set of baseline designs is very strong, and all the designs fall very close to a single line, clustered in three groups. Any baseline design with adaptive washout (free trailing edge) has lift slopes between 0.035 and 0.037, any overly-stiff sharp downward forces at the leading and trailing edges, the latter of which exacerbates the effect of the airfoil reflex. Despite this, the membrane inflation increases the camber of the wing and thus the lift, by 6.5% over the rigid wing. 0.04 1.5 0.03 0.02 1 / -0.5 0.0 1 0 0 o -0.01 N_ .... -0.5 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 x/c X/C p~ Figure 7-11. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology designs, a = 12, reflex wing. As discussed above, several disparate deformation mechanisms contribute to the high lift of the MAV design located by the aeroelastic topology optimizer (middle column, Figure 7-10). First, the membrane inflation towards the leading edge increases lift via cambering, similar to the PR wing (the pressure distributions over the two wing structures are identical through x/c = 0.25). The main trailing edge batten structure is then depressed downward along the trailing edge (due to the reflex) for wash-in, while the forward portion of this structure is pushed upwards. This structure essentially swivels about the inflection point of the wing's airfoil, a deformation which is able to further increase the size of the membrane cambering, and is only possible because the laminate is free-floating within the membrane skin. It can also be seen (from the left side of Figure 7-11 in particular) that the local bending/twisting of this batten obtained by maximizing Nx and setting Ny to zero. Peak efficiency is found with a slack membrane: this corresponds to minimum drag, which is not shown. It should be mentioned however, that if a design goal is to maximize the lift slope (or minimize the pitching moment slope for stability), a BR wing is most likely a poor choice. Opposite trends are found for the PR wing (Figure 6-3 and Figure 6-4): increasing the pre- tension decreases CL,, increases Cma, and increases L/D. Similar to before, added wing stiffness decreases the adaptive inflation of the wing skin, and results tend towards that of a rigid wing. Without the directional influence of the battens and the trailing edge stress correction needed for the BR wing, the PR wing surfaces in Figure 6-3 are very smooth, and converge monotonically for high pre-stress. CLax Cma L/D 0.054015 4.9 / 20 -0.016 20 4.85 .1 20 0.052 0 4 0 10 -0.017 10 48 20 10 0 0 N 20 10 00 N 20 10 0 0N N N N Y y Figure 6-3. Computed tailoring of pre-stress resultants (N/m) in a PR wing, a = 12. As before, the PR wing is more sensitive to pre-tension in the spanwise direction than the chordwise direction. The slack membrane wing inflates to 5% of the chord: maximizing tension in the chord direction (with none in the span direction) drops this value to 3%, though the opposite case drops the value to 1.5%. This is probably due to the fact that the chord of the membrane skin is about twice as long as it's span. The sensitivity of a pressurized rectangular membrane to a directional pre-stress is inversely-proportional to its length in the same direction, as indicated by solutions to Eq. (4-4). Though the L/D of the PR wing is equally affected by pre- CHAPTER 7 AEROELASTIC TOPOLOGY OPTIMIZATION The conceptual design of a wing skeleton essentially represents an aeroelastic topology optimization problem. Conventional topology optimization is typically concerned with locating the holes within a loaded homogenous structure, by minimizing the compliance [16]. This work details the location of holes within a carbon fiber wing shell, holes which will then be covered with a thin, taut, rubber membrane skin. In other words, the wing will be discretized into a series of panels, wherein each panel can be a carbon fiber laminated shell or an extensible latex rubber skin. Rather than compliance, a series of aerodynamic objective functions can be considered, including L/D, CL, CD, CL., Cma, etc. While the two wing topologies discussed in the preceding section (batten- and perimeter- reinforced wings) have been shown to be effective at load alleviation via streamlining and load augmentation via cambering, respectively, both designs have deficiencies. The BR wing experiences membrane inflation from in-between the battens towards the leading edge (Figure 5- 1), cambering the wing and contradicting the load alleviating effects of the adaptive washout at the trailing edge. Furthermore, the unconstrained trailing edge is only moderately effective at adaptive geometric twist, as the forces in this region are very small (Figure 5-20). If re-curve is built into the wing section, the forces in this area may push the trailing edge downward, actually increasing the incidence, and thus the loads. The PR wing, being a simpler design, is more effective in its intended purpose (adaptive cambering for increased lift and static stability), but the drastic changes in wing geometry at the carbon fiber/membrane interfaces towards the leading and trailing edges of the membrane skin are aerodynamically inefficient. Large membrane inflations are also seen to lead to potentially unacceptable drag penalties as well. All of these deficiencies can be remedied via the tailoring fairly fine structural grid is needed to resolve topologies on the order of those seen in Figure 7-1. The fine grid will, of course, increase the computational cost associated with solving the set of FEA equations, as the number of variables in the optimization algorithm is proportional to the number of finite elements. The wing is discretized into a set of quadrilaterals, which represent the density variables: 0 or 1. These quadrilaterals are used as panels for the aerodynamic solver, and broken into two triangles for the finite element solver, as shown in Figure 7-2. As in Figure 7-1, the wing topology at the root, leading edge, and wing tip is fixed as carbon fiber, to maintain some semblance of an aerodynamic shape capable of sustaining lift. The wing topology in the figure is randomly distributed. / desink domain Figure 7-2. Sample wing topology (left), aerodynamic mesh (center), and structural mesh (right). Further complications are associated with the fact that these variables are binary integers: 1 if the element is a carbon fiber ply, 0 if the element is latex membrane. Several binary optimization techniques (genetic algorithms [109], for example) are impractical for the current problem, due to the large number of variables, but also due to the extremely large computational cost associated with each aeroelastic function evaluation. A fairly standard technique for topology optimization problems classifies the density of each element as continuous, rather than binary [16]. Intermediate densities can then be penalized (implicitly or otherwise) to push the design towards a pure carbon fiber/membrane distribution, with no "porous" material. towards the leading edge and at the wing tip is also removed. Further iterations see topological changes characterized by intersecting threads of membrane material that grow across the surface, leaving behind "islands" of carbon fiber. These structures aren't connected to the laminate wing, but are imbedded within the membrane skin. 4.2 4.15 4.1 4.05 4.-5 / explicit 4 --- penalty 4 IJ - "3 added 3.95 carbon fiber 3.9 initial design 3.85 *D nimembrane 3.8 0 50 100 150 iteration Figure 7-5. Convergence history for maximizing L/D, a = 30, reflex wing. These results indicate two fundamental differences between the designs in Figure 7-1 and those computed via aeroelastic topology optimization. The first is the presence of"islands"; these designs can be built, but the process is significantly more complicated than with a monolithic wing skeleton. Such structures could be avoided with a manufacturability constraint/objective function (such as discussed by Lyu and Saitu [161]), but the logistics of such a metric (as above, with the trailing edge reinforcement constraint) are difficult to formulate. Furthermore, the aeroelastic advantages of free-floating laminate structures are significant, as will be discussed below. A second difference is the fact that the designs of Figure 7-1 are composed entirely from thin strips of carbon fiber embedded within the membrane, while the veracity of the former beneficial comparison requires a viscous flow solver to ascertain the actual height of the low-pressure spike at x/c = 0.68. The batten-reinforced design of Figure 7-12 is substantially more effective with the cambered wing, than with the reflex wing. As discussed above, reflex in the wing pushes the trailing edge down, limiting the ability of the battens to washout for load reduction. This can also be seen by comparing the airfoil shapes between Figure 7-13 and Figure 7-11: the cambered wing shows a continuous increase in the deformation from leading to trailing edge, while most of the deformation in the reflex wing is at the flexible membrane/carbon fiber interface. Aft of this point, deformation is relatively constant to the trailing edge. The 1.60 of washout in the cambered BR wing decreases the load throughout most of the wing and decreases the lift by 8.5% (compared to the rigid wing), but, as before, the load- alleviating design located by the topology optimizer (right column, Figure 7-12) is superior. Similar to above, the design utilizes a series of disconnected carbon fiber structures, oriented parallel to the flow, and extending to the trailing edge. The structures are spaced far enough apart to allow for some local membrane inflation, but this cambering only increases the loads towards the trailing edge. The discontinuous wing surface forces a number of high-pressure spikes on the upper surface, notably at x/c = 0.2 and 0.6. This, in combination with the substantial adaptive washout at the trailing edge, decreases the lift by 13.6% over the rigid wing and by 5.6% over the BR wing. Three of the wing topologies discussed above (minimum CL,, minimum drag, and minimum pitching moment slope, all optimized for a reflex wing at 30 angle of attack) are built and tested in the closed loop wing tunnel, as seen in Figure 7-14. Though the aeroelastic model relies on a sizable state of pre-stress in the membrane skin to remain bounded, all three of the 4.1 I I I I I 0.04 -0.039 3.9 C 3.8 .... . Lj 0.038 0 20 40 60 80 100 120 iteration Figure 7-16. Convergence history for maximizing L/D and minimizing CL,, 6 = 0.5, a = 3, reflex wing. The multi-objective results of Figure 7-16 can be directly compared to the single-objective results of Figure 7-5, where only L/D must be improved. For the latter, L/D can be increased to 4.17, with the inclusion of trailing edge battens for adaptive wash-in, and an unconstrained membrane skin towards the leading edge for cambering via inflation. This is a load-augmenting design, and as such the lift slope is very high: 0.040. In order to strike an adequate compromise between the two designs, the multi-objective optimizer leaves the trailing edge battens, but fills the membrane skin at the leading edge with a disconnected carbon fiber structure. The L/D of this design obviously degrades (4.05), but the lift slope is much shallower (0.038), as desired. The Pareto front for this same trade-off (maximum L/D and minimum CLa) is given in over the inflated shape: such a deceleration results in a pressure spike. Aft of these spikes, the pressure is slightly lower in the membrane skin than over each batten (due to the adaptive camber), driving the flow into the membrane patches. This is a very small effect (mildly visible in the streamlines) for the current case, but can be expected to play a large role with potential flow separation, where the chordwise velocities are very small [154]. ... .. ....... .. . -80 -30 20 70 120 -80 -30 20 70 120 -80 -30 20 70 120 Figure 5-16. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), a = 0. For the PR wing (Figure 5-16), the pressure spike is stronger, and exists continuously along the membrane interface. A significant percentage of this spike is directed axially, increasing the drag (as seen in Figure 5-13). The adaptive inflation causes an aft-ward shift in the pressure recovery location of each flexible wing section. The longer moment arm increases the nose-down pitching moment about the leading edge (Figure 5-14), which is the working mechanism behind the benevolent longitudinal static stability properties of the PR wing. Furthermore, the aerodynamic twist increases the adverse pressure gradient over the membrane portion of the wing: some flow now separates as it travels down the inflated shape, further increasing the drag (as compared to the rigid wing). Similar results are given for the lower/pressure side of the three wings at 0 angle of attack in Figure 5-17. The flow beneath the rigid wing is dominated by an adverse pressure gradient towards the leading edge, causing a large separation bubble underneath the wing camber. This [129] Barlow, J., Rae, W., Pope, A., Low-Speed Wind Tunnel Testing, Wiley, New York, NY, 1999. [130] Fleming, G., Bartram, S., Waszak, M., Jenkins, L., "Projection Moire Interferometry Measurements of Micro Air Vehicle Wings," SPIE Paper 4448-16. [131] Burner, A., Fleming, G., Hoppe, J., "Comparison of Three Optical Methods for Measuring Model Deformation," AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10-13, 2000. [132] Sutton, M., Turner, J., Bruck, H., Chae, T., "Full Field Representation of the Discretely Sampled Surface Deformation for Displacement and Strain Analysis," Experimental Mechanics, Vol. 31, No. 2, 1991, pp. 168-177. [133] Schreier, H., Braasch, J., Sutton, M., "Systematic Errors in Digital Image Correlation caused by Intensity Interpolation," Optical Engineering, Vol. 39, No. 11, 2000, pp. 2915-2921. [134] Sutton, M., McFadden, C., "Development of a Methodology for Non-Contacting Strain Measurements in Fluid Environments Using Computer Vision," Optics andLasers in Engineering, Vol. 32, No. 4, 2000, pp. 367-377. [135] Albertani, R., Stanford, B., Sytsma, M., Ifju, P., "Unsteady Mechanical Aspects of Flexible Wings: an Experimental Investigation Applied to Biologically Inspired MAVs," European Micro Air Vehicle Conference and Flight Competition, Toulouse, France, September 17-21, 2007. [136] Batoz, J., Bathe, K., Ho, L., "A Study of Three-Node Triangular Plate Bending Elements," International Journalfor Numerical Methods in Engineering, Vol. 15, No. 12, 1980, pp. 1771- 1812. [137] Cook, R., Malkus, D., Plesha, M., Witt, R., Concepts andApplications ofFinite Element Analysis, Wiley, New York, NY, 2002. [138] Reaves, M., Horta, L., Waszak, M., Morgan, B., "Model Update of a Micro Air Vehicle (MAV) Flexible Wing Frame with Uncertainty Quantification," NASA Technical Memorandum, TM 213232, 2004. [139] Isenberg, C., The Science of Soap Films and Soap Bubbles, Dover, New York, NY, 1992. [140] Pujara, P., Lardner, T., "Deformations of Elastic Membranes Effect of Different Constituitive Relations," Zeitschrift fir Angewandte Mathematik und Physik, Vol. 29, No. 2, 1978, pp. 315- 327. [141] Small, M., Nix, W., "Analysis of the Accuracy of the Bulge Test in Determining the Mechanical Properties of Thin Films," Journal ofMaterials Research, Vol. 7, No. 6, 1992, pp. 1553-1563. [142] Pauletti, R., Guirardi, D., Deifeld, T., "Argyris' Natural Finite Element Revisited," International Conference on Textile Composites and Inflatable Structures, Stuttgart, Germany, October 2-5, 2005. the combination of a poorly-constrained trailing edge and unsteady vortex shedding will lead to a large-amplitude flapping vibration, similar to that discussed by Argentina and Mahadevan [62]. Wind tunnel testing of this wing is given in Figure 7-4 at 13 m/s; the critical speed of flapping vibration is approximately 3 m/s. 6 - - - 1.2 5- 0.8- 3 0.4 0.2OT 0 0 0.2 0.4 0.6 0 0.5 1 CD CL Figure 7-4. Measured loads of an inadequately reinforced membrane wing, U. = 13 m/s. As expected, the measured lift of the membrane wing is significantly less than that measured from the rigid wing in the wind tunnel: the poorly-supported wing cannot sustain the flight loads, while the large amplitude vibrations levy a substantial drag penalty. Even a mild amount of trailing edge reinforcement (such as that seen in the upper left of Figure 7-1) will prevent this behavior, but formulating a constraint that will push the aeroelastic topology optimizer away from wing designs with a poorly-reinforced trailing edge is difficult, and is not included. This section only serves to highlight one significant shortcoming of the aeroelastic model used here, and to diminish the perceived optimality of certain wing topologies. Adjoint Sensitivity Analysis As the number of variables in the aeroelastic system (essentially the density of each element) will always outnumber the number of constraints and objective functions, a sensitivity constituitive relationships provide the same numerical solutions up to deformations on the order of 30% of the radius, a figure well above the deflections expected on a membrane wing. Geometric nonlinearity implies that the deformation is large enough to warrant finite strains, and that the direction of the non-conservative pressure loads significantly changes with deformation. Eq. (4-4) is still valid, only now the stress resultants depend upon the state of pre-stress, as well as in-plane stretching, which in turn depends on the out-of-plane displacement. Furthermore, the rotation of the membrane is no longer well-approximated by the derivative of w, rendering the equilibrium equation nonlinear. Three displacement degrees of freedom are required per node (u, v, w), rather than the single w used above. Finite element implementation of such a model is described by Small and Nix [141] and Pauletti et al. [142]. The strain pseudo-vector within each element is given as: Ss=s + L =Bo Xe + BL.Xe (4-5) where Eo and EL represent the division of the linear (infinitesimal) and nonlinear contributions to the Green-Lagrange strain, Xe is a vector of the degrees of freedom in the elements (three displacements at the three nodes), and Bo and BL are the appropriate strain-displacement matrices (the latter of which depends upon the nodal displacements) [143]. The pre-stress (if any) can be included into the model in one of two methods. First, they can be simply added to the stresses computed my multiplying the strain vector of Eq. (4-5) through the constituitive matrix. This may cause problems if the imposed pre-stress distribution does not exactly satisfy equilibrium conditions, or if there is excessive curvature in the membrane skin: the membrane will deform, even in the absence of an external force. A second option is to use the pre-stresses in a finite element implementation of Eq. (4-4), then add the resulting stiffness matrix and force vectors to the nonlinear terms. For a flat structure is minimal: the deformation along this structure is largely linear down the wing. The intersection of this linear trend with the curved inflated membrane shape produces a cusp in the airfoil. The small radius of curvature forces very large velocities, resulting in the lift spike at 46% of the chord. This combination of wash-in and cambering leads to a design which out-performs the lift of the PR wing by 5.6%, but the former effect is troubling. The wash-in essentially removes the reflex from the airfoil (as does the aerodynamic twist of the PR wing), an attribute originally added to mitigate the nose-down pitching moment. This fact leads to two important ideas. First, thorough optimization of a single design metric is ill-advised for micro air vehicle design, as other aspects of the flight performance will surely degrade. Its inclusion here is only meant to emphasize the relationship between aeroelastic deformation and flight performance, and show the capabilities of the topology optimization. A better approach is the multi-objective scheme discussed below. Secondly, if the design goal is a single-minded maximization of lift, a reflex airfoil is a poor choice compared to a singly-curved airfoil, a shape which the topology optimizer strives to emulate through aeroelastic deformations. Furthermore, if the design metric is an aerodynamic force or moment, passive shape adaptations need not be used at all: simply compute the optimal wing shape from the bottom row of Figure 7-10, and build a similar rigid wing. Mass restrictions prevents such a strategy in traditional aircraft design (though a similar idea can be seen in the jig-shape approach [163], where wing shape is optimized, followed by identification of the internal structure which allows for deformation into this shape), but two layers of carbon fiber can adequately hold the intended shape without a stringent weight penalty. However, if the design metric is an aerodynamic derivative (gust rejection or longitudinal static stability, for Deformation measurements of a membrane wing under propwash indicate unsymmetrical (about the root) wing shapes, a phenomenon which diminishes with higher angles of attack and dynamic pressures [47]. Stults et al. use laser vibrometry to measure the modal parameters (shapes, damping, frequency spectra) of a BR wing, which are then fed into a computational model for simulation of static and dynamic deformations in both steady freestream and a gust [93]. Aeroelastic Tailoring Although aeroelastic tailoring is generally defined as the addition of directional stiffness into a wing structure so as to beneficially affect performance [11], this has traditionally meant the use of unbalanced composite plates/shells. Despite the use of such laminated materials on many MAV wing frames [94], there does not appear to be any tailoring studies on fixed micro air vehicle wings. Some investigators have applied the concept to the design of flapping omithopter wings [95] [96], where a bend/twist coupling can vary the twist-induced incidence of a wing to improve thrust. Conventional tailoring studies can also be found applied to a larger class of unmanned aerial vehicles [97] [98]. The latter study by Weisshaar et al. uses laminate tailoring to improve the lateral control of a vehicle with an aspect ratio of 3. With ailerons, a wing tailored with adaptive wash-in is shown to improve roll performance and roll-reversal speed, though wash-out is preferred for a leading edge slat [98]. In addition to conventional laminate-based tailoring, drastic changes in the performance of a membrane wing are attainable by altering the pre-tension distribution within the extensible membrane. Holla et al. [79], Fink [87], Smith and Shyy [72], Murai and Maruyama [68], and Ormiston [15] all note the enormous impact that membrane pre-tension has on aerodynamics: for the two-dimensional case, higher pre-tension generally pushes flexible wing performance to that of a rigid wing. For a three-dimensional wing, the response can be considerably more complex, depending on the nature of the membrane reinforcement. Well-reported effects of increasing the flows over a membrane wing are studied by Lian and Shyy [8] (who correlate the frequency spectrum of the vibrating membrane wing to the vortex shedding). Three-Dimensional Wings Three complicating factors can arise with the simulation of a three-dimensional membrane wing, rather than the planar case [76]. First, the tension is not constant (in space or direction), but is in a state of plane-stress. Secondly, the wing geometry can vary in the spanwise direction, and must be specified. Finally, the membrane may possess a certain degree of orthotropy [59]. Most importantly, analytical solutions cannot generally be found. Simplifying assumptions to this problem are given by Sneyd et al. [77] (triangular planform) and Ormiston [15] (rectangular sailingg. Sneyd et al. reduce both the aerodynamics and the membrane deformation to two- dimensional phenomena, where the third dimension is felt through a trailing edge cable. Ormiston assumes both spanwise and chordwise deformation (but not aerodynamics), and is able to effectively decouple the two modules by using only the first term of a Fourier series to describe the inflated wing shape. Boudreault uses a higher-fidelity vortex lattice solver, but also prescribes the wing shape, here using cubic polynomials [78]. Holla et al. [79] use an iterative procedure to couple a double lattice method to a structural model, but assume admissible mode shapes to describe the deformation of a rectangular membrane clamped along the perimeter. The stress in the membrane is assumed to be always equal to the applied pre-stress (inextensibility, which overwhelms the nonlinearities in the membrane mechanics). A similar framework is used by Sugimoto in the study of circular membrane wings, where the wing shape is completely determined by a linear finite element solver [80]. Jackson and Christie couple a vortex lattice method to a nonlinear structural model for the simulation of a triangular membrane wing. Comparisons between a rigid wing, a membrane wing fixed at the trailing edge, and one with a free trailing edge elucidate the tradeoffs in lift exploited. A comprehensive numerical review of the design space is provided with a full factorial designed experiment of the three aforementioned variables. This data is then used to optimize six aerodynamic variables, as well as compromises between each. The six designs resulting from the single-objective optimizations are built and tested in the wind tunnel: five show improvements over the baseline designs, one has a similar response. While the flexible wing structures have been shown to effectively alter the flow fields over a MAV wing, aeroelastic topology optimization (Chapter 7) can be used to improve on the shortcomings of the previously-considered baseline designs. Results are superior to those computed via tailoring, as the number of variables is much larger: the wing is discretized into a series of panels, each of which can be membrane or carbon fiber laminate. The computational cost is severe: hundreds of iterations are expected for convergence, and a sensitivity analysis of the coupled aeroelastic system must be conducted. The optimization is able to identify a series of interesting designs, emphasizing the relationships between flight condition, airfoil, design metric, and wing topology. For load alleviation, the algorithm fills the membrane skin with a number of disconnected laminate structures. The structure is flexible enough to washout at the trailing edge, but the patches of exposed membrane skin are not large enough to inflate and camber the wing. Such a design has less drag and a shallower lift curve than the batten-reinforced wing. For load augmentation, the topology optimizer utilizes a combination of cambering, wash-in, and wing surface geometry cusps to increase the lift over the perimeter-reinforced wing. As a wing design optimized for a single metric is of minor usefulness, the topology optimizer is expanded to minimize a convex combination of two metrics for computation of the Pareto front. Three such designs are built and tested in the wind tunnel, confirming the computed superiority over the baseline wings. and root. When Xo = 0.1, the optimizer is unable to fill in enough space with laminates to prevent membrane inflation. Of the three designs, this is the least tractable from a manufacturing point of view as well. Xo = 1, CD = 0.131 Xo = 0.5, CD = 0.125 Xo = 0.1, CD = 0.128 CDrigid = 0.134 Figure 7-7. Affect of initial design upon the optimal CD topology, a = 12, reflex wing. The dependency of the optimal topology (maximum lift) upon both angle of attack and airfoil shape are given in Figure 7-8, for both a reflex (left two plots) and a cambered wing (right two plots). For the wing with trailing edge reflex, the optimal lift design looks similar to that found in Figure 7-5: trailing edge battens that extend no farther up the wing than the half-chord, a spanwise member that coincides with the inflection point of the airfoil, and unconstrained membrane skin towards the leading edge, where the forces are largest. The optimizer has realized that it can maximize lift by both cambering the wing through inflation at the leading edge, and forcing the trailing edge battens downward for wash-in. This latter deformation is only possible due to the reflex (negative camber) in this area, included to offset the nose-down pitching moment of the remainder of the "flying wing", and thus allow for removal of a horizontal stabilizer due to size restrictions. Increasing the angle of attack from 3 to 120 shows no significant difference in the wing topology, slightly increasing the length of the largest batten. At the lower angle of attack, up to 22% increase in lift is indicated through topology optimization. at the root first, and is confined (at this angle) to the in-board portion of the wing. This may be due to the steeper pressure gradients at the root, or an interaction with the tip vortex system [5]. The reattached flow aft of the bubble (and the resulting pressure distribution) must be viewed with a certain amount of suspicion. Such a reattachment is known to be turbulent process [25], and no such module is included in the CFD (or even, to the author's knowledge, exists for complex three-dimensional flows). Unsteady vortex shedding may accompany the bubble as well [8], though time-averaging of vortex shedding is known to compare well with steady measurements of a single stationary bubble [18]. The augmented incidence has considerably increased the strength of the wing-tip vortex swirling system over the rigid wing. The size of the vortex core is larger (indicative of the expected increase in induced drag [27]), and the low pressure cells at the wing tip are very evident [3]. Figure 5-18. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), a = 15. As expected, the aeroelastic effects of the BR and PR wings are more predominate at 15 in Figure 5-18. For the BR wing, the three high-pressure lobes over the membrane/carbon fiber interface are larger. Significant pressure-redistribution over the membrane stretched between the outer batten and the wing tip can be seen as well. Adaptive washout slightly decreases the intensity of the separation bubble, but has no noticeable effect on the pressure distribution at the trailing edge of the upper BR wing surface. At 15, the aerodynamic twist of the PR wing is string in two dimensions) with geometric twist, as the structure alone cannot sustain a flight load in a stable manner. A second option involves considering an elastic sheet with some bending/flexural stiffness. A large variety of work can be found in the literature concerning two- dimensional flexible beams in flow. For problems on a MAV scale, work tends to focus on flags and organic structures such as leaves, seaweed, etc. Fitt and Pope [58] derive an integro-differential flag equation for the shape of a thin membrane with bending stiffness in unsteady inviscid flow, considering both a hinged and a clamped leading edge boundary condition. Argentina and Mahadevan [62] solve a similar problem, and are able to predict a critical speed that marks the onset of an unstable flapping vibration, noting that the complex instability is similar to the resonance between a pivoting airfoil in flow and a hinged-free beam vibration. Over-prediction of the unstable flapping speed (when compared to experimental data) leads to the possibility of a stability mechanism wherein skin friction induces tension in the membrane. Alben et al. [63] discuss the streamlining of a two-dimensional flexible filament for drag reduction. In particular, they are able to show that the drag on a filament at high angles of attack decreases from the rigid Uo2 scaling to Uo4/3. Early work in the study of membrane wings without bending stiffness is given by Voelz [64], who describes the classical two-dimensional sail equation: an inextensible membrane with slack, fixed at the leading and trailing edges, immersed in incompressible, irrotational, inviscid steady flow. Using thin airfoil theory, along with a small angle of attack assumption, Voelz is able to derive a linear integro-differential equation for the shape of the sail as a function of incidence, freestream velocity, and slack ratio. Various numerical solution methods are available, including those by Thwaites [65] (eigenfunction methods) and Nielsen [66] (Fourier series methods), to solve for lift, pitching moments, and membrane tension. of MAVs are reported in the literature. Mueller and DeLaurier cite aspect ratio as the most important design variable, followed by wing planform and Reynolds number. Free stream turbulence intensity and trailing edge geometry are reported to be non-factors, and Reynolds number is only important near stall [39]. Flow visualization experiments by Gursul et al. [40] on swept, non-slender, low aspect ratio wings find the presence of primary and secondary vortices, with stagnant flow regions outboard of the former. Vortex merging and other unsteady interactions within the shear layers are found to be highly dependent on Reynolds number. Kaplan et al. [37] report a fluctuation in the location of the vortex core off of a semi- elliptical wing at 8,000 Reynolds number. Numerical simulations and flow visualization by Tang and Zhu [6] of an accelerating elliptical wing show an unstable interaction between a longitudinal secondary separated vortex and the tip vortices. This destabilization (for angles of attack above 11) causes the tip vortex system to swing back and forth along the wing, leading to bilateral asymmetry problems in roll. The authors also note a stationary separated vortex (rather than the customary shedding) for angles above 33, possibly due to the vertical components of the tip vortices. Cosyn and Vierendeels [41], Brion et al. [42], and Stanford et al. [43], discuss numerical wing modeling of lift and drag for comparison with wind tunnel experiments: the lack of a three- dimensional turbulent-transition model is generally cited as the reason for poor correlation at higher angles of attack. Results documenting the aerodynamics of a complete micro air vehicle (wing with fuselage, stabilizers, propellers, etc) are scarce: wind tunnel experimentation by Zhan et al. document longitudinal and lateral stability as a function of vertical stabilizer placement and wing sweep [44]. Similar stability data is given by Ramamurti et al. [45] for a MAV wing with counter-rotating propellers. match between measured and predicted membrane deformation for the taut case in Figure 4-5 is very good, despite the fact that the numerical pre-strains were presumed uniform. The error resulting from a uniform pre-stress assumption can be estimated with the following method. The pre-strain distribution throughout a flat circular membrane is considered a normally-distributed random variable: each finite element has a different pre-strain. The linear membrane model of Eq. (4-4) is then used to compute the displacement at the center of the membrane due to a hydrostatic pressure. The same membrane is then given a constant pre-strain distribution (the average value of the randomly-distributed pre-strain), and the central deflection is recomputed for comparison purposes. Monte Carlo simulations are then used to estimate the average error at the membrane center, for a given coefficient of variation of the pre-strain. The results of the Monte Carlo simulation are given in Figure 4-7. Each data point is the percentage error between the central displacement computed with a non-homogenous random pre-strain, and that with a constant pre-strain. Each error percentage is the average of 500 finite element simulations. The radius of the circle is 57.15 mm, the thickness is 0.12 mm, the elastic modulus is 2 MPa, the Poisson's ratio is 0.5, and the hydrostatic pressure is fixed at 200 Pa. The mean pre-strain is 0.05, and the standard deviation is decided by the COV of each data point's abscissa. Nonlinear membrane modeling is not used. The smoothing nature of the Laplacian operator in Eq. (4-4) is very evident: even in the presence of 30% spatial pre-strain variability, the error in assuming a constant pre-strain is still less than 5%. On one hand, the error in Figure 4-7 is probably under-predicted, as strain cannot truly be a spatially-random variable: on a local scale measured strain may seem random, but on a global scale it must satisfy the compatibility equations [146]. Both of these scale-trends are evident in Figure 4-6. On the other hand, Figure 4-7 represents the worst case scenario, as nonlinear membrane effects will dilute the importance devised to handle the membrane skin pre-tension. The estimated validity range of each model is discussed. I detail the deformation patterns, flow structures, and aerodynamic characterization of a series of baseline flexible and rigid MAV wings, obtained both numerically and experimentally for comparison purposes. Once the predictive capability of the aeroelastic model is well- verified, these data sets are studied to uncover the working mechanisms behind the passive shape adaptation and their associated aerodynamic advantages. I then use a non-standard aeroelastic tailoring study to identify the optimal wing type and structural composition for a given objective function, as well as various combinations thereof. Wing types are limited to rigid, batten-reinforced, and perimeter-reinforced designs; structural composition variables include anisotropic membrane pre-tension and laminate lay-up schedule. Multi-objective optimization is conducted using a design of experiments approach, with a series of aerodynamic coefficients and derivatives as metrics. The tailoring concludes with experimental validation of the performance of selected optimal designs. Finally, I formulate a computational framework for aeroelastic topology optimization of a membrane micro air vehicle wing. A gradient-based search is used, with analytically computed sensitivities of the same aerodynamic metrics as used above. The optimal wing topology is discussed as a function of flight condition, grid density, initial guess, and design metric. I optimize a convex combination of two conflicting objective functions to construct the Pareto front, with a demonstrated superiority over the baseline wing structures employed in the tailoring study. As before, the work concludes with experimental validation of the performance of selected optimal designs. studies considered above, but the greater generality of an aeroelastic topology scheme (due to the larger number of variables) would suggest better potential improvements in aerodynamic performance. Furthermore, such an undertaking can potentially be followed by an aeroelastic tailoring study of the optimal topology for further improvements, as discussed by Krog et al. [115]: topology optimization to locate a good design, followed by sizing and shape optimization. A flexible MAV wing topological optimization procedure has some precedence in early micro air vehicle work by Ifju et al. [10], with an array of successfully flight tested designs shown in Figure 7-1. Each of these designs consists of a laminated leading edge, wing tip, and wing root; a series of thin strips of carbon fiber are imbedded within the concomitant membrane skin. Both the BR and PR wings are present, along with slight variations upon those themes. Ifju et al. [10] qualitatively ranks these wing structures based upon observations in the field and pilot-reported handling qualities: a crude trial and error process led to the batten-reinforced design as a viable candidate for MAV flight. Figure 7-1. Wing topologies flight tested by Ifju et al. [10]. Several challenges are associated with the optimization procedure considered here. First, a law of SIMP is still useful for the current application, as demonstrated in Figure 7-3. Both linear and nonlinear material interpolations are given for the lift computation, and the wing topology is altered uniformly. For the linear interpolation (i.e., without SIMP), the aeroelastic response is a weak function of the density until X becomes very small (-0.001), when the system experiences a very sharp change as X is further decreased to 0. This is a result of the large stiffness imbalance between the carbon fiber laminates and the membrane skin, and the fact that lift is a direct function of the wing's compliance (the inverse of the weighted sum of the two disparate stiffness matrices in Eq. (7-1)). The inclusion of a nonlinear penalization power (p = 5), spreads the response evenly between 0 and 1. Aeroelastic topology optimization with linear material interpolation experiences convergence difficulties, as the gradient-based technique struggles with the nearly- disjointed design space; a penalization power of 5 is utilized for the remainder of this work. 0.63 0.62 p = 5 0.61 0.6 0.59 T 0 0.2 0.4 0.6 0.8 1 X Figure 7-3. Effect of linear and nonlinear material interpolation upon lift. The results from Figure 7-3 suggest a number of other potential difficulties with an aeroelastic topology optimization scheme. First, the sensitivity of the aeroelastic response to element density is zero for a pure membrane wing (X = 0), as can be inferred from Eq. (7-1). As such, using a pure membrane wing as an initial guess for optimization will not work, as the design with battens oriented perpendicular to the flow (or the carbon fiber wing) has a slope between 0.038 and 0.039, and the PR wing has a lift slope of 0.041. The strong data correlation is in sharp contrast to the results of Figure 7-17 for the reflex wing, where the baseline structures are well-distributed through the design space. This emphasizes the large role that the doubly-curved airfoil can play in producing many different types of aeroelastic deformation, providing greater freedom to the designer and better compromise designs. Despite this, the magnitude of the variability is higher for the cambered wing, as the forces are generally larger: CL, can be varied by 14.5% for the reflex wing in Figure 7-17, but by 26.4% for the cambered wing in Figure 7-21. These numbers can be increased further with the use of nonlinear membrane structures, but deformations must be kept at a moderate level to preserve the fidelity of the linear finite element model in the current work. As wing structures with high lift and shallow lift slopes are rare, the set of baseline designs lies close to the Pareto front in Figure 7-21. None are superior however, in terms of individual metrics or Pareto optimality. The designs located by the topology optimizer to maximize lift and maximize lift slope are almost identical, though disparate designs can be obtained with a reflex wing, as noted above. The PR wing is very effective for cambered wings at higher angles of attack, and lies close to these two optimums. The slight convexity in the Pareto front produces two designs with the sought-after higher lift and lower lift slope than the homogenous carbon fiber wing. The topology highlighted in Figure 7-21 increases the lift coefficient from 0.842 to 0.876 and decreases the lift slope from 0.038 to 0.036, and is found from an equal weighting of the two metrics (6 = 0.5). Wing displacements and pressure distributions for selected wings along the Pareto front of Figure 7-21 are given in Figure 7-22, for a cambered wing at 120 angle of attack. Corresponding [143] Lian, Y., Shyy, W., Ifju, P., Verron, E., "A Computational Model for Coupled Membrane-Fluid Dynamics," AIAA Fluid Dynamics Conference and Exhibit, St. Louis, MO, June 24-26, 2002. [144] Wu, B., Du, X., Tan, H., "A Three-Dimensional FE Nonlinear Analysis of Membranes," Computers and Structures, Vol. 59, No. 4, 1996, pp. 601-605. [145] Campbell, J., "On the Theory of Initially Tensioned Circular Membranes Subjected to Uniform Pressure," Quarterly Journal of Mechanics andAppliedA it/h1eutii \, Vol. 9, No. 1, 1956, pp. 84-93. [146] Mase, G., Mase, G., Continuum Mechanics for Engineers, CRC Press, Boca Raton, FL, 1999. [147] Stanford, B., Boria, F., Ifju, P., "The Validity Range of Pressurized Membrane Models with Varying Fidelity," Society for Experimental Mechanics, Springfield, MA, June 4-6, 2007. [148] Tannehill, J., Anderson, D., Pletcher, R., Computational Fluid Mechanics andHeat Transfer, Taylor and Francis, Philadelphia, PA, 1997. [149] Shyy, W., Computational Modelingfor Fluid Flow and Interfacial Transport, Elsevier, Amsterdam, The Netherlands, 1994. [150] Thakur, S., Wright, J., Shyy, W., "STREAM: A Computational Fluid Dynamics and Heat Transfer Navier-Stokes Solver: Theory and Applications," Streamline Numerics, Inc., Gainesville, FL, 2002. [151] Lewis, W., Tension Structures: Form and Behavior, Thomas Telford Ltd, London, UK, 2003. [152] Kamakoti, R., Lian, Y., Regisford, S., Kurdila, A., Shyy, W., "Computational Aeroelasticity Using a Pressure-Based Solver," Computer Methods in Engineering and Sciences, Vol. 3, No. 6, 2002, pp. 773-790. [153] Sytsma, M., "Aerodynamic Flow Characterization of Micro Air Vehicles Utilizing Flow Visualization Methods," Masters Thesis, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 2006. [154] Hepperle, M., "Aerodynamics of Spar and Rib Structures," MHAeroTools Online Database, http://www.mh-aerotools.de/airfoils/ribs.htm, March 2007. [155] Giirdal, Z., Haftka, R., Hajela, P., Design and Optimization of Laminated Composites Materials, Wiley, New York, NY, 1999. [156] Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., "A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 2002, pp. 182-197. [157] Antony, J., Design of Experiments for Engineers and Scientists, Butterworth-Heinemann, Boston, MA, 2003. wings are constructed with a slack membrane. This is to ensure similarity between the three wings (pre-stress is very difficult to control), and also to compare the force and moment data to the baseline membrane data acquired above (Figure 5-12 Figure 5-15). Figure 7-14. Wing topology optimized for minimum CL, built and tested in the wind tunnel. Results are given in Figure 7-15, for a longitudinal a-sweep between 0 and 300. All three structures located by the topology optimizer show marked improvements over the baseline experimental data, validating the use of a low fidelity aeroelastic model (vortex lattice model coupled to a linear membrane solver) as a surrogate for computationally-intensive nonlinear models. With the exception of very low (where deformations are small) and very high angles of attack (where the wing has stalled), the optimized designs consistently out-perform the baselines. As discussed above, this is not expected to be true for L/D, where design strategies vary strongly with incidence (Figure 7-9). It should also be noted that the three optimized designs in Figure 7-15 provide shallower lift slopes, less drag, and steeper pitching moment slopes, respectively, than the experimental data gathered from the designs utilizing aeroelastic tailoring (Figure 6-15 Figure 6-18). This confirms the idea that topology optimization can out-perform tailoring of the baseline MAV wings, as the former has a larger number of variables to work with. The two techniques need not be mutually exclusive: having located suitable wing topologies, the designs can be subjected to a Multiple solutions are found to exist at small angles of attack with a finite slack ratio: approaching 0 from negative angles provides a negatively-cambered sail, though the opposite is true if this mark is approached from a positive value. The sail is uncertain as to which side of the chord-line it should lie [65], a phenomenon which ultimately manifests itself in the form of a hysteresis loop [15]. Variations on this problem are considered by Haselgrove and Tuck [67], where the trailing edge of the membrane is attached to an inextensible rope, thereby introducing a combination of adaptive aerodynamic and geometric twist. Increasing the length of the rope is seen to improve static stability, but decrease lift. Membrane elasticity is included in the work of Murai and Maruyama [68], Jackson [69], and Sneyd [70], indicating a nonlinear CL-a relationship as strains develop within the membrane at high incidence. Viscous flow models are employed in the work of Cyr and Newman [71] and Smith and Shyy [72]. The latter cites viscous effects as having much more influence on the aerodynamics of a sail wing than the effects of the assumptions made with linear thin airfoil theory. Specifically, inviscid solutions tend to over-predict lift at higher angles of attack (or large slack ratios), due to a loss of circulation caused by viscous effects about the trailing edge. A comparison of lift and tension versus angle of attack with experimental data (provided by Newman and Low [73], among others) yields mixed results; surprisingly, the lift is over- predicted by the viscous flow model, yet the tension is under-predicted. Smith and Shyy also note a substantial discrepancy in the available experimental data in reported values of slack ratios, sail material properties, and Reynolds numbers, which may play a role in the mixed comparisons [74]. Comparison of numerical and experimental data for two- dimensional sails is also discussed by Lorillu et al. [75], who report satisfactory correlation for the flow structures and deformed membrane shape. Unsteady laminar-turbulent transitional AEROELASTIC ANALYSIS AND OPTIMIZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By BRET KENNEDY STANFORD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 average diameter (less than 0.5 mm) and is generally not connected to another speckle pattern; the pattern should not provide significant reinforcement to the latex. If information concerning the state of pre-strain in the skin is desired, a picture of the un-stretched latex sheet is taken for future use as a reference in the VIC system. The latex is then appropriately stretched about a frame (or not at all if a slack membrane is desired), and adhered to the upper carbon fiber wing surface (which must be painted white) with spray glue. After the glue has dried, the excess latex is trimmed away. A picture of the resultant wind tunnel model is given in Figure 3-3. Figure 3-3. Speckled batten-reinforced membrane wing with wind tunnel attachment. inflation among the battens. This demonstrates the degree to which the adaptive washout at the trailing edge depends on the bending/twisting of the leading edge laminate (where the forces are very high, seen in Figure 5-18), and also explains why tailoring the thickness of the battens, discussed above, has only a minor effect upon the aerodynamics. CLa ma L/D 0.052 -001 5.5 0.048 0.048410 -0.015 10 5 20 N 20 N 20 N 10 0 x 10 0 x 10 0 Nx N N N y y y 1 layer 2 layers 3 layers Figure 6-10. Computed full factorial design of a BR wing, a = 12. S 0 0.01 0.02 0.03 0.04 0.05 Figure 6-11. Computed BR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), a = 12. This inability of the slack membrane wing to alleviate the flight loads decreases the efficiency, but surprisingly, has little effect on the stability derivatives. One possible reason for this is the negative deformations at the trailing edge of the three-layer MAV wing. The stiffer wing adheres closely to the original, rigid wing shape, which contains reflex (negative camber) at the trailing edge. The negative forces in this area push the membrane downward, increasing the wing camber. Increasing the stiffness of the plain weave laminate may convert the BR wing from a structure with adaptive washout to one with progressive de-cambering, leaving the rTF topology optimization is apt to utilize two-dimensional laminate structures. After 112 iterations in Figure 7-5, the optimization has largely converged (with only minimal further improvements in L/D), but some material with intermediate densities remains towards the leading edge of the wing. Many techniques exist for effectively interpreting gray level topologies [162]; the explicit penalty of Eq. (7-2) is used here. Surprisingly, the L/D sees a further increase with the addition of this penalty, contrary to the conflict between performance and 0-1 convergence reported by Chen and Wu [158]. The explicit penalty does not significantly alter the topology, but merely forces all of the design variables to their limits, as intended. The final wing skeleton has three trailing edge battens (one of which is connected to a triangular structure towards the center of the membrane skin), and a fourth batten oriented at 450 to the flow direction. The structure shows some similarities to a wing design in Figure 7-1 (third row, first column), and appears to be a topological combination of a BR and a PR wing, with both battens and membrane inflation towards the leading edge. The optimized topology increases the L/D by 9.5% over the initial design, and (perhaps more relevant, as the initial intermediate density design does not technically exist) by 10.2% over the rigid wing. The affect of mesh density is given in Figure 7-6, for a reflex wing at 120 angle of attack, with L/D maximization as the objective function. The 30x30 grid, for example, indicates that 900 vortex panels (and 1800 finite elements) cover each semi-wing. As the leading edge, root, and wing tip of each wing are fixed as carbon fiber, 480 density design variables are left for the topology optimization. One obvious sign of adequate convergence is the efficiency of the rigid wing, with only a 0.44% difference between that computed on the two finer grids. The three optimal wing topologies are similar, with three distinct carbon fiber structures imbedded within the membrane skin: two extend to the trailing edge and the third resides towards the leading the angle of attack is being increased or decreased can lead to hysteresis [24]) are all cited by Young and Horton [2] as highly influential on the formation of a bubble. Furthermore, the flow will only reattach to the surface if there is enough energy to maintain circulating flow against dissipation [25]. An extensive survey of low Reynolds number (3-104 5-105) airfoils is given by Carmichael [17] (there are quite a few others, as reviewed by Shyy et al. [26]). The study finds that, for the lower end of tested Reynolds numbers, the laminar separated flow does not have time to reattach to the surface. Above 5-104, the flow will reattach, forming a long separation bubble over the wing. At the upper end of the range of Reynolds numbers discussed by Carmichael, the size of the bubble decreases, generally resulting in a decrease in form drag. Increasing the angle of attack generally enhances the turbulence in the flow, which can also prompt quicker reattachment and shorter bubbles [8]. The length of the separation bubble can generally be inferred from the plateau-like behavior of the pressure distribution: the flow speeds up before the bubble (dropping the pressure), and slows down after the bubble [27]. This description is a time-averaged scenario: in an unsteady sense, the inflectional velocity profile across the separation bubble can develop inviscid Kelvin-Helmhotlz instabilities and cause the shear layer to roll up. This leads to periodic vortex shedding and the required matching downstream [18], and can cause the separation bubble to move back and forth [28]. Further work detailing low Reynolds number flow over rigid airfoils can be found by Nagamatsu [29], Masad and Malik [30], and Schroeder and Baeder [31]. Low Aspect Ratio Wings Early work in low aspect ratio aerodynamics was sparked by an inability to fit experimental data with linear aerodynamics theories, as reported by Winter [32] for aspect ratios between 1.0 and 1.25. The measured lift is typically higher than predicted (similar to vortex lift drag [8], but is not included in the model. The magnitude of the pitching moment is slightly over-predicted by the CFD, though the data still falls within the error bars, the slopes match well, and the onset of nonlinear behavior (due to the low aspect ratio [3]) is well-predicted. The CFD is also able to predict the onset of stall (via a loss of lift) at about 210, but loses its predictive capability in the post-stall regime, as the flow is known to be highly unsteady. 1,4 1,4 1.2 1.2 1 1 0.8 0.8 0.6 06 , 0.4 data 0.4 data 0.2 CFD 0.2 CFD 0 ------ VLM 0 VLM -0.2 ... .......... .... ............-. 2 -0.2 ...................... -10 0 10 20 30 0 0.2 0.4 0.6 ac CD 0.1 6- 0. -0.1 4 E -0.2 2 data 2 -0.3 C-- VLM 0 / CFD -0.4 ------ VLM -0.5 F -2 ' -0.2 0 0.2 0.4 0.6 0.8 1 12 1.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 CL CL Figure 4-11. Computed and measured aerodynamic coefficients for a rigid MAV wing, Re = 85,000. The vortex lattice method is accurate at low angles of attack, but the slope is under- predicted (possibly due to an inability to compute the low-pressure cells at the wing tips, similar to vortex lift discrepancies seen in delta wings [27]), and the wing never stalls. The drag predicted by the vortex lattice method is necessarily augmented by a non-zero CDo (estimated response vector r: r =(u z F ') (7-3) where u is the solution to the system of finite element equations (composed of both displacements and rotations) at each free node, z is the shape of the flexible wing, and r is the vector of unknown horseshoe vortex circulations. The coupled system of equations G(r) is then: K-u-Q-T G(r)= z-zo-P.u =0 (7-4) SC--L The first row of G is the finite element analysis: K is the stiffness matrix assembled from the elemental matrices in Eq. (7-1), and appropriately reduced based upon fixed boundary conditions along the wing root. Q is an interpolation matrix that converts the circulation of each horseshoe vortex into a pressure, and subsequently into the transverse force at each free node. The second row of G is a simple grid regeneration analysis: Zo is the original (rigid) wing shape, and P is a second interpolation matrix that converts the finite element state vector into displacements at each free and fixed node along the wing. The third row of G is the vortex lattice method. C is an influence matrix depending solely on the wing geometry (computed through the combination of Eqs. (4-11) and (4-12)), and L is a source vector depending on the wing's outward normal vectors, the angle of attack, and the free stream velocity. Convergence of this system can typically be obtained within 25 iterations, and is defined when the logarithmic error in the wing's lift coefficient is less than -5. One potential shortcoming of this aeroelastic model can be seen in Figure 7-3, where the computed lift of a wing with no carbon fiber in the design domain (X = 0) is larger than the lift generated by the rigid wing (X = 1). This is due to a combination of membrane cambering towards the leading edge, and a depression of the trailing edge reflex region. In reality, however, 0.01 I0.1 numerical experimental 0.005 0.050 -0.005 0 005 -0.01 -0.01 -1 -0.5 0 0.5 1 2y/b Figure 5-4. Baseline BR shear strain (Exy), a = 15. Normalized out-of-plane displacements for the perimeter-reinforced wing are given in Figure 5-5. Deformations are slightly larger than with the BR wing (6%), and are dominated by the membrane inflation between the carbon fiber leading and trailing edges. The membrane apex occurs approximately in the middle of the membrane skin, despite the pressure gradient over the wing. This location is a function of angle of attack, as the peak will move slightly forward with increased incidence [74], [43]. The carbon fiber wing tip twists less than previously, thought to be a result of the fact that the wingtip is not free in a PR configuration, but attached to the trailing edge by the laminate perimeter. Some bending of the leading edge at the root can also be seen, but not in the BR wing (Figure 5-1). Correspondence between model and experiment is suitable, with the model again under-predicting wing deformation, but accurately locating the apex. Slight asymmetries in the measured wing profile (also evident in the BR wing) are probably a result of manufacturing errors (particularly in the application of the membrane skin tension), and not due to flow problems in the wind tunnel. As the amount of unconstrained membrane is greater in a PR wing than in a BR wing, chordwise strains (Figure 5-6) are much larger as well: peak stretching (3%) is located at the membrane/carbon fiber boundary towards the leading edge, as before. The magnitude and size of this high-extension lobe is over-predicted by the model. Both model and experiment show a region of compressive strain aft of this lobe, towards the trailing edge. This is a Possion strain material distribution) can be achieved through a nonlinear power law interpolation. This technique is known as the solid isotropic material with penalization method, or SIMP [105]. For the two-material wing considered above (membrane or carbon fiber), the stiffness matrix Ke of each finite element in Figure 7-2 can be computed as: Ke= (K.(1-P)-Km).XP+Km+P.K (7-1) where Kp and Km are the plate and membrane elements, respectively (the latter with zeros placed within rows and columns corresponding to bending degrees of freedom). 3 is a small number used to prevent singularity in the pure membrane element (due to the bending degrees of freedom), and Xe is the density of the element, varying from 0 (membrane) to 1 (carbon fiber). p is the nonlinear penalization power (typically greater than 3). A common criticism of this power law approach is that intermediate densities do not actually exist. This is a particular problem for the current application, where each element is either carbon fiber or membrane rubber. The physics of these two elements is completely different, as the carbon fiber is inextensible yet has resistance to bending and twisting, while the opposite is true for the latex. An equal combination of these two (equivalent to stating that the density within an element is 0.5), while computationally conceivable, is not physically possible. The wing topology will not represent a real structure until the density of each element is pushed to 1 (carbon fiber) or 0 (membrane). The power law's effectiveness as an implicit penalty is predicated upon a volume constraint: intermediate densities are unfavorable, as their stiffness is small compared to their volume [16]. No such volume constraint is utilized here, due to an uncertainty upon what this value should be. Furthermore, for aeronautical applications it is typically desired to minimize the mass of the wing itself, as discussed by Maute et al. [118]. Regardless, the nonlinear power (4-2) Ao where T is a matrix which transforms each element from a local coordinate system to a global system, Bp is the appropriate strain-displacement matrix [137], and Ao is the un-deformed area of the triangular element. Kp is a 9x9 matrix whose components reflect the out-of-plane displacement w and two in-plane rotations at the three nodes. Similarly, in-plane stretching of the laminates (a secondary concern, but necessarily included), is given by: NL Ap =ZQkhk (4-3) k=1 where Ap is a laminate matrix relating three in-plane stress resultants to three strains. Expressions similar to Eq. (4-2) are then formulated to compute Km, the 6x6 finite element stiffness matrix governing in-plane displacements u and v at the three nodes. Km and Kp are then combined to form the complete 15x15 shell stiffness matrix of each element, KI. Drilling degrees of freedom are not included. Though some wing designs may use un-symmetric laminates, coupling between in-plane and out-of-plane motions is not included. Loads Model Validation/Estimation. The following method is used to both validate the model presented above, and identify the relevant material properties of the laminates. A series of weights are hung from a batten-reinforced wing (with 2 layers of bi-directional carbon fiber oriented 450 to the chord line and 1 layer uni-directional battens, but no membrane skin) at nine locations: the two wing tips, the trailing edges of the six battens, and the leading edge, as shown in Figure 4-2. VIC is used to measure the resulting wing displacements. A linear curve is fit through the load-displacement data of all nine points due to all nine loadings. The slopes of these curves are used to populate the influence matrix in Table 4-1: the diagonal gives the motion KP = T f B T DP.Bpn dA.T T and experiment is adequate up to w/R = 0.22 (slightly lower than the value given by Pujara and Lardner [140]), when the model begins to under-predict the inflated shape. Hyperelastic effects appear after this point: Hooke's law over-predicts the stress for a given strain level (Figure 4-4), and thus the membrane's resistance to a transverse pressure. For the case with membrane pre-tension, VIC is used to measure the pre-strain in the membrane skin (applied radially [147]), the average of which is then used for finite element computations. The mean pre-strain is 0.044, with a coefficient of variation of 3.1%. For this case, the linear model now has a small range of validity, up to w/R = 0.15. Prior to this deformation level, linear and nonlinear models predict the same membrane inflation. The response then becomes nonlinear, due to the advent of finite strains, but also because a relevant portion of the uniform non-conservative pressure is now directly radially, rather than vertically. The nonlinear model and experiment now diverge at w/R = 0.3: the addition of a pre-tension field increases the range of validity of both the linear and the nonlinear membrane models. Skin Pre-tension Considerations A state of uniform membrane pre-tension, though numerically convenient [15] [80] [14], is essentially impossible to actually fabricate on a MAV wing. One reason is that the latex sheets used on the MAVs in this work are not much wider than the wingspan, subjecting the state of pre-stress to end-effects. This may perhaps be remedied with larger sheets and a biaxial tension machine, which hardly seems worth the effort for MAV construction. Another problem is the fact that the wing is not a flat surface. Even if a state of uniform pre-tension were attainable, it cannot be transferred to the wing without significant field distortions, particularly due to the camber in the leading edge. A typical pre-strain field is given in Figure 4-6, as measured by the VIC system off of a BR wing in the chordwise direction. The contour on the left is the pre-strain field after the spray adhesive has dried, but before the latex surrounding the wing has been de- ACKNOWLEDGMENTS Thank you to Dr. Peter Ifju, who offered his guidance on a countless number of research projects, and still let me work on the ones he thought were dumb. Unquestionably the coolest, smartest, most up-beat professor I've ever been around. Thank you to Dr. Rick Lind, for consistently pointing out when I need to shave, or get a haircut, or more frequently, both. Thank you to Dr. Roberto Albertani, for sharing with me his passion for all things wind tunnel related, and for sharing his equipment up at the REEF. Thank you to Dr. Raphael Haftka and Dr. David Bloomquist for serving on my committee and sitting through my long, scientifically-questionable presentations without complaining. Thank you to Dr. Dragos Viieru, for imparting me with his vast knowledge of CFD. Thank you to Dr. Wei Shyy, for all his help my first few semesters of grad school. A final thank you to all the people who hung around the labs I worked in. Frank Boria, who helped teach me the real names for various tools and hardware, which had previously been known to me only as shiny metal things. A thanks-in-advance to Frank for taking all of my future phone calls concerning mortgages, insurance, child rearing, etc, no matter how distraught and hysterical they may be. Mujahid Abdulrahim, for discussing with me the ethics of returning a rental car completely caked in mud, and going in reverse through a drive-thru. Wu Pin, for relating countless unintentionally funny and creepy stories that I'll never forget, despite my best efforts. I'll always wonder how you got into this country. [158] Chen, T., Wu, S., "Multiobjective Optimal Topology Design of Structures," Computational Mechanics, Vol. 21, No. 6, 2998, pp. 483-492. [159] Haftka, R., Girdal, Z, Elements of Structural Optimization, Kluwer, Dordrecht, The Netherlands, 1992. [160] Maute, K., Nikbay, M., Farhat, C., "Coupled Analytical Sensitivity Analysis and Optimization of Three-Dimensional Nonlinear Aeroelastic Systems," AIAA Journal, Vol. 39, No. 11, 2001, pp. 2051-2061. [161] Lyu, N., Saitou, K., "Topology Optimization of Multicomponent Beam Structure via Decomposition-Based Assembly Synthesis," Journal of Mechanical Design, Vol. 127, No. 2, 2005, pp. 170-183. [162] Hsu, M., Hsu, Y., "Interpreting Three-Dimensional Structural Topology Optimization Results," Computers and Structures, Vol. 83, No. 4, 2005, pp. 327-337. [163] Rohl, P., Schrage, D., Mavris, D., "Combined Aerodynamic and Structural Optimization of a High-Speed Civil Transport Wing," AIAA Structures, Structural Dynamics, andMaterials Conference, New Orleans, LA, April 18-21, 1995. wing (Figure 6-6) and a PR wing (Figure 6-7) at 120 angle of attack. As before, the membrane skin is slack. Aeroelastic trends are expected to repeat every 900, and will be symmetric about the line 01 = 02. This latter point is only true because bending-extension coupling in non- symmetric laminates is ignored, though the effect of its inclusion would be very small as the wing is subjected mostly to normal pressure forces. CLax Cmm L/D -0.0115 5.4 90 45 0 0 01 90 45 0 0 1 90 45 0 0 0 9 2 2 02 Figure 6-6. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a BR wing, a = 12. For the BR wing, efficiency is maximized and the lift slope is minimized when the fibers make 450 angles with the chord and span directions. Static stability is improved when fibers align with the chord. The response surface of the two stability derivatives are very noisy, suggesting possible finite differencing errors, and all three surfaces in Figure 6-6 show little variation (only Cma of the BR wing can be varied by more than 5%). Unlike any of the tailoring studies discussed above, the PR wing shows the same overall trends and optima as the BR wing. The surfaces for the PR wing, however, are much smoother but have less overall variation. Of the sampled laminate designs, [15]2 and [75]2 will exhibit the greatest bend-twist coupling, yet neither are utilized by the membrane wings. This fact, along with the similarity between the PR and the BR surfaces, suggest that the orientation of a plain weave laminate with two layers is too stiff to have much impact on the aerodynamics, which is dominated by membrane inflation/stretching. The use of bi-directional plain weave is not the most effective [31] Schroeder, E., Baeder, J., "Using Computational Fluid Dynamics for Micro Air Vehicle Airfoil Validation and Prediction," AIAA Applied Aerodynamics Conference, Toronto, Canada, June 6- 9, 2005. [32] Winter, H., "Flow Phenomena on Plates and Airfoils of Short Span," NACA Technical Report, TR 539, 1935. [33] Sathaye, S., Yuan, J., Olinger, D., "Lift Distributions on Low-Aspect-Ratio Wings at Low Reynolds Numbers for Micro-Air-Vehicle Applications," AIAA Applied Aerodynamics Conference and Exhibit, Providence, RI, Aug. 16-19, 2004. [34] Pellettier, A., Mueller, T., "Low Reynolds Number Aerodynamics of Low Aspect Ratio Thin/Flat/Cambered-Plate Wings," Journal ofAircraft, Vol. 37, No. 5, 2000, pp. 825-832. [35] Bartlett, G., Vidal, R., "Experimental Investigation of Influence of Edge Shape on the Aerodynamic Characteristics of Low Aspect Ratio Wings at Low Speeds," Journal of Aeronautical Sciences, Vol. 22, No. 8, 1955, pp. 517-533. [36] Polhamus, E., "A Note on the Drag Due to Lift of Rectangular Wings of Low Aspect Ratio," NACA Technical Report, TR 3324, 1955. [37] Kaplan, S., Altman, A., 01, M., "Wake Vorticity Measurements for Low Aspect Ratio Wings at Low Reynolds Numbers," Journal ofAircraft, Vol. 44, No. 1, 2007, pp. 241-251. [38] Viieru, D., Albertani, R., Shyy, W., Ifju, P., "Effect of Tip Vortex on Wing Aerodynamics of Micro Air Vehicles," Journal ofAircraft, Vol. 42, No. 6, 2005, pp. 1530-1536. [39] Mueller, T., DeLaurier, J., "Aerodynamics of Small Vehicles," AnnualReview ofFluid Mechanics, Vol. 35, No. 35, 2003, pp. 89-111. [40] Gursul, I., Taylor, G., Wooding, C., "Vortex Flows Over Fixed-Wing Micro Air Vehicles," AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 14-17, 2002. [41] Cosyn, P., Vierendeels, J., "Numerical Investigation of Low-Aspect-Ratio Wings at Low Reynolds Numbers," Journal ofAircraft, Vol. 43, No. 3, 2006, pp. 713-722. [42] Brion, V., Aki, M., Shkarayev, S., "Numerical Simulation of Low Reynolds Number Flows Around Micro Air Vehicles and Comparison against Wind Tunnel Data," AIAA Applied Aerodynamics Conference, San Francisco, CA, June 5-8, 2006. [43] Stanford, B., Sytsma, M., Albertani, R., Viieru, D., Shyy, W., Ifju, P., "Static Aeroelastic Model Validation of Membrane Micro Air Vehicle Wings," AIAA Journal, Vol. 45, No. 12, 2007, pp. 2828-2837. [44] Zhan, J., Wang, W., Wu, Z., Wang, J., "Wind-Tunnel Experimental Investigation on a Fix-Wing Micro Air Vehicle," Journal ofAircraft, Vol. 43, No. 1, 2006, pp. 279-283. C om posite L am inated Shells......................................... ............................................52 M em brane M modeling ................. .................................. ................ .. ............. 56 Skin Pre-tension Considerations..................... ....... .............................. 62 F lu id S olv ers .................................................................................................................... 66 V ortex Lattice M methods ............................................................................. 66 Steady N avier-Stokes Solver ........................................... ....... ...................... 67 Fluid M odel Com prisons and V alidation ........................................... .....................70 A eroelastic Coupling .............. ......... ............................. ............ 72 M moving G rid T technique ......................................................................... ...................72 N um erical Procedure ..................................................... ........ .. ...... .... 73 5 BASELINE WING DESIGN ANALYSIS....................................................................... 75 W ing D eform ation .................................................................................... .. ................ .. 7 5 Aerodynamic Loads ....................................................... .. ....................... 83 F low Structures....................................................88 6 A ER O ELA STIC TA IL O R IN G .............................................................. ......... .................97 OFAT Simulations ............... .............................. ............................. 98 M em brane P re-T pension ......................................................................... ....................99 Single Ply L am inmates .................................................... .. ........................ 102 D double Ply L am inmates ......................... ............................................. 103 Batten Construction .......... ............... ..... ................ ...... .......... .. ........ .... 105 Full F actorial D designed E xperim ent......... ................. ...................................... ...............107 Experimental Validation of Optimal Design Performance ............................................116 7 AEROELASTIC TOPOLOGY OPTIMIZATION.................................... ............... 122 C om putational F ram ew ork ......... .. ................... ...................................... ........................125 M material Interpolation .......... ............................................. .... ......... 125 Aeroelastic Solver .................................... ... .. ......... ....... .... 128 A djoint Sensitivity A nalysis.................................................................................... 130 O ptim ization P rocedure............................................................................... ........... 133 Single-O objective O ptim ization ...................... .. .. ......... .. ........................ ......................134 M ulti-O objective O ptim ization ...................................................................... ..................149 CONCLUSIONS AND FUTURE WORK ................................. .................................161 R E F E R E N C E S .........................................................................167 B IO G R A PH IC A L SK E T C H ...................... .. .. ......... .. ............................... ......................... 179 6 convexity of the Pareto fronts. Only compromises between 2 objective functions are considered in this work. The corresponding performance of each design is given in Table 6-2. The value in each cell is predicated upon the label at the top of each column; the performance of the second objective function (row-labeled) is found in the cell appropriately located across the diagonal. Table 6-1. Optimal MAV design array with compromise designs on the off-diagonal, a = 12: design description is (wing type, Nx, Ny, number of plain weave layers). max L/D min mass max CL min CD min Cma max CL. min CL. max L/D BR,0,0,1L BR,0,0,1L PR,10,0,1L BR,0,0,1L BR,20,0,3L PR,20,0,2L BR,0,0,1L min mass BR,0,0,1L PR,20,20,1L PR,0,0,1L BR,0,0,1L PR,0,10,1L PR,0,10,1L BR,20,10,1L max CL PR,10,0,1L PR,0,0,1L PR,0,0,1L BR,0,10,1L PR,0,0,1L PR,0,0,1L BR,20,10,1L min CD BR,0,0,1L BR,0,0,1L BR,0,10,1L BR,0,0,1L BR,20,0,3L BR,10,20,3L BR,10,0,1L min Cma BR,20,0,3L PR,0,10,1L PR,0,0,1L BR,20,0,3L PR,0,10,1L PR,0,10,1L BR,20,0,3L max CL. PR,20,0,2L PR,0,10,1L PR,0,0,1L BR,10,20,3L PR,0,10,1L PR,0,10,1L BR,0,20,3L min CLa BR,0,0,1L BR,20,10,1L BR,20,10,1L BR,10,0,1L BR,20,0,3L BR,0,20,3L BR, 20,10,1L Table 6-2. Optimal MAV design performance array, a = 12: off-diagonal compromise design performance is predicated by column metrics, not rows. max L/D min mass (g) max CL min CD min Cma max CL minCLl max L/D 5.49 4.36 0.780 0.112 -0.015 0.054 0.047 min mass 5.49 4.10 0.817 0.112 -0.019 0.057 0.043 max CL 4.84 4.18 0.817 0.145 -0.018 0.056 0.043 min CD 5.49 4.36 0.716 0.112 -0.014 0.052 0.045 min Cma 5.05 4.16 0.817 0.134 -0.019 0.057 0.049 max CL. 4.90 4.16 0.817 0.141 -0.019 0.057 0.050 min CLa 5.49 4.31 0.673 0.119 -0.015 0.049 0.043 For reference purposes, the design performance of the rigid wing (at 120 angle of attack) is: L/D = 4.908, mass = 6.36 grams, CL = 0.6947, CD = 0.1415, Cm, = -0.0147, and CL, = 0.0507. As above, at no point does the rigid wing represent an optimum design (compromise or otherwise). The compromise between minimizing the lift slope, and maximizing the lift slope is identified by located the design closest to the normalized CL, of 0.5. This is found by a BR wing design with peak pre-tension normal to the battens to limit adaptive washout, but no pre-tension in the chordwise direction to allow for camber and lift via inflation. Both BR and PR wings are equally-represented throughout the design array, with the exception of designs requiring load alleviation: all compromises involving drag or lift slope minimization utilize a BR wing. The acoustic disturbances (noise emitted from the turbulent boundary layer along the tunnel walls, the wind tunnel fan, etc. [24]). Mounting techniques are also presumed to cause an incorrect relationship between Reynolds number and the zero-lift angle of attack among several sets of published data [122]. Sensitivity is another concern, particularly in drag force measurements which may be as low as 0.025 N (computed for a wing with a chord of 100 mm and a Reynolds number of 5-104). An electrical resistance strain gage sting balance is typically used for force and moment measurements. While strain gages typically provide the greatest sensitivity and simplicity, they are also prone to temperature drift, electromagnetic interference, creep, and hysteresis. An internal Aerolab 01-15 6-component strain gage sting balance is used for force/moment measurements in the current work. Wind tunnel models are mounted to the sting balance by a simple jaw mechanism. Each of the six channels is in a full Wheatstone-bridge configuration, with five channels dedicated to forces, and one to a moment. Two forces are coincident with the vertical plane of the model (traveled during an a-sweep), two are in the plane normal to the previous (traveled during a P-sweep), one force is in the axial direction, and the moment is dedicated to roll. Data acquisition is done with a NI SCXI 1520 8 channel programmable strain gage module with full bridge configuration, 2.5 excitation volts, and a gain of 1000. Other modules included in the system are a SCXI 1121 signal conditioner, 1180 feed- through with 1302 breakout and 1124 D/A module. A NI 6052 DAQPad firewire provides A/D conversion, multiplexing, and the PC connection. For a given flight condition, the output signals from the six components are sampled at 1000 Hz for 2 seconds. The average of this data is sent to one module for calculation of the relevant aerodynamic coefficients, and the standard deviation of the data is stored for further uncertainty analysis. 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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AEROELASTIC ANALYSIS AND OPTIMIZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By Bret Kennedy Stanford May 2008 Chair: Peter Ifju Major: Aerospace Engineering Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separation erodes wing efficiency, the wings are susceptible to rolling instabilities, wind gusts can be the same size as the flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divided into two categories: load-alleviating or load-augmenting. This depends on the wing's topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms camberingg, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical designed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs. 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[22] Kellogg, M., Bowman, J., "Parametric Design Study of the Thickness of Airfoils at Reynolds Numbers from 60,000 to 150,000," AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 5-8, 2004. [23] Laitone, E., "Wind Tunnel Tests of Wings at Reynolds Numbers Below 70,000," Experiments in Fluids, Vol. 23, No. 5, 1997, pp. 405-409. [24] Mueller, T., "The Influence of Laminar Separation and Transition on Low Reynolds Number Airfoil Hysteresis," Journal ofAircraft, Vol. 22, No. 9, 1985, pp. 763-770. [25] Gad-el-Hak, M., "Micro-Air-Vehicles: Can They be Controlled Better?" Journal ofAircraft, Vol. 38, No. 3, 2001, pp. 419-429. [26] Shyy, W., Lian, Y., Tang, J., Viieru, D., Liu, H., Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, New York, NY, 2008. [27] Katz, J., Plotkin, A., Low-SpeedAerodynamics, Cambridge University Press, Cambridge, UK, 2001. [28] Lian, Y., Shyy, W., Viieru, D., Zhang, B., "Membrane Wing Mechanics for Micro Air Vehicles," Progress in Aerospace Sciences, Vol. 39, No. 6, 2003, pp. 425-465. [29] Nagamatsu, H., "Low Reynolds Number Aerodynamic Characteristics of Low Drag NACA 63- 208 Airfoil," Journal ofAircraft, Vol. 18, No. 10, 1981, pp. 833-837. [30] Masad, J., Malik, M., "Link Between Flow Separation and Transition Onset," AIAA Journal, Vol. 33, No. 5, 1995, pp. 882-887. All nodes that lie on the wing root are constrained appropriately as necessitated by wing symmetry. Figure 4-1. Unstructured triangular mesh used for finite element analysis, with different element types used for PR and BR wings. Composite Laminated Shells Discrete Kirchhoff triangle plate elements [136] are use to model the bending/twisting behavior of the carbon fiber areas of the wings: leading edge, root, perimeter, and battens. Due to the comparative stiffness of these materials, linear behavior is assumed. The orthotropy of the plates is introduced by the flexural stiffness matrix of the laminates, Dp, relating three moments (two bending, one twisting) to three curvatures: NL D = Qk-(h/12 + hk Z) (4-1) k=1 where NL is the number of layers in the laminate, hk is the thickness of the kth ply, Zk is the normal distance from the mid-surface of the laminate to the mid-surface of the kth ply, and Qk is the reduced constituitive matrix of each ply, expressed in global coordinates. Qk depends upon the elastic moduli in the 1 and 2 directions (equal for the bi-directional laminate, but not so for the uni-directional) El and E2, the Poisson's ratio v12, and the shear modulus G12. The finite element stiffness matrix pertaining to bending/twisting is then found to be: [74] Smith, R., Shyy, W., "Computation of Aerodynamic Coefficients for a Flexible Membrane Airfoil in Turbulent Flow: A Comparison with Classical Theory," Physics ofFluids, Vol. 8, No. 12, 1996, pp. 3346-3353. 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[80] Sugimoto, T., "Analysis of Circular Elastic Membrane Wings," Transactions of the Japanese Society ofAerodynamics andSpace Sciences, Vol. 34, No. 105, 1991, pp. 154-166. [81] Charvet, T., Hauville, F., Huberson, S., "Numerical Simulation of the Flow Over Sails in Real Sailing Conditions," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, No. 1, 1996, pp. 111-129. [82] Schoop, H., Bessert, N., Taenzer, L., "On the Elastic Membrane in a Potential Flow," International Journal for Numerical Methods in Engineering, Vol. 41, No. 2, 1998, pp. 271- 291. [83] Stanford, B., Albertani, R., Ifju, P., "Static Finite Element Validation of a Flexible Micro Air Vehicle," Experimental Mechanics, Vol. 47, No. 2, 2007, pp. 283-294. [84] Ferguson, L., Seshaiyer, P., Gordnier, R., Attar, P., "Computational Modeling of Coupled Membrane-Beam Flexible Wings for Micro Air Vehicles," AIAA Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, April 23-26, 2007. [85] Smith, R., Shyy, W., "Incremental Potential Flow Based Membrane Wing Element," AIAA Journal, Vol. 35, No. 5, 1997, pp. 782-788. [86] Heppel, P., "Accuracy in Sail Simulation: Wrinkling and Growing Fast Sails," High Performance Yacht Design Conference, Auckland, New Zealand, December 4-6, 2002. [87] Fink, M., "Full-Scale Investigation of the Aerodynamic Characteristics of a Model Employing a Sailwing Concept," NASA Technical Report, TR 4062, 1967. section and increasing the flow velocity (and the coefficients) in the vicinity of the model [128]. Wake blockage occurs when the flow outside of the model's wake must increase, in order to satisfy the flow continuity in a closed tunnel. In an open freestream, the velocity outside of the wake would be equal to the freestream velocity. The effect of wake blockage is proportional to the wake size, and therefore proportional to drag [3]. Streamline curvature blockages are the effect of the tunnel walls on the streamlines around the model. The streamlines are compressed, increasing the effective camber and lift [129]. Such corrections generally decrease both lift and drag, while the pitching moment is made less negative, with percentage changes on the order of 2-3%. Finally, flexibility effects within the wind tunnel setup must be accounted for. These effects are primarily caused by the elasticity of the internal strain gage sting balance; under load the wind tunnel model will pitch up via a rigid body rotation. Visual image correlation (described below) is used to measure the displacement at points along the wing known to be nominally rigid (specifically, the sting balance attachment points along the wing root). This data then facilitates the necessary transformations and translations of the wing surface, and is used to correct for the angle of attack. Aa is a positive monotonically increasing function of both lift and dynamic pressure, and can be as large as 0.70 at high angles of attack [47]. Visual Image Correlation Wind tunnel model deformation measurements are a crucial experimental tool towards understanding the role of structural composition upon aerodynamic performance of a MAV wing. The flexible membrane skin generally limits applications to non-contacting optical methods, several of which have been reported in the literature. Galvao et al. [89] use stereo photogrammetry for displacement measurements of a membrane wing, with a reported resolution between 35 and 40 [tm. Data is available at discrete markers placed along the wing. Projection compromise between conflicting metrics, filling in the trade-off curves. While thorough exploration of this aeroelastic tailoring design space can provide a fundamental understanding of the relationship between spatial stiffness distribution and aerodynamic performance in a flexible MAV wing, further steps towards generality can be achieved by removing the constraint that the wing structures must utilize a BR or a PR design. Topology optimization is typically used to find the location of holes within a homogenous structure, by minimizing compliance under a constraint upon the volume fraction [16]. Here it is used to find the location of membrane skin within a carbon fiber skeleton that will optimize a given aerodynamic objective function. This work will be able to highlight wing topologies with superior efficacy to those designs considered above (for example, a wing with better gust suppression qualities than the BR wing), as well as designs that strike a compromise between conflicting metrics (for example, a topological combination of the BR and the PR wings). While the results may be more rewarding than those obtained from tailoring, aeroelastic topology optimization is significantly more complex. Tailoring requires 5-10 sizing and stiffness variables, but the topology optimization may utilize thousands of variables: the wing is divided into a series of panels, each of which may be membrane or carbon fiber. This necessitates a gradient-based algorithm, while evolutionary algorithms or response surface approaches are feasible for the former problem. Both aeroelastic tailoring and topology optimization are effective tools for exploiting the passive shape adaptation of flexible MAV wings, but the computational cost is prohibitive. It is not uncommon for aeroelastic optimization studies to require hundreds, or even thousands, of function evaluations. Numerical modeling of flexible MAV wings is very challenging and expensive: flow separation, transition, and reattachment [17], vortex shedding and pairing [18], approach described below), an unconstrained Fletcher-Reeves conjugate gradient algorithm [159] is employed. Step size is kept constant, at a reasonably small value to preserve the fidelity of the sensitivity analysis. The upper and lower bounds of each design variable (1 and 0) are preserved by restricting the step size such that no density variable can leave the design space, forced to lie on the border instead. In order to increase the chances of locating a global optimum (rather than a local optimum), each optimization is run with three distinct initial designs: Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1. A pure membrane wing (Xo = 0) cannot be considered for the reasons discussed above. Six objective functions are considered: maximum lift, minimum drag, maximum L/D, maximum CL,, minimum CL,, and minimum Cma. Flight speed is kept constant at 13 m/s, but both 3 and 12 angles of attack are considered, with a Aa of 1 for finite differences. Both the reflex airfoil seen in Figure 5-16 and a singly-curved airfoil are used, though aspect ratio, planform, and peak camber are unchanged. The stiffness of the carbon fiber laminates is as computed by Figure 4-3, and the pre-stress of the membrane is fixed in both the chordwise and spanwise directions at 7 N/m. No correction is applied to the free trailing edge, as such a computation would render the pre-stress in this location very small, leading to unbounded behavior of the linear membrane model. The circular radius rmin for the mesh-independent filter is fixed at 4% of the chord. Single-Objective Optimization A typical convergence history of the aeroelastic topology optimizer can be seen in Figure 7-5, for a reflex wing at 3 angle of attack, with a maximum L/D objective function. The initial guess is an intermediate density of 0.5. Within 4 iterations, the optimizer has removed all of the carbon fiber adjacent to the root of the wing, with the exception of the region located at three- quarters of the chord, which corresponds to the inflection point of the reflex airfoil. The material unusually steep (an instability discussed by Ormiston [15], among others), as the strains in the skin are low enough to allow for large changes in camber. Greenhalgh and Curtiss conduct wind tunnel testing to study the effect of planform on a membrane wing; only a parabolic planform is capable of sustaining flight loads without the aid of a trailing edge support member [88]. Galvao et al. [89] conduct tests on a membrane sheet stretched between two rigid posts, at Reynolds numbers between 3-104 and 105. The results show a monotonic increase in membrane camber with angle of attack and dynamic pressure, up to stall, as well as the aforementioned steep lift slopes. De-cambering of the wing as the pressure on the upper surface increases due to imminent flow separation is seen to ameliorate the stall behavior, as compared to a rigid plate. Flow visualization of a batten-reinforced membrane MAV wing exhibits a weaker wing tip vortex system than rigid wings [13], possibly due to energy conservation requirements [90]. Parks measures the vortex core of a BR wing 5% to 15% higher above the wing than for the rigid case, though the flexible wing is seen to have a denser core-distribution of velocity for moderate angles of attack [91]. Gamble and Reeder [92] measure the flow structures resulting from interactions between propeller slipstream and a BR wing. The rigid wing spreads the axial component of the propwash further along the wing (resulting in a higher measured drag), whereas the membrane wing can absorb the downwash and upwash. A region of flow separation is measured at the root of the rigid wing, significantly larger and stronger than that measured from the membrane wing; the superiority decreases with larger Reynolds numbers. Albertani et al. [9] detail loads measurements of both BR and PR wings, with dramatic improvements in longitudinal static stability of both membrane wings over their rigid counterpart. The BR wing has a noticeably smoother lift behavior in the stalled region, though neither deforms into a particularly optimal aerodynamic shape: both incur a drag penalty. attention is now turned to the computed flow structures. No experimental validation is available for this work, though whenever possible the results will be correlated to data in the previous two sections, or results in the literature. Experimental flow visualization work for low aspect ratios and low Reynolds number is given by Tang and Zhu [6] and Kaplan et al. [37]. Work done explicitly on MAV wings is given by Gursul et al. [40], Parks [91], Gamble and Reeder [92], and Systma [153]. The pressure distributions and flow structures are given in Figure 5-16 at 00 angle of attack for the upper/suction wing surface of all three baseline wing designs. The plotted streamlines reside close to the surface, typically within the boundary layer. For the rigid wing, a high pressure region is located close to the leading edge, corresponding to flow stagnation. This is followed by pressure recovery (minimum pressure), located approximately at the camber of each rigid wing section. Pressure recovery is followed by a mild adverse pressure gradient, which is not strong enough to cause the flow to separate. A further decrease in Reynolds number has been shown to cause mild flow separation over the top surface for 00 however [14], [153]. A small locus of downward forces are present over the negatively-cambered region (reflex) of the airfoil, helping to offset the nose-down pitching moment of the remainder of the rigid MAV wing, as discussed above. The reflex can also help improve the wing efficiency, compared to positively-cambered wings [55]. There is positive lift of this wing at 00 (Figure 5-12), resulting in a mild tip vortex swirling system. The low pressure cells at the wing tip are not yet evident. Aeroelastic pressure redistributions of the upper surface of the BR wing are seen in the form of three high-pressure lobes at the carbon fiber/membrane boundary interface towards the leading edge. The membrane inflation in between each batten (Figure 5-1) results in a slight tangent discontinuity in the wing surface. This forces the flow to slow down and redirect itself 4. The range of flyable (statically stable) CG locations is generally only a few millimeters long, which represents a strenuous weight management challenge [9]. These problems, along with a broad range of dynamics and control issues, can be alleviated through the appropriate use of wing shape adaptation. Active morphing mechanisms have been successfully used on a small class of unmanned air vehicles [1], but the limited energy budgets and size constraints of micro air vehicles make such an option, at present writing, infeasible. As such, the current work is restricted to passive shape adaptation. Passive shape adaptation can be successfully built into a MAV wing through the use of a flexible membrane skin [10]. The basic structure of these vehicles is built around a composite laminate skeleton. Bi-directional graphite/epoxy plain weave or uni-directional plies are usually the materials of choice, due to durability, low weight, high strength, and ease of fabrication: all qualities well-exploited in the aviation industry [11]. The carbon fiber skeleton is affixed to an extensible membrane skin, of which several choices are available: latex, silicone, plastic sheets, or polyester [12]. The distribution of carbon fiber and membrane skin in the wing determines the aeroelastic response, and is demonstrated by two distinct designs. The first utilizes thin strips of uni-directional carbon fiber imbedded within the membrane skin, oriented in the chordwise direction (Figure 1-1). The trailing edge of the batten-reinforced (BR) design is unconstrained, and the resulting nose-down geometric twist of each flexible wing section should alleviate the flight loads: decrease in CD, decrease in CL,, delayed stall (as compared to a rigid wing) [13]. b Figure 1-1. Batten-reinforced membrane wing design. Op- _74 [88] Greenhalgh, S., Curtiss, H., "Aerodynamic Characteristics of a Flexible Membrane Wing," AIAA Journal, Vol. 24, No. 4, 1986, pp. 545-551. [89] Galvao, R., Israeli, E., Song, A., Tian, X., Bishop, K., Swartz, S., Breuer, K., "The Aerodynamics of Compliant Membrane Wings Modeled on Mammalian Flight Mechanics," AIAA Fluid Dynamics Conference and Exhibit, San Francisco, CA, June 5-8, 2006. [90] Pennycuick, C., Lock, A., "Elastic Energy Storage in Primary Feather Shafts," Journal of ExperimentalBiology, Vol. 64, No. 3, 1976, pp. 677-689. [91] Parks, H., "Three-Component Velocity Measurements in the Tip Vortex of a Micro Air Vehicle," Masters Thesis, School of Engineering and Management, Air Force Institute of Technology, Wright Patterson Air Force Base, OH, 2006. [92] Gamble, B., Reeder, M., "Experimental Analysis of Propeller Interactions with a Flexible Wing Micro Air Vehicle," AIAA FluidDynamics Conference and Exhibit, San Francisco, CA, June 5- 8, 2006. [93] Stults, J., Maple, R., Cobb, R., Parker, G., "Computational Aeroelastic Analysis of a Micro Air Vehicle with Experimentally Determined Modes," AIAA AppliedAerodynamics Conference, Toronto, Canada, June 6-9, 2005. [94] Ifju, P., Ettinger, S., Jenkins, D., Martinez, L., "Composite Materials for Micro Air Vehicles," Society for the Advancement of Material and Process Engineering Annual Conference, Long Beach, CA, May 6-10, 2001. [95] Frampton, K., Goldfarb, M., Monopoly, D., Cveticanin, D., "Passive Aeroelastic Tailoring for Optimal Flapping Wings," Proceedings of Conference on Fixed, Flapping, and Rotary Wing Vehicles at Very Low Reynolds Numbers, South Bend, IN, June 5-7, 2000. [96] Snyder, R., Beran, P., Parker, G., Blair, M., "A Design Optimization Strategy for Micro Air Vehicles," AIAA Structures, Structural Dynamics, andMaterials Conference, Honolulu, HI, April 23-26, 2007. [97] Allen, M., Maute, K., "Probabilistic Structural Design of UAVs under Aeroelastic Loading," AIAA "Unmanned Unlimited" Conference, San Diego, CA, September 15-18, 2003. [98] Weisshaar, T., Nam, C., Batista-Rodriguez, A., "Aeroelastic Tailoring for Improved UAV Performance," AIAA Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, April 20-23, 1998. [99] Garrett, R., The Symmetry of Sailing: The Physics of Sailingfor Yachtsmen, Adlard Coles, Dobbs Ferry, NY, 1996. [100] Eden, M., The Magnificent Book of Kites: Explorations in Design, Construction, Enjoyment, andFlight, Sterling Publishing, New York, NY, 2002. design won't change. Secondly, two local optima exist in the design space of Figure 7-3, which may prevent the gradient-based optimizer from converging to a 0-1 material distribution. To counteract this problem, an explicit penalty on intermediate densities is added to the objective function, as discussed by Chen and Wu [158]: N, R. sin(X, .7r) (7-2) 1=i1 where R is a penalty parameter appropriately sized so as not to overwhelm the aerodynamic performance of the wing topology. This penalty is only added when and if the aeroelastic optimizer has converged upon a design with intermediate densities, as will be discussed below. Aeroelastic Solver Due to the large number of expected function evaluations (- 200) needed to converge upon an optimal wing topology, and the required aeroelastic sensitivities (computed with an adjoint method), a lower-fidelity aeroelastic model (compared to that utilized in Chapters 5 and 6) must be used for the current application. An inviscid vortex lattice method (Eq. (4-11)) is coupled to a linear orthotropic plate model and a linear stress stiffening membrane model (Eq. (4-4)). The latter module is perfectly valid in predicting membrane inflation as long as the state of pre-stress is sufficiently large, as seen in Figure 4-5. Furthermore, in-plane stretching of the laminate is ignored; only out-of-plane displacements (as well as in-plane rotations in the laminate) are computed over the entire wing. The vortex lattice method is reasonably accurate as well, despite the overwhelming presence of viscous effects within the flow. As seen in Figure 4-11, the lift slope is consistently under-predicted due to an inability to model the large tip vortices [3], and the drag is under- predicted at low and high angles of attack due to separation of the laminar boundary layer [4]. Aeroelastic coupling is facilitated by considering the system as defined by a three field a = 3, CL = 0.312 = 12, CL = 0.675 a = 3', CL = 0.561 a = 12', CL = 0.947 CL,rigid = 0.254 CL,rigid = 0.604 CL,rigid = 0.487 CL,rigid = 0.842 Figure 7-8. Affect of angle of attack and airfoil upon the optimal CL topology. For the cambered wing (singly-curved airfoil, right two plots of Figure 7-8), the lift over the rigid wing is, as expected, much larger than found in the reflex wings, but adequate stability becomes critical. With the removal of the negatively-cambered portion of the airfoil, most of the forces generated over this wing will be positive, and the topology optimizer can no longer gain additional lift via wash-in. Imbedding batten structures in the trailing edge will now result in washout, surely decreasing the lift. As such, the optimizer produces a trailing edge member that outlines the planform and connects to the root (similar to the perimeter-reinforced wing designs), restraining the motion of the trailing edge and inducing an aerodynamic twist. Unlike the PR wing, this trailing edge reinforcement does not extend continuously from the root to the tip, instead ending at 65% of the semi-span. This is then followed by a trailing edge batten that extends into the membrane skin, similar to the designs seen for the reflex wing in Figure 7-8. Why such a configuration should be preferred over the PR wing design for lift enhancement will be discussed below. As before, increasing the angle of attack has little bearing on the optimal topology, again increasing the size of the trailing edge batten. A potential increase in lift by 15% over the rigid wing is indicated at the lower angle of attack. Similar results are given in Figure 7-9, with L/D maximization as the topology design metric. Presumably due to the conflictive nature of the ratio, the wing topology that maximizes L/D is a strong function of angle of attack. For the reflex wing at lower angles, the optimal their presence is peculiar. Basic membrane inflation mechanics indicates large extension at the boundaries rather than compression [145] (as is computed by the model). The compression may be membrane wrinkling (which, again, is not evident from Figure 5- 5, or may be an error in the VIC strain computations, potentially caused by the large displacement gradients in this area of the wing. A third possibility is that the VIC is measuring a bending strain at this point, where the radius of curvature is close to zero. The latex skin, though modeled as a membrane, does have some (albeit very small) bending resistance due to its finite thickness. The anti-symmetric shear strain field (Figure 5-8) shows good correspondence between model and experiment, with accurate computations in-board, but slight under- predictions of the high shear closer to the wingtip. x 10-3 numerical experimental 0.02 20 0 ^0 --0.01- -1 -0.5 0 0.5 1 2 /b Figure 5-7. Baseline PR spanwise strain (cyy), a = 15. numerical experimental 0.01 0.01 -0.010 -1 -0.5 0 0.5 1 2y/b Figure 5-8. Baseline PR shear strain (Exy), a = 15. The aerodynamic twist (camber and camber location) and geometric twist angle section has an inner dimension of 0.84 m on each side and is 2.44 m deep. The velocity range is between 2 and 45 m/s, and the maximum Reynolds number is 2.7 million. The flow is driven via a two-stage axial fan with an electric motor powered by three-phase 440 V at 60 Hz. The controller is operated remotely with appropriately dedicated data acquisition software, wherein the driving frequency is based upon a linear scaling of an analog voltage input. Suitable flow conditions are achieved through hexagonal aluminum honeycomb cell, high-porosity stainless steel screens, and turning vane cascades within the elbows of the closed loop. Centerline turbulence levels are measured on the order of 0.2%. Optical glass window access is available on the sidewalls and the ceiling. A Heise model PM differential pressure transducer rated at 12.7 cm and 127 cm of water (with a manufacturer-specified 0.002% sensitivity and a 0.01% repeatability) is used to measure the pressure difference from a pitot-static tube mounted within the test section, whose stagnation point is located at the center of the section's entrance. The Heise system is capable of measuring wind speeds up to 45 m/s. A four-wire resistance temperature detector is mounted to the wall of the test section for airflow temperature measurements. Strain Gage Sting Balance Several outstanding issues exist with measuring the aerodynamic loads from low Reynolds number flyers. Several such airfoils are known to exhibit hysteresis loops at high angles of attack. If the flow does not reattach to the wing surface (typically for lower Reynolds numbers below 5-104 [17]) counterclockwise hysteresis loops in the lift data may be evident; the opposite is true if a separation bubble exists via reattachment [24]. Adequate knowledge of such a loop is obviously important as it effects vehicle control problems via stall and spin recovery. As described by Marchman [122], the size of the hysteresis loop measured in a wind tunnel can be incorrectly decreased by poor flow quality: large freestream turbulence intensity levels or experimental test-bed for flexible MAVs? What performance metrics should be compared between numerical and experimental results for sufficient model validation? Upon suitable validation of the aeroelastic model, two optimization studies are developed: tailoring and topology optimization. Considering the former, with a given spatial distribution of laminated carbon fiber and membrane skin throughout the wing, what is the optimal chordwise and spanwise membrane pre-tension and carbon fiber laminate lay-up schedule? For the aeroelastic topology optimization studies, with a given membrane pre-tension and laminate orientation, what is the optimal distribution of carbon fiber and membrane skin throughout the wing? What performance metrics should be optimized? As these metrics will surely conflict, what multi-objective optimization schemes are appropriate for computation of the Pareto front? Can the numerically-indicated optimal wing design structures be built and tested, and will the experimental results also indicate superiority over similarly-tested baseline designs? Dissertation Outline This work begins with a detailed literature review of micro air vehicle aerodynamics (low Reynolds number flows, low aspect ratio wings, unsteady flow phenomena), aeroelasticity (membrane sailwings, flexible filaments), and optimization (rigid wing airfoil and planform optimization, tailoring). I review the literature pertaining to topology optimization as well, with a particular emphasis upon aeronautical and aeroelastic applications. I then discuss the apparatus and procedures used for experimental characterization of the membrane micro air vehicle wings. This includes a low-speed closed loop wind tunnel, a high sensitivity sting balance, and a visual image correlation system. Information is also given detailing wing fabrication and preparation. I summarize the computational framework, including both linear and nonlinear structural finite element models. Three-dimensional viscous and inviscid flow solvers are formulated, along with aeroelastic coupling and ad hoc techniques with one another as the angle of attack increases. This decrease in tip vortex strength is also seen in Figure 5-14: the nonlinear aerodynamics (from the low pressure cells at the tip) is evident in the pitching moments of the rigid and BR wings, while the PR curve is very linear. On the underside of the rigid wing at 15 angle of attack (Figure 5-19), the increased incidence provides for completely attached flow behavior. The pressure gradient is largely favorable, smoothly accelerating the flow from leading to trailing edge. From the previous four figures it can be seen that separated flow over the bottom surface gradually attaches for increasing angles of attack, while attached flow over the upper surface gradually separates (eventually leading to wing stall). As time-averaged flow separation is likely to be unsteady vortex shedding [18]: this explains the aforementioned membrane vibration amplitudes that decrease to a quasi-static behavior, then increase through the a-sweep [135]. Figure 5-19. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), a = 15. Load alleviation on the lower surface of the BR wing is evidenced by a decrease in the high-pressure regions associated with camber, and a growth of the suction region at the trailing edge (the latter presumably due to a decrease in the local incidence). A high-pressure lobe also develops at the trailing edge of the membrane panel between the carbon fiber root and the inner batten. At higher angles, this region of the membrane does not locally inflate; it merely stretches between the two laminates, acting as a hinge. The adaptive inflation of the PR wing causes a membrane/carbon fiber interface is largest with the compromise design. The severity of the surface cusp (and the concomitant lift spike) increases with decreasing 6, emphasizing its usefulness as a lift-augmentation device. As discussed above, the severity of this spike is certainly over-predicted by the inviscid flow solver, though similar trends are seen using Navier- Stokes solvers for wings with tangent discontinuities Figure 5-21. 0.06 2.5 0.05 2 - 0.04 ,A 1.5 0.03 0.02 0.5 0, 0.01 0 0 -0.5 ....... 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 x/c x/c Figure 7-23. Deformations and pressures along 2y/b = 0.58 for designs that trade-off between CL and CL,, a = 12, cambered wing. that optimize a single objective function are of limited value; of greater importance is the array of designs that lie along the Pareto optimal front. This front is a trade-off curve comprised of non-dominated designs, one of which can be selected based on additional considerations not included in the optimization: manufacturability, flight specifications (duration, payload), etc. Computation of the Pareto front is costly when gradient-based search routines are used for optimization, typically involving successive optimization runs with various convex combinations of the objective functions. The success of this technique depends on the convexity of the Pareto optimal front. More efficient methods for computing the Pareto front are available if evolutionary algorithms or response surface methods are employed. Problem Statement The static aeroelasticity of membrane micro air vehicle wings represents the intersection of several rich aerodynamics and mechanics problems; numerical modeling can be very challenging and expensive. Furthermore, MAVs are beset with many detrimental flight issues, and are very difficult to fly: systematic numerical optimization schemes can be used to offset these problems, improving flight duration, gust suppression, or static stability. Many optimization studies can be considered for MAVs; the current work utilizes aeroelastic optimization, which will require hundreds of costly function evaluations to adequately converge to an optimal design. As the feasibility of such a scheme relies on a moderate computational cost, what is the lowest fidelity aeroelastic model that can be appropriately used? Model development requires extensive experimental validation, and several challenges exist here as well. The forces generated by a MAV wing are very small, and highly-sensitive instrumentation is needed. For deformation measurements, only vision-based non-contacting methods are appropriate. What particular components are required to construct an adequate separated flow is largely confined to the in-board portions of the wing. Flow reattaches slightly aft of the quarter-chord, after which the pressure gradient is favorable. The flow accelerates beneath the negatively-cambered portion of the rigid wing: this decreases the local pressures, further offsetting the nose-down pitching moment. The pressure distribution on the lower surface is not greatly affected by the tip vortices, previously noted by Lian et al. [28]. S-130 -S2 -34 14 -130 -82 -34 14 -130 -82 -34 14 Figure 5-17. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), a = 0. For the BR wing (Figure 5-17), slight undulations in the pressure distribution are indicative of the membrane inflation in between the battens. This causes the opposite of what is seen on the upper wing: flow is slightly packed towards the battens [154], though the effect is minor, as before. The adaptive aerodynamic twist of the PR membrane wing pushes the bulk of the separated flow at the leading edge towards the root, and induces further separation beneath the inflated membrane shape, as the air flows into the cavity against an adverse pressure gradient. The location of maximum pressure is increased and pushed aft-ward to coincide with the apex of the inflated membrane, increasing both the lift and the stability. Flow structures over the upper surface at 150 angle of attack are given in Figure 5-18. At this higher incidence, the adverse pressure gradient is too strong for the low Reynolds number flow, and a large separation bubble is present at the three-quarter chord mark of the rigid wing. Despite the nose-up geometric twist built into the wing (7 at the tip, Figure 5-9), flow separates Gyllhem et al. [46] reports that the presence of a fuselage, motor, and stabilizers surprisingly improves the computed maximum lift and stall angle (compared to simulations with just the wing), but increases the drag as well. Experimental work by Albertani [47] finds just the opposite: a decrease in lift of the entire vehicle, but less of a penalty when passive shape adaptation is built into the wing. Waszak et al. [13] are able to show significant improvements in efficiency if a streamlined MAV fuselage is used. Rigid Wing Optimization Though the main scope of the current work is to improve the aerodynamic qualities of fixed micro air vehicle wings through the judicious use of aeroelastic membrane structures, much successful work has been done with multidisciplinary optimization of the shape, size, and components of a rigid MAV wing. These studies must often make use of low fidelity models due to the large number of function evaluations required for a typical optimization run, and may not be able to capture the complicated flow physics described above. Nonetheless, insight into the relationship between sets of sizing/shape variables and a given objective function can still be gained. Early work is given by Morris [48], who finds the smallest vehicle that will satisfy given constraints throughout a theoretical mission, using several empirical and analytical expressions for the performance evaluation. Rais-Rohani and Hicks investigate a similar problem, using a vortex lattice method (for computations of aerodynamic performance and stability, along with propulsion and weight modules) and an extended interior penalty function method to reduce the size of a biplane MAV [49]. 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Greater variation with laminate thickness is seen for non-zero pre-tensions, particularly when Nx = 20 N/m and Ny = 10 N/m. If a single layer of carbon fiber is used, this data point represents the minimum lift slope. Like the double-layered laminates studied above, the BR wing removes the camber due to membrane inflation (and thus the lift) at the leading edge with high chordwise stiffness, and allows for adaptive washout with low spanwise stiffness perpendicular to the battens. Such a design has biological inspiration: the bone-reinforced membrane skins of pterosaurs [101] and bats [102] both have larger chordwise stiffness. For low levels of pre-tension, decreasing the number of plain weave layers increases the efficiency; if the membrane is highly-tensioned, the opposite is true. The L/D objective function is optimized with a one-layer slack membrane BR wing. It can also be seen in Figure 6-10 that for high levels of membrane pre-tension, there is little computed difference between 2 and 3 layer laminates. Similar data is given in Figure 6-12 and Figure 6-13, for a PR wing. The single-layer PR wing exhibits a substantial amount of adaptive washout, owning to deflection of the weak carbon fiber perimeter. Two and three-layer laminates remove this feature completely, forcing the wing into a pure aerodynamic twist. Regardless of the load alleviation along the trailing edge, the steepest lift and moment curves are found with single-layer laminates, as the weak carbon fiber reinforcement intensifies the cambering of the membrane wing. The two-dimensional equivalent to this case is a sailing with the trailing edge attached to a flexible support. Well-known solutions to this problem indicate that increasing the flexibility of the support improves the static stability [67], a trend re-iterated in Figure 6-12. None of the PR aerodynamic metrics or the displacement contours show a substantial difference between two and three layer laminates. For the thicker laminates, increasing the A second design leaves the interior of the membrane skin unconstrained, while the perimeter of the skin is sealed to a thin curved strip of carbon fiber (Figure 1-2). The perimeter-reinforced (PR) wing deformation is closer in nature to an aerodynamic twist. Both the leading and the trailing edges of each membrane section are constrained by the relatively-stiff carbon fiber. The positive cambering (inflation) of the wing should lead to an increase in CL and a decrease (more negative) in Cma [14]. Figure 1-2. Perimeter-reinforced membrane wing design. While both of these wing structures can adequately perform their intended tasks (load alleviation for the BR wing, load enhancement for the PR wing), several sizing/stiffness variables exist within both designs, leading to an aeroelastic tailoring problem. Conventional variables such as the laminate fiber orientation [11] can be considered, but the directional stiffness induced by varying the pre-tension within the membrane skin may play a larger role within the aeroelastic response [15]. Systematic optimization of a single design metric will typically lead to a wing structure with poor performance in other important aspects. For example, tailoring the PR wing structure for maximum static stability may provoke an unacceptable drag penalty, and vice-versa. Of the aerodynamic performance metrics considered here (lift, drag, efficiency, static stability, gust suppression, and mass) many are expected to conflict, as with most engineering optimization applications. Formal multi-objective optimization procedures can be used to tailor flexible MAV wing designs that strike an adequate CHAPTER 1 INTRODUCTION Motivation The rapid convergence of unmanned aerial vehicles to continually smaller sizes and greater agility represents successful efforts along a multidisciplinary front. Technological advances in materials, fabrication, electronics, propulsion, actuators, sensors, modeling, and control have all contributed towards the viable candidacy of micro air vehicles (MAVs) for a plethora of tasks. MAVs are, by definition, a class of unmanned aircraft with a maximum size limited to 15 cm, capable of operating speeds of 15 m/s or less. Ideally, a MAV should be both inexpensive and expendable, used in situations where a larger vehicle would be impractical or impossible, to be flown either autonomously or by a remote pilot. Military and defense opportunities are perhaps easiest to envision (in the form of over-the-hill battlefield surveillance, bomb damage assessment, chemical weapon detection, etc.), though MAVs could also play a significant role in environmental, agriculture, wildlife, and traffic monitoring applications. MAVs are notoriously difficult to fly; an expected consequence of a highly maneuverable and agile vehicle that must be flown either remotely or by autopilot [1]. The aerodynamics are beset by several unfavorable flight issues: 1. The operational Reynolds number for MAVs is typically between 104 and 105. Flow over the upper wing surface can be characterized by massive flow separation, a possible turbulent transition in the free shear layer, and then reattachment to the surface, leaving behind a separation bubble [2]. Such flow structures typically result in a loss of lift, and an increase in drag, and a drop in the overall efficiency [3]. 2. The low aspect ratio wing (on the order of unity) promotes a large wing tip vortex swirling system [4], which interferes with the longitudinal circulation of the wing [5]. Entrainment of the aforementioned separated flow can lead to tip vortex destabilization [6]; the resulting bilateral asymmetry may be the cause of the rolling instabilities known to plague MAV flight. 3. Sudden wind gusts may be of the same order of magnitude as the vehicle flight speed (10-15 m/s). Maintaining smooth controllable flight can be difficult [7] [8]. stresses in both directions, the two aerodynamic derivatives in Figure 6-3 have a significantly muted response to Nx. Such a result has noteworthy ramifications upon a multi-objective optimization scenario. The longitudinal static stability is optimal for a slack membrane wing, but the wing efficiency at this data point is poor. Maximizing Nx and setting Ny to zero greatly improves the lift-to-drag ratio (only 0.2% less than the true optimum found on this surface), with a negligible loss in static stability. N = 0 N/m N = I t N/m Ny = 20 N/m NO = 0 N/m N, = 10 N/nmi Nx = 20 N/mn 0 0.01 0.02 0.03 0.04 0.05 Figure 6-4. Computed PR wing deformation (w/c) with various pre-tensions, a = 12. Single Ply Laminates The same aerodynamic metrics are given in Figure 6-5 as a function of the ply angle (with respect to the chord line) for a set of wings with a single layer of bi-directional carbon fiber at the wing root, leading edge, and perimeter (for the PR wing only). The membrane wing is slack. Due to the plain weave nature of the laminate, all trends are periodic every 900. Only fiber orientations of 0, 45, and 900 automatically satisfy the balance constraint [155]. For the PR wing, changing the fiber angle has a minor effect on the aeroelastic response, and optima are moire interferometry requires no such marker placement (a fringe pattern is projected onto the wing surface), and the resulting data set is full-field. However, displacement resolutions reported by Fleming et al. [130] are relatively poor (250 itm), the dual-camera system must be rotated during the a-sweep, and only out-of-plane data is available, making strain calculations (if needed) impossible. Burner et al. [131] discuss the use of photogrammetry, projection moire interferometry, and the commercially available OptotrakTM system. The authors find no single technique suitable for all situations, and that a cost-benefit tradeoff study may be required. Furthermore, the methods need not be mutually exclusive, as situations may arise wherein they can be used in combination. For the current work, a visual image correlation system (VIC), originally developed by researchers at the University of South Carolina [132], is used to measure wing geometry, displacements, and plane strains. The underlying principle of VIC is to calculate the displacement field by tracking the deformation of a subset of a random speckle pattern applied to the specimen surface. The random pattern is digitally acquired by two cameras before and after loading. The acquisition of images is based on a stereo-triangulation technique, as well as the computing of the intersection of two optical rays: the stereo-correlation matches the two 2-D frames taken simultaneously by the two cameras to reconstruct the 3-D geometry. The calibration of the two cameras (to account for lenses distortion and determine pixel spacing in the model coordinates) is the initial fundamental step, which permits the determination of the corresponding image locations from views in the different cameras. Calibration is done by taking images (with both cameras) of a known fixed grid of black and white dots. Temporal matching is then used: the VIC system tries to find the region (in the image of the deformed specimen) that maximizes a normalized cross-correlation function corresponding to distributions for the baseline BR and rigid wings are given in Figure 5-9. The rigid wing is characterized by positive (nose-up) twist and a progressive de-cambering toward the wingtip. The carbon fiber inboard portion of the BR wing exhibits very similar wing twist to the rigid wing. Past 2y/b = 0.3 however, both model and experiment show that the membrane wing has a near-constant decrease in twist of 2-3: adaptive washout. Though this geometric twist dominates the behavior of the BR wing, the membrane also exhibits some aerodynamic twist. This occurs predominately in the latex between the battens, about 1% of the chord in magnitude. The location of this camber has large variations: some portions of the wing are pushed back from 25% (rigid) to 75% (membrane), as shown by both model and experiment. Shifting the camber aft-ward on low Reynolds number wings is one method to hinder flow separation through control of the pressure gradient [27], and may play a role in the BR wing's delayed stall as well. 8f 0.08 A 0.8 6 exp. / exp. --- 0.06 / .6 num. S.........-----d- 0.04 ri g \: nu o 0.6 2 \ num 0.4 S 0.02 --------- rigid 0 0 i.........g 0.2 -1 0 1 -1 0 1 -1 0 1 2y/b 2y/b 2y/b Figure 5-9. Baseline BR aerodynamic and geometric twist distribution, a = 15. The aerodynamic and geometric twist distributions for the baseline PR and rigid wings are given in Figure 5-10. Membrane inflation adaptively increases the camber by as much as 4%, though this figure is slightly under-predicted by the model. The location of this camber is shifted aft-ward, though not as much as with the BR wing. The flexible laminate used for the wing skeleton pushes the location of the camber at the root slightly forward. Like the BR wing deformation, shape changes over the PR wing are a mixture of both aerodynamic and geometric computational cost of such an undertaking is large, and adequate location of the front is not ensured for non-convex problems (such as seen in Figure 6-14). The objective function is now: g =(l-). fl i1 mm +6. 2 .2m.. (7-16) f -f f -f, flmax l,mm ) f2,max 2,mm . where 6 is a weighting parameter that varies between 0 and 1, and fi and f2 are the two objective functions of interest. These functions are properly normalized, with the minimum and maximum bounds computed from the single-objective optimizations (optimizing with 6 set as 0 or 1). Eq. (7-16) is cast as a minimization problem, and the sign of f1 and f2 is set accordingly. As before, the objective function can be augmented with the explicit penalty of Eq. (7-2) as needed. Typical convergence history results are given in Figure 7-16, for simultaneous maximization of L/D and minimization of the lift slope. The weighting parameter 6 is set to 0.5, for an equal convex combination of the two variables. The values given for CL, (- 0.4) are smaller than experimentally measured trends (- 0.5, from Table 5-1), as the inviscid solver is unable to predict the vortex lift from the tip vortex swirling system [27]. Beginning with an intermediate density (Xo = 0.5), the optimizer is able to decrease the convex combination (g) from 0.7 to 0.3, using similar techniques seen above. All of the carbon fiber material adjacent to the root, leading edge, and wingtip is removed. Intersecting streams of membrane material grow across the wing, leaving behind disconnected carbon fiber structures. The lift-to-drag ratio monotonically converges after 25 iterations, while the lift slope requires 70 iterations to converge to a minimum value. An explicit penalty on intermediate densities is employed at the 80 iteration mark, providing a moderate decrease in the combination objective function. The lift-to-drag ratio is improved as well through the penalty, though the lift slope suffers. As before, the penalty only serves to force the density variables to 0 or 1, and does not significantly alter the wing topology. Contributions 1. Develop a set of variable-fidelity aeroelastic models for low Reynolds number, low aspect ratio membrane micro air vehicle wings. 2. Develop a highly-sensitive non-intrusive experimental test-bed for model deformation and flight loads. 3. Optimization-based system identification of the wing structure's material properties. 4. Experimental aeroelastic model validation of flight loads and wing deformation. 5. Optimize multiple flight metrics by tailoring membrane pre-tension and laminate orientation. 6. Develop computational framework for topology optimization of membrane wings, with an analytical sensitivity analysis of the coupled aeroelastic system. 7. Able to provide scientific insight into the relationship between optimal wing flexibility, flow structures, and the resulting beneficial effects upon flight loads and efficiency. 8. Experimental validation of the superiority of selected optimal designs over baselines. membrane pre-tension include: decrease in drag [89], decrease in CL, [15], linearized lift behavior [72], increase in the zero-lift angle of attack [68], and more abrupt stalling patterns [89]. Ormiston details aeroelastic instabilities in terms of the ratio of spanwise to chordwise pre- tension [15]. Adequate control of membrane tension has long been known as a crucial concern to sailors in order to efficiently exploit wind power [99]. Tension-control is similarly important to the performance and agility of fighter kites: a wrinkled membrane surface will send the kite into an unstable spin. When pointed in the desired direction, pulling the control line tenses and deforms the kite, which thus attains forward velocity [100]. Biological inspiration for aerodynamic tailoring of membrane tension can be seen in the wing structures of pterosaurs and bats. In addition to membrane anisotropy (pterosaur wings have internal fibrous reinforcement to limit chordwise stretching [101], while bat wings skins are measured to be 100 times stiffer in the chordwise direction than the spanwise [102]), the tension can be controlled through a single digit (pterosaurs) [59], or varied throughout the wing via multiple digits (bats) [103]. Work formally implementing membrane tension as a variable for optimizing aerodynamic performance is very rare. Levin and Shyy [104] study a modified Clark-Y airfoil with a flexible membrane upper surface, subjected to a varying freestream velocity. Response surface techniques are used to maximize the power index averaged over a sinusoidal gust cycle, with membrane thickness variation, elastic modulus, and pre-stress used as variables. The maximum power index is found to coincide with the lower bound placed upon pre-stress, though lift and efficiency are also seen to be superior to a rigid wing. Topology Optimization The basics of topology optimization are given by Bendsoe and Sigmund [16] and Zuo et al. [105]: the design domain is discretized, and the relative density of each element can be 0 or 1. channel, as well as potential interactions (second-order interactions are not included) in both single and multiple load configurations. Further information on the calibration of strain gage sting balances for micro air vehicle measurements is given by Mueller [126] and Albertani [47]. Uncertainty Quantification Two types of uncertainty are thought to contribute to the eventual error bounds of the sting balance data. The resolution error is indicative of a measurement device's resolution limit: for example, the inclinometer used to measure the pitch of a model can measure angles no finer than 0.1, an uncertainty that can be propagated through the equations to find its theoretical effect on the aerodynamic coefficients using the Kline-McClintock technique [127]. The following resolution errors are used: 3 Pa of dynamic pressure from the Heise, 1.2-10-7 V from the output voltage of the strain gages (estimated from the quantization error of the 16-bit DAQ cards), 0.001 m2 from wing area measurements, and 0.002 m from chord length measurements. The second source is the precision error, a measure of the repeatability of a measurement. This is well quantified by the standard deviation of the voltage signals from 2000 samples at each angle of attack, as described above. Uncertainty bounds are computed with a squared sum of the resolution and precision errors (where the latter is magnified by Student's t at 95% confidence and infinite degrees of freedom). The precision of the strain gage signals is found to contribute the most error to the aerodynamic coefficients, particularly in the stalled regions. Typical uncertainty percentages are 5% for CL, 7% for CD, 9% for L/D, and 20% for Cm. Theses values can be expected to double during stall. Wind Tunnel Corrections Corrections are applied to the coefficients of lift, drag, and pitching moment based upon wind tunnel blockage, and model flexibility effects. The solid blockage effect is due to the presence of the model within the wind tunnel, thus decreasing the effective area of the test thrust will conflict: thrust relies on wing twist via deformation for thrust generation, while lift is dependent upon the leading edge vortex, which can be disrupted by excessive deformations. This requires successive optimizations of a convex combination of the two weighted metrics to fill out the trade-off curve (assuming that this Pareto front is convex). The optimal design can then be selected from this front based upon metrics not considered in the formal optimization: trim requirements, manufacturability, etc. The flow structures that develop over flapping wing systems are very complicated, unsteady vortex driven flows. Navier-Stokes solvers can adequately handle these phenomena, but the computational cost may be prohibitive. Topology optimization of flapping wings may require lower-fidelity aerodynamic methods for effective navigation through the design space. Finally, the aeroelastic topology optimization of both the fixed and flapping wings can be followed by a tailoring study for additional improvements to the flight performance. This is a standard optimization process: topology optimization, interpretation of the results to form an engineering design, followed by sizing and shape optimization (or in this case, tailoring). Both laminate thickness/orientation and membrane pre-tension can be used, as above. Membrane pre- tension is difficult to control however, and will relax at the un-reinforced borders of the wing, leading to a pre-tension gradient. Anisotropic membranes (through imbedded elastic fibers or crinkled/pleated geometries) are an attractive alternative for directional wing skin stiffness. The excess area of the skin may also be a useful variable. As the number of variables in a tailoring study is relatively small (-10), gradient-free global optimizers such as evolutionary algorithms or response surface techniques may become applicable. 6-5 Computed tailoring of laminate orientation for single ply bi-directional carbon fiber, a = 12 ......... ....................................................................... ...... ................ 103 6-6 Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a B R w ing, a = 12 ........ ................................................................ ........ .. .... ........104 6-7 Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a P R w in g a = 12 ................. ..... .. ........................................ ..................... .......... 10 5 6-8 Computed tailoring of batten construction in a BR wing, a = 12 ......................................106 6-9 Computed normalized out-of-plane displacement (left) and differential pressure (right) at x/c = 0.5, for various BR designs, a = 12 ...................................... ...............107 6-10 Computed full factorial design of a BR wing, a = 12 ..................................................109 6-11 Computed BR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), a = 12 ............................................ ......... .............. 109 6-12 Computed full factorial design of a PR wing, a = 12................................................ 111 6-13 Computed PR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), a = 12 ................. ......... .......... ............... .. 111 6-14 Computed design performance and Pareto optimality, a = 12........................................113 6-15 Experimentally measured design optimality over baseline lift ...................................118 6-16 Experimentally measured design optimality over baseline pitching moments ................19 6-17 Experimentally measured design optimality over baseline drag ................................. 120 6-18 Experimentally measured design optimality over baseline efficiency .............................120 7-1 W ing topologies flight tested by Ifju et al. [10] ....................................... ............... 123 7-2 Sample wing topology (left), aerodynamic mesh (center), and structural mesh (right).......124 7-3 Effect of linear and nonlinear material interpolation upon lift.............................................127 7-4 Measured loads of an inadequately reinforced membrane wing, U, = 13 m/s .................... 130 7-5 Convergence history for maximizing L/D, a = 30, reflex wing................ .................135 7-6 Affect of mesh density upon optimal L/D topology, a = 120, reflex wing......................137 7-7 Affect of initial design upon the optimal CD topology, a = 120, reflex wing....................138 7-8 Affect of angle of attack and airfoil upon the optimal CL topology ................................139 paneled grid, and placing a horseshoe vortex upon each panel. Each horseshoe vortex is comprised of a bound vortex (which coincides with the quarter-chord line of each panel), and two trailing vortices extending downstream. Each vortex filament creates a velocity whose magnitude is assumed to be governed by the Biot-Savart law [27]. Furthermore, a control point is placed at the three-quarter-chord point of each panel. The velocity induced at the mth control point by the nth horseshoe vortex is: m,n Cm,n n Vm,n Cm,n Wm,n Cm,n .F (4-11) where u, v, and w are the flow velocities in Cartesian coordinates, F is the vortex filament strength, and C' are influence coefficients that depend on the geometry of each horseshoe vortex and control point combination. The complete induced velocity at each control point is the sum of the contributions from each horseshoe vortex, resulting in a linear system of equations. The strength of each vortex must be found so that the resulting flow is tangent to the surface of the wing: the wing becomes a streamline of the flow. This requirement is enforced at each control point by: {U, cos(a)+um vm U, .sin(a)+wm)}VF(xm,ym,zm)= 0 (4-12) where Uo, is the free-stream velocity, a is the angle of attack, and F(x,y,z) = 0 is the equation of the surface of the wing. Inserting the relevant terms of Eq. (4-11) into Eq. (4-12) provides a linear system of equations for the filament strength of each horseshoe vortex. Micro air vehicle simulations that utilize a vortex lattice method are typically forced to do so by the computational requirements of optimization (as is the case in the current work). Examples can be seen in the work ofNg and Leng [52], Sloan et al. [53], and Stanford et al. [61]. Steady Navier-Stokes Solver The three-dimensional incompressible Navier-Stokes equations, written in curvilinear Torres [3] uses a genetic algorithm to minimize a weighted combination of payload, endurance, and agility metrics, with various discrete (wing and tail planform) and continuous (aspect ratio, propeller location, angle of attack, etc) variables. Aerodynamic analysis is provided by a combination of experimental data, analytical methods, and interpolation techniques. The author cites convergence problems stemming from the discrete variables. Genetic algorithms are also used in the work of Lundstrom and Krus [51] and Ng et al. [52]. The latter indicates that these algorithms are more suited for the potentially disjointed design spaces presented by MAV optimization efforts. A comparison between a genetic algorithm and gradient-based sequential quadratic programming used to design winglets for a swept wing MAV indicates the superiority of the former, with a vortex lattice method used for aerodynamic analysis. However, a genetic algorithm may only be feasible for lower fidelity tools, due to the large number of function evaluations required for convergence. Higher fidelity aerodynamics tools (namely, thin-layer or full Navier-Stokes equation solvers) are employed in recent studies. For example, a combined 2-D thin layer Navier-Stokes model and a 3-D panel method is used by Sloan et al. [53], who use the outcome to construct a response surface to optimize the wing geometry for minimum power consumption. As above, the study reveals the superiority of thin wings, and finds that optimal airfoil shapes are insensitive to aspect ratio. Lian et al. [54] use a full Navier-Stokes solver to maximize the lift-to- drag ratio of a rigid MAV wing subject to various lift and wing convexity constraints, with sequential quadratic programming search methods. Efficiency improvements are feasible by decreasing the camber at the root and increasing at the tip, thereby decreasing the amount of flow separation. Improvements are found to be more substantial at moderate angles of attack. Given the computational complexities associated with MAV simulation, several research Gomes and Suleman [120] use a spectral level set method to maximize aileron reversal speed by reinforcing the upper skin of a wing torsion box via topology optimization. Maute and Reich [106] optimize the topology of a compliant morphing mechanism within an airfoil, by considering both passive and active shape deformations. The authors are able to locate superior optima with this aeroelastic topology optimization approach, as compared to a jig-shape approach: optimizing the aerodynamic shape, and then locating the mechanism that leads to such a shape. At present, there is no research pertaining to aeroelastic topology optimization of membrane wings, or micro air vehicle wings. Biological inspiration for this concept can be found in the venation of insect wings however. For example, a pleated grid-like venation can be seen in dragonfly wings, posteriorly curved veins in fly wings, and a fan-like distribution of veins in the locust hindwing [121]. On the whole, the significance of this variation in wing stiffness distribution between species is not well understood. extends the length of the membrane skin. The wing is flexible enough to adaptively washout, but the remaining patches of membrane skin are not large enough to inflate and camber. S= 1.0 8 = 0.8 8 = 0.6 8 = 0.4 6 = 0.0 L/D =3.81 L/D 3.97 L/D 4.06 L/D 4.15 L/D 4.18 w/c -0.01 0 0.01 0.02 ACP -1.5 -1 -0.5 0 0.5 1 1.5 Figure 7-18. Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom) for designs that trade-off between L/D and CL,, a = 30, reflex wing. Gradually adding weight to the L/D design metric removes the structures from the leading edge of the membrane skin, leaving batten-like structures at the trailing edge of the wing. The former transition allows the membrane to inflate and camber the wing, while the latter provides wash-in through depression of the trailing edge. The cambering membrane inflation does not grow monotonically with decreasing 6, but the trailing edge deformation does: from 0.250 of washout to 0.750 of wash-in. The size of the depressed trailing edge portion also grows in size. Decreasing 6 shifts the lift penalty (pressure spike on the upper surface) forward towards the membrane/carbon fiber interface, and the lift spike (due to the surface geometry cusp at the leading edge of the batten structures) aft-ward. However, the design that maximizes L/D (6 = 0) has no spike, with a smooth pressure and displacement profile aft of the lift penalty towards the leading edge. This may be indicative of the detrimental effect the airfoil cusp has on drag. (and thus not a compressive stress), but the stress in this region does become slightly negative for higher angles of attack. Erroneous computation of compressive membrane stresses indicates the need for a wrinkling module. Though wrinkles in the membrane skin are not obviously visible in the VIC measurements (possibly an unsteady process averaged out with multiple images), wrinkling towards the onset of stall is a well-known membrane wing phenomena [87]. As before, no appreciable strain is measured or computed in the carbon fiber areas of the wing. 0.06 0.06 06 numerical experimental 0.0 0.04 0,04 0.04 At 0.04 4 0,02 0 -1 -0.5 0 0.5 1 2./b Figure 5-5. Baseline PR normalized out-of-plane displacement (w/c), a = 15. 0.03 numerical experimental 0.03 0.02 ( 0.02 00.0 0.01 0 0 05 1 x/c Figure 5-6. Baseline PR chordwise strain (exx), a = 15. Peak spanwise stretching (Figure 5-7) occurs at the membrane carbon fiber interface towards the center of the wing root, and is well predicted by the model. The computed strain field erroneously shows a patch of negative Poisson strain towards the leading edge, due to the high chordwise strains in this area. One troubling aspect of the measured spanwise strains is the areas of negative strains along the perimeter of the membrane skin: namely on the sidewalls towards the root and the wingtip. Such strains have been measured in previous studies [9], but Figure 7-17, along with the performance of the 20 baseline MAV wing designs (Figure 7-1), and the design located by the single-objective topology optimizer to maximize CL,. All results are for a reflex wing at 3 angle of attack. Focusing first on the baseline wings, the BR and PR wings represent the extremes of the group in terms of lift slope, as expected. The homogenous carbon fiber wing has the lowest L/D (implying that for a reflex wing at this flight condition, any aeroelastic deformation will improve efficiency, regardless of the type), while a MAV design with 2 trailing edge battens as the largest L/D. 0.0421 0.041 0.04 \ a E S 0.039 [ 0 O 0.038 0.037 0* baseline S' Pareto front max CL deign 0.036 3.7 3.8 3.9 4 4.1 4.2 L/D Figure 7-17. Trade-off between efficiency and lift slope, a = 3, reflex wing. The aeroelastic topology optimization produces a set of designs that significantly out- perform the baselines, in terms of individually-considered metrics (maximum and minimum lift slope, maximum L/D), and multiple objectives: all of the baselines are removed from the computed Pareto front. The optimized designs lay consistency closer to the fictional utopia point as well, which for Figure 7-17 is at (4.18, 0.0366). The entirety of the Pareto front is not A quantitative summary of the last four figures is given in Table 5-1, for all three baseline wings at 6 angle of attack. Experimental error bounds are computed as described above. Aerodynamic sensitivities (as well as the pitching moment about the aerodynamic center) are found with a linear fit through the pre-stall angles of attack. Error bounds in these slopes are computed with Monte Carlo simulations. Computed lift, drag, and pitching moments consistently fall within the measured error bars (the latter of which are exceptionally large), though pitching moments are significantly under-predicted (10-30%). Sensitivities are also under-predicted, though still fall within the large error bars associated with pitching moment slopes. With the exception of L/D of a PR wing, trends between different wing structures are well-predicted by the aeroelastic model. Table 5-1. Measured and computed aerodynamic characteristics, a = 6. S CL I CD num. exp. error (%) num. exp. error (%) rigid 0.396 0.384 0.024 3.10 0.070.0.069 + 0.007 1.15 BR 0.381 0.382 + 0.024 -0.16 0.067 0.069 + 0.007 -3.04 PR 0.465 0.495 + 0.031 -5.98 0.085 0.076 + 0.009 11.61 Cm I L/D A num. exp. error (%) num. exp. error (%) rigid -0.084 -0.063 0.033 -32.81 5.64 5.49 + 0.69 2.72 BR -0.087 -0.073 + 0.034 -19.39 5.70 5.49+ 0.68 3.77 PR -0.138 -0.131 0.042 -5.64 5.49 6.49+ 0.87 -15.36 I CL. Cm,AC num. exp. error (%) num. exp. error (%) rigid 0.049 0.051 0.003 -5.26 0.013 0.016 0.018 BR 0.044 0.048 0.004 -9.35 0.006 -0.001 + 0.020 PR 0.052 0.057 0.004 -9.21 -0.008 -0.015 + 0.026 Cma I dCm/dCL num. exp. error (%) num. exp. error (%) rigid -0.012 -0.010 + 0.004 -11.65 -0.246 -0.199 0.086 -23.07 BR -0.011 -0.009 0.004 -17.97 -0.244 -0.185 0.098 -31.88 PR -0.014 -0.013 0.006 6.01 -0.280 -0.229 0.105 -22.17 Flow Structures Having established sufficient confidence in the static aeroelastic membrane wing model, [58] Fitt, A., Pope, M., "The Unsteady Motion of Two-Dimensional Flags With Bending Stiffness," Journal ofEngineering Mechanics, Vol. 40, No. 3, 2001, pp. 227-248. [59] Wilkinson, M, "Sailing the Skies: the Improbable Aeronautical Success of the Pterosaurs," Journal ofExperimental Biology, Vol. 210, No. 10, 2007, pp. 1663-1671. [60] Bisplinghoff, R., Ashley, H., Halfman, R., Aeroelasticity, Dover, Mineola, NY, 1955. [61] Stanford, B., Abdulrahim, M., Lind, R., Ifju, P., "Investigation of Membrane Actuation for Roll Control of a Micro Air Vehicle," Journal ofAircraft, Vol. 44, No. 3, 2007, pp. 741-749. [62] Argentina, M., Mahadevan, L., "Fluid-Flow-Induced Flutter of a Flag," Proceedings of the National Academy of Science: Applied A///hei'/ntiL %, Vol. 102, No. 6, 2005, pp. 1829-1834. [63] Alben, S., Shelley, M., Zhang, J., "How Flexibility Induces Streamlining in a Two-Dimensional Flow," Physics ofFluids, Vol. 16, No. 5, 2004, pp. 1694-1713. [64] Voelz, K., "Profil und Auftrieb Eines Segels," Zeitschriftf'ur Angewandte Mathematik und Mechanik, Vol. 30, 1950, pp. 301-317. [65] Thwaites, B., "The Aerodynamic Theory of Sails," Proceedings of the Royal Society ofLondon, Vol. 261, No. 1306, 1961, pp. 402-422. [66] Nielsen, J., "Theory of Flexible Aerodynamic Surfaces," Journal ofApplied Mechanics, Vol. 30, No. 3, 1963, pp. 435-442. [67] Haselgrove, M., Tuck, E., "Stability Properties of the Two-Dimensional Sail Model," Society of Naval Architects and Marine Engineers New England Sailing Yacht Symposium, New London, CN, January 24, 1976. [68] Murai, H., Murayama, S., "Theoretical Investigation of Sailwing Airfoils Taking Account of Elasticities," Journal ofAircraft, Vol. 19, No. 5, 1982, pp.385-389. [69] Jackson, P., "A Simple Model for Elastic Two-Dimensional Sails," AIAA Journal, Vol. 21, No. 1, 1983, pp.153-155. [70] Sneyd, A., "Aerodynamic Coefficients and Longitudinal Stability of Sail Aerofoils," Journal of Fluid Mechanics, Vol. 149, No. 7, 1984, pp.127-146. [71] Cyr, S., Newman, B., "Flow Past Two-Dimensional Membrane Aerofoils with Rear Separation," Journal of Wind Engineering andIndustrial Aerodynamics, Vol. 63, No. 1, 1996, pp. 1-16. [72] Smith, R., Shyy, W., "Computational Model of Flexible Membrane Wings in Steady Laminar Flow," AIAA Journal, Vol. 33, No. 10, 1995, pp. 1769-1777. [73] Newman, B., Low H., "Two-Dimensional Impervious Sails: Experimental Results Compared with Theory," Journal ofFluidMechanics, Vol. 144, 1984, pp. 445-462. and wing tip vortex formation/destabilization [6] are all known to occur within the flow over low aspect ratio wings at low Reynolds numbers. Structurally, the membrane skins used for MAV construction are beset by both geometric and material nonlinearities [19]; the orthotropic nature of the carbon fiber laminates must also be computed. At present, no numerical model exists which can accurately predict all of the three dimensional unsteady features of an elastic MAV wing (flow transition being a particular challenge [8]). As such, an important step in aeroelastic optimization of MAV wings is careful development of lower-fidelity numerical models. Both inviscid vortex lattice methods and laminar Navier-Stokes solvers are investigated, along with linear and nonlinear membrane finite elements (only static aeroelastic models are considered here). In light of the low-fidelity tools that must be used (to maintain computational cost at a reasonable level), a second important step in aeroelastic optimization is extensive experimental model validation. Three levels of model validation are employed: validation of the structural response of individual components of the membrane wing, validation of the aeroelastic behavior of various baseline membrane wings, and validation of the superiority of the computed optimal wings (found either through tailoring or topology optimization) over the baselines. Further complications arise from complex objective functions. As discussed above, gust response is an important performance metric for micro air vehicles [7], but systematic optimization would require an unsteady model, non-homogenous incoming flow, and subsequent time integration. Similarly, delaying the onset of wing stall would require several sub-iterations to locate the stalling angle. Both objectives can be reasonably replaced by a minimization of the lift slope, which is more amenable to a systematic optimization. Finally, the computational complexity is further exacerbated by the multi-objective nature of the problem. Wing structures slack membrane), a design whose lift slope is 8% higher than the minimum possible lift slope, 18% lower than the maximum possible lift slope, and whose pitching moment slope is 34% higher than the minimum possible moment slope. Most of the dominated solutions do not lie far from the Pareto front, indicative of the fact that all of the objective functions are obtained by integrating the pressure and shear distributions over the wing. Substantial variations in the CFD state variables can be obtained on a local level through the use of wing flexibility (Figure 5-21, for example), but integration averages out these deviations. It can also be seen that two of the three Pareto fronts in Figure 6-14 are non-convex. As such, techniques which successively optimize a weighted sum of the two objective functions (convex combination) to fill in the Pareto front will not work; more advanced schemes, such as elitist-based evolutionary algorithms [156], must be used. ........- ,.-0.01 .... .. ...... 8 ritid 0.055 A BR -0.012- a Do a PR A A -0.014 0.05 r Pareto front UA Pareto front 2 U A max L/D, min C max L/D, max C -0.016- & a E--0--0 : P0 Pareto front 1 n rigid 0.045 max lD. min C ,- -0.018 a A BR -0.02 4.6 4.8 5 5.2 5.4 5.6 4.6 4.8 5 5.2 5.4 5.6 L/D L/D Figure 6-14. Computed design performance and Pareto optimality, a = 12. Having successfully implemented the designed experiment, the typical next step is to fit the data with a response surface, a technique used by Sloan et al. [53] and Levin and Shyy [104] for MAV work. Having verified the validity of the surrogate, it can then be used as a relatively inexpensive objective function for optimization. Such a method is not used here for several means of introducing bend-twist coupling in a laminate. The fact that the two fiber directions within the weave are perpendicular automatically satisfies the balance constraint at angles such as 45. This would not be the case if plies of uni-directional carbon fiber are utilized, but this is prohibitive in MAV fabrication for the following reason. Curved, unbalanced, potentially non- symmetric thin uni-directional laminates can experience severe thermal warpage when removed from the tooling board, retaining little of the intended shape. CLa Cma L/D 0.055 -0.0168 4.85 90 / 90 90 0.054 9 / 90 9 -0.0169 4.8 / 4 0.053 45 45 45 90 0 1, 90 45 0 01 90 45 0 1 90 45 0 0 2 02 02 Figure 6-7. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a PR wing, a = 12. Batten Construction Computed lift slope and efficiency of a BR wing at 120 angle of attack is given in Figure 6- 8 as a function of the number of layers in each batten. The thickness of each batten can be varied independently, though the number of layers is limited to three, resulting in 27 possible designs. As before, the membrane skin is slack, and a two-layer plain weave at 450 makes up the remainder of the wing. The normalized out-of-plane displacement and differential pressure coefficients along the chord-station x/c = 0.5 for 4 selected designs is given in Figure 6-9. As expected, the wing with three one-layer battens has the most adaptive washout, which provides the shallowest lift slope, but also the best lift-to-drag ratio. Additional plies, regardless of which batten they are added to, monotonically decreases the efficiency. The same technique can be used to increase CL,, except for combinations of stiff battens towards the wing root and a spanwise pre-tension provides steeper lift and pitching moment curves; the system has a low sensitivity to chordwise pre-tension. This may be due to the membrane skin's shape: its chord is much greater than its span, as discussed above. CL Cna L/D 5 0.056 20 -0.016 20 20 0.054 4.8 0.052 10 -0.018 10 4.6 10 20 10 0 0 Nx 20 10 0 0 Nx 20 10 0 0 x N N N y y Y S1 layer -A-- 2 layers 3 layers Figure 6-12. Computed full factorial design of a PR wing, a = 12. 0 0.01 0.02 0.03 0.04 0.05 0.06 Figure 6-13. Computed PR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), a = 12. For one-layer laminates, no clear trend between CL,, Cma, and pre-tension (chordwise or spanwise) emerges. Whereas the thicker laminates prefer a slack membrane wing to optimize longitudinal static stability, the one-layer wing optimizes this metric when 10 N/m is applied in the span direction. The reflex in the airfoil shape may again be the reason for this. The mild amount of spanwise pre-tension enforces the intended reflex in the membrane skin, and the downward forces depress the membrane skin (seen in Figure 6-4). Slight increases in angle of attack increases the inflation camber towards the leading edge, but decreases the reflex at the trailing edge, resulting in a significant restoring moment. The efficiency of thick-laminate PR throughout a membrane skin affect the aeroelastic response? No model currently exists that can accurately predict such aeroelasticity (the three- dimensional transition is the biggest numerical hurdle), and so the current work utilizes a series of low-fidelity aeroelastic models for efficient movement through the design space: vortex lattice methods and laminar Navier-Stokes solvers are coupled to linear and nonlinear structural solvers, respectively (detailed in Chapter 4). Due to the lower-fidelity nature of the models (despite which, the computational cost of this coupled aeroelastic simulation is very large), experimental model validation is required. Such characterization is conducted in a low speed closed loop wind tunnel. Aerodynamic forces and moments are measured using a strain gage sting balance with an estimated resolution of 0.01 N. Structural displacement and strain measurements are made with a visual image correlation system; a calibrated camera system is mounted over the test section, as discussed in Chapter 3. Chapter 5 provides a detailed analysis of the flow structures, wing deformation, and aerodynamic loads of a series of baseline membrane MAV wings. At small angles of attack, the low Reynolds number flow beneath a MAV wing separates across the leading edge camber, the flow over the upper surface is largely attached, and the tip-vortex swirling system is weak. The opposite is true has the incidence is increased: the bubble on the upper surface grows, eventually leading to stall. The lift curves of the low aspect ratio wings are typically shallow, with a large stalling angle. Low pressure cells deposited on the upper surface of the wing tip by the vortex swirling grow with angle of attack, adding nonlinearities to the lift and moment trends. The structural deformation of a batten-reinforced wing has two main trends: the forces towards the leading edge are very large, and induce membrane inflation in-between the battens. This increases the camber over the wing, and thus the lift. A second trend comes from the free CHAPTER 3 EXPERIMENTAL CHARACTERIZATION As will be extensively discussed below, numerical modeling of flexible MAV wings, while conducive to optimization studies, is very challenging: at the present time, no model exists which can accurately predict all of the unsteady flow phenomenon known to occur over a micro air vehicle. As such, experimental model validation is required to instill confidence in the employed models, highlight numerical shortcomings, and provide additional aeroelastic wing characterization. All of the aerodynamic characterization experiments discussed in this work are run in a closed-loop wind tunnel, a diagram of which can be seen in Figure 3-1. Only longitudinal aerodynamics are of interest, and only a-sweep capability is built into the test setup. Mounting Bracket " VC Cameras Test Section Incoming Flow Sting Speckled Balance MAV Wing Figure 3-1. Schematic of the wind tunnel test setup. Closed Loop Wind Tunnel The test facility used for this work is an Engineering Laboratory Design, Inc. (ELD) 407B closed-loop wind tunnel, with the flow loop arranged in a horizontal configuration. The test system (and the accompanying nonlinear lift and moment curves [3]), as well as the laminar flow separation against an adverse pressure gradient [2]. Similar laminar, steady flow computations for low Reynolds number flyers can be found in the work of Smith and Shyy [72], Viieru et al. [38], and Stanford et al. [43]. 11 1111.111 Figure 4-10. Detail of structured CFD mesh near the wing surface. In order to handle the arbitrarily shaped geometries of a micro air vehicle wing with passive shape adaptation, the Navier-Stokes equations must be transformed into generalized curvilinear coordinates: (x,y,z), rl(x,y,z), ((x,y,z). This transformation is achieved by [148]: Ix Iy Lz f1i f2 f13 fx ]y z 21 f22 f23 (4-13) x Cy z _f31 A32 A33 wherefj are metric terms, and J is the determinant of the transformation matrix: J =(,) (4-14) Using the above information, the steady Navier-Stokes equations can then be written in three-dimensional curvilinear coordinates [149]. The continuity equation and u-momentum equation are presented here in strong conservative form, with the implication that the v- and w- momentum equations can be derived in a similar manner. wing. Similar experimental work [9] at lower speeds also show early stall, again indicating the sensitivity of Reynolds number to stall. At angles of attack below 100, the BR wing has very similar lift characteristics to the rigid wing, a fact also noted in the work of Lian et al. [28]. This is thought to be due to two offsetting characteristics of a wing with both aerodynamic and geometric twist [67]: the inflation in between each batten increases the lift, while the adaptive washout at the trailing edge decreases the lift. Both of these deformations can be seen in Figure 5-1. At higher angles of attack, the load alleviation from the washout dominates the deformation, and decreases both the lift and the lift slope, as indicated by both model and experiment. Delayed stall is not present in the measurements (though, as with the PR wing, has been measured at lower Reynolds numbers [9]), and numerical BR wing modeling cannot be taken past 20 due to aforementioned problems with the moving boundary. Figure 5-13 shows drag coefficients through the a-sweep, with good experimental validation of the model. As before, the drag of the rigid and the BR wings are very similar for modest angles of attack. Above 100 the load alleviation at the trailing edge decreases the drag, a streamlining effect [63]. It should be noted however that for a given value of lift, the BR wing actually has slightly more drag than a rigid wing [9]. Regardless of whether the comparative basis is lift or angle of attack, the PR wing has a drag penalty over the rigid wing. This is in part due to the highly non-optimal airfoil shape of each membrane wing section: Figure 5-5 shows the tangent discontinuity of the wing shape at the membrane/carbon fiber interface towards the leading edge. Excessive inflation may also induce additional flow separation. Longitudinal pitching moments (measured about the leading edge) are given as a function of lift for the three baseline designs in Figure 5-14. Of the three, the PR wing is not statically similar. This is largely due to the linear pitching moment behavior previously noted on the PR wings, possibly due to membrane inflation interference with the tip vortices [14]. Despite the measured improvements over the baseline PR wing, the data indicates that longitudinal control beyond stall (- 280) may not be possible [27]. Interestingly, the same wing design theoretically minimizes the moment slope and maximizes the lift slope, but only the former metric is considerably improved over the baseline. -0.1 -0.2 -0.3 --- rigid 6-- BR -0.4 PR -*- min C -0.5 ma 0 10 20 30 a Figure 6-16. Experimentally measured design optimality over baseline pitching moments. Similar validation results are given in Figure 6-17 and Figure 6-18, for the minimization of drag and maximization of L/D. Both metrics are optimized by wing design (BR,0,0, 1L). The drag is consistently lower than the three baseline designs up to 200. Accurate drag data for micro air vehicles at low speeds is very difficult to measure, largely due to resolution issues in the sting balance [34]. Questionable data typically manifests itself through atypically low drag. Regardless, the veracity of the data from the optimal wing in Figure 6-17 may be confirmed by the identical results at the bottom of the drag bucket with the rigid wing, where deformation is very small. The data also compares very well with computed results. Unlike the baseline BR wing, the optimal design has less drag at a given angle of attack and at a given value of lift (the latter of which is visible in the drag polar, which is not shown). Past 200, the optimal design |

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PAGE 1 1 AEROELASTIC ANALYSIS AND OPTIMI ZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By BRET KENNEDY STANFORD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Bret Kennedy Stanford PAGE 3 3 To my family, despite making fun of me for being in school for so long To Angel, despite making fun of me for a bunch of other reasons And to Fatty too, for only scratching me wh en I really deserve it, which is often PAGE 4 4 ACKNOWLEDGMENTS Thank you to Dr. Peter Ifju, who offered his guidance on a countless number of research projects, and still let m e work on the ones he thought were dumb. Unque stionably the coolest, smartest, most up-beat professor Ive ever been around. Thank you to Dr. Rick Lind, for consistently po inting out when I need to shave, or get a haircut, or more frequently, both. Thank you to Dr. Roberto Albertani, for shari ng with me his passion for all things wind tunnel related, and for sharing his equipment up at the REEF. Thank you to Dr. Raphael Haftka and Dr. David Bloomquist for serv ing on my committee and sitting through my long, scientifically-que stionable presentations without complaining. Thank you to Dr. Dragos Viieru, for impar ting me with his vast knowledge of CFD. Thank you to Dr. Wei Shyy, for all his help my first few semesters of grad school. A final thank you to all the people who hung aro und the labs I worked in. Frank Boria, who helped teach me the real names for various tools and hardware, which had previously been known to me only as shiny metal things. A thanks-in-advance to Frank for taking all of my future phone calls concerning mortgages, insuran ce, child rearing, etc, no matter how distraught and hysterical they may be. Mujahid Abdulrahim, for discussing with me the ethics of returning a rental car completely caked in mud, and going in reverse through a drive-thru. Wu Pin, for relating countless unintentionally funny and creepy stories that Ill never forget, despite my best efforts. Ill always wonder how you got into this country. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................12 CHAP TER 1 INTRODUCTION..................................................................................................................13 Motivation...............................................................................................................................13 Problem Statement.............................................................................................................. ....18 Dissertation Outline........................................................................................................... .....19 Contributions..........................................................................................................................21 2 LITERATURE REVIEW.......................................................................................................22 Micro Air Vehicle Aerodynamics.......................................................................................... 22 Low Reynolds Number Flows.........................................................................................22 Low Aspect Ratio Wings.................................................................................................23 Low Reynolds Number Low Aspect Ratio Interactions............................................... 24 Rigid Wing Optimization....................................................................................................... 26 Micro Air Vehicle Aeroelasticity........................................................................................... 28 Two-Dimensional Airfoils...............................................................................................28 Three-Dimensional Wings............................................................................................... 31 Aeroelastic Tailoring.......................................................................................................... ....34 Topology Optimization.......................................................................................................... .35 3 EXPERIMENTAL CHARACTERIZATION........................................................................ 39 Closed Loop Wind Tunnel...................................................................................................... 39 Strain Gage Sting Balance...................................................................................................... 40 Uncertainty Quantification.............................................................................................. 43 Wind Tunnel Corrections................................................................................................43 Visual Image Correlation....................................................................................................... .44 Data Procession...............................................................................................................47 Uncertainty Quantification.............................................................................................. 48 Model Fabrication and Preparation........................................................................................ 49 4 COMPUTATIONAL FRAMEWORK AND VALIDATION............................................... 51 Structural Solvers............................................................................................................. .......51 PAGE 6 6 Composite Laminated Shells........................................................................................... 52 Membrane Modeling....................................................................................................... 56 Skin Pre-tension Considerations......................................................................................62 Fluid Solvers...........................................................................................................................66 Vortex Lattice Methods................................................................................................... 66 Steady Navier-Stokes Solver........................................................................................... 67 Fluid Model Comparisons and Validation......................................................................70 Aeroelastic Coupling........................................................................................................... ...72 Moving Grid Technique..................................................................................................72 Numerical Procedure....................................................................................................... 73 5 BASELINE WING DESIGN ANALYSIS............................................................................. 75 Wing Deformation..................................................................................................................75 Aerodynamic Loads................................................................................................................83 Flow Structures.......................................................................................................................88 6 AEROELASTIC TAILORING..............................................................................................97 OFAT Simulations..................................................................................................................98 Membrane Pre-Tension...................................................................................................99 Single Ply Laminates..................................................................................................... 102 Double Ply Laminates................................................................................................... 103 Batten Construction.......................................................................................................105 Full Factorial Designed Experiment..................................................................................... 107 Experimental Validation of Opti mal Design Performance................................................... 116 7 AEROELASTIC TOPOLOGY OPTIMIZATION............................................................... 122 Computational Framework................................................................................................... 125 Material Interpolation.................................................................................................... 125 Aeroelastic Solver......................................................................................................... 128 Adjoint Sensitivity Analysis.......................................................................................... 130 Optimization Procedure................................................................................................. 133 Single-Objective Optimization............................................................................................. 134 Multi-Objective Optimization.............................................................................................. 149 CONCLUSIONS AND FUTURE WORK .................................................................................. 161 REFERENCES............................................................................................................................167 BIOGRAPHICAL SKETCH.......................................................................................................179 PAGE 7 7 LIST OF TABLES Table page 4-1 Experimental influence matrix (m m /N) at points labeled in Figure 4-2................................55 4-2 Numerical influence matrix (mm/ N) at points labele d in Figure 4-2 ..................................... 56 5-1 Measured and computed aerodynamic characteristics, = 6 ................................................ 88 6-1 Optimal MAV design array with co m promise designs on the off-diagonal, = 12: design description is (wing type, Nx, Ny, number of plain weave layers)....................... 115 6-2 Optimal MAV design performance array, = 12: off-diagonal com promise design performance is predicated by column metrics, not rows................................................. 115 PAGE 8 8 LIST OF FIGURES Figure page 1-1 Batten-reinforced membrane wing design.............................................................................. 14 1-2 Perimeter-reinforce d m embrane wing design......................................................................... 15 3-1 Schematic of the wind tunnel test setup................................................................................. 39 3-2 Quantification of the reso lution error in the VIC system ....................................................... 48 3-3 Speckled batten-reinforced membra ne wing with wind tunnel attachm ent........................... 50 4-1 Unstructured triangular mesh used for fi nite elem ent analysis, with different element types used for PR and BR wings........................................................................................52 4-2 Computed deformations of a BR wing skeleton due to a poin t lo ad at the wing tip (left) and the leading edge (right)...............................................................................................54 4-3 Compliance at various locations along the wing, due to a point load at those locations ....... 56 4-4 Uni-axial stretch test of a latex rubber membrane.................................................................. 61 4-5 Circular membrane response to a uniform pressure............................................................... 61 4-6 Measured chordwise pre-strains in a BR wi ng before the tension is released from the latex (left), and after (right)............................................................................................... 63 4-7 Monte Carlo simulations: error in the com puted m embrane deflection due to a spatiallyconstant pre-strain distribution assumption.......................................................................65 4-8 Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and shear (right) in a BR wing, corrected at the trailing edge for a uniform pre-stress resultant of 10 N/m............................................................................................................66 4-9 CFD comput ational dom ain.................................................................................................. ..68 4-10 Detail of structured CF D m esh near the wing surface.......................................................... 69 4-11 Computed and measured aerodynamic co efficients for a rigid MAV wing, Re = 85,000 ... 71 4-12 Iterative aeroela stic convergence of m embrane wings, = 9.............................................. 74 5-1 Baseline BR normalized out-of-plane displacement (w/c), = 15 ....................................... 76 5-2 Baseline BR chordwise strain ( xx), = 15........................................................................... 77 5-3 Baseline BR spanwise strain ( yy), = 15.............................................................................77 PAGE 9 9 5-4 Baseline BR shear strain ( xy), = 15................................................................................... 78 5-5 Baseline PR normalized out-of-plane displacement (w/c), = 15 ........................................ 79 5-6 Baseline PR chordwise strain ( xx), = 15............................................................................ 79 5-7 Baseline PR spanwise strain ( yy), = 15.............................................................................80 5-8 Baseline PR shear strain (xy), = 15.................................................................................... 80 5-9 Baseline BR aerodynamic and geom etric twist distribution, = 15.....................................81 5-10 Baseline PR aerodynamic and geometric twist distribution, = 15 ...................................82 5-11 Aerodynamic and geometric twist at 2y/b = 0.65................................................................. 83 5-12 Baseline lift coefficients: numer ical (left), experim ental (right).......................................... 84 5-13 Baseline drag coefficients: nume rical (left), experim ental (right)....................................... 86 5-14 Baseline pitching moment coefficients: num erical (l eft), experimental (right)................... 87 5-15 Baseline wing efficiency: numeric al (left), experim ental (right)......................................... 87 5-16 Pressure distributions (Pa) and stream lines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 0.................................................................................. 90 5-17 Pressure distributions (Pa) and streamlin es on the lower surface of a rigid (left), BR (center), and PR wing (right), = 0.................................................................................. 91 5-18 Pressure distributions (Pa) and stream lines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 15................................................................................92 5-19 Pressure distributions (Pa) and streamlin es on the lower surface of a rigid (left), BR (center), and PR wing (right), = 15................................................................................94 5-20 Section normal force coefficients, and pressure coefficients (2y/b =0.5), = 0 ............... 96 5-21 Section normal force coefficients, and pressure coefficients (2y/b =0.5), = 15 .............. 96 6-1 Computed tailoring of pre-st ress resultants (N/ m) in a BR wing, = 12............................. 99 6-2 Computed BR wing deformation (w/c) with various pre-tensions, = 12 ......................... 100 6-3 Computed tailoring of pre-stress resultants (N/ m) in a PR wing, = 12............................ 101 6-4 Computed PR wing deformation (w/c) with various pre-tensions, = 12 .......................... 102 PAGE 10 10 6-5 Computed tailoring of lam inate orientation for single pl y bi-directional carbon fiber, = 12....................................................................................................................................103 6-6 Computed tailoring of lami nate orientations for two plies of bi-directional carbon fiber in a BR wing, = 12 .......................................................................................................104 6-7 Computed tailoring of lami nate orientations for two plies of bi-directional carbon fiber in a PR wing, = 12 .......................................................................................................105 6-8 Computed tailoring of batten construction in a BR wing, = 12 .......................................106 6-9 Computed normalized out-of-plane displaceme nt (left) and different ial pressure (right) at x/c = 0.5, for various BR designs, = 12 ...................................................................107 6-10 Computed full fact orial design of a BR wing, = 12 .......................................................109 6-11 Computed BR wing deformation (w/c) with one layer of plain weave (left), tw o layers (center), and three layers (right), = 12......................................................................... 109 6-12 Computed full factor ial design of a PR wing, = 12 ........................................................111 6-13 Computed PR wing deformation (w/c) with one layer of plain weave (left), tw o layers (center), and three layers (right), = 12......................................................................... 111 6-14 Computed design perfor m ance and Pareto optimality, = 12..........................................113 6-15 Experimentally measured design optimality over baseline lift.......................................... 118 6-16 Experimentally measured design optimality over baseline pitching moments.................. 119 6-17 Experimentally measured design optimality over baseline drag........................................ 120 6-18 Experimentally measured design optimality over baseline efficiency............................... 120 7-1 Wing topologies flight te sted by Ifju et al. [10] ................................................................... 123 7-2 Sample wing topology (left), aerodynamic me sh (center), and structural m esh (right).......124 7-3 Effect of linear and nonlinear material interpolation upon lift ............................................. 127 7-4 Measured loads of an inade quately reinforced m embrane wing, U = 13 m/s.................... 130 7-5 Convergence history for maximizing L/D, = 3, reflex wing ............................................ 135 7-6 Affect of mesh density upon optimal L/D topology, = 12, reflex wing ........................... 137 7-7 Affect of initial design upon the optimal CD topology, = 12, reflex wing....................... 138 7-8 Affect of angle of attack and airfoil upon the optimal CL topology..................................... 139 PAGE 11 11 7-9 Affect of angle of attack and airfoil upon the optim al L/D topology................................... 140 7-10 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for baseline and optimal topology designs, = 12, reflex wing......................141 7-11 Deformations and pressures along 2y/b = 0.58 for baseline and optim al topology designs, = 12, reflex wing............................................................................................ 142 7-12 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for baseline and optimal topology designs, = 12, cambered wing............... 145 7-13 Deformations and pressures along 2y/b = 0.58 for baseline and optim al topology designs, = 12, cambered wing..................................................................................... 146 7-14 Wing topology optim ized for minim um CL built and tested in the wind tunnel............... 148 7-15 Experimentally measured forces a nd m oments for baseline and optimal topology designs, reflex wing.........................................................................................................149 7-16 Convergence history for ma xi mizing L/D and minimizing CL = 0.5, = 3, reflex wing..................................................................................................................................151 7-17 Trade-off between e fficiency and lift slop e, = 3, reflex wing........................................ 152 7-18 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for designs that trade-off between L/D and CL = 3, reflex wing................154 7-19 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between L/D and CL = 3, reflex wing.............................................................................................155 7-20 Trade-off between drag and pitching moment slope, = 12, reflex wing ........................ 156 7-21 Trade-off between lift and lift slope, = 12, cam bered wing........................................... 157 7-22 Normalized out-of-plane displacements (top) and differential pressure coefficients (bottom ) for designs that trade-off between CL and CL = 12, cambered wing.......... 159 7-23 Deformations and pressures along 2y/b = 0.58 for designs that trade-off between CL and CL = 12, cambered wing..................................................................................... 160 PAGE 12 12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AEROELASTIC ANALYSIS AND OPTIMI ZATION OF MEMBRANE MICRO AIR VEHICLE WINGS By Bret Kennedy Stanford May 2008 Chair: Peter Ifju Major: Aerospace Engineering Fixed-wing micro air vehicles are difficult to fly, due to their low Reynolds number, low aspect ratio nature: flow separa tion erodes wing efficiency, the wi ngs are susceptible to rolling instabilities, wind gusts can be the same size as th e flight speed, the range of stable center of gravity locations is very small, etc. Membrane aeroelasticity has been identified has a tenable method to alleviate these issues. These flexible wing structures are divide d into two categories: load-alleviating or load-augme nting. This depends on the wings topology, defined by a combination of stiff laminate composite members overlaid with a membrane sheet, similar to the venation patterns of insect wings. A series of well-validated variable-fidelity static aeroelastic models are developed to analyze the working mechanisms (cambering, washout) of membrane wing aerodynamics in terms of loads, wing deformation, and flow structures, for a small set of wing topologies. Two aeroelastic optimization schemes are then discussed. For a given wing topology, a series of numerical de signed experiments utilize tailoring of laminate orientation and membrane pre-tension. Further generality can be obtained with aeroelastic topology optimization: locating an optimal distribution of laminate shells and membrane skin throughout the wing. Both optimization schemes consider several design metrics, optimal compromise designs, and experimental validation of superiority over baseline designs. PAGE 13 13 CHAPTER 1 INTRODUCTION Motivation The rapid co nvergence of unmanned aerial vehicles to continually smaller sizes and greater agility represents successful efforts along a mu ltidisciplinary front. Technological advances in materials, fabrication, electroni cs, propulsion, actuators, sensors, modeling, and control have all contributed towards the viable candidacy of micro air vehicles (MAVs) for a plethora of tasks. MAVs are, by definition, a class of unmanned ai rcraft with a maximum size limited to 15 cm, capable of operating speeds of 15 m/s or less Ideally, a MAV should be both inexpensive and expendable, used in situations wh ere a larger vehicle would be im practical or impossible, to be flown either autonomously or by a remote pilot. Military and defens e opportunities are perhaps easiest to envision (in the form of over-the-hill battlefield surveillance, bomb damage assessment, chemical weapon detection, etc.), tho ugh MAVs could also play a significant role in environmental, agriculture, wildlife, and traffic monitoring applications. MAVs are notoriously difficult to fly; an e xpected consequence of a highly maneuverable and agile vehicle that must be flow n either remotely or by autopilot [1]. The aerodynamics are beset by several unfavorable flight issues: 1. The operational Reynolds num ber for MAVs is typically between 104 and 105. Flow over the upper wing surface can be characterized by massi ve flow separation, a possible turbulent transition in the free shear layer, and then reattachment to the surface, leaving behind a separation bubble [2]. Such flow structures typically resu lt in a loss of lift, and an increase in drag, and a drop in the overall efficiency [3]. 2. The low aspect ratio wing (on the order of unity) prom otes a large wing tip vortex swirling system [4], which interferes with the l ongitudinal circulation of the wing [5]. Entrainment of the aforem entioned separated flow can lead to tip vortex destabilization [6]; the resulting bilateral asymmetry m ay be the cause of the rolling instabiliti es known to plague MAV flight. 3. Sudden wind gusts may be of the same order of magnitude as the vehicle flight speed (10-15 m/s). Maintaining smooth contro llable flight can be difficult [7] [8]. PAGE 14 14 4. The range of flyable (statically stable) CG lo cations is generally only a few millimeters long, which represents a strenuous weight management challenge [9]. These problem s, along with a broad range of dynamics and control issues, can be alleviated through the appropriate use of wi ng shape adaptation. Active morphing mechanisms have been successfully used on a small class of unmanned air vehicles [1], but the limited energy budgets and size constraints of micro air ve hicles m ake such an option, at present writing, infeasible. As such, the current work is restrict ed to passive shape adaptation. Passive shape adaptation can be successfully built into a MAV wing through the use of a flexible membrane skin [10]. The basic structure of these vehicles is built around a com posite laminate skeleton. Bi-directional graphite/epoxy pl ain weave or uni-directi onal plies are usually the materials of choice, due to durability, low we ight, high strength, and ease of fabrication: all qualities well-exploited in the aviation industry [11]. The carbon fiber skeleton is affixed to an extensible m embrane skin, of which several choices are available: latex, s ilicone, plastic sheets, or polyester [12]. The distribution of carbon fiber and me mbrane skin in the wing determ ines the aeroelastic response, and is demonstrated by two dis tinct designs. The first ut ilizes thin strips of uni-directional carbon fiber imbe dded within the membrane ski n, oriented in the chordwise direction ( Figure 1-1). The trailing edge of the batte n-reinforced (BR) design is unconstrained, and the resulting nose-down geom etric twist of each flexible wing section should alleviate the flight loads: decrease in CD, decrease in CL delayed stall (as compared to a rigid wing) [13]. Figure 1-1. Batten-reinf orced membrane wing design. PAGE 15 15 A second design leaves the interior of the me mbrane skin unconstrained, while the perimeter of the skin is sealed to a thin curved strip of carbon fiber ( Figure 1-2). The perimeter-reinforced (PR) wing defor mation is closer in nature to an aerodynamic twis t. Both the leading and the trailing edges of each membrane section are cons trained by the relatively-stiff carbon fiber. The positive cambering (inflation) of the wi ng should lead to an increase in CL and a decrease (more negative) in Cm [14]. Figure 1-2. Perimeter-reinf orced membrane wing design. While both of these wing structures can ade quately perform their intended tasks (load alleviation for the BR wing, load enhancement for the PR wing), several sizing/stiffness variables exist within both desi gns, leading to an aeroelastic tailoring problem. Conventional variables such as the la minate fiber orientation [11] can be considered, but the directional stiffness induced by varying the pre-tension within the m embrane skin may play a larger role within the aeroelastic response [15]. Systematic optimization of a single design metric will typically lead to a wing structure with poor pe rform ance in other important aspects. For example, tailoring the PR wing structure for maximum static stability may provoke an unacceptable drag penalty, and vice-versa. Of the aerodynamic performance metrics considered here (lift, drag, efficiency, static stability, gust suppression, and mass) many are expected to conflict, as with most engineering optimi zation applications. Formal multi-objective optimization procedures can be used to tailor flex ible MAV wing designs that strike an adequate PAGE 16 16 compromise between conflicting metrics, filling in the trade-off curves. While thorough exploration of this aeroel astic tailoring design space can provide a fundamental understanding of the relationship between spatial stiffn ess distribution and aerodynamic performance in a flexible MAV wi ng, further steps towards generality can be achieved by removing the constraint that the wing structures must utilize a BR or a PR design. Topology optimization is typically used to find the location of holes within a homogenous structure, by minimizing compliance unde r a constraint upon the volume fraction [16]. Here it is used to find the location of m embrane skin w ithin a carbon fiber skeleton that will optimize a given aerodynamic objective function. This work will be able to highlight wing topologi es with superior efficacy to those designs considered above (for example, a wing with be tter gust suppression quali ties than the BR wing), as well as designs that strike a compromise between conflicting metrics (for example, a topological combination of the BR and the PR wings ). While the results may be more rewarding than those obtained from tailoring, aeroelast ic topology optimization is significantly more complex. Tailoring requires 5-10 sizing and sti ffness variables, but the topology optimization may utilize thousands of variables: the wing is divi ded into a series of panels, each of which may be membrane or carbon fiber. This necessitates a gradient-based algorithm, while evolutionary algorithms or response surface approaches are feasible for the former problem. Both aeroelastic tailoring and topology optimi zation are effective tools for exploiting the passive shape adaptation of flexible MAV wings, but the computational cost is prohibitive. It is not uncommon for aeroelastic optimization studies to require hundreds, or even thousands, of function evaluations. Numerical modeling of fl exible MAV wings is very challenging and expensive: flow separation, tr ansition, and reattachment [17], vortex shedding and pairing [18], PAGE 17 17 and wing tip vortex formation/destabilization [6] are all known to occur within the flow over low aspect ratio wings at low Reynol ds numbers. Structurally, the membrane skins used for MAV construction are beset by both geometric and material nonlinearities [19]; the orthot ropic nature of the carbon fiber lam inates must also be computed. At present, no numerical model exists whic h can accurately predict all of the three dimensional unsteady features of an elastic MAV wing (flow transition being a particular challenge [8]). As such, an important step in aeroe las tic optimization of MAV wings is careful development of lower-fidelity numerical models Both inviscid vortex lattice methods and laminar Navier-Stokes solvers are investigated, al ong with linear and nonlinear membrane finite elements (only static aeroelastic models are considered here). In light of the low-fidelity tools that must be used (to maintain computational cost at a reasonable level) a second important step in aeroelastic optimization is extensive experime ntal model validation. Three levels of model validation are employed: validation of the struct ural response of indi vidual components of the membrane wing, validation of the aeroelastic beha vior of various baseline membrane wings, and validation of the superiority of the computed optimal wings (found either through tailoring or topology optimization) over the baselines. Further complications arise from complex objec tive functions. As discussed above, gust response is an important performan ce metric for micro air vehicles [7], but systematic optim ization would require an unsteady model, non-homogenous incoming flow, and subsequent time integration. Similarly, delaying the onset of wing stall would require several sub-iterations to locate the stalling angle. Both objectives can be reasonably replaced by a minimization of the lift slope, which is more amenable to a syst ematic optimization. Finally, the computational complexity is further exacerbated by the multi-objective nature of the problem. Wing structures PAGE 18 18 that optimize a single objective function are of limited value; of greater importance is the array of designs that lie along the Pareto optimal front. This front is a trade-off curve comprised of non-dominated designs, one of which can be selected based on additional considerations not included in the optimization: manufacturability, flight specifications (duration, payload), etc. Computation of the Pareto front is costly when gradient-based search routines are used for optimization, typically involving successive optimization runs with various convex combina tions of the objective functions. The success of this technique depends on the convexity of the Pa reto optimal front. More efficient methods for computing the Pareto front are available if evol utionary algorithms or response surface methods are employed. Problem Statement The static aeroelas ticity of membrane micro air vehicle wi ngs represents th e intersection of several rich aerodynamics and mechanics problems ; numerical modeling can be very challenging and expensive. Furthermore, MAVs are beset w ith many detrimental flight issues, and are very difficult to fly: systematic numerical optimization schemes can be used to offset these problems, improving flight duration, gust supp ression, or static stability. Ma ny optimization studies can be considered for MAVs; the curren t work utilizes aero elastic optimization, which will require hundreds of costly function evaluations to adequa tely converge to an optimal design. As the feasibility of such a scheme relies on a moderate computational cost, what is the lowest fidelity aeroelastic model that ca n be appropriately used? Model development requires extensive experimental valid ation, and several challenges exist here as well. The forces generated by a MAV wing are very small, and highly-sensitive instrumentation is needed. For deformation measurements, only visi on-based non-contacting methods are appropriate. What particular com ponents are required to construct an adequate PAGE 19 19 experimental test-bed for flexible MAVs? What performance metrics should be compared between numerical and experimental re sults for sufficient model validation? Upon suitable validation of the aeroelastic m odel, two optimization studies are developed: tailoring and topology optimization. Considering the former, with a given spatial distribution of laminated carbon fiber and membrane skin thro ughout the wing, what is the optimal chordwise and spanwise membrane pre-tension and carbon fiber laminate lay-up schedule? For the aeroelastic topology optimization studies, with a given membrane pre-tension and laminate orientation, what is the optimal distribution of carbon fiber and membrane skin throughout the wing? What performance metrics should be optim ized? As these metrics will surely conflict, what multi-objective optimization schemes are appropr iate for computation of the Pareto front? Can the numerically-indicated optimal wing design structures be built and tested, and will the experimental results also indi cate superiority over similarl y-tested baselin e designs? Dissertation Outline This work begins with a detailed literature review of m icro air ve hicle aerodynamics (low Reynolds number flows, low aspect ratio wings unsteady flow phenomena), aeroelasticity (membrane sailwings, flexible filaments), and optimization (rigid wing airfoil and planform optimization, tailoring). I review the literatur e pertaining to topology optim ization as well, with a particular emphasis upon aeronautical and aeroelastic applications. I then discuss the apparatus and procedures us ed for experimental ch aracterization of the membrane micro air vehicle wings. This incl udes a low-speed closed loop wind tunnel, a high sensitivity sting balance, and a visual image correlation system. Information is also given detailing wing fabrication and pr eparation. I summarize the com putational framework, including both linear and nonlinear structural finite elem ent models. Three-dimensional viscous and inviscid flow solvers are formulated, along w ith aeroelastic coupling and ad hoc techniques PAGE 20 20 devised to handle the membrane skin pre-tension. The estimated validity range of each model is discussed. I detail the deformation patterns, flow stru ctures, and aerodynamic characterization of a series of baseline flexible a nd rigid MAV wings, obtained both numerically and experimentally for comparison purposes. Once the predictive capa bility of the aeroelastic model is wellverified, these data sets are st udied to uncover the working mech anisms behind the passive shape adaptation and their associated aerodynamic advantages. I then use a non-standard aeroela stic tailoring study to iden tify the optimal wing type and structural composition for a given objective function, as well as va rious combinations thereof. Wing types are limited to rigid, batten-reinforced and perimeter-reinforced designs; structural composition variables incl ude anisotropic membrane pre-tensi on and laminate lay-up schedule. Multi-objective optimization is c onducted using a design of experiments approach, with a series of aerodynamic coefficients and derivatives as metrics. The tailoring concludes with experimental validation of the perfor mance of selected optimal designs. Finally, I formulate a computa tional framework for aeroelast ic topology optimization of a membrane micro air vehicle wing. A gradient-based search is used, with analytically computed sensitivities of the sa me aerodynamic metrics as used above The optimal wing topology is discussed as a function of flight condition, grid density, initial guess, and design metric. I optimize a convex combination of two conflicting objective functions to construct the Pareto front, with a demonstrated superiority over the base line wing structures em ployed in the tailoring study. As before, the work concludes with e xperimental validation of the performance of selected optimal designs. PAGE 21 21 Contributions 1. Develop a set of variable-fidel ity aeroelastic m odels for lo w Reynolds number, low aspect ratio membrane micro air vehicle wings. 2. Develop a highly-sensitive non-intrusive experi mental test-bed for model deformation and flight loads. 3. Optimization-based system identification of the wing structures material properties. 4. Experimental aeroelastic model validation of flight loads and wing deformation. 5. Optimize multiple flight metrics by tailoring memb rane pre-tension and laminate orientation. 6. Develop computational framework for topology optimization of membrane wings, with an analytical sensitivity analysis of the coupled aeroelastic system. 7. Able to provide scientific insight into the re lationship between optimal wing flexibility, flow structures, and the resulting beneficial effects upon flight loads and efficiency. 8. Experimental validation of the superiority of selected optimal designs over baselines. PAGE 22 22 CHAPTER 2 LITERATURE REVIEW Micro Air Vehicle Aerodynamics A long history of flight te sting, com putational modeli ng, and wind tunnel work has generally pushed the design methodology of succe ssful fixed wing MAVs to a thin, cambered, low aspect ratio flying wing. Maximizing the wing area for a gi ven size constraint obviously leads to a low aspect ratio design. Further desi re to minimize the size of a MAV negates the use of horizontal stabilizers, repla ced with a reflex airfoil for longi tudinal static stability, wherein negative camber present towards the trailing edge helps offset the longitudinal pitching moment of the remainder of the wing. The superiority of thin wings for MAV applications can be shown by both three dimensional inviscid simulations [20] and two-dimensional viscous simulations [21] [22], where the drop in the adverse pressure gradients increas es the lift and decreases the drag towards stall. Similar tools, as well as wind tunnel testing, indi cates the advantage of cambered wings over flat plates [5]; beyond the obvious increase in lift, higher lift-to-drag ratios are reported by Laitone [23]. Much work has also been done on locating suitable MAV planform shapes. Torres identifies the inverseZimmerman as ideal, based upon size restrictions, required angle of attack, and drag performance; th e optimum shifts to an elliptical shape as the aspect ratio is increased [3]. Low Reynolds Number Flows Low Reynolds num ber laminar flow is likely to separate against an adverse pressure gradient aft of the pressure re covery location (velocity peak) on the upper wing surface, even for fairly low angles of attack. The formation of a turbulent boundary la yer aft of a separation bubble is a very mutable process: Reynolds numbe r, pressure distributions, airfoil geometry, surface roughness, turbulence intensity, acoustic noise, wall heating, and -direction (whether PAGE 23 23 the angle of attack is being increase d or decreased can lead to hysteresis [24]) are all cited by Young and Horton [2] as highly influential on the formati on of a bubble. Furtherm ore, the flow will only reattach to the surface if there is enough energy to maintain circulating flow against dissipation [25]. An extensive survey of low Reynolds num ber (34 55) airfoils is given by Carmichael [17] (there are quite a few othe rs, as reviewed by Shyy et al. [26]). The study finds that, for the lower end of tested Reynolds num be rs, the laminar separated flow does not have time to reattach to the surface. Above 54, the flow will reattach, forming a long separation bubble over the wing. At the upper end of the range of Reynolds numbers discussed by Carmichael, the size of the bubble decreases, genera lly resulting in a decrease in form drag. Increasing the angle of attack ge nerally enhances the turbulence in the flow, which can also prompt quicker reattach ment and shorter bubbles [8]. The length of the separation bubble can generally be inferred from the plateau-like behavior of the pressure distribution: the flow speeds up before the bubble (dropping the pressu re), and slows down after the bubble [27]. This description is a time-averaged scenario: in an unsteady se nse, the inf lectional velocity profile across the separation bubble can develop inviscid Kelvin-Helmho tlz instabilities and cause the shear layer to roll up. This leads to periodic vortex sh edding and the required matching downstream [18], and can cause the separation bubble to m ove back and forth [28]. Further work detailing low Reynolds num ber flow over rigid airfoils can be found by Nagamatsu [29], Masad and Malik [30], and Schroeder and Baeder [31]. Low Aspect Ratio Wings Early work in low aspect ratio aerody namics was sparked by an inability to fit experimental data with linear aerodyna mics theories, as reported by Winter [32] for aspect ratios between 1.0 and 1.25. The m easured lift is typically higher than predicte d (similar to vortex lift PAGE 24 24 discrepancies seen on delta wings [27]), as the strong wing tip vortices interfere with the longitudinal wing circulation. Th e most obvious indication of such an interaction is the high stalling angles of low aspect ratio wings, wher e the downward m omentum of the tip vortices can keep the flow attached to the upper wing surf ace. Experimental work by Sathaye et al. [33] using an array of pressure ports was able to confirm the devia tion of the lift distribution from elliptic wing theory. Lian et al. [28] report a computed dip in the previous ly constant pressure coefficients over the upper wing surface at 75% of the sem i-span for high angles of attack, but only minor changes on the bottom surface at the wing tip. These low pr essure cells at the wing tip will grow in intensity and spread inward towards the root as th e angle of attack (and t hus the strength of the swirling system) is increased [34]. The cells are a nonlinear cont r ibution to the wings lift; their growth with angle of attack increases CL with angle of attack as well. Torres [3] gives a general cutoff between a lin ear and a nonlinear CLrelationship at an aspect ratio of 1.25. Low aspect ratio corrections to the lift pr edicted by linear theory (among many) are given by Bartlett and Vidal [35], while Polhamus [36] is able to collapse the meas ured profile drag data at various aspect ratios to a single curve th rough the use of an effective twodimensional lift coefficient. Further experimental work is given by Kaplan et al. [37], who use measurements of the trailing vortex structure off of low aspect ratio flat plates for adequate com parison with force balance measurements. The authors indicate that the nonlinear lift curves may also be caused by a loss of leading edge suction, a nd a rotation of the force vector into a flow-normal direction. Viieru et al. [38] discusses the use of endplates to temp er the induced drag fr om the tip vortices, with reported improvements in the lift-to-drag ra tio at small to moderate angles of attack. Low Reynolds Number Low As pect Ratio Interactions Several interactions betw een the low aspect ratio and low Reynolds number aerodynamics PAGE 25 25 of MAVs are reported in the lite rature. Mueller and DeLaurier ci te aspect ratio as the most important design variable, followed by wing pl anform and Reynolds number. Free stream turbulence intensity and trailing edge geometry are reported to be non-factors, and Reynolds number is only important near stall [39]. Flow visualization e xperim ents by Gursul et al. [40] on swept, non-slender, low aspect ratio wings find the presence of prim ary and secondary vortices, with stagnant flow regions outboard of the former. Vortex merging and other unsteady interactions within the shear layers are found to be highly de pendent on Reynolds number. Kaplan et al. [37] report a fluctuation in the locati on of the vortex core off of a se mielliptical wing at 8,000 Reynolds number. Numerical simulations and flow visualization by Tang and Zhu [6] of an accelerating elliptical wing show an unstable interaction between a longitudinal secondary separated vortex and the tip vortices. This destabilization (for angles of attack above 11) causes the tip vor tex system to swing back and fo rth along the wing, leading to bilateral asymmetry problems in roll. The author s also note a stationary separated vortex (rather than the customary shedding) for angles above 33 possibly due to the vertical components of the tip vortices. Cosyn and Vierendeels [41], Brion et al. [42], and Stanford et al. [43], discuss numerical wing m odeling of lift and drag for comparison with wind tunnel experiments: the lack of a threedimensional turbulent-transition model is genera lly cited as the reason for poor correlation at higher angles of attack. Results documenting th e aerodynamics of a complete micro air vehicle (wing with fuselage, stabilizers, propellers, etc) are scarce: wi nd tunnel experimentation by Zhan et al. document longitudinal and la teral stability as a f unction of vertical stabilizer placement and wing sweep [44]. Similar stability data is given by Ramamurti et al. [45] for a MAV wing with counter-rotating propellers. PAGE 26 26 Gyllhem et al. [46] reports that the presence of a fuselage, m otor, and stabilizers surprisingly improves the computed maximum lift a nd stall angle (compared to simulations with just the wing), but increases the drag as well. Experimental work by Albertani [47] finds just the opposite: a decrease in lif t of the entire vehicle, but less of a penalty when passive shape adaptation is built into th e wing. Waszak et al. [13] are able to show significant improvements in efficiency if a stream lined MAV fuselage is used. Rigid Wing Optimization Though the m ain scope of the current work is to improve the aerodynamic qualities of fixed micro air vehicle wings through the judicious use of aeroelastic membrane structures, much successful work has been done with multidiscip linary optimization of the shape, size, and components of a rigid MAV wing. These studies must often make use of low fidelity models due to the large number of function evaluations required for a typical optimization run, and may not be able to capture the complicated flow physics described above. Nonetheless, insight into the relationship between sets of sizing/shape va riables and a given objective function can still be gained. Early work is given by Morris [48], who finds the smallest ve hicle tha t will satisfy given constraints throughout a theoretical mission, using several empirica l and analytical expressions for the performance evaluation. Rais-Rohani and Hicks investigate a similar problem, using a vortex lattice method (for com putations of aerodynamic performance and stability, along with propulsion and weight modules) a nd an extended interior penalty function method to reduce the size of a biplane MAV [49]. Kajiwara and Haftka empha size the unconventional need for sim ultaneous design of the aerodynamic and the cont rol systems at the micro air vehicle scale, due to limited energy budgets [50]. PAGE 27 27 Torres [3] uses a genetic algorithm to minimi ze a weighted com bination of payload, endurance, and agility metrics, with various di screte (wing and tail planform) and continuous (aspect ratio, propeller location, angle of attack, etc) variable s. Aerodynamic analysis is provided by a combination of experimental data, analytical methods and interpolation techniques. The author cites convergence problems stemming fr om the discrete variables. Genetic algorithms are also used in the work of Lundstrm and Krus [51] and Ng et al. [52]. The latter indicates that thes e algorithms are more su ited for the potentially disjointed design spaces presented by MAV optimization efforts. A comparison between a genetic algorithm and gradient-based sequential quadratic programming used to design winglets for a swept wing MAV indicates the superiority of th e former, with a vortex lattice method used for aerodynamic analysis. However, a genetic algorithm may only be feasible for lo wer fidelity tools, due to the large number of function evalua tions required for convergence. Higher fidelity aerodynamics tools (namely, th in-layer or full Navier-Stokes equation solvers) are employed in recent studies. For ex ample, a combined 2-D thin layer Navier-Stokes model and a 3-D panel method is used by Sloan et al. [53], who use the outcome to construct a response su rface to optimize the wing geometry for minimum power consumption. As above, the study reveals the supe riority of thin wings, and finds that optimal airfoil shapes are insensitive to aspect ratio. Lian et al. [54] use a full Navier-Stokes so lv er to maximize the lift-todrag ratio of a rigid MAV wing subject to vari ous lift and wing convexity constraints, with sequential quadratic programming search methods Efficiency improvements are feasible by decreasing the camber at the root and increasing at the tip, thereby decreasi ng the amount of flow separation. Improvements are found to be more s ubstantial at moderate angles of attack. Given the computational complexities associated with MAV simulation, several research PAGE 28 28 efforts use wind tunnel hardware-i n-the-loop for optimization. Load measurements from a sting balance are fed into an optimizer as the objective function or constraints. Genetic or other types of evolutionary algorithms are invariably used, as a sensitivity analysis requires finite difference computations which are easily dist orted by experimental error. Examples with MAVs are given by Boria et al. [55] (optimize lift and efficiency w ith airfoil morphing), Hunt et al. [56] (optimize the forward velocity and efficiency of an ornith opter, with flapping rate and tail position as variables), and Day [57] (planform optimiza tion of a wing with vari able f eather lengths). Micro Air Vehicle Aeroelasticity The role of aeroelasticity in the study of membrane micro ai r vehicle wings differs greatly from conventional aircraft. Wh ile certain aeroelastic instabi lities do exist (t ypically involving the lift slope approaching infinity [15], unstable flapping of a poorly constrained trailing edge [58], or luffing [59]), classical problems like torsional divergence and flutter have little bearing on MAV design, due to the low aspect ratio natu re of the wings and the sm all operating dynamic pressures [60]. Great savings are available in the for m of load redistribution however, as mentioned above: potential improvements in lift, dr ag, stall, and longitudina l static stability can all be obtained. Lateral cont rol improvements are also obta inable with membrane wings [1] [61]. Furtherm ore, chordwise bending of a wing section (aerodynamic twist) can often be ignored in conventional aircraft (except, for example, when constructed from laminates with many off-axis plies [11]), but such deformation is very prev alent in low aspect ratio m embrane wings. Two-Dimensional Airfoils The aeroelastic m embrane structure is domi nated by three-dimensional structural and aerodynamic effects, but much useful insight can be gained from two-dimensional simulations and experiments. Such endeavors are obviously easier to undertake for PR-type membrane wings, but three-dimensional reinforcement must be taken into account for a pure membrane (or PAGE 29 29 string in two dimensions) with geometric twist, as the structure alone cannot sustain a flight load in a stable manner. A second option involve s considering an elastic sheet with some bending/flexural stiffness. A large variety of work can be found in the literature concerning twodimensional flexible beams in flow. For problem s on a MAV scale, work tends to focus on flags and organic structures such as leaves, seaweed, etc. Fitt and Pope [58] derive an integro-differential fl ag equation for the shap e of a thin membrane with bending stiffness in unsteady inviscid flow, consider ing both a hinged and a clamped leading edge boundary condition. Argentina and Mahadevan [62] solve a similar problem and are able to predict a critical speed that marks the onset of an unstable flapping vibration, noting that the complex instability is similar to the resonance between a pivoting airfoil in flow and a hinged-free beam vibration. Over-predicti on of the unstable flapping speed (when compared to experimental data) leads to the possibility of a stability mechanism wherein skin friction induces tension in the membrane. Alben et al. [63] discuss the streamlining of a two-dim ensional flexible filament fo r drag reduction. In particular, they are able to show that the drag on a filament at high angles of attack decreases from the rigid U 2 scaling to U 4/3. Early work in the study of membrane wings without bending stiffness is given by Voelz [64], who describes the classical twodimensional sail equation: an inextensible membrane with slack, fixed at the leading and trai ling edges, immersed in incompressible, irrotational, inviscid steady flow. Using thin airfoil theory, along with a small angle of attack assumption, Voelz is able to derive a linear integro-differential equati on for the shape of the sail as a function of incidence, freestream velocit y, and slack ratio. Various num erical solution methods are available, including those by Thwaites [65] (eigenfunction methods) and Nielsen [66] (Fourier series m ethods), to solve for lift, pitc hing moments, and membrane tension. PAGE 30 30 Multiple solutions are found to exist at small a ngles of attack with a finite slack ratio: approaching 0 from negative angles provides a negatively-cambered sa il, though the opposite is true if this mark is approached from a positive valu e. The sail is uncertain as to which side of the chord-line it should lie [65], a phenomenon which ultimately manifests itself in the form of a hysteresis loop [15]. Variations on this problem are considered by Haselgrove and Tuck [67], where the trailing edge of the m embrane is attached to an inextensible rope, thereby introducing a combination of adaptive aerodynamic and geometric twist. Increasing the le ngth of the rope is seen to improve static st ability, but decrease lift. Membrane elasticity is included in the work of Murai and Maruyama [68], Jackson [69], and Sneyd [70], indicating a nonlinear CLrelationship as strains de velop within the membrane at high incidence. Viscous flow models ar e employed in the work of Cyr and Newman [71] and Sm ith and Shyy [72]. The latter cites vi scous effects as having m uch more influence on the aerodynamics of a sail wing than th e effects of the assumptions ma de with linear thin airfoil theory. Specifically, inviscid solutions tend to ov er-predict lift at higher angles of attack (or large slack ratios), due to a loss of circulation caused by viscous effects a bout the trailing edge. A comparison of lift and tension versus angle of attack with experimental data (provided by Newman and Low [73], among others) yields mixed result s; surprisingly, the lift is overpredicted by the viscous flow m odel, yet the tension is under-predicted. Smith and Shyy also note a substantial discrepa ncy in the available experimental data in reported values of slack ratios, sail material properties, and Re ynolds numbers, which may play a role in the mixed comparisons [74]. Comparison of numerical and experim ental data for twodimensional sails is also di scussed by Lorillu et al. [75], who report satisf actory correlation for the flow structures and defor med membrane shape. Unsteady laminar-turbulent transitional PAGE 31 31 flows over a membrane wing are studied by Lian and Shyy [8] (who correlate the frequency spectrum of the vibrating membra ne wing to the vortex shedding). Three-Dimensional Wings Three com plicating factors can arise with the simulation of a three-dimensional membrane wing, rather than the planar case [76]. First, the tension is not constant (in space or direction), but is in a state of plane-stress Secondly, the wing geom etry can vary in the spanwise direction, and must be specified. Finally, the membra ne may possess a certain degree of orthotropy [59]. Most im portantly, analytical so lutions cannot generally be found. Simplifying assumptions to this problem are given by Sneyd et al. [77] (triangular planform) and Ormiston [15] (rectangular sailwing). Sneyd et al. reduce both the aerodyna m ics and the membrane deformation to twodimensional phenomena, where the third dimens ion is felt through a trailing edge cable. Ormiston assumes both spanwise and chordwis e deformation (but not aerodynamics), and is able to effectively decouple the two modules by using only the firs t term of a Fourier series to describe the inflated wing shape. Boudreault uses a higher-fidelity vortex lattice solver, but also prescribes the wing shape, he re using cubic polynomials [78]. Holla et al. [79] use an iterative procedure to couple a double latt ice m ethod to a structural mode l, but assume admissible mode shapes to describe the deformation of a recta ngular membrane clamped along the perimeter. The stress in the membrane is assumed to be always equal to the applied pre-stress (inextensibility, which overwhelms the nonlinearities in the membrane mechanics). A similar framework is used by Sugimoto in the study of circular membra ne wings, where the wing shape is completely determined by a linear finite element solver [80]. Jackson and Christie couple a vortex lattice m ethod to a nonlinear structural model for the simulation of a triangular membrane wing. Co mparisons between a rigid wing, a membrane wing fixed at the trailing edge, and one with a fr ee trailing edge elucidate the tradeoffs in lift PAGE 32 32 between adaptive camber and adaptive washout [76]. Charvet et al. [81] study the effect of nonhom ogenous incoming flow (vertical wind gradients and gusts) on a flexible sail. Schoop et al. use a nonlinear membrane stress-strain relationship (hyperelasticity ) with a vortex lattice solver for simulation of a flat rectangular membrane wing [82]. Lian et al. [28] compute the unsteady aeroelasticity of a batten-reinforced m embrane micro air vehicle wing, with a nonlinear h yperelastic solver and a turbul ent viscous flow solver, using thin plate splines as an interf acing technique. Battens are simu lated with a dense membrane. The results indicate self-exciting membrane vi bration on the order of 100 Hz, with a maximum wing speed about 2% of the freestr eam, though overall aerodynamics ar e similar to that of a rigid wing prior to stall. Stanford and Ifju [14] discuss steady laminar ae roelasticity of a perim eterreinforced membrane micro air vehicle wing, and are able to show the exp ected increase in lift and stability. Significant drag penalties are seen to arise w ith increasing Reynolds numbers, though the opposite is true for the rigid wing. Complexities involving membrane wing models with both membranes and elastic shells (such as batten reinforcement) can be found in the work of Stanford et al. [83] (linear mechanics) and Ferguson et al. [84] (nonlinear). Higher-o rdered m embrane modeling with wrinkling (the loss of one or more principle stresses) as pertai ning to membrane wings is given in the work of Smith and Shyy [85] and Heppel [86]. A large volum e of work can be found dealing with experimental characterization of membrane wings. Early wind tunnel work by Fink [87] details a full-scal e investigation of an 11.5 aspect ratio sail wing with a rigid leading edge, wingtip, and root, and a cabled trailing edge. The defor mation is reported to be fairly sm ooth prior to stall, but visible rippling develops along the membrane at the onset of stall. At low angles of attack, the slope of the lift curve is PAGE 33 33 unusually steep (an instability discussed by Ormiston [15], among others), as the strains in the skin are low enough to allow for large changes in cam ber. Greenhalgh and Curtiss conduct wind tunnel testing to study the effect of planform on a membrane wing; only a parabolic planform is capable of sustaining flight loads without the aid of a trailing edge support member [88]. Galvao et al. [89] conduct tests on a membrane sheet stretched between two rigid posts, at Reynolds numbers between 34 and 105. The results show a monotonic increase in membrane camber with angle of attack and dynamic pressu re, up to stall, as well as the aforementioned steep lift slopes. De-cambering of the wing as the pressure on the upper surface increases due to imminent flow separation is seen to ameliorate the stall behavior, as compared to a rigid plate. Flow visualization of a batten-reinforced membrane MAV wing exhibits a weaker wing tip vortex system than rigid wings [13], possibly due to energy conservation requirem ents [90]. Parks m easures the vortex core of a BR wing 5% to 15% higher above the wing than for the rigid case, though the flexible wing is s een to have a denser core-distrib ution of velocity for moderate angles of attack [91]. Gamble and Reeder [92] measure the flow st ructu res resulting from interactions between propeller slipstream and a BR wing. Th e rigid wing spreads the axial component of the propwash further along the wi ng (resulting in a higher measured drag), whereas the membrane wing can absorb the downwash and upwash. A region of flow separation is measured at the root of the rigid wing, signifi cantly larger and stronge r than that measured from the membrane wing; the superiority d ecreases with larger Reynolds numbers. Albertani et al. [9] detail loads measurements of both BR and PR wings, with dramatic im provements in longitudinal static stability of both membrane wings over their rigid counterpart. The BR wing has a noticeably smoothe r lift behavior in th e stalled region, though neither deforms into a particularly optimal aerodynamic shape: both incur a drag penalty. PAGE 34 34 Deformation measurements of a membrane wing under propwash indicate unsymmetrical (about the root) wing shapes, a phenomenon which diminish es with higher angles of attack and dynamic pressures [47]. Stults et al. use la ser vibrom etry to measure th e modal parameters (shapes, damping, frequency spectra) of a BR wing, which are then fed into a computational model for simulation of static and dynamic deforma tions in both steady freestream and a gust [93]. Aeroelastic Tailoring Although aeroelastic tailoring is generally defi ned as the addition of directional stiffness into a wing structure so as to beneficially affect perform ance [11], this has traditionally meant the use of unbalanced com posite plates/shells. Despite the use of such laminated materials on many MAV wing frames [94], there does not appear to be an y tailoring studies on fixed m icro air vehicle wings. Some investigators have applied the concept to the design of flapping ornithopter wings [95] [96], where a bend/twist coup ling can vary the twist-indu ced incidence of a wing to im prove thrust. Conventional tailoring studies ca n also be found applied to a larger class of unmanned aerial vehicles [97] [98]. The latter study by Weisshaar et al. uses lam inate tailoring to improve the lateral control of a vehicle with an aspect ratio of 3. With ailerons, a wing tailored with adaptive wash-in is shown to im prove roll performance and roll-reversal speed, though wash-out is preferred for a leading edge slat [98]. In addition to conventional lam inate-based tail oring, drastic changes in the performance of a membrane wing are attainable by altering the pr e-tension distribution within the extensible membrane. Holla et al. [79], Fink [87], Smith and Shyy [72], Murai and Maruyama [68], and Or miston [15] all note the enormous im pact that m embrane pre-tension has on aerodynamics: for the two-dimensional case, higher pre-tension generally pushes flex ible wing performance to that of a rigid wing. For a three-dimensional wing, th e response can be considerably more complex, depending on the nature of the membrane reinfor cement. Well-reported e ffects of increasing the PAGE 35 35 membrane pre-tension include: decrease in drag [89], decrease in CL [15], linearized lift behavior [72], increase in the zero -lift angle of attack [68], and more abrupt stalling patterns [89]. Ormiston details aeroelastic in stabilities in terms of the ratio of spanwise to chordwise pretension [15]. Adequate control of m embrane tension has long been known as a crucia l concern to sailors in order to efficiently exploit wind power [99]. Tension-control is sim ilarly important to the performance and agility of fighter kites: a wrinkled membrane su rface will send the kite into an unstable spin. When pointed in the desired dire ction, pulling the control line tenses and deforms the kite, which thus attains forward velocity [100]. Biological in spiration for aerodynam ic tailoring of membrane tension can be seen in the wing structures of pterosaurs and bats. In addition to membrane anisotropy (p terosaur wings have internal fibrous reinforcement to limit chordwise stretching [101], while bat wings skins are measured to be 100 times stiffer in the chordwise direction than the spanwise [102]), the tension can be c ontrolled through a single digit (ptero saurs) [59], or varied throughout the wing via m ultiple digits (bats) [103]. Work for mally implementing membrane tension as a variable for optimizing aerodynamic performance is very rare. Levin and Shyy [104] study a modified Clark-Y airfoil with a flexible m embrane upper surface, subjected to a vary ing freestream velocity. Response surface techniques are used to maximize the power index averaged over a sinusoidal gust cycle, with membrane thickness variation, elastic modulus, and pre-stress used as variables. The maximum power index is found to coincide with the lower bound placed upon pre-st ress, though lift and efficiency are also seen to be superior to a rigid wing. Topology Optimization The basics of topology optim ization are given by Bendse and Sigmund [16] and Zuo et al. [105]: the design domain is discretiz ed, and the relative density of each elem ent can be 0 or 1. PAGE 36 36 Traditionally, this is done on a structure with static loads by minimizing the compliance under an equality constraint upon the volume fraction, thou gh recent work can be seen in the design of compliant mechanisms [106] and channel flows [107] as well. Solving the problem with strictly discre te variables is rare; Beckers [108] uses a dual method to solve the large-scale discrete problem while Deb and Goel [109] use a genetic algorithm. Th is latter option, though attractive, requires a very large number of function evaluations even for a sm all number of variables. The topology optimization problem is typically solved using the SIMP approach (solid isotropic material with penalization): the density of each element is allowed to vary continuously between 0 and 1. A nonlinear pow er-law interpolation provides an implicit penalty which pushes the density to 0 or 1: intermediate densit ies are unfavorable, as their stiffness is small compared to their volume [16]. An adjoint sensitivity analysis of the discrete system is required to compute the sensitivity of the compliance (or other objective functions ) with respect to each element density, as the number of variables is much larger than the number of constraints [110]. A m esh-independent filter upon the gr adients is also typically empl oyed, in order to limit the minimum size of the structure and eliminate chec kerboards. Computatio n of the topological Pareto trade-off curve can be done us ing a multi-objective genetic algorithm [109], or by success ively optimizing a weighted sum of conflicting objectives [111]. Aeronautical applications are given by Borrvall and Petersson [112], who divide a com putational domain into either fluid or solid walls to find the minimum drag profile of submerged bodies in Stokes flows. Pingen et al. [107] solve a similar problem, using a lattice Boltzm ann method as an approximation to the Navi er-Stokes equations. Drag is minimized by a football shape (with front and back angles of 90) at low Reynolds num bers (where reducing surface area is important), and a symmetric airfoil at higher Reynolds numbers (where PAGE 37 37 streamlining is more important). Several examples can be found in the literature pertaining to compliance minimization of a flexible aircraft structure. F light loads are typically obtained from an aerodynamics model, but the redistribution of these loads with wing de formation (aeroelasticity) is not included. Balabanov and Haftka [113] optimize the internal structur e of a transport wing, using a ground structure approach (the dom ain is filled with interconnected trusse s, and the cross-sectional area of each is a design variable [16]) for compliance minimization. Eschenauer and Olhoff [114] optim ize the topology of an internal wing rib und er both pull-up load maneuvers and internal tank pressures, using a bubble method. Krog et al. [115] also optimize the topology of wing box ribs, and discuss m ethods for interpretation of th e results to form an engineering design, followed by sizing and shape optimization. Luo et al. [116] compute the optimal topology of an entire aerodynam ic missile body, considering both stat ic loads and natura l frequencies. Santer and Pellegrino [117] replace the leading edge of a wing section with a com pliant morphing mechanism, which is subjected to topology optimization. Rather than a compliancebased objective function, the authors use airf oil efficiency, but as above, do not include aeroelastic load redistribution. Such an aeroela stic topology optimization is an under-served area in the literature. Maute and Allen [118] consider the topological layout of stiffeners within a swept wing, using a three-dim ensional Euler solver coupled to a linear finite element model. Results from an adjoint sensitivity analysis of the three-field coupl e aeroelastic system [110] [119] are fed into an augmented Lagrangian optim izer to minimize mass with constraints upon the lift, drag, and wing displacement. The author s are able to demonstrate the superiority of designs computed with aeroelastic topology optim ization, rather than considering a constant pressure distribution. PAGE 38 38 Gomes and Suleman [120] use a spectral level set method to maximize aileron reversal speed by reinforcing the upper skin of a wing torsion box via topology optim ization. Maute and Reich [106] optimize the topology of a compliant m orphing mechanism within an airfoil, by considering both passive and active shape deformati ons. The authors are ab le to locate superior optima with this aeroelastic topology optimiza tion approach, as compared to a jig-shape approach: optimizing the aerodynamic shape, and then locating the mechanism that leads to such a shape. At present, there is no research pertai ning to aeroelastic t opology optimization of membrane wings, or micro air vehicle wings. Bi ological inspiration for this concept can be found in the venation of insect wings however. Fo r example, a pleated grid-like venation can be seen in dragonfly wings, posterior ly curved veins in fly wings, and a fan-like distribution of veins in the locust hindwing [121]. On the whole, the signifi cance of this variation in wing stiffness distribution between sp ecies is not well understood. PAGE 39 39 CHAPTER 3 EXPERIMENTAL CHARACTERIZATION As will be e xtensively discussed below, numerical modeling of flexible MAV wings, while conducive to optimization studies, is very challenging: at the present time, no model exists which can accurately predict all of the unsteady fl ow phenomenon known to occur over a micro air vehicle. As such, experimental model validation is required to instill confidence in the employed models, highlight numerical shortcomings and provide additional aeroelastic wing characterization. All of the aerodynamic characteriz ation experiments discussed in this work are run in a closed-loop wind tunnel, a diagram of which can be seen in Figure 3-1. Only longitudinal aerodynam ics are of interest, and only -sweep capability is built into the test setup. Figure 3-1. Schematic of the wind tunnel test setup. Closed Loop Wind Tunnel The test facility us ed for this work is an Engineering Laboratory Design, Inc. (ELD) 407B closed-loop wind tunnel, with the flow loop arranged in a horizont al configuration. The test Wind Tunnel Test Section Incoming Flow Mounting Bracket VIC Cameras Model Arm Sting Balance Speckled MAV Wing PAGE 40 40 section has an inner dimension of 0.84 m on each side and is 2.44 m deep. The velocity range is between 2 and 45 m/s, and the maximum Reynolds number is 2.7 million. The flow is driven via a two-stage axial fan with an electric motor pow ered by three-phase 440 V at 60 Hz. The controller is operated remotely with appropriately dedicated da ta acquisition software, wherein the driving frequency is based upon a linear scaling of an analog voltage input. Suitable flow conditions are achieved through hexagonal alumin um honeycomb cell, high-porosity stainless steel screens, and turning vane cascades with in the elbows of the closed loop. Centerline turbulence levels are measured on the order of 0.2%. Optical glass wi ndow access is available on the sidewalls and the ceiling. A Heise model PM differential pressure trans ducer rated at 12.7 cm and 127 cm of water (with a manufacturer-specified 0.002% sensitiv ity and a 0.01% repeatability) is used to measure the pressure difference from a pitot-stat ic tube mounted within the test section, whose stagnation point is located at the center of the sec tions entrance. The Heise system is capable of measuring wind speeds up to 45 m/s. A four-wire re sistance temperature de tector is mounted to the wall of the test section for airflow temperature measurements. Strain Gage Sting Balance Several outstanding issues exis t with m easuring the aerodynami c loads from low Reynolds number flyers. Several such airfoils are known to exhibit hysteresis l oops at high angles of attack. If the flow does not reattach to the wing surface (typi cally for lower Reynolds numbers below 54 [17]) counterclockwise hysteresis loops in the lift data m ay be evident; the opposite is true if a separation bubble exists via reattachment [24]. Adequate knowledge of such a loop is obviously important as it effects vehicle control problem s via stall and spin recovery. As described by Marchman [122], the size of the hysteresis loop m easured in a wind tunnel can be incorrectly decreased by poor flow quality: large freestream turbulence intensity levels or PAGE 41 41 acoustic disturbances (noise emitted from the tu rbulent boundary layer along the tunnel walls, the wind tunnel fan, etc. [24]). Mounting techniques are also presum ed to cause an incorrect relationship between Reynolds num ber and the zero-lift angle of attack among several sets of published data [122]. Sensitiv ity is another concern, particularly in drag force measurements which may be as low as 0.025 N (computed for a wing with a c hord of 100 mm and a Re ynolds number of 54). An electrical resistance strain gage sting bala nce is typically used for force and moment measurements. While strain gages typically provide the greatest sensitivity and simplicity, they are also prone to temperature drift, electroma gnetic interference, cree p, and hysteresis. An internal Aerolab 01-15 6-component strain gage sting balance is used for force/moment measurements in the current work. Wind tunnel models are mounted to the sting balance by a simple jaw mechanism. Each of the six channe ls is in a full Wheatst one-bridge configuration, with five channels dedicated to forces, and one to a moment. Two forces are coincident with the vertical plane of the mo del (traveled during an -sweep), two are in th e plane normal to the previous (traveled during a -sweep), one force is in the ax ial direction, and the moment is dedicated to roll. Data acquisition is done with a NI SCXI 15 20 8 channel programmable strain gage module with full bridge configurati on, 2.5 excitation volts, and a gain of 1000. Other modules included in the system ar e a SCXI 1121 signal conditioner, 1180 feedthrough with 1302 breakout and 1124 D/A module A NI 6052 DAQPad firewire provides A/D conversion, multiplexing, and the PC connection. For a given flight condition, the output signals from the six components are sampled at 1000 Hz for 2 seconds. The average of this data is sent to one module for calculation of the relevant aerodynamic coefficients, and the standard deviation of the data is stored for further uncer tainty analysis. Signals from each channel are PAGE 42 42 recorded before and after a testing sequence, w ith no airflow through the tunnel, to provide an estimation of the overall drift. The sting balance is mounted to a custom-fab ricated aluminum model arm within the wind tunnel (seen in Figure 3-1). A U-shape is built into the arm so that the structure curves well behind the model and aerodynamic interactions are minimal. The arm extends through a hole in the wall of the test section, and is then attach ed to a gearbox and a brushless servomotor for pitching control. The motor is run by a single ax is motion controller; a high precision US Digital absolute encoder, connected to an SCXI 1121 module provides angle of attack feedback. Pitching rates are on the order of 1 /s. For a given flight condition, the aforementioned instrumentation (the Heise and thermocouple connected through an RS232, and the sting balance) is used to measure the pressure, temperat ure, and voltage signals. A set of tare voltages (obtained prior to the test, with no flow through th e tunnel) are subtracted from the sting balance data, which is then filtered through the calibra tion matrix, and normalized by the subsequent computations of flow velocity and air dens ity. The numerous systems described above are integrated to allow for completely automated wind tunnel testing for force/moment data, along with a LABview GUI written for user inputs of th e wing geometry, angle of attack array, and the commanded wind speed. Standard procedures [123] are used to calib rate the sting balance down to an adequate sensitivity: 0.01N in drag (though s till just 40% of the m inimum gi ven above). Such a resolution is comparable to that found in the work of Pelletier and Mueller [34], but superior precision is used by Kochersberg er and Abe [124] and Moschetta and Thipyopas [125]. The calibration m atrix is determined through the use of known we ights applied at contro l points in specified directions. This calibration is ab le to predict the relationship be tween load and signal for a given PAGE 43 43 channel, as well as potential interactions (sec ond-order interactions ar e not included) in both single and multiple load configurations. Furthe r information on the calibration of strain gage sting balances for micro air vehicle measurements is given by Mueller [126] and Albertani [47]. Uncertainty Quantification Two types of uncertainty are t hought to contribute to the eventu al error bounds of the sting balance d ata. The resolution error is indicativ e of a measurement devi ces resolution limit: for example, the inclinometer used to measure the p itch of a model can measure angles no finer than 0.1, an uncertainty that can be propagated through the equations to find its theoretical effect on the aerodynamic coefficients usi ng the Kline-McClintock technique [127]. The following resolution errors are used: 3 Pa of dyna m ic pressure from the Heise, 1.2-7 V from the output voltage of the strain gages (estimated from the quantization error of the 16-bit DAQ cards), 0.001 m2 from wing area measurements, and 0.002 m from chord length measurements. The second source is the precision error, a measure of the repeatability of a me asurement. This is well quantified by the standard deviation of the voltage signals from 2000 samples at each angle of attack, as described above. Uncertainty bounds are computed with a squared sum of the resolution and precision errors (w here the latter is magnified by Students t at 95% confidence and infinite degrees of freedom). The precision of the strain gage signals is found to contribute the most error to the aerodynamic coefficients, pa rticularly in the stal led regions. Typical uncertainty percentages are 5% for CL, 7% for CD, 9% for L/D, and 20% for Cm. Theses values can be expected to double during stall. Wind Tunnel Corrections Corrections are app lied to the coefficients of lift, drag, and pitching moment based upon wind tunnel blockage, and model flex ibility effects. The solid bl ockage effect is due to the presence of the model within the wind tunnel, th us decreasing the effective area of the test PAGE 44 44 section and increasing the flow velocity (and th e coefficients) in the vicinity of the model [128]. Wake blockage occurs when the flow outside of the m odels wake must increase, in order to satisfy the flow continuity in a closed tunnel. In an open freestream, the velocity outside of the wake would be equal to the freestream velocity. The effect of wa ke blockage is proportional to the wake size, and therefore proportional to drag [3]. Streamline curvature blockages are the effect of the tunnel walls on the stream lines around the model. The streamlines are compressed, increasing the effective camber and lift [129]. Such corrections gene rally decrease both lift and drag, while the pitching mom ent is made less negative, with per centage changes on the order of 2-3%. Finally, flexibility effects within the wind tunnel setup must be accounted for. These effects are primarily caused by the elasticity of the internal strain gage sting balance; under load the wind tunnel model will pitch up via a rigid body rotation. Visual image correlation (described below) is used to measure the displacement at points al ong the wing known to be nominally rigid (specifically, the sting balance attachment points al ong the wing root). This data then facilitates the necessary transformations and translations of the wing surface, and is used to correct for the angle of attack. is a positive monotonically incr easing function of both lift and dynamic pressure, and can be as large as 0.7 at high angles of attack [47]. Visual Image Correlation W ind tunnel model deformation measurements are a crucial experime ntal tool towards understanding the role of structural com position upon aerodynamic performance of a MAV wing. The flexible membrane skin generally limits applications to non-contacting optical methods, several of which have been repor ted in the literature. Galvao et al. [89] use stereo photogrammetry for disp lacement measurements of a membrane wing, with a reported resolution between 35 and 40 m. Data is available at discrete ma rkers placed along the wing. Projection PAGE 45 45 moir interferometry requires no such marker placement (a fringe pattern is projected onto the wing surface), and the resulting data set is full -field. However, displacement resolutions reported by Fleming et al. [130] are relatively poor (250 m ), the dual-camera system must be rotated during the -sweep, and only out-of-plane data is av ailable, making strain calculations (if needed) impossible. Burner et al. [131] discuss the use of phot ogramm etry, projection moir interferometry, and the commercially available OptotrakTM system. The authors find no single technique suitable for all situa tions, and that a cost-benefit tradeoff study may be required. Furthermore, the methods need not be mutually ex clusive, as situations may arise wherein they can be used in combination. For the current work, a visual image correlation system (VIC), originally developed by researchers at the University of South Carolina [132], is used to measure wing geom etry, displacements, and plane strains. The underlying principle of VIC is to calcu late the displacement field by tracking the deformation of a subset of a random speckle pa ttern applied to the sp ecimen surface. The random pattern is digitally acquired by two cameras before and after loading. The acquisition of images is based on a stereo-triangulation techniqu e, as well as the computing of the intersection of two optical rays: the stereo -correlation matches the two 2-D frames taken simultaneously by the two cameras to reconstruct the 3-D geometry. The calibration of the two cameras (to account for lenses distortion and determine pixel spaci ng in the model coordinates) is the initial fundamental step, which permits the determina tion of the corresponding image locations from views in the different cameras. Calibration is done by taking images (with both cameras) of a known fixed grid of black and white dots. Temporal matching is then used: the VIC syst em tries to find the region (in the image of the deformed specimen) that maximizes a norma lized cross-correlation function corresponding to PAGE 46 46 a small subset of the reference image (taken when no load is applied to the structure) [132]. The im age space is iteratively swept by the parameters of the cross-correlation function, to transform the coordinates from the original reference frame to coordinates within the deformed image. An originally square subset in the un-deformed image can then be mapped to a subset in the deformed image. As it is unlikely that the de formed coordinates will directly fall onto the sampling grid of the image, accurate grey-value interpolation schemes [133] are implemented to achieve op timal sub-pixel accuracy without bias. This procedure is repeated for a large number of subsets to obtain full-field data. In order to capture the three-dimensional feat ures and deformation of a wind tunnel model, twin synchronized cameras, each looking from a di fferent viewing angle, are installed above the wind tunnel ceiling, as can be seen in Figure 3-1. As the cameras must remain stationary through the experim ent (to preser ve the information garnered fr om the calibration procedure), a mounting bracket is constructed to straddle the tunnel, and prevent the transmission of vibration. Optical access into the test section is through an optical glass ceiling. The results of conducting visual image correlation tests with a glass inte rface between the cameras and the specimen have been studied, with litt le benign effects reported [134]. Furthermore, th e ca meras are initially calibrated through the window to ensure minima l distortion. Two 250 W lamps illuminate the model, enabling the us e of exposure times of 5 to 10 ms. The twin cameras are connected with a PC via an IEEE 1394 firewire cable, and a specialized unit is used to synchronize the camera triggers for instantaneous shots. A standard acquisition board installed in the computer carries out the digitalization of the images, and the image processing is carried out by custom soft ware, provided by Correlated Solutions, Inc. Typical data results that can be obtained from the VIC system consist of the geometry of the PAGE 47 47 surface in discrete x, y, and z coordinates (where the origin is located at the centroid of the speckled area of interest, and the outward norma l points towards the cameras, by default), and the corresponding displacements along the wing ( u, v, and w). The VIC system places a grid point every N pixels, where N is user defined. A final post-processing option involves calculating the in-plane strains ( xx, yy, and xy). This is done my mapping the displacement field onto an unstructured triangular mesh, and conducting the appropriate numerical differentiation (the complete defini tion of finite strains is used). Data Procession The objectiv e of most of the wi nd tunnel tests given in the re mainder of this work is to determine the deformation of the wings under steady aerodynamic loads, at different angles of attack and free stream velocities, while simulta neously acquiring aerodynamic force data. Each angle of attack requires a separate wind-off reference image: failure to do so will inject rigid body motions (as the body moves seque ntially from one angle of attack to the next) into the displacement fields. If each reference image ta ken for VIC is of the fully assembled wing, the amount of pre-strain in the wing is not include d in the measured strain field, but only those caused by the aerodynamic loads. This condition needs to be carefully considered in the evaluation of the results, since the areas of rela xation of the pre-existing tension will generate areas of virtual compression w ithin the skin. The thin me mbrane cannot support a genuine compressive stress (it will wrinkle), but ne gative Poisson strains are possible. An alternative procedure uses the un-stretched sheet of latex rubber (prior to adhesion on the wing) as a reference image. This provides the state of pre-stra in in the membrane, as well as the absolute strain field duri ng wind tunnel testing, but makes the displacement fields very difficult to interpret and is not used here. The pre-strain data is merely recorded (with a separate set of reference and deformed images), but not used as a reference for further aerodynamic PAGE 48 48 testing. As mentioned above, the acquired displ acement field will be composed of both elastic wing deformation and rigid body motion/translations originating from the sting balance, the latter of which must be filtered out. The computed strains are unaffected by these motions. Uncertainty Quantification In order to estim ate the resolution error of th e VIC system, a simple ad hoc experiment is conducted. A known displacement field is applied to a structure, and then compared with the field experimentally determined by way of imag e correlation. A thin latex membrane is stretched and fixed to a rigid aluminum ring w ith a diameter of 100 mm. The center of the membrane circle is then indented with a rigid steel bar with a s pherical head of 8 mm diameter. The bar is moved against the membrane by a micrometer with minimum increments of 0.25 mm. Results, in terms of the error between commanded displacement (via the micrometer) and the measured displacement at the apex of the membra ne profile, directly beneath the axis of the indentation bar, are given in Figure 3-2. Figure 3-2. Quantification of the re s olution error in the VIC system. Three different VIC setups are shown: 0.5, 3, and 10 speckles per millimeter of membrane, the latter of which corresponds to 2 pixels per speckle (with half the membrane in view). As PAGE 49 49 expected, the error is smallest for the finest speckle pattern, whose readings randomly oscillate about zero, with a peak error of 0.018 mm (0.6%). The coarse r speckling patterns randomly oscillate at an offset error of 0.04 mm, with a peak error of 0.077 mm (2.2%). This places the resolution error for the VIC system between 10 and 20 m, about twice the resolution reported for the photogrammetry system [89]. Though not explicitly discussed here, the strain resolution is estim ated to be between 500 and 1000 (a non-dimensional parameter independent of speckle size), a high value (compared to strain gages, for example) owing to the differentiation methods used. Model Fabrication and Preparation Only the wing (152 mm wingspan, 124 mm root chord, 1.25 aspect ratio) of the MAVs seen in Figure 1-1 and Figure 1-2 is considered in this wor k. The cam ber at the root is 6.8% (at x/c = 0.22), the reflex at the root is -1.4% (at x/c = 0.86) and 7 of positive geometric twist (nose up) is built into the wingtip. The MAV wing has 7 of dihedral between 2y/b = 0.4 and the wingtip. The fuselage, stabilizers, and prope ller are omitted from both computations and experiments. The leading edge, inboard portion of the wing, and perimeter (of the PR wings) are constructed from a bi-directio nal plain weave carbon fiber laminate with 3000 fibers/tow, preimpregnated with thermoset epoxy. The battens (for the BR wings) are built from uni-directional strips of carbon fiber. These materials are plac ed upon a tooling board (appropriately milled via CNC) and cured in a convection oven at 260 F for four hours. A wind tunnel attachment (to be fastened to the aforementioned jaw mechanism) is bonded along the root of the wing between x/c = 0.25 and 0.8. The latex rubber skin adhered to this wi ng surface is 0.12 mm thick, and approximately isotropic. A random speckle pattern is applied to the la tex sheet with flat black spray paint, and then coated with a layer of du lling spray. Each paint speckle, while relatively brittle, has a small PAGE 50 50 average diameter (less than 0.5 mm) and is genera lly not connected to an other speckle pattern; the pattern should not provide sign ificant reinforcement to the la tex. If information concerning the state of pre-strain in the skin is desired, a picture of the un-stretched latex sheet is taken for future use as a reference in the VIC system. Th e latex is then appropria tely stretched about a frame (or not at all if a slack membrane is desired), and adhered to the upper carbon fiber wing surface (which must be painted white) with spray glue. After the glue has dried, the excess latex is trimmed away. A picture of the resultant wind tunnel model is given in Figure 3-3. Figure 3-3. Speckled batten-re inforced membrane wing with wind tunnel attachment. PAGE 51 51 CHAPTER 4 COMPUTATIONAL FRAMEWORK AND VALIDATION Several d ifficulties are associated with modeling the passive shape adaptation of a flexible micro air vehicle wing. From a fluid dynamics standpoint, the low aspect ratio wing (1.25) forces a highly three-dimensional flow field, and the low Reynolds number (105) implies strong viscous effects such as flow separation, transiti on, and potential re attachment. Structurally, the mechanics of the rubber membrane inflation are i nherently nonlinear, a nd the orthotropy of the thin laminated shells used for the wing sk eleton is dependent on the plain weave fiber orientation. Further difficulties arise with the inclusion of pretension within the membrane. Only static aeroelasticity is considered here. Several computational membrane wing studies have included unsteady effects [28] [81], and are thus able to study phenom ena such as vortex shedding [18], membrane vibration (unstable [62] or otherwise), unsteady interactions between the separated flow and the tip vortices [6], and wind gusts [8]. Past wind tunnel work, however, has indicated that MAV me m brane inflation is essentially qausi-static for a large range of angles of attack up to stall [135], and that adequate predictive capability still exists for those flight conditions with obvious unsteady features [43]. Structural Solvers The unstructured m esh used for finite element analysis can be seen in Figure 4-1. 2146 nodes are used to describe the surface of th e sem i-wing, connected by 4158 three-node triangle elements. The same mesh is used for both ba tten and perimeter-reinforced computations, by using different element-identificatio n techniques, as seen in the fi gure. Greater local effects are expected in the membrane areas of the wing, and the mesh density is altered accordingly. Nodes that lie along the wing root between x/c = 0.25 and 0.8 are given zero displacement/rotation boundary conditions, to emulate the restrictiv e effect of the wind tunnel attachment ( Figure 3-3). PAGE 52 52 All nodes that lie on the wing root are constr ained appropriately as necessitated by wing symmetry. Figure 4-1. Unstructured triangular mesh used for finite element analysis, with different element types used for PR and BR wings. Composite Laminated Shells Discrete Kir chhoff triangle plate elements [136] are use to model the bending/twisting behavior of the carbon fiber areas of the wings: le ading edge, root, perim eter, and battens. Due to the comparative stiffness of th ese materials, linear behavior is assumed. The orthotropy of the plates is introduced by the flexural stiffness matrix of the laminates, Dp, relating three moments (two bending, one twisting) to three curvatures: NL 32 p kkkk k1h12hz DQ (4-1) where NL is the number of layers in the laminate, hk is the thickness of the kth ply, zk is the normal distance from the mid-surface of th e laminate to the mid-surface of the kth ply, and Qk is the reduced constituitive matrix of each ply, expressed in global coordinates. Qk depends upon the elastic moduli in the 1 and 2 directions (equal for the bi-directional laminate, but not so for the uni-directional) E1 and E2, the Poissons ratio 12, and the shear modulus G12. The finite element stiffness matrix pertaining to bending/twisting is then found to be: PAGE 53 53 oTT pppp AA KTBDBT d (4-2) where T is a matrix which transforms each element from a local coordinate system to a global system, Bp is the appropriate strain-displacement matrix [137], and Ao is the un-deformed area of the triangular element. Kp is a 9x9 matrix whose compone nts reflect the out-of-plane displacement w and two in-plane rotations at the three nodes. Similarly, in-plane stretching of the lamina tes (a secondary con cern, but necessarily included), is given by: NL p kk k1h AQ (4-3) where Ap is a laminate matrix relating three in-pla ne stress resultants to three strains. Expressions similar to Eq. (4-2) are then formulated to compute Km, the 6x6 finite element stiffness matrix governing in-plane displacements u and v at the three nodes. Km and Kp are then combined to form the complete 15x15 shell stiffness matrix of each element, Ke. Drilling degrees of freedom are not included. T hough some wing designs may use un-symmetric laminates, coupling between in-plane a nd out-of-plane motions is not included. Loads Model Validation/Estimation. The following method is used to both validate the model presented above, and identify the relevant mate rial properties of the la minates. A series of weights are hung from a batten-reinforced wing (w ith 2 layers of bi-directional carbon fiber oriented 45 to the chord line a nd 1 layer uni-directional battens, but no membrane skin) at nine locations: the two wing tips, the tr ailing edges of the six battens, and the leading edge, as shown in Figure 4-2. VIC is used to measure the resu lting wing dis placements. A linear curve is fit through the load-displacement data of all nine poi nts due to all nine load ings. The slopes of these curves are used to popul ate the influence matrix in Table 4-1: the diagonal gives the motion PAGE 54 54 of a wing location due to a force at that lo cation; the off-diagonals represent indirect relationships. Figure 4-2. Computed deforma tions of a BR wing skeleton due to a point load at the wing tip (left) and the leading edge (right). A genetic algorithm is then used for system identification. The six variables are the material parameters: E1, 12, and G12 of both the plain weave and the battens. E2 is assumed to be equal to E1 for the plain weave, and equal to 10 MPa fo r the uni-directional battens. This latter value has little bearing on the re sults, as the 1 direc tion corresponds with the axis of the batten. The objective function is the sum of the squared error between the diagonals of the computed and the measured influence matrix. The error terms are appropriately normalized before summation and off-diagonal components are not cons idered in the optimization. For the genetic algorithm, the population size is 20, the elitism count is 2, reproduction is via a two-point crossover function with a 0.8 cro ssover fraction, and a uniform mu tation function is used with a 0.01 mutation rate. Convergence is adequately achieved after 30 itera tions, with each function evaluation call requiring a single finite element analysis. The resulting numerical influence matrix is given in Table 4-2. This matrix is symmetric, whereas th e experimental matrix is slightly un -symmetric, probably due to manufacturing errors. PAGE 55 55 For the plain weave, E1 = 34.8 GPa, 12 = 0.41, and G12 = 2.34 GPa. For the uni-directional battens, E1 = 317.2 GPa, 12 = 0.31, and G12 = 1.05 GPa. The model correctly predicts the very stiff leading edge (point 1), and the negative influence it has on the remainder of the wing (as shown on the right of Figure 4-2). The rest of the points alon g the wing positively influence one another. Errors between the two matrices are typically on the order of 5-10%; the numerical wing is generally stiffer than the actual wing. As expected, the weakest battens are the longest, found towards the root (points 8 and 9). The wingtips (points 2 and 3) generally have the greatest indirect influence on the rest of the wing (as shown on the left of Figure 4-2). Force-displacem ent trends at th e nine locations along the wing, due to loads at those points (the diagonal terms in the matrices) are given in Figure 4-3, showing a suitable match between model and experim ent. With the exception of th e leading edge, two data points are given for each load level, corresponding to the data from the left and right sides of the wing. Higher fidelity methods for system identification of a ca rbon fiber MAV skeleton are given by Reaves et al. [138], who utilize model update techniques with uncertainty quantification m ethods. This is largely done due to the uncertainty in the lami nate lay-up, predominat ely in ply overlapping regions within the skeleton, which is not an issue for the current work. Table 4-1. Experimental influence matrix (mm/N) at points labeled in Figure 4-2. 1 2 3 4 5 6 7 8 9 1 1.58 -2.90 -2.97 -3.00 -3.07 -3.47 -3.54 -3.05 -3.03 2 -2.93 104.65 3.94 50.85 5.88 36.47 7.65 26.06 9.99 3 -3.07 3.67 118.468.18 52.24 9.39 38.29 13.33 29.22 4 -3.68 49.05 5.69 329.118.71 44.14 10.51 33.05 13.23 5 -3.78 5.68 50.15 9.17 366.9211.89 45.19 14.98 33.68 6 -4.33 36.41 7.92 44.63 10.68 547.5013.41 38.46 15.23 7 -4.37 7.77 36.74 11.46 44.35 13.28 513.8317.40 38.19 8 -4.75 24.30 9.02 30.15 11.24 34.68 12.63 757.00 15.81 9 -4.76 9.22 25.09 12.34 30.30 14.32 35.51 17.65 742.25 PAGE 56 56 Table 4-2. Numerical influence ma trix (mm/N) at points labeled in Figure 4-2. 1 2 3 4 5 6 7 8 9 1 1.49 -2.92 -2.92 -3.61 -3.61 -4.24 -4.24 -4.60 -4.60 2 -2.92 107.96 3.36 54.49 5.05 33.10 6.80 20.84 8.13 3 -2.92 3.36 107.965.05 54.49 6.80 33.10 8.13 20.84 4 -3.61 54.49 5.05 312.006.78 39.30 8.51 25.09 9.73 5 -3.61 5.05 54.49 6.78 312.008.51 39.30 9.73 25.09 6 -4.24 33.10 6.80 39.30 8.51 584.2610.14 28.81 11.18 7 -4.24 6.80 33.10 8.51 39.30 10.14 584.2611.18 28.81 8 -4.60 20.84 8.13 25.09 9.73 28.81 11.18 776.48 11.95 9 -4.60 8.13 20.84 9.73 25.09 11.18 28.81 11.95 776.48 Figure 4-3. Compliance at vari ous locations along the wing, due to a point load at those locations. Membrane Modeling In the m odeling of thin, elastic membranes (w ith no resistance to a bending couple), three basic options are available. If there exists a significant pre-strain field throughout the sheet, linear modeling is possible by assuming inextensib ility: the pre-strain overwhelms the strains that develop as a result external loading. As th ese strains grow in magnit ude (or if the membrane is originally slack) a nonlinear model must be used, as the membranes resistance depends upon the loading (geometric nonlinearity). However, a linear constituitive relationship is still typically PAGE 57 57 valid up to a point, after which the membrane b ecomes hyperelastic (mater ial nonlinearity), and the stress-strain relationship changes with increasing load. Linear Modeling. Geometric stress stiffening provide s a relationship be tween in-plane forces and transverse deflection [137], and is indicative of a stru ctures reluctance to change its state of stress. For an initially flat membrane with a transverse pressure, the constitutive equation is: xxxxyxyyyyNw,2Nw,Nw,p0 (4-4) where w is the out-of-plane displacement (as above), Nx, Nxy, and Ny are the in-plane pre-stress resultants, and p is the applied pressure field. For an isotropic stress field with no shear, this equation reduces to the we ll known Poissons equation [139]. This model assumes that the displacem ent along the membrane is purely out-of-plane; thus the membrane is inextensible in response to a pressure field (althoug h extensibility is needed to appl y the initial pre-stress field). The resulting finite element model is fairly inexpensive, as each node has only one degree of freedom, and standard direct linear solvers can be used. This model is thought to be accurate for small pressures, small displacements, and larg e pre-stresses. Though it is not expected that the MAV wing displacements will be particularly large (typically less than 10% of the root chord), it is expected that a slack membrane skin may provide many aerodynamic advantages. As the solution to Eq. (4-4) becomes unbounded as the pre-stress approaches zero, higher fidelity models will also be pursued for the current work MAV wing simulations with linear membrane models can be found in the work of Stanford and Ifju [14], Thwaites [65], and Sugimoto [80]. Nonlinear Modeling. The nonlinear membrane modeling di scussed in this section will incorporate geometric nonlinearities, but Hookes law is assumed to still be valid. For the inflation of a circular me mbrane, Pujara and Lardner [140] show that linear and hyperelastic PAGE 58 58 constituitive relationships provide the same numerical solutions up to deformations on the order of 30% of the radius, a figur e well above the deflections expected on a membrane wing. Geometric nonlinearity im plies that the deformation is large enough to warrant finite strains, and that the direction of the non-cons ervative pressure loads significantly changes with deformation. Eq. (4-4) is still valid, only now the stress resultants depend upon the state of pre-stress, as well as in-plane stretching, which in turn depends on the out-of-plane displacement. Furthermore, the rotation of the membrane is no longer well-appr oximated by the derivative of w, rendering the equilibrium equation nonlinear. Three displacement degrees of freedom are required per node (u, v, w), rather than the single w used above. Finite element implementation of such a model is described by Small and Nix [141] and Pauletti et al. [142]. The strain pseudo-vector with in each elem ent is given as: oLoeLeBXBX (4-5) where o and L represent the division of the linear (inf initesimal) and nonlinea r contributions to the Green-Lagrange strain, Xe is a vector of the degrees of freedom in the elements (three displacements at th e three nodes), and Bo and BL are the appropriate stra in-displacement matrices (the latter of which depends upon the nodal displacements) [143]. The pre-stress (if any) can be included into the m odel in one of two methods. Fi rst, they can be simply added to the stresses computed my multiplying the strain vector of Eq (4-5) through the constituitive matrix. This may cause problems if the imposed pre-stress di stribution does not exactly satisfy equilibrium conditions, or if there is excessive curvature in the membrane skin: the membrane will deform, even in the absence of an external force. A second option is to use the pre-stresses in a finite element implementation of Eq. (4-4), then add the resulting stiffness matrix and for ce vectors to the nonlinear terms. For a flat PAGE 59 59 membrane with uniform pre-stress, the two met hods are identical. The internal force in each element Pe can be computed from the principle of virtual work: oT eoLeemwe VV PTBBXXATKX d (4-6) where Am is the linear constitutive elastic matrix of the membrane, Am is the stress pseudovector within each element, and Kw is the stiffness matrix represen tation of Eq. (4-4), containing only terms related to the out-of-plane displacement w. The tangential stiffness matrix Ke is then the sum of the geometric K, constituitive Kc, external Kext and pre-stress stiffness matrix Kw: oTT VV KTGMGTd (4-7) oT T coLeemoLee VV KTBBXXABBXXTd (4-8) extee KFX (4-9) where G is a matrix linking the nodal degrees of fr eedom to a displacement gradient vector [144], M is a stress matrix whose elements can be found in [137], and Fe is the external force vector. Computation of the skew-symmetric external stiffness matrix is given in [142]. Fe must be written in the unknown de formed configuration: T epA/3 FTIIIn (4-10) where A is the deformed area of the triangle, p is the uniform pressure over the element, I is the identity matrix, and n is the unit normal vector to the defo rmed triangular finite element. The resulting non-linear set of equations is solved with Newtons recurrence formula [142]. The above m ethod essentially separates the line ar and nonlinear stiffness contributions. If the pre-stress in the membrane is very large, Kw will overwhelm its nonlinear counterparts, and membrane response will be essentially linear fo r small pressures and displacements. Continued PAGE 60 60 inflation will transition from linear to nonlinear response [145]. In the event of a slack m embrane, the membranes initial response to a pre ssure will have an infinite slope until strains develop and provide stiffness. Numerous memb rane wing models use some variant of the geometrically nonlinear model de scribed above: Stanford et al. [43], Ormiston [15], Smith and Shyy [72], Jackson and Christie [76], and Levin and Shyy [104]. Inflatable Diaphragm Validation. In order to validate the above m embrane models, the material properties of the latex are first identified with a uni-axial tension test. The test specimen has a width of 20 mm, a length of 120 mm, and a thickness of 0.12 mm. The latex rubber sheets are formed in a rolling process, implying an orthotropy, though specimens cut from different orientations yield very similar results. VIC is used to monitor both the extensional and the Poisson strains: data is sample d at 50 pixel locations within the membrane strip, and then averaged. The resulting data can be seen in Figure 4-4, and is used to identity the linear elastic modulus and the Poissons ratio. A linear fit through the stress-strain curve results in a m odulus of 2 MPa; the nonlinear stress-softening beha vior for higher strains is the hallmark of hyperelasticity [146]. The Poissons ratio for small strains is 0.5, a result of the materials incom pressibility. Using these material parameters to construct the constitutive matrix Am, the finite element model can be appropriately validated with the Hencky test [144]: a flat circular membrane (with or without p re-tension), clamped along its boundary, and subjected to a uniform pressure [145]. A 57.15 mm radius is chosen in order to em ul ate the length scale of a micro air vehicle. Although the problem is axisymmetric, a full circul ar mesh is used for numerical computations. Experimentally, VIC is used to monitor the sh ape of the membrane, while a Heise pressure transducer measures the pressure within a chamber, to the top of which the membrane sheet is PAGE 61 61 fixed. Results are given in Figure 4-5, in terms of the displacement of the membrane center (norm alized by the radius) versus pressure. Figure 4-4. Uni-axial stretch test of a latex rubber membrane. Figure 4-5. Circular membrane response to a uniform pressure. Two cases are considered: a slack membrane, a nd a taut membrane. Computational results from both the linear and the nonlinear models fo rmulated above are given. As expected, the response of the slack membrane to an applied pr essure is at first unbounded, but becomes finite with the advent of the extensional strains. The linear model is useless for a slack membrane (unbounded), but the nonlinear mode l can predict this behavior. The correlation between model PAGE 62 62 and experiment is adequate up to w/R = 0.22 (slightly lower than the value given by Pujara and Lardner [140]), when the model begins to under-predic t the inflated shap e. Hyperelastic effects appear after this point: Hookes law over-predicts the stress for a given strain level ( Figure 4-4), and thus th e membranes resistance to a transverse pressure. For the case with membrane pre-tension, VIC is used to measure the pre-strain in the membrane skin (applied radially [147]), the average of which is then used for finite element com putations. The mean pre-strain is 0.044, with a coefficient of variation of 3.1%. For this case, the linear model now has a small range of validity, up to w/R = 0.15. Prior to this deformation level, linear and nonlinear models predict the same membrane inflation. The response then becomes nonlinear, due to the advent of finite stra ins, but also because a relevant portion of the uniform non-conservative pressure is now directly radially, rather than vertically. The nonlinear model and experiment now diverge at w/R = 0.3: the addition of a pre-tension field increases the range of validity of both the linear and the nonlinear membrane models. Skin Pre-tension Considerations A state of uniform membrane pre-tension, though numerically convenient [15] [80] [14], is essentially impossible to actually fabricate on a M AV wing. One reason is that the latex sheets used on the MAVs in this work are not much wider than the wingspan, subjecting the state of pre-stress to end-effects. This may perhaps be remedied with larger sheets and a biaxial tension machine, which hardly seems worth the effort for MAV construction. Another problem is the fact that the wing is not a flat surface. Even if a state of unifo rm pre-tension were attainable, it cannot be transferred to the wing without significant fi eld distortions, particularly due to the camber in the leading edge. A typical pre-strain field is given in Figure 4-6, as measured by the VIC system off of a BR wing in the chordwise direc tion. The contour on the left is the pre-strain field after the spray adhesive has dried, but before the late x surrounding the wing has been de- PAGE 63 63 pinned from the stretching frame (as discu ssed above). The contour on the right of Figure 4-6 is the pre-strain after the excess la tex has been trimmed away. Figure 4-6. Measured chordwise pr e-strains in a BR wing before the tension is released from the latex (left), and after (right). The pre-strains measured from the carbon fiber areas of the wing (leading edge, root, battens) are meaningless, as the shell mechanics largely govern the response in these areas. The large extensional strains (~12%) at the leading ed ge are indicative of the fact that the wing skeleton is flattened against the membrane until the spray glue dries. At this point, the wing is allowed to re-camber, causing the latex adhered to its top surface to stre tch. The anisotropic nature of the pre-tension field is very evident, with strains ranging from between 4% to 9% on the left semi-wing and slightly higher on the righ t. Furthermore, when the surrounding latex is de-pinned from its frame the membrane at the trai ling edge contracts, leaving an area of almost no tension (right side of Figure 4-6). This is a result of th e B R wings free trailing edge, and would not be a problem with a perimeter-reinforced wing. One numerical solution to such a problem is to interpolate the data of Figure 4-6 onto the finite elem ent grid, and compute the pre-stress w ithin each element, as discussed by Stanford et al. [43]. This method, though accurate, would re quire an exp erimental VIC analysis in conjunction with every numerical analysis; not a cost -effective method for thorough exploration of the design space. Eq. (4-4) however, is a natural smoothing operator [139]; simply averaging the pre-strains for the co mputations, though crude, can in some cases be relatively accurate. The PAGE 64 64 match between measured and predicted membrane deformation for the taut case in Figure 4-5 is very good, d espite the fact that the numerical pre-strains were presumed uniform. The error resulting from a uniform pre-stress assumption can be estimated with the following method. The pre-strain distribution throughout a flat ci rcular membrane is considered a normally-distributed random variable: each finite element has a different pre-strain. The linear membrane model of Eq. (4-4) is then used to compute the displacemen t at the center of the membrane due to a hydrostatic pressure. The same membrane is then given a constant pre-strain distribution (the average value of the randomly-distributed pre-st rain), and the central deflection is recomputed for comparison purposes. Monte Ca rlo simulations are then used to estimate the average error at the membrane center, for a give n coefficient of variation of the pre-strain. The results of the Monte Carlo simulation are given in Figure 4-7. Each data point is the percentage error betw een the central displa cem ent computed with a non-homogenous random pre-strain, and that with a constant pre-strain. Each error percentage is the average of 500 finite element simulations. The radius of the circle is 57.15 mm, the thickness is 0.12 mm, the elastic modulus is 2 MPa, the Poissons ratio is 0.5, and the hydrostatic pressure is fixed at 200 Pa. The mean pre-strain is 0.05, and the standard devia tion is decided by the COV of each data points abscissa. Nonlinear membrane modeling is not used. The smoothing nature of the Laplacian operator in Eq. (4-4) is very evident: even in th e presence of 30% spatial pre-strain variability, the error in assuming a constant pre-strain is still less than 5%. On one hand, the error in Figure 4-7 is probably under-predicted, as strain canno t truly b e a spatially-random variable: on a local scale measured strain may seem random, but on a global scale it must satisfy the compatibility equations [146]. Both of these scale-trends are evident in Figure 4-6. On the other hand, Figure 4-7 represen ts the worst case scenario, as nonlin ear membrane effects will dilute the importance PAGE 65 65 of the pre-tension [145], whatever its distributio n throughout the m embrane skin. Figure 4-7. Monte Carlo simulations: error in the computed membrane deflection due to a spatially-constant pre-strain distribution assumption. Though the above results indicate the appropr iateness of using a constant membrane prestress for MAV wing computations (despite an inability to reproduce this in the laboratory), the tension relaxation at the free trailing edge of the BR wing (seen in Figure 4-6) should be corrected fo r. Regardless of the amount of pr e-tension placed in a batten-reinforced membrane, the pre-stress traction normal to the free trailin g edge will always be zero, producing a stress gradient. This can be accounted for in the following manner: 1. Specify the pre-stress field within the membrane skin (uniform or otherwise). 2. Compute the traction due to this pre-stress along the outward normal, at each edge in a membrane finite element that coincides with a free surface. 3. Apply a transverse pressure along each edge, equal and opposite to th e computed traction. 4. Compute the resulting stress fiel d (while holding the carbon fi ber regions of the wing rigid), add this field to the prescribed stress in st ep 1, and use the result as the new pre-stress resultant field for aeroelastic computations. The resulting pre-stress field will be very small along the free edge, and approach the original specified value deeper into the wing towards the leading edge, as can be seen in Figure 4-8. For this exam ple, about a fourth of the membrane area is affected by the free edge, while the remainder retains a pre-stress close to th e prescribed value (a result validated by Figure 4-6). PAGE 66 66 As mentioned above, this pre-stress correction only needs to be applied for simulations of a batten-reinforced wing. Figure 4-8. Computed pre-stress resultants (N/m) in the chordwise (left), spanwise (center), and shear (right) in a BR wing, corrected at th e trailing edge for a uniform pre-stress resultant of 10 N/m. Fluid Solvers As discussed above, several viscous effects dom inate the flow about a micro air vehicle wing: laminar separation, turbulent transition an d reattachment, periodic shedding and pairing, and three-dimensional flow via wing tip vortex sw irling. An inviscid flow solved such as the vortex lattice method is unable to predict any of thes e effects (drag, in particular, will be severely underestimated), but its small co mputational expense is attractiv e. Solving the steady NavierStokes equations represents a substantial increase in cost, but an equally large step forward in predictive capability. Some asp ects of the flow (namely, turb ulent transition over a separation bubble, and subsequent shedding), still cannot be predicted with the methods presented here. Vortex Lattice Methods This sec tion briefly describes a well-developed family of methods for predicting the steady lifting flow and induced drag over a thin wing at small angles of attack. The continuous distribution of bound vorticity over the wing is approximated by discretizing the wing into a PAGE 67 67 paneled grid, and placing a horseshoe vortex upon each panel. Each horseshoe vortex is comprised of a bound vortex (which coincides with the quarter-chord line of each panel), and two trailing vortices extending downstream. Each vortex filament creates a velocity whose magnitude is assumed to be governed by the Biot-Savart law [27]. Furthermore, a control point is placed at the three-quarter-cho rd point of each panel. The velocity induced at the mth control point by the nth horseshoe vortex is: xyz m,nm,nnm,nm,nnm,nm,nnuCvCwC (4-11) where u, v, and w are the flow velocities in Cartesian coordinates, is the vortex filament strength, and Ci are influence coefficients that depend on the geometry of each horseshoe vortex and control point combination. The complete induc ed velocity at each control point is the sum of the contributions from each horseshoe vortex resulting in a linear system of equations. The strength of each vortex must be found so that the resulting flow is tangent to the surface of the wing: the wing becomes a streamline of the flow. This requirement is enforced at each control point by: mm mmmmUcosuvUsinwFx,y,z0 (4-12) where U is the free-stream velocity, is the angle of attack, and F( x,y,z) = 0 is the equation of the surface of the wing. Inserting the relevant terms of Eq. (4-11) into Eq. (4-12) provides a linear system of equations for the filament streng th of each horseshoe vortex. Micro air vehicle simulations that utilize a vortex lattice method ar e typically forced to do so by the computational requirements of optimization (as is the case in the current work). Examples can be seen in the work of Ng and Leng [52], Sloan et al. [53], and Stanford et al. [61]. Steady Navier-Stokes Solver The three -dimensional incompressible Navier-S tokes equations, written in curvilinear PAGE 68 68 coordinates, are solved for the steady, laminar flow over a MAV wing. As before, the fuselage, stabilizers, and propeller are not taken into account. The computational domain can be seen in Figure 4-9, with the MAV wing enclosed within. Inlet and outlet boundaries are m arked by the flow vectors; velocity is specifi ed at the inlet, and a zero-pressure boundary condition is enforced at the outlet. The configuration shown in Figure 4-9 is for simulations at a model inclination of 0 angle of attack. Fo r non-zer o angles, the lower and upper surf aces will also see a mass flux, rather than re-meshing the wing itself. The sidewalls are modeled as slip walls, and thus no boundary layer forms. The MAV wing its elf is modeled as a no-slip surface. Figure 4-9. CFD computational domain. Because no flow is expected to cross the root -chord of the wing (unsteady effects that may lead to bilateral asymmetry [6] are not included; nor is prop eller slipstream [45]), symmetry is exploited by modeling only half of the computational domain (the plane of symmetry is also modeled as a slip wall). A detailed view of the resulting structured mesh (the nodes that lie on the plane of symmetry and the MAV wing) is given in Figure 4-10. 210,000 nodes fill half of the com putational domain, with 1300 nodes on the wing surface. This is a multi-block grid, with four patches coinciding with the upper an d lower wing surfaces. The wing itself has no thickness. Such a flow model should be able to adequately predict the strong tip vortex swirling 8c z x y 11c 12c 6c PAGE 69 69 system (and the accompanying nonlinear lift and moment curves [3]), as well as the laminar flow separation against an ad verse pressure gradient [2]. Similar laminar, steady flow computations for low Reynolds num ber flyers can be found in the work of Smith and Shyy [72], Viieru et al. [38], and Stanford et al. [43]. Figure 4-10. Detail of structured CFD mesh near the wing surface. In order to handle the arbitrarily shaped geometries of a micro air vehicle wing with passive shape adaptation, the Navier-Stokes equations must be transformed into generalized curvilinear coordinates: (x,y,z), (x,y,z), (x,y,z). This transformation is achieved by [148]: xyz111213 xyz212223 xyz3132331 J f ff f ff f ff (4-13) where fij are metric terms, and J is the determinant of the transformation matrix: x,y,z J ,, (4-14) Using the above information, the steady Navie r-Stokes equations can then be written in three-dimensional curvilinear coordinates [149]. The continuity equation and u-m omentum equation are presented here in st rong conservative form, with the implication that the vand wmomentum equations can be derived in a similar manner. PAGE 70 70 UVW0 (4-15) 1112 13 21 22 23 31 32 33 11 21 31UuVuWu quququ J ququququququ JJ ppp fff (4-16) where is the density, p is the pressure, is the viscosity, qij are parameters dictated by the transformation (expressions can be found in [149]), and U, V, and W are the contravariant velocities, given by the flux through a contro l surface norm al to the corresponding curvilinear directions: 111213 212223 313233Uuvw Vuvw Wuvw fff fff fff (4-17) In order to numerically solve the above e quations, a finite volume formulation is employed, using both Cartesian and contravariant velo city components [148]. The latter can evaluate the flux at the cell faces of the structur ed grid and enforce the conservation of m ass. A second order central difference operator is used for computations involving pressure and diffusive terms, while a second order upw ind scheme handles all convective terms [150]. Fluid Model Comparisons and Validation Validation of both the linear vortex lattice method and the nonlinear CFD is given in Figure 4-11, in terms of lift, drag, and longitudinal pitching m oments (measured about the leading edge) at 13 m/s. Pre-stall, the CFD mode l is able to accurately predict lift and drag within the experimental error bars of the measured data. Drag is consistently over-predicted at higher angles of attack; turbulent reattachment of separated flow is know to decrease the profile PAGE 71 71 drag [8], but is not included in the model. The magnitude of the pitching mom ent is slightly over-predicted by the CFD, though the data still falls within the error bars, the slopes match well, and the onset of nonlinear behavior (due to the low aspect ratio [3]) is well-predicted. The CFD is also able to predic t the onset of stall (via a loss of lift) at about 21 but loses its predictive capability in the post-stall regime, as th e flow is known to be highly unsteady. Figure 4-11. Computed and measured aerodynami c coefficients for a rigid MAV wing, Re = 85,000. The vortex lattice method is accurate at low angles of attack, but the slope is underpredicted (possibly due to an inability to comput e the low-pressure cells at the wing tips, similar to vortex lift discrepancies seen in delta wings [27]), and the wing never stalls. The drag predicted by the vortex lattice method is necessarily augmented by a non-zero CDo (estimated PAGE 72 72 from the experimental data), and is moderately accurate up to 10 angle of attack. After this point, the inviscid drag is under-p redicted due to massive flow separation over the wing. No significant differences can be seen between th e pitching moments predicted by the CFD and the vortex lattice method, until the af orementioned nonlinear behavior appears, which the inviscid solver cannot predict. Aeroelastic Coupling Transfer of data between a structured CFD mesh and an unstructured FEA mesh is done with simple polynomial interpolation. If information from grid A is to be in terpolated to grid B, the element from grid A (triangular for the FE A mesh, quadrilateral for the CFD mesh) is found which is closest to each node from grid B. Ex cept for nodes that lie on the wing border, these elements will enclose their corresponding nodes. Polynomial shape functions for the desired variable are formulated to descri be its distribution within the element, and then the value at the node is solved for. Such a method is found to ad equately preserve the integrated forces and the strain energies from one mesh to a nother, and is fairly inexpensive. The un-deformed (due to aerodynamic loads) wing shape technically depends upon the membrane pre-tension. This shape could be f ound with Eq. (4-4), sett ing the pressure source terms to zero, and letting the wing shape at nodes upon the carbon fiber-latex boundary be prescribed displacement boundary conditions. Such a scheme should result in slight concavities along the membrane surface [151]. This effect is considered sm all, however, and is ignored for the current work. Shear stress over the wing is also not included in the aeroelastic coupling. Moving Grid Technique For aeroelastic computations using the Navier-Stokes flow solver, a re-meshing algorithm is needed to perturb the stru ctured grid surrounding the flex ible wing (no such module is required when a vortex lattice method is used, as all of the nodes lie on the wing surface). For PAGE 73 73 the current work, a moving grid routine based upon the master-slave concept is used to maintain a point-matched grid block interface, preserve grid quality, and prevent grid cross-over. Master nodes are defined as grid points that lie on the moving surface (the wing surface of the micro air vehicle, in this case), while the slave nodes cons titute the remaining grid points. A slave nodes nearest surface point is defi ned as its master node, and its movement is given by: ssmmxxxx (4-18) where xs is the location of the slave node, xm is the location of its master node, the tilde indicates a new position, and is a Gaussian distribution decay function: 222 smsmsm 222 mmmmmmxxyyzz exp-min500, xxyyzz (4-19) where is small number to avoid division by zero, and is a stiffness coefficient; larger values of promote a more rigid-body movement. Further information concerning this technique is given by Kamakoti et al. [152]. Numerical Procedure The steady fluid structure interaction of a MAV wing is computed as follows: 1. If computations involve a batt en-reinforced wing, correct for the membrane pre-tensions at the free trailing edge. 2. Solve for the aerodynamic pressures over the wi ng, using either the steady Navier-Stokes equations or the tangency conditi on of the vortex lattice method. 3. Interpolate the computed pressures from the flow solver grid to the FEA grid. 4. Solve for the resulting wing displacement using either the linear or the nonlinear membrane/carbon fiber model. 5. Interpolate the displacement onto the MAV wing of the flow solver grid. 6. If nonlinear CFD models are ut ilized, re-mesh the grid using the master/slave scheme. 7. Repeat steps 2-6 until suitable convergence is achieved: less than 0.1% change in the lift. PAGE 74 74 Less than ten iterations are usually adequa te at modest angles of attack (3< <18). Typical results are given in Figure 4-12 for the lift and efficiency of both a BR and a PR wing, com puted with a Navier-Stokes flow solver and a nonlinear membrane solver. The lift of the PR wing monotonically converges (lift increases camber, which furthe r increases lift), while the history of a BR wing is staggere d (lift decreases wing twist, d ecreasing lift). For the nonlinear modules, step 2 requires between 150 and 250 sub-it erations, while step 4 can typically converge within 20 sub-iterations. For the linear modules, the equations of state can be solved for directly. Figure 4-12. Iterative aeroelastic convergence of membrane wings, = 9. PAGE 75 75 CHAPTER 5 BASELINE WING DESIGN ANALYSIS Three bas eline micro air vehicle wing designs are considered in this section: first, a completely rigid wing. Secondly, a batten-reinfo rced wing with no pre-tension in the membrane skin, two layers of bi-directiona l carbon fiber at ply angles of 45 to the chord line (at the root and leading edge), and one laye r of uni-directional carbon fiber (f ibers aligned in the chordwise direction) for the battens. Third, a perime ter-reinforced wing with no pre-tension in the membrane skin and two layers of bi-directional carbon fiber at ply an gles of 45 to the chord line (at the root, leading edge, and perimeter). As a large number of function evaluations are not required for this strictly analysis section, all numerical results are comput ed with the higher-fidelity met hods discussed above: the steady Navier-Stokes solver and the nonlinear membrane solver. Furthermore, all results are found at U = 15 m/s, a value towards the upper range of MAV flight. A higher velocity is chosen to emphasize aeroelastic deformations. For a given flight condition, 10 VIC images are taken of the deformed wing (at 1 Hz) and averaged together Sting balance results are, as discussed, sampled at 1000 Hz for 2 seconds, and then averaged. Wing Deformation Numerical and experimental outof-plane displacements, normalized by the root chord, are given in Figure 5-1 for a BR wing, along with a section of the data at x/c = 0.5, at 15 angle of attack. As expected, the prim ary mode of wing deformation is a positive deflection of the trailing edge, resulting in a nose-down twist of each flexible wing section. Deformations are relatively small (~5%, or 6.2 mm), though still ha ve a significant effect upon the aerodynamics. The membrane inflates from between the battens (clearly seen in the section plot) towards the leading edge, but at the traili ng edge the wing shape is more homogenous and smooth, and no PAGE 76 76 distinction between batten and me mbrane can be made. This is presumably due to the pressure gradient, with very high forces at the leading edge which diss ipates down the wing. The carbon fiber wing tips, though several orders of magnitude stiffer than the membrane, shows appreciable twisting, indicative of the large suction forces from the tip vortex. Co rrelation between model and experiment is acceptable, with the model s lightly under-predicting the adaptive washout, and over-predicting the local membrane inflation betw een the battens. Wing shapes and magnitudes match well with time-averaged results reported by Lian et al. [28]. Figure 5-1. Baseline BR normalized out-of-plane displacement (w/c), = 15. Chordwise strains for the same case as above can be seen in Figure 5-2. The directional stif fness of the battens generally prevents signif icant stretching in the ch ordwise direction. The model predicts appreciable strain (1.4%) at the carbon fiber/membrane interface towards the leading edge (due to inflation), almost no strain near the mid-chord region, and negative Poisson strains at the trailing edge. St rains in the carbon fiber regions, while computed, are much smaller than the membrane strains, and cannot be discerned in Figure 5-2. The measured chordwise strain is very sm all and noisy, with no evid ent differences between the carbon fiber and membrane regions. Much of the measured fiel d lies below the systems strain resolution (~1000 ). Several noise spikes are also evident in th is strain field, while the displacement field in Figure 5-1 has none; the strain diffe rentiation procedure is m ore se nsitive to experimental noise than the displacement temporal matching. PAGE 77 77 Figure 5-2. Baseline BR chordwise strain ( xx), = 15. A better comparison between model and experiment is given in Figure 5-3, with the spanwise strains. These extensions are essentia lly a product of the change in distance between the battens as they deform. Both m ode l and experiment indicate a peak in yy between the inner batten and the carbon fiber root, though the model indicates this maximum towards the leading edge (~1.2%), while the measuremen t places it farther aft. Strain concentrations at the trailing edge are visible in both fields. Though stil l noisy, the VIC systems spanwise strain can differentiate between battens and membrane. Suit able model validation is also seen in shear ( Figure 5-4). Both peaks and distributions of th e anti-sym metric shear are well predicted. The tips of the battens at the traili ng edge cause a shear concentration, typically of opposite sign to the strain in the rest of the memb rane segments between each batten. Figure 5-3. Baseline BR spanwise strain ( yy), = 15. PAGE 78 78 Figure 5-4. Baseline BR shear strain ( xy), = 15. Normalized out-of-plane displacements for th e perimeter-reinforced wing are given in Figure 5-5. Deformations are s ligh tly larger than with the BR wing (6%), and are dominated by the membrane inflation between the carbon fiber l eading and trailing edges. The membrane apex occurs approximately in the middle of the membra ne skin, despite the pre ssure gradient over the wing. This location is a function of angle of att ack, as the peak will move slightly forward with increased incidence [74], [43]. The carbon fiber wing tip twists less than previously, thought to be a resu lt of the fact that the wingtip is not free in a PR configurati on, but attached to the trailing edge by the laminate perimeter. Some bendi ng of the leading edge at the root can also be seen, but not in the BR wing ( Figure 5-1). Correspondence betw een m odel and experiment is suitable, with the model again under-predicting wing deformation, but accurately locating the apex. Slight asymmetries in the measured wing profile (also evident in the BR wing) are probably a result of manufacturing errors (particularly in the app lication of the membrane skin tension), and not due to flow problems in the wind tunnel. As the amount of unconstrained membrane is greater in a PR wing than in a BR wing, chordwise strains ( Figure 5-6) are much larger as well: peak stretching (3%) is located at the m embrane/carbon fiber boundary towards the leading edge, as before. The magnitude and size of this high-extension lobe is over-predicted by the model. Both model and experiment show a region of compressive strain aft of this lobe, towards the trailing edge. This is a Possion strain PAGE 79 79 (and thus not a compressive stress), but the stress in this region does become slightly negative for higher angles of attack. Errone ous computation of compressive membrane stresses indicates the need for a wrinkling module. T hough wrinkles in the membrane sk in are not obviously visible in the VIC measurements (possibly an unsteady pr ocess averaged out with multiple images), wrinkling towards the onset of stall is a well-known membrane wing phenomena [87]. As before, no appreciable strain is m easured or computed in the carbon fiber areas of the wing. Figure 5-5. Baseline PR normalized out-of-plane displacement (w/c), = 15. Figure 5-6. Baseline PR chordwise strain ( xx), = 15. Peak spanwise stretching ( Figure 5-7) occurs at the membrane carbon fiber interface towards the cente r of the wing root, and is well predicted by the model. The computed strain field erroneously shows a patch of negative Poisson strain towards the leading edge, due to the high chordwise strains in this area. One troubling aspect of the measured spanwise strains is the areas of negative strains along the perimeter of the membrane skin: namely on the sidewalls towards the root and the wingtip. Such stra ins have been measured in previous studies [9], but PAGE 80 80 their presence is peculiar. Basic membrane infl ation mechanics indicates large extension at the boundaries rather than compression [145] (as is computed by the model). The com pression may be membrane wrinkli ng (which, again, is not evident from Figure 55, or m ay be an error in the VIC strain computations, potentially caused by the large displacement gradients in this area of the wing. A third possibility is that the VIC is measuring a bending strain at this point, wher e the radius of curvature is clos e to zero. The latex skin, though modeled as a membrane, does have some (albeit ve ry small) bending resistance due to its finite thickness. The anti-symmetric shear strain field ( Figure 5-8) shows good correspondence between m odel and experiment, with accurate computations in-board, but slight underpredictions of the high shear closer to the wingtip. Figure 5-7. Baseline PR spanwise strain ( yy), = 15. Figure 5-8. Baseline PR shear strain ( xy), = 15. The aerodynamic twist (camber and camber location) and geometric twist angle PAGE 81 81 distributions for the baseline BR and rigid wings are given in Figure 5-9. The rigid wing is characterized by positive (nose-up ) twist and a progressive de-cam bering toward the wingtip. The carbon fiber inboard portion of the BR wing exhibits very sim ilar wing twist to the rigid wing. Past 2y/b = 0.3 however, both model and experiment show that the membrane wing has a near-constant decrease in twis t of 2-3: adaptive washout. Though this geometric twist dominates the behavior of the BR wing, the memb rane also exhibits some aerodynamic twist. This occurs predominately in the latex between th e battens, about 1% of the chord in magnitude. The location of this camber has large variations: some portions of the wing are pushed back from 25% (rigid) to 75% (membrane), as shown by bot h model and experiment. Shifting the camber aft-ward on low Reynolds number wings is one me thod to hinder flow separation through control of the pressure gradient [27], and may play a role in th e BR wing's delayed stall as well. Figure 5-9. Baseline BR aerodynamic and geometric twist distribution, = 15. The aerodynamic and geometric tw ist distributions fo r the baseline PR and rigid wings are given in Figure 5-10. Membrane inflation adaptively increases the camber by as much as 4%, though this figure is slightly under-p redicted by the m odel. The loca tion of this camber is shifted aft-ward, though not as much as with the BR wing. The flexible laminate used for the wing skeleton pushes the location of the camber at the root slightly forward. Like the BR wing deformation, shape changes over the PR wing ar e a mixture of both aerodynamic and geometric PAGE 82 82 twist (though the former dominates). The lamina ted perimeter deflects upward farther than the leading edge, resulting in a slight nose down twist. This is as much as 2 at the wingtips, slightly under-predicted by the model. Figure 5-10. Baseline PR aerodynamic and geometric twist distribution, = 15. Wing twist and camber throughout the entire -sweep are given in Figure 5-11, at a flexible win g section at 2y/b = 0.65. The master slave moving grid algorithm [152] fails with BR wings at angles of attack higher than 20 : the steep d isplacement gradients between the carbon fiber root and the membrane skin leads to excessive shearing within the CFD mesh surrounding the wing. The nose-down twist of both the BR and the PR wing increase monotonically with angle of attack, thought the former is obviously much larger. Experimentally measured BR wing twist has a linear trend (up to stall at about 22) with while the numerical curve is more nonlinear, and under-predicts twist at moderate angles. Both model and experiment demonstrate a moderate increase in camber of th e BR wing, with a linear trend in up to stall. After stall, the camber of the BR wing increases subs tantially, from 5% to 8%. The camber of the PR wing is much larger than the rigid wing, even at low and negative angles of attack. This is due to the lack of pre-tension: even a moderate amount of force will cause substantial deformations [147]. Both measurements and simulations of the PR wing are PAGE 83 83 difficult at lower angles than shown in Figure 5-11: the membrane is equally apt to lie on eithe r side of the chordline [65], and steady-state solutions dont exist. PR wing cam ber variations with angle of attack are nonlinear (the develo pment of finite strains cause a 1/3 power law response to the applied load [72]), and are slightly under-p redicted by the m odel. The location of this camber in a PR wing m oves somewhat forward for modest angles, while the BR wing sees a significant aft-ward shif t at the onset of stall. Both of these camber location trends are well-predicted by the model. Experimental error bars for camber, though not shown here, are on the order of 10% at low angles, less than 2% at moderate angles, and upwards of 20% in the stalled region [43]. This stems not from uncertainty but from unsteady membrane vibration, possibly due to vortex sh edding as discussed by Lian and Shyy [8]. Figure 5-11. Aerodynamic and ge ometric twist at 2y/b = 0.65. Aerodynamic Loads Lift coefficients (both measured and predicted) throughout the -sweep, with no model yaw, are given in Figure 5-12, for the three baseline wing designs discussed above. For all six data sets, lift slopes are very lo w (~0.05/ about half of the va lue for two-dimensional airfoils [27]) as expected from low aspect ratio wings T he downward momentum from the tip vortices helps mitigate the flow separation, delaying stall to relatively high angles (18-22). Focusing first on the rigid wing, mild nonl inearities can be seen in th e lift curve. Both model and PAGE 84 84 experiment indicate an increase in the slope by 25% between 0 and 15 angle of attack. This is presumably due to a growth in the low pressure cells at the wing tips of the low aspect ratio wing [3]. Such nonlinearities should become more pr evalent for lower aspect ratios than considered here (1.25). Model and experim ent show good ag reement for the lift over the rigid wing prior to stall. At stall (where the static models predictive capability is questionable due to unsteady flow separation [18] and tip vortices [6]) the model slightly under-pr edicts the stalling angle and CL,max; the loss of lift is more severe in the experimental data. Figure 5-12. Baseline lift coefficients: numerical (left), experimental (right). The adaptive inflation/cambering of the PR wi ng substantially increases the lift and the lift slope as compared to the rigid wing. The lift curve of the PR wing is less nonlinear than the rigid wing. This may be due to the nonlinear cambering seen in Figure 5-11, which is known to decrea se the lift slope [15] and can offset the growth of th e tip vortices. Drastic chan ges in the lift characteristics at low angles due to hysteresis effects [65], and a gradua l onset of stall [89] are not evid ent in either the num erical or the experimental da ta, perhaps because a relevant portion of the wing is not composed of the flex ible membrane. The mo del significantly underpredicts CL,max of the PR wing, and erroneously computes that the wing stalls before the rigid PAGE 85 85 wing. Similar experimental work [9] at lower speeds also show ear ly stall, again indicating the sensitivity of Reynolds num ber to stall. At angles of attack below 10, the BR wing has very similar lift charact eristics to the rigid wing, a fact also noted in the work of Lian et al. [28]. This is thought to be due to two offsetting characteristics of a wing with both aerodynam ic and geometric twist [67]: the inflation in between each batten increas es the lift, while the adaptive was hout at the trailing edge decreases the lift. Both of these defor mations can be seen in Figure 5-1. At higher angles of attack, the load alleviation from the washout dominates the deformation, and decreases both the lift and the lift slope, as indicate d by both model and experiment. De layed stall is not present in the measurements (though, as with the PR wing, has been measured at lower Reynolds numbers [9]), and num erical BR wing modeling cannot be taken past 20 due to aforementioned problems with the moving boundary. Figure 5-13 shows drag co efficients through the -sweep, with good experim ental validation of the model. As before, the drag of the rigid and the BR wings are very similar for modest angles of attack. Above 10 the load alleviation at the tr ailing edge decreases the drag, a streamlining effect [63]. It should be noted however that for a given value of lift, the BR wing actually has slightly m ore drag than a rigid wing [9]. Regardless of whether the comparative basis is lift or angle of attack, the PR wing has a dr ag penalty over the rigid wi ng. This is in pa rt due to the highly non-optimal airfoil sh ape of each membrane wing section: Figure 5-5 shows the tang ent discontinuity of the wing shape at the membrane/carbon fiber interface towards the leading edge. Excessive inflation may also induce additional fl ow separation. Longitudinal pitching moments (measured about the leading edge) are given as a function of lift for the three baseline designs in Figure 5-14. Of the three, the PR wing is not statically PAGE 86 86 stable (based upon a negative Cm,AC), and the hinged trailing edge portion (seen in Figure 1-1 and Figure 1-2) must be used for trimmed flight. Prior to stall, both the aeroelastic model and the experim ent indicate very similar behavior be tween the rigid and the BR wing, with mild nonlinearities in the moment curves. This is os tensibly due to tip vortex growth, as before [3]. Figure 5-13. Baseline drag coefficients: numerical (left), experimental (right). The PR wing has a 15% lower pitching moment sl ope than the rigid wing. This is a result of the membrane inflation, which shifts the pressure recovery towards the trailing edge, adaptively increasing the strength of the nose-do wn (restoring) pitching moment with increases in lift and Steeper Cm slopes indicate larger static margins: stability concerns are a primary target of design improvement from one generation of micro air vehicles to the next. The static margin of a MAV is generally only a few m illimeters long; properly fitting all the microcomponents on board can be difficult. Furthermor e, the PR wing displays a greater range of linear Cm behavior, possibly due to the fact that the adaptive membrane inflation quells the strength of the low pressure cells [13]. Finally, L /D characteristics are given in Figure 5-15, as a function of lift. For low angles of attack (and lift), the three wings perform similarly. At higher angles of attack (prior to stall), the PR wing has the highest efficiency. The model incorrectly computes the BR wing to have PAGE 87 87 the best L/D for a small range of modest lift values. Correlation between model and experiment is generally acceptable for the rigid and BR wings, though the L/D of the PR wing is significantly under-predicted by the model, owing mostly to poor lift prediction at these angles ( Figure 5-12). The camber of the PR wing is subsequently under-predicted as well ( Figure 511), and m ay be a result of membrane vibration [89]. At no point does either model or experim ent indicate that the rigid wing has the best efficiency; pe rhaps surprising, given the fact that neither wing deforms into a particularly optimal airfoil shape. Figure 5-14. Baseline pitching mo ment coefficients: numerical (l eft), experimental (right). Figure 5-15. Baseline wing efficiency: num erical (left), experimental (right). PAGE 88 88 A quantitative summary of the last four figures is given in Table 5-1, for all three baseline wings at 6 angle of attack. Experimental error bounds are co mputed as described above. Aerodynamic sensitivities (as well as the pitc hing moment about the aerodynamic center) are found with a linear fit th rough the pre-stall angles of attack. Error bounds in these slopes are computed with Monte Carlo simulations. Co mputed lift, drag, and pitching moments consistently fall within the measured error bars (the latter of which are exceptionally large), though pitching moments are significantly under-pr edicted (10-30%). Sensitivities are also under-predicted, though still fall with in the large error bars asso ciated with pitching moment slopes. With the exception of L/D of a PR wing, trends between differe nt wing structures are well-predicted by the aeroelastic model. Table 5-1. Measured and computed aerodynamic characteristics, = 6. CL CD num. exp. error (%)num. exp. error (%) rigid 0.396 0.384 0.024 3.10 0.070 0.069 0.007 1.15 BR 0.381 0.382 0.024 -0.16 0.067 0.069 0.007 -3.04 PR 0.465 0.495 0.031 -5.98 0.085 0.076 0.009 11.61 Cm L/D num. exp. error (%)num. exp. error (%) rigid -0.084 -0.063 0.033-32.81 5.64 5.49 0.69 2.72 BR -0.087 -0.073 0.034-19.39 5.70 5.49 0.68 3.77 PR -0.138 -0.131 0.042-5.64 5.49 6.49 0.87 -15.36 CL Cm,AC num. exp. error (%)num. exp. error (%) rigid 0.049 0.051 0.003 -5.26 0.013 0.016 0.018 BR 0.044 0.048 0.004 -9.35 0.006 -0.001 0.020 PR 0.052 0.057 0.004 -9.21 -0.008 -0.015 0.026 Cm dCm/dCL num. exp. error (%)num. exp. error (%) rigid -0.012 -0.010 0.004-11.65 -0.246 -0.199 0.086 -23.07 BR -0.011 -0.009 0.004-17.97 -0.244 -0.185 0.098 -31.88 PR -0.014 -0.013 0.0066.01 -0.280 -0.229 0.105 -22.17 Flow Structures Having established sufficient confidence in th e static aeroelastic membrane wing model, PAGE 89 89 attention is now turned to the computed flow stru ctures. No experimental validation is available for this work, though whenever possible the results wi ll be correlated to data in the previous two sections, or results in the literat ure. Experimental flow visuali zation work for low aspect ratios and low Reynolds number is given by Tang and Zhu [6] and Kaplan et al. [37]. Work done explicitly on MAV wings is given by Gursul et al. [40], Parks [91], Gamble and Reeder [92], and Systm a [153]. The pressure distributions and flow structures are given in Figure 5-16 at 0 angle of attack for the upper/suction wing surface o f all three baseline wing desi gns. The plotted streamlines reside close to the surf ace, typically within the boundary layer. For the rigid wing, a high pressure region is located close to the leading ed ge, corresponding to flow stagnation. This is followed by pressure recovery (minimum pressure), located approximately at the camber of each rigid wing section. Pressure recovery is followe d by a mild adverse pressure gradient, which is not strong enough to cause the flow to separate. A further decrease in Reynolds number has been shown to cause mild flow separation over the top surface for 0 however [14], [153]. A sm all locus of downward forces are present over the negatively-cambered region (reflex) of the airfoil, helping to offset the nose-down pitchi ng moment of the remainder of the rigid MAV wing, as discussed above. The reflex can also help improve the wing efficiency, compared to positively-cambered wings [55]. There is positive lif t of this wing at 0 (Figure 5-12), resulting in a m ild tip vortex swirling system. The low pre ssure cells at the wing tip are not yet evident. Aeroelastic pressure redistributions of th e upper surface of the BR wing are seen in the form of three high-pressure lobes at the ca rbon fiber/membrane boundary interface towards the leading edge. The membrane infl ation in between each batten ( Figure 5-1) result s in a s light tangent discontinuity in the wing surface. This fo rces the flow to slow down and redirect itself PAGE 90 90 over the inflated shape: such a d eceleration results in a pressure spike. Aft of these spikes, the pressure is slightly lower in the membrane sk in than over each batten (due to the adaptive camber), driving the flow into the membrane patches. This is a very small effect (mildly visible in the streamlines) for the current case, but can be expected to play a la rge role with potential flow separation, where the chordw ise velocities are very small [154]. Figure 5-16. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 0. For the PR wing (Figure 5-16), the pressure spike is stronger, and exists continuously along the m embrane interface. A significant per centage of this spike is directed axially, increasing the drag (as seen in Figure 5-13). The adaptive inflation causes an aft-ward shift in the pressure recovery location of each flexible wing section. The longer mom ent arm increases the nose-down pitching moment about the leading edge ( Figure 5-14), which is the working m echanism behind the benevolent longitudinal sta tic stability properties of the PR wing. Furthermore, the aerodynamic twist increases th e adverse pressure gradient over the membrane portion of the wing: some flow now separates as it travels down the inflated shape, further increasing the drag (as comp ared to the rigid wing). Similar results are given for the lower/pressure si de of the three wings at 0 angle of attack in Figure 5-17. The flow beneath the rigid wing is dom inated by an adverse pressure gradient towards the leading edge, causing a large separa tion bubble underneath the wing camber. This PAGE 91 91 separated flow is largely confined to the in-board portions of the wing. Flow reattaches slightly aft of the quarter-chord, after wh ich the pressure gradient is favorable. The flow accelerates beneath the negatively-cambered portion of the rigid wing: this decreases the local pressures, further offsetting the nose-down pitching moment The pressure distribution on the lower surface is not greatly affected by the tip vortices, previously noted by Lian et al. [28]. Figure 5-17. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), = 0. For the BR wing ( Figure 5-17), slight undulatio ns in the pressure distribution are indicative of the m embrane inflation in be tween the battens. This causes the opposite of what is seen on the upper wing: flow is slightly packed towards the battens [154], though the effect is minor, as before. The adaptive aerodynam ic twist of the PR membrane wing pushes the bulk of the separated flow at the leading edge towards the root, and induces furthe r separation beneath the inflated membrane shape, as the air flows into the cavity against an adverse pressure gradient. The location of maximum pressure is increased and pushed aft-ward to coincide with the apex of the inflated membrane, increasing both the lift and the stability. Flow structures over the upper surface at 15 angle of attack are given in Figure 5-18. At this higher incidence, the adve rse pressure gradient is too strong for the low Reynolds num ber flow, and a large separation bubble is present at the three-quarter c hord mark of the rigid wing. Despite the nose-up geometric twist bu ilt into the wing (7 at the tip, Figure 5-9), flow separates PAGE 92 92 at the root first, and is confined (at this angle) to the in-board portion of the wing. This may be due to the steeper pressure gradients at the root or an interaction with the tip vortex system [5]. The reattach ed flow aft of the bubble (and th e resulting pressure distribution) must be viewed with a certain amount of suspicion. Such a reattachment is known to be turbulent process [25], and no such module is included in the CFD (or even, to the authors knowledge, exists for co mplex three-dimensional flows) Unsteady vortex shedding may accompany the bubble as well [8], though time-averaging of vortex sh edding is known to com pare well with steady measurements of a single stationary bubble [18]. The augmented incidence has consider ably increased the strength of the wingtip vortex swirling system over the rigid wing. The size of the vortex core is larger (indicativ e of the expected increase in induced drag [27]), and the low pressure cells at th e wing tip are very evident [3]. Figure 5-18. Pressure distributions (Pa) and streamlines on the upper surface of a rigid (left), BR (center), and PR wing (right), = 15. As expected, the aeroelastic effects of the BR and PR wings are more predominate at 15 in Figure 5-18. For the BR wing, the three highpressure lobes over the m embrane/carbon fiber interface are larger. Significant pressure-redistr ibution over the membrane stretched between the outer batten and the wing tip can be seen as well. Adaptive washout slightly decreases the intensity of the separation bubble, but has no noticeab le effect on the pressu re distribution at the trailing edge of the upper BR wing surface. At 15, the aerodynamic twist of the PR wing is PAGE 93 93 considerably larger than before, as is the resu lting pressure spike at the membrane-carbon fiber interface. Despite the large adverse pressure gradient in this region, flow does not separate (though it has been noted in other studies [153]). The inflated m embrane shape of the PR wing pus hes the bulk of the fl ow separation closer to the wing root. Some of this separated flow reattaches to the wing and travels into the wake, while the rest travels spanwise. This flow is attr acted either by the low pressures associated with the adaptive cambering, or by the low pressures at the core of the tip vortex. Some of these separated streamlines are entrained into the swirling system, an interaction that has been shown to cause potential bilateral instab ilities for high angles of attack [6]. This effect, not seen in the rigid or BR wings, obviously cannot be further studi ed in this work, due to both the symm etry and the steady assumptions made in the solver. It can also be seen that the passive shape adaptation decreases the magnitude of the low pressure cells at the wing-tip, by 9% for the BR wing and 13% for the PR wing, compared to the rigid case. This indicates that the induced drag is decreased with flexibility, though this is only a re-distribution of the total drag. Two possible e xplanations exist for the decrease in tip vortex strength. The mechanical strain energy in the inflated membrane skin may be removing energy from the vortex swirling system [90]. For the PR wing, the inflated membrane shape may act as a barrier to the tip vo rtex formation, preventi ng the full swirling development at the wing-tip. A similar effect is demonstrated in the work of Viieru et al. [38] by the use of endplates installed on a rigid MAV wing. Wh ereas the endplates are able to decrease induced drag only at moderate an gles (afterwards the tip vortices incr ease in strength to overwhelm the geometrical presence of the endplates), the phenomena demonstrated in Figure 5-18 is effective at all angles: both the size of the m embrane barrier and the st rength of the vortex swir ling grow in conjunction PAGE 94 94 with one another as the angle of at tack increases. This decrease in tip vortex strength is also seen in Figure 5-14: the nonlinear aerodyna m ics (from the low pressure ce lls at the tip) is evident in the pitching moments of the rigid and BR wings, while the PR curve is very linear. On the underside of the rigid wi ng at 15 angle of attack ( Figure 5-19), the increased incidence provides for com pletely attached flow behavior. The pressure gradient is largely favorable, smoothly accelerating the flow from leading to trailing edge. From the previous four figures it can be seen that separated flow over the bottom surface gradually attaches for increasing angles of attack, while attached flow over the upper surface gradually separates (eventually leading to wing stall). As time-aver aged flow separation is likely to be unsteady vortex shedding [18]: this explains the aforementione d m embrane vibration amplitudes that decrease to a quasi-static beha vior, then increase through the -sweep [135]. Figure 5-19. Pressure distributions (Pa) and streamlines on the lower surface of a rigid (left), BR (center), and PR wing (right), = 15. Load alleviation on the lower surface of th e BR wing is evidenced by a decrease in the high-pressure regions associated with camber, a nd a growth of the suction region at the trailing edge (the latter presumably due to a decrease in the local incidence). A high-pressure lobe also develops at the trailing edge of the membrane panel between the carbon fi ber root and the inner batten. At higher angles, this region of the memb rane does not locally inflate; it merely stretches between the two laminates, acting as a hinge. The adaptive inflation of the PR wing causes a PAGE 95 95 significant redirection of the flow vectors beneat h the wing, but does not induce flow separation. Two sharp pressure drops are seen beneath the wing: one as the flow accelerates into the inflated membrane shape, and the second as the flow accelerates out of from the membrane and underneath the re-curved area of the wing. Further delineation of the flow structures over the three base line wings can be seen in Figure 5-20 ( = 0 ) and Figure 5-21 ( = 15 ), with the sectional normal force coefficient and the pressure coefficients over a flexible span sta tion (2y/b = 0.5) of the wing. For the rigid wing at 0, the sectional normal force peaks at 2y/b = 0.9 (due to the decreasi ng local chord length of the Zimmerman planform, but also the low pressu re cells left by the ti p vortices) and then experiences a sharp drop at the tip, as necessitated by the low thickness of the wing. Grid resolution and errors from interpolating the pressures from the cell centers to the nodes [150] prevent this curve from reaching the correct va lue of zero. No significant differences arise between the computed cn of the BR and rigid wings at 0, as previously indicated by the similar aerodynamic loads ( Figure 5-12). The adaptive inflati on of the PR wing increases the norm al force over most of the wing, incl uding the stiff carbon fiber root. Turning now to the pressure coefficients at 0 ( Figure 5-20), both the BR and the PR wings experience a pressure spike over the upper surface at 2y/b = 0.2, corresponding to the m embrane inflation. Outside of this location, pressure redistribution over the BR wing is negligible. The PR wing shows an aft-ward shif t in the high-lift forces over both the upper and lower surfaces. Adaptive inflation is also seen to increase the se verity of the adverse pr essure gradient (leading to the flow separation seen in Figure 5-16), and exacerbate the pre ssu re gradient reversal over the reflex portion of the wing. At 15 angle of attack ( Figure 5-21), the BR wing is more e ffective, able to allev iate the PAGE 96 96 load over the majority of the wing. An evaluati on of the pressure coefficients at this angle indicates that the majority of this reduction in lift occurs towards the trailing edge of the bottom surface, where the suction forces are increased. Both flexible wing pressure spikes over the upper surface are intensifie d at the higher angle, with the PR wings approaching the strength of the leading edge stagnation pressu re. Sharp pressure drops are also visible on the underside of the wing, as the flow accelerat es into the membrane cavity. All three wings show a mild pressure plateau associated with separation [27]; the plateaus of the fl exib le wings are shifted towards the trailing edge. Figure 5-20. Section normal force coefficients, and pressure coeffi cients (2y/b =0.5), = 0. Figure 5-21. Section normal force coefficien ts, and pressure coeffi cients (2y/b =0.5), = 15. PAGE 97 97 CHAPTER 6 AEROELASTIC TAILORING The static aeroelastic m odeling algorithm de tailed above (using the Navier-Stokes flow solver and the nonlinear membrane solver) can el ucidate accurate quantita tive dependencies of a variety of parameters (CL, CD, Cm, L/D, CL Cm mass) upon the wing structure. Having first studied the general effect of wing topology (b atten-reinforced and perimeter-reinforced membranes, as well as rigid wings), attention is now turned to structural sizing/str ength variables within the BR and PR wings. Results from the previous section show that the membrane skins inflation/stretching dominates the aeroelastic behavior, indicating the importance of the pretension in the membrane skin. Pre-stress resultan ts in the spanwise and chordwise directions are both considered as variables. With the excepti on of the free trailing e dge correction of the BR wings detailed above, the pre-tens ion is constant throughout the wi ng. The laminate orientation and number of plies used to construct the plai n weave carbon fiber areas of the wing can be varied as well. Finally, the num ber of layers in each batten of the BR wing can be altered, though the orientation will be fixed so that the fibers run parallel to the chord line. The sizing/strength parameters listed above leads to an optimization framework with 9 variables, if the number of layers in each of the three battens are permitted to differ, and the wing type (BR, PR, rigid) is considered a variable as well. Some of the variables are discrete, others continuous. The variables are not entirely independent either: the fiber orientation of the second bi-directional plain weave ply is m eaningless if only a single ply is used. Genetic algorithms are well-suited to problems with a mixed intege r-continuous formulation, can handle laminate stacking sequence designs without a set number of layers (with the use of addition and deletion modules [155]), are a cost-effec tive m ethod of solving multi-objective problems [156], and can navigate dis jointed design spaces [52]. The computational cost of a genetic algorithm is PAGE 98 98 prohibitive however, typically requiring thousands of function evaluations for suitable convergence; a single simulation using the static aeroelastic model desc ribed above takes 2-3 hours of processor time on a Compaq Alpha workstation. A viable alternative is a designed experiment: the computational cost is lower, and provides an effective investigation of the desi gn space. For this work, one-factor-at-a-time (OFAT) numerical tests are run to establish the effect of various structural parameters upon the relevant aerodynamics. The three baseline wing designs used above will represent the nominal wing designs (2 layers of plain weave at 45, one layer batt ens, slack membrane). Having identified the structural variables that displa y the greatest sensitivity within the system, a fullfactorial designed experiment [157] will be run on a reduced set of variables. This data set can then be used to identify the optim al wing type and structural composition for a given objective function. Designs that strike a compromise between two objective functions are considered as well. The work concludes with experimental wind tunnel validation of the performance of selected optimal designs. OFAT Simulations The schedule of OFAT simulations is as fo llows: a 6-level full factorial design is conducted for the chordwise and spanwise pre-stre ss resultants, 6 simulations for the orientation of a single laminate of plain-weave, a 6-level fu ll factorial design for th e orientations of a twolayer laminate plain weave, and a 3-level full fact orial design for the number of layers used in the three battens. Pre-stress resultants are bounde d by 0 N/m (slack membrane) and 25 N/m (axial batten buckling can be computed for a distributed axial force equivalent to 31 N/m of pre-stress resultant in the membrane). The latter value corresponds to roughly 10% pre-strain. Plies of plain weave carbon fiber are limited to two la yers, while battens are limited to three. PAGE 99 99 Membrane Pre-Tension Computed aerodynamic derivatives (CL and Cm ) and efficiency (L/D) are given as a function of the pre-stress resultants in the chordwise (Nx) and spanwise (Ny) directions for a BR wing in Figure 6-1. The corresponding norma lized wing disp lacement is given in Figure 6-2 for a subset of the data m atrix. All results ar e computed at 12 angl e of attack, aerodynamic derivatives are computed with a finite differe nce between 11 and 12. In a global sense, increasing the pre-tension in the BR wing increases CL decreases Cm and decreases L/D. The increased membrane stiffness prevents effec tive adaptive washout (and the concomitant load alleviation), and the wing perfor mance tends towards that of a ri gid wing. At 12 for a rigid wing, CL = 0.0507, Cm = -.0143, and L/D = 4.908. Overall sensitivity of the aerodynamics to the membrane pre-tension can be large for the derivatives (up to 20%), though less so for the wing efficiency (less than 5%, presumably due to the conflictive nature of the ratio). Figure 6-1. Computed tailoring of prestress resultants (N/m) in a BR wing, = 12. The BR wing is very sensitive to the pre-stress in the spanwise direction, but less so to stiffness in the chordwise direction. This is seen in Figure 6-2: the slack membrane wing has a trailing edge deflection of 2.5% of the root chord. Maxim izing the spanwise pre-tension (with the other direction slack) drops this value to 1% while the opposite scenario drops the value to only 1.9%. This is due to the directional sti ffness of the battens (which depend on compliance PAGE 100 100 normal to their axis for movement), but also the trailing edge stre ss correction detailed above. Despite the global trend toward s a rigid wing with increased pre-tension, the changes are not monotonic. A wing design with a minimum lif t slope (for gust rejection, improved stall performance, etc.) is found, not with a complete ly slack wing, but a wing with a mild amount of stiffness (10 N/m) in the chord direction, and none in the span direction. Such a tactic removes the aforementioned conflicting sources of aeroelastic lift in a BR wing. Th e pre-stress correction eliminates most of the stiffne ss at the trailing edge (allowin g for adaptive washout and load alleviation), but retains the chordwise stiffn ess towards the leading edge, as seen in Figure 4-8. The m embrane inflation in this area is thus decreased, along with th e corresponding increase in lift due to camber. Figure 6-2. Computed BR wing deformation (w/c) with various pre-tensions, = 12. Maximizing CL (for efficient pull-up maneuvers, for example), is found by maximizing Ny and setting Nx to zero; this eliminates the adaptive wa shout, but retains the inflation towards the leading edge. Conversely, maximizing CL with a constraint on the acceptable L/D might be PAGE 101 101 obtained by maximizing Nx and setting Ny to zero. Peak efficiency is found with a slack membrane: this corresponds to minimum drag, which is not shown. It should be mentioned however, that if a design goal is to maximize the lift slope (or minimize the pitching moment slope for stability), a BR wing is most likely a poor choice. Opposite trends are found for the PR wing ( Figure 6-3 and Figure 6-4): increasing the pretension d ecreases CL increases Cm and increases L/D. Similar to before, added wing stiffness decreases the adaptive inflation of the wing skin, and results tend towards that of a rigid wing. Without the directional influence of the battens a nd the trailing edge stress correction needed for the BR wing, the PR wing surfaces in Figure 6-3 are very smooth, and converge m onotonically for high pre-stress. Figure 6-3. Computed tailoring of prestress resultants (N/m) in a PR wing, = 12. As before, the PR wing is more sensitive to pr e-tension in the spanwise direction than the chordwise direction. The slack membrane wing in flates to 5% of the chord: maximizing tension in the chord direction (with none in the span direction) drops this value to 3%, though the opposite case drops the value to 1.5%. This is pr obably due to the fact that the chord of the membrane skin is about twice as long as its span. The sensitiv ity of a pressurized rectangular membrane to a directional pre-stress is inverselyproportional to its length in the same direction, as indicated by solutions to Eq. (4-4). Though the L/D of the PR wing is equally affected by pre- PAGE 102 102 stresses in both directions, th e two aerodynamic derivatives in Figure 6-3 have a significantly muted respo nse to Nx. Such a result has noteworthy ramifications upon a multi-objective optimization scenario. The longitudinal static stab ility is optimal for a slack membrane wing, but the wing efficiency at this da ta point is poor. Maximizing Nx and setting Ny to zero greatly improves the lift-to-drag ratio (only 0.2% less than the true optimum found on this surface), with a negligible loss in static stability. Figure 6-4. Computed PR wing deformation (w/c) with various pre-tensions, = 12. Single Ply Laminates The same aerodynamic metrics are given in Figure 6-5 as a function of the ply angle (with respect to the chord line) for a set of wings with a single layer of bi-directional carbon fiber at the wing root, leading edge, and perim eter (for the PR wing only). The membrane wing is slack. Due to the plain weave nature of the laminate, all trends are periodic every 90. Only fiber orientations of 0, 45, a nd 90 automatically satisf y the balance constraint [155]. For the PR wing, changing the fiber angle has a m inor effect on the aeroel astic response, and optima are PAGE 103 103 mostly located at either 45 (where spanwise bendi ng is largest) or 90 (whe re it is smallest). This indicates that the PR wing, whose planfo rm is dominated by membrane skin, can only take advantage of different laminates inasmuch as the spanwise bending can increase or decrease the aerodynamic membrane twist/cambering. On the other hand, the BR wing relies mostly upon geometric twist ( Figure 5-11), which can be provided from unbalanced laminate s via bend-twist coupli ng; the concept behind traditional aeroelastic tailoring [11]. Of the 7 data points shown in Figure 6-5, orientations less than 45 cause the wing to wash-in, while angles greater than 45 cause washout, the latter of which m inimizes CL of a BR wing, as expected. Using la minate wash-in to counter the load alleviation of the membrane washout (at 15 ) optimizes the wing efficiency. Aerodynamic sensitivity of the BR wing to laminate orientation is also larger than th at seen in the PR wing because the carbon fiber skeleton is less constrai ned. The wing tip of the BR wing (where the forces can be large, due to the tip vortices seen in Figure 5-18) is not connected to the trailing edge via a perim eter strip. Figure 6-5. Computed tailoring of laminate orientation for single ply bi-directional carbon fiber, = 12. Double Ply Laminates Computed aerodynamic derivatives (CL and Cm ) and efficiency (L/D) are given as a function of the ply orientations ( 1 and 2) of the two layers of bi-directional plain weave in a BR PAGE 104 104 wing ( Figure 6-6) and a PR wing ( Figure 6-7) at 12 angle of atta ck. As before, the m embrane skin is slack. Aeroelastic trends are expected to repeat every 90, and will be symmetric about the line 1 = 2. This latter point is only true because bending-extension coupling in nonsymmetric laminates is ignored, though the effect of its inclusion would be very small as the wing is subjected mostly to normal pressure forces. Figure 6-6. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a BR wing, = 12. For the BR wing, efficiency is maximized and the lift slope is minimized when the fibers make 45 angles with the chord and span directio ns. Static stability is improved when fibers align with the chord. The response surface of the two stability derivatives are very noisy, suggesting possible finite differencin g errors, and all three surfaces in Figure 6-6 show little variation (only Cm of the BR wing can be varied by more th an 5%). Unlike any of the tailoring studies discussed above, the PR wing shows the sa me overall trends and optima as the BR wing. The surfaces for the PR wing, however, are much smoother but have less overall variation. Of the sampled laminate designs, [15]2 and [75]2 will exhibit the greatest bend-twist coupling, yet neither are utilized by the membrane wings. This fact, along with the similarity between the PR and the BR surfaces, suggest that th e orientation of a plain weave laminate with two layers is too stiff to have much im pact on the aerodynamics, which is dominated by membrane inflation/stretching. The use of bi-d irectional plain weave is not the most effective PAGE 105 105 means of introducing bend-twist coup ling in a laminate. The fact that the two fiber directions within the weave are perpendicular automatically sa tisfies the balance constraint at angles such as 45. This would not be the ca se if plies of uni-dir ectional carbon fiber are utilized, but this is prohibitive in MAV fabrication for the followi ng reason. Curved, unbalanced, potentially nonsymmetric thin uni-directional laminates can experience severe thermal warpage when removed from the tooling board, retaining little of the intended shape. Figure 6-7. Computed tailoring of laminate orientations for two plies of bi-directional carbon fiber in a PR wing, = 12. Batten Construction Computed lift slope and effi ciency of a BR wing at 12 a ngle of attack is given in Figure 68 as a function of the number of la yers in each batten. The thickness of each batten can be varied independently, though the num ber of layers is limited to three, re sulting in 27 possible designs. As before, the membrane skin is slack, and a two-layer plain weav e at 45 makes up the remainder of the wing. The normalized out-of-plane displacement and differential pressure coefficients along the chordstation x/c = 0.5 for 4 sele cted designs is given in Figure 6-9. As expected, the wing with three one-layer battens has the m ost adaptive washout, which provides the shallowest lift slope, but also the best lift-to-drag rati o. Additional plies, regardless of which batten they are added to, monotonically decrea ses the efficiency. The same technique can be used to increase CL except for combinations of stiff battens towards the wing root and a PAGE 106 106 thin outer batten at the wing tip (331 and 332, fo r example, where the battens are numbered from inner to outer and the integers indicate the numbe r of layers), which can cause the lift slope to decrease from these peaks. Design 223 show s the steepest lift slope of the wings in Figure 6-8. Figure 6-8. Computed tailoring of batten construction in a BR wing, = 12. The undulations in the differential pressure due to local membrane inflation from in between the battens are clearly visible in Figure 6-9. Low pressure regions on the upper surface of the m embrane skin and high pressure on the lo wer surface (which slightly re-directs the flow towards the battens [154]) results in the four high-lift lobes over the in fl ated membrane skin. This inflation can be controlled in obvious ways: wing displacement is larger for design 111 than design 333, throughout the entire le ngth of the wing section in Figure 6-9. The wing defor mation of design 123 is comparable to desi gn 111 towards the root of the wing, but tapers off towards the wingtip, where it resembles design 333. In some cases, redistributing the batten sizes causes a trade-off between local inflati on and spanwise bending. The displacement of design 123 is less than design 111 between 24% and 60% of the semispan, but the local inflation between the stiffer battens is higher, causi ng greater redistribution of the flow and high differential pressures. A similar comparison can be made between designs 321and 111 towards PAGE 107 107 the wingtip. Overall changes in the aerodynamics due to batten tailoring are relatively small however, with 5% possible variability in CL and 1.5% in L/D. Though not shown, the static stability of the BR wing can be vari ed by 10% with batten tailoring. Figure 6-9. Computed normalized out-of-plane displacement (lef t) and differential pressure (right) at x/c = 0.5, for various BR designs, = 12. Full Factorial Designed Experiment Of the structural sizing/str ength parameters discussed above, spanwise membrane pretension, chordwise pre-tension, the number of layers of bi-dir ectional plain weave carbon fiber, and the wing type (BR, PR, rigid) are considered in a designed experiment. As stated above, the aeroelasticity of the MAV wing is dominated by the membrane inflation, and the laminate stacking is a seconda ry effect (though Cm of a BR wing is moderately sensitive to fiber orientation and batten thickness). The number of layers of plain weave carbon fiber, though not explicitly discussed above, is included due to interesting discrepancies between laminate tailoring with one layer ( Figure 6-5) and tailoring with two layers ( Figure 6-6 and Figure 6-7). For this s tudy, the number of layers in each ba tten is fixed at one, a nd all plain weaves are oriented at 45 to the chordline. A three-level, three-variable full factorial designed experiment is implemented for each PAGE 108 108 membrane wing. Only 1, 2, or 3 layers of carbon fiber are permitted, while pre-tension resultant (chordwise or spanwise) is restricted to 0, 10 or 20 N/m. More than 3 layers is excessively stiff and heavy; 1 layer may not be able to withstand flight loads or survive a crash. The upper cap on pre-tension is, as discussed above meant to prevent batten buckli ng. Each full factorial design array requires 27 simulations for each membra ne wing, a number which must be doubled to obtain finite difference approximations of the lift and moment derivatives in angle of attack. Including the two data points needed for the rigid wing, 110 computationally expensive simulations are required. While a full factorial matrix is not the most economical choice for a designed experiment (a central-composite design is an adequate fraction of the full factorial, for example [157]), the uniform sampling will provide the best qualitative insight into the membrane wing tailoring. All 27 data points for the BR wing are given in Figure 6-10, in terms of CL Cm and L/D. The data points for a two-layer laminate are identical to those seen in Figure 6-1. The computed norm alized out-of-plane displacement of the BR wi ng with a slack membrane can be seen in Figure 6-11, with one, two, and th ree layers of plain weave carbon fiber. These designs are the three found on the z-axis of Figure 6-10. The variability in the aerodynam ics with the three design variables is substantial: 22% in the lift slope, 54% in the pitching moment slope, and 16% in efficiency. As above, increasing th e pre-tension in the BR wing increases CL decreases Cm and decreases L/D, though the tre nd is not monotonic. No prevalent trend exists for the number of plain weave layers, demonstr ating strong interacti ons with the membrane pre-tension. For a slack BR membrane wing (Nx = Ny = 0), increasing the number of plain weave layers significantly decreases the deformation of the wi ng tip and the adaptive washout at the trailing edge. As seen in Figure 6-11, a three-layer BR wing is most ly characterized m y local membrane PAGE 109 109 inflation among the battens. This demonstrates the degree to which the adaptive washout at the trailing edge depends on the bending /twisting of the leading edge la minate (where the forces are very high, seen in Figure 5-18), and also explains why ta ilo ring the thickness of the battens, discussed above, has only a minor effect upon the aerodynamics. Figure 6-10. Computed full f actorial design of a BR wing, = 12. Figure 6-11. Computed BR wing deformation (w/c ) with one layer of plain weave (left), two layers (center), and three layers (right), = 12. This inability of the slack membrane wing to alleviate the flight loads decreases the efficiency, but surprisingly, has little effect on the stability derivatives. One possible reason for this is the negative deformations at the trailing edge of the three-layer MAV wing. The stiffer wing adheres closely to the origin al, rigid wing shape, which contai ns reflex (negative camber) at the trailing edge. The negative forces in this area push the membrane downward, increasing the wing camber. Increasing the stiffness of the plain weave laminate may convert the BR wing from a structure with adaptive washout to one with progressive decambering, leaving the PAGE 110 110 longitudinal stability deriva tives relatively unchanged. Greater variation with laminate thickness is seen for non-zero pre-te nsions, particularly when Nx = 20 N/m and Ny = 10 N/m. If a single layer of carbon fiber is used, this data point represents the minimum lift slope. Like th e double-layered laminates studied above, the BR wing removes the camber due to membrane inflati on (and thus the lift) at the leading edge with high chordwise stiffness, and allows for ad aptive washout with low spanwise stiffness perpendicular to the battens. Such a design has biological inspiration: the bone-reinforced membrane skins of pterosaurs [101] and bats [102] both have larger chordwise stiffness. For low levels of pre-tension, decreasing the n umber of plain weave layers increases the efficiency; if the membrane is highly-tensioned, the opposite is true. The L/D objective function is optimized with a one-layer slack membrane BR wing. It can also be seen in Figure 6-10 that for high levels of membrane pre-tension, there is little com puted difference between 2 and 3 layer laminates. Similar data is given in Figure 6-12 and Figure 6-13, for a PR wing. The single-layer PR wing exhibits a substantial am ount of adaptive washout, owning to deflection of the weak carbon fiber perimeter. Two and three-layer laminates remove this feature completely, forcing the wing into a pure aerodynamic twist. Regardless of the load alleviation along the trailing edge, the steepest lift and moment curves are found with single-layer laminates, as the weak carbon fiber reinforcement intensifies the cambering of the me mbrane wing. The two-dimensional equivalent to this case is a sailwing with the trailing edge attached to a flexible support. Well-known solutions to this problem indicate that increasing the flexibility of the support improves the static stability [67], a trend re-iterated in Figure 6-12. None of the PR aerodynam ic metrics or the displacement contours show a substantial difference between two and three layer laminates. For the thicker laminates, increasing the PAGE 111 111 spanwise pre-tension provides steeper lift and pitching moment curves; the system has a low sensitivity to chordwise pre-tension. This may be due to the membrane skins shape: its chord is much greater than its span, as discussed above. Figure 6-12. Computed full f actorial design of a PR wing, = 12. Figure 6-13. Computed PR wing deformation (w/c) with one layer of plain weave (left), two layers (center), and three layers (right), = 12. For one-layer laminates, no clear trend between CL Cm and pre-tension (chordwise or spanwise) emerges. Whereas the thicker lamina tes prefer a slack membrane wing to optimize longitudinal static stability, the one-layer wing optimizes this metr ic when 10 N/m is applied in the span direction. The reflex in the airfoil shape may again be the reason for this. The mild amount of spanwise pre-tension enforces the in tended reflex in the membrane skin, and the downward forces depress the membrane skin (seen in Figure 6-4). Slight in creases in angle of atta ck increases the inflation camber towards the leading edge, but decreases the reflex at the trailing edge, resulting in a significant restoring moment. The efficiency of thick-laminate PR PAGE 112 112 wings is equally degraded by chordwise and span wise pre-tensions. The opposite is true for single-layers, where L/D can actually be improved with less tension. The above data is recompiled in Figure 6-14, which plots th e perform ance of the 27 BR designs, the 27 PR designs, and the rigid wing, in terms of the lift slope and pitching moment slope as a function of L/D. As seen many ti mes in the above plots, the various objective functions conflict: tailori ng a wing structure for longitudinal st atic stability may induce a severe drag penalty, for example. No wing design exists (typically) that will optimize all of the relevant performance metrics, and compromise designs mu st be considered. The set of compromise solutions fall on the design spaces Pareto optimal front [156]. A Pareto optimal solution is nondom inated: no solution exists within the data set that out performs the Pareto optimal solution in all of the performance metrics. Three Pareto fronts are given in Figure 6-14. The first details the tradeoff between m aximizing L/D while minimizing CL The second gives the tradeoff between maximizing L/D while maximizing CL and the final front is a trad eoff between maximizing L/D while minimizing Cm It may be beneficial for a MAV wi ng to have a very steep lift slope (for efficient pull-up maneuvers, for example) or very shallow (for gust rejection), so both are included. All three of these objective functions could be used to compute a common Pareto front, but visualization of the resulting hype rsurface would be difficult. Furthermore, maximizing CL and minimizing Cm evolve from similar mechanisms, and seldom conflict. The overlap between BR wings and PR wings in Figure 6-14 is minimal, with the latter design typically having higher effi ciency and sh allow lift and mo ment slopes. The rigid wing lies close to the interface between the two membrane wing types, but is not Pareto optimal. The basic performance tradeoffs are readily visible: peak L/D is 5.49 (a single-layer BR wing with a PAGE 113 113 slack membrane), a design whose lift slope is 8% higher than the minimum possible lift slope, 18% lower than the maximum possible lift slope, and whose pitching moment slope is 34% higher than the minimum possible moment slope. Most of the dominated solutions do not lie far from the Pareto front, indicative of the fact that all of the objective functions are obtained by integrating the pr essure and shear distributions over the wing. Substantial variations in the CFD state variables can be obtained on a local level through the use of wing flexibility ( Figure 5-21, for example), but in tegration averages out thes e deviations. It can also be seen th at two of the three Pareto fronts in Figure 6-14 are non-convex. As such, techniques which success ively optimize a weighted sum of the two objective functions (convex combination) to fill in the Pareto front will not work; more advanced schemes, such as elitist-based evolut ionary algorithms [156], must be used. Figure 6-14. Computed design performance and Pareto optimality, = 12. Having successfully implemented the designed e xperiment, the typical ne xt step is to fit the data with a response surface, a technique used by Sloan et al. [53] and Levin and Shyy [104] for MAV work. Having verified the valid ity of the surrogate, it can then be used as a relatively inexpensive objective function for optimization. Such a method is not used here for several PAGE 114 114 reasons. First, half of the desi gn variables (wing type and laminate thickness) are discrete, which as discussed by Torres [3], can cause convergence problem s in conventional optim ization algorithms. Second, nonlinear cu rve fitting is likely required (membrane wing performance asymptotically approaches that of a rigid wi ng for increased pre-tension), and the moderate number of data points (only 9 for each wing type and laminate thickness) wont provide enough information for an adequate fit. Finally, su ch a method may result in an optimal pre-stress resultant of 5.23 N/m, for example. As discusse d above, the actual applica tion of pre-tension to a membrane MAV wing is an inexact science, an d such resolution could never be produced in the laboratory (or more importantly, the field) with any measure of repeatability or accuracy. A more practical approach is to simply treat th e pre-tension as a discrete variable: taut (~20 N/m), moderate (~10 N/m), and slack (0 N/m). Figure 6-10 and Figure 6-12 now represent an enum eration-type optimization, wherein every possi ble design is tested. Optimal wing designs in terms of 7 objective functions (maximum L/D, minimum mass, maximum lift, minimum drag, minimum pitching moment slope, maximum lif t slope and minimum lift slope) are located among the 55 available data points, and given along the diagonal of the design array in Table 61. Results from the OFAT tests above are not included. Satisfactory compromise designs are f ound by first normalizing design performance between 0 and 1, and then locating a utopia point. This utopia point is a (typically) fictional design point which would simultaneously optimize both objective functions. In the design tradeof between L/D and Cm in Figure 6-14, the utopia point is (5.49, -0.0189). An adequate com promise is the Pareto optimal design which lies closest to the utopia point; these are listed in the off-diagonal cells in Table 6-1. This method is found to give a better compromise than optim izing a convex combination of the two obj ective functions, presumably due to the non- PAGE 115 115 convexity of the Pareto fronts. Only compromi ses between 2 objective f unctions are considered in this work. The corresponding performance of each design is given in Table 6-2. The value in each cell is predicated u pon the label at the top of each column; the performance of the second objective function (row-labeled) is found in the cell appropriately located across the diagonal. Table 6-1. Optimal MAV design array with compromise designs on the off-diagonal, = 12: design description is (wing type, Nx, Ny, number of plain weave layers). max L/D min mass max CL min CD min Cm max CL min CL max L/D BR,0,0,1L BR,0,0,1L PR,10,0,1LBR,0,0,1L BR,20,0,3LPR,20,0,2L BR,0,0,1L min mass BR,0,0,1L PR,20,20,1L PR,0,0,1L BR,0,0,1L PR,0,10,1LPR,0,10,1L BR,20,10,1L max CL PR,10,0,1L PR,0,0,1L PR,0,0,1L BR,0,10,1LPR,0,0,1LPR,0,0,1L BR,20,10,1L min CD BR,0,0,1L BR,0,0,1L BR,0,10,1L BR,0,0,1L BR,20,0,3LBR,10,20,3L BR,10,0,1L min Cm BR,20,0,3L PR,0,10,1L PR,0,0,1L BR,20,0,3L PR,0,10,1L PR,0,10,1L BR,20,0,3L max CL PR,20,0,2L PR,0,10,1L PR,0,0,1L BR,10,20,3LPR,0,10,1L PR,0,10,1L BR,0,20,3L min CL BR,0,0,1L BR,20,10,1L BR,20,10,1LBR,10,0,1LBR,20,0,3LBR,0,20,3L BR, 20,10,1L Table 6-2. Optimal MAV design performance array, = 12: off-diagonal compromise design performance is predicated by column metrics, not rows. max L/D min mass (g)max CL min CD min Cm max CL min CL max L/D 5.49 4.36 0.780 0.112 -0.015 0.054 0.047 min mass 5.49 4.10 0.817 0.112 -0.019 0.057 0.043 max CL 4.84 4.18 0.817 0.145 -0.018 0.056 0.043 min CD 5.49 4.36 0.716 0.112 -0.014 0.052 0.045 min Cm 5.05 4.16 0.817 0.134 -0.019 0.057 0.049 max CL 4.90 4.16 0.817 0.141 -0.019 0.057 0.050 min CL 5.49 4.31 0.673 0.119 -0.015 0.049 0.043 For reference purposes, the design performance of the rigid wing (at 12 angle of attack) is: L/D = 4.908, mass = 6.36 grams, CL = 0.6947, CD = 0.1415, Cm = -0.0147, and CL = 0.0507. As above, at no point does the rigid wing re present an optimum design (compromise or otherwise). The compromise between minimizing th e lift slope, and maximizing the lift slope is identified by located the design closest to the normalized CL of 0.5. This is found by a BR wing design with peak pre-tension normal to the batte ns to limit adaptive washout, but no pre-tension in the chordwise direction to allow for camber a nd lift via inflation. Both BR and PR wings are equally-represented throughout the de sign array, with the excepti on of designs requiring load alleviation: all compromises involving drag or lift slope minimization utilize a BR wing. The PAGE 116 116 majority of the optimal designs use a single laye r of plain weave carbon fiber to take the most advantage of wing flexibility. A single la yer slack BR wing can minimize drag through streamlining [63], for example, as a significant portion of the wing is deform ed ( Figure 6-11). A few designs use 3 layers; onl y one design uses 2 layers. A few comprom ise wing designs coincide with the utopia point: a one-layer BR wing with a slack membrane maximizes L/D and minimizes the drag. A one-layer PR wing with no pretension in the chordwise directi on and 10 N/m in the spanwise dire ction provides the steepest lift slope and pitching moment slope. Most compro mise designs improve both objective functions, compared to the rigid wing, but the system par ticularly struggles to maximize both L/D and lift (above results indicate that efficiency improvements are driven by drag reduction), and to maximize lift and minimize the lift slope. The conflictive nature of the objective functions means that l ooking at designs that strike a reasonable compromise between three or more aer odynamic metrics is of minor usefulness. It should be noted however, that the design that lies closest to the utopia poin t of all 7 objectives shown in Table 6-1 is a 2-layer BR wing with a slack m embrane in the chordwise direction, and 10 N/m of pre-tension spanwise, similar to the design that lies closest to the normalized CL of 0.5, as discussed above. Finally, mass minimizati on is obviously afforded with a single layer of plain weave carbon fiber: membrane pre-tensi on then provides moderate and insignificant deviations from this value, by changing the amount of latex used over the MAV wing. Experimental Validation of Optimal Design Performance The design results from the single-objectiv e optimization studies (the diagonal of Table 61, with the exception of the m inimum mass design) ar e fabricated and tested in the wind tunnel. Only loads are measured through the -sweep, for comparison with the experimental data from the three baseline wings designs in Chapter 5. As discussed above, each of these wing designs PAGE 117 117 utilize a single layer of plain weave carbon fiber. Two layers are typically used for MAVs of this scale. Despite the extremely compliant natu re of the wings (which is precisely why they were located as optimal), all designs are able to withstand flight loads in the wind tunnel without buckling. Whether they can withstand maneuver loads or strong gusts is still unknown however, as is their ability to endure a flight crash without breaking. Some wing designs display subs tantial leading edge vibrati on at very low and negative angles of attack (presumably due to the vor tex shedding from the separation bubble seen in Figure 5-17), though deformation is observed to be quasi-static above 3 and prior to stall. The required pre-stress resultants are co nverted into pre-strains using Hookes law, and applied to a square of latex rubber by uniformly stretching each side. VIC is used to confirm the pre-strain levels, with spatial coefficients of variation between 10 and 20%, similar to data given in Figure 4-6 and by Stanford et al. [43]. Results f or lift-related optima are given in Figure 6-15. The design that maximizes CL (PR,0,0,1L) produces more lift than the base line PR wing up to 10 angle of attack, though within the error bars (not shown, but on the or der of 5%). Above this angle the wing shows a premature stall: CL, max is much lower than measured from the baseline PR wing, bearing closer similarities to the rigid wing. The vibrati on and buffeting typically seen over MAV wings towards stall is obviously magnified for these compliant designs; the coupling between the shedding and the wing vibration may contribute to the loss of lift, as demonstrated in the work of Lian and Shyy [8]. The upward deformation of the si ngle layer trailing edge perim eter is substantial (as seen in Figure 6-13), and the resulting adaptiv e washout may also play a role. Sim ilar results are seen for the wing design that maximizes CL (PR,0,10,1L), though in this case the lift slope is nearly identical to that measured from the baseline PR wing up to 10, after which PAGE 118 118 premature stall occurs. This benign stall behavior is not necessarily detrimental [49], though unintended by the num erical model, caused by optim izing at a single angle of attack with a steady aeroelastic solver. Figure 6-15. Experimentally measured design optimality over baseline lift. The optimizer is considerably more successful when minimizing CL with design (BR,20,10,1L), as seen in Figure 6-15. The BR wings used in these tests are qualitatively observed to have sm aller vibrati on amplitudes, compared to the PR wings, at very low and very high angles of attack. At low a ngles of attack, the lift of the optimal design is smaller than both the baseline rigid and BR wings, though the lift slope is comparable. For moderate angles, no significant differences are evident. After 10 however, the optimal design shows a clear drop in lift slope, a very flat stalling region, and stalling angle delayed by 3 over the baseline designs. CL, max is measured to be 9% less than th at measured for the baseline BR wing. Experimental validation results for the wing de sign minimizing the pitching moment slope (PR,0,10,1L) is given in Figure 6-16. As before, performance of the baseline PR wing and the optim al design are comparable up to 13. Above this angle, and through the stalling region, the optimal design has a steeper slope than the base line PR wing. At these angles, the nose-down pitching moment is stronger than that seen in th e baseline BR and rigid wings, but the slope is PAGE 119 119 similar. This is largely due to the linear pitc hing moment behavior prev iously noted on the PR wings, possibly due to membrane inflati on interference with the tip vortices [14]. Despite the m easured improvements over the baseline PR wing, the data indicates that longitudinal control beyond stall (~ 28) may not be possible [27]. Interestingly, the sa m e wing design theoretically minimizes the moment slope and maximizes the lift slope, but only the former metric is considerably improved over the baseline. Figure 6-16. Experimentally measured design optimality over baseline pitching moments. Similar validation results are given in Figure 6-17 and Figure 6-18, for the minimization of drag and m aximization of L/D. Both metrics are optimized by wing design (BR,0,0,1L). The drag is consistently lower than the three baseline designs up to 20. Accurate drag data for micro air vehicles at low speeds is very difficult to measure, largely due to resolution issues in the sting balance [34]. Questionable data typically manife sts itself through atypically low drag. Regardless, the ver acity of the data from the optimal wing in Figure 6-17 may be confirmed by the iden tical results at the bottom of the drag bu cket with the rigid wing, where deformation is very small. The data also compares very well with computed results. Unlike the baseline BR wing, the optimal design has less drag at a given a ngle of attack and at a given value of lift (the latter of which is visible in the drag polar, wh ich is not shown). Past 20, the optimal design PAGE 120 120 shows more of a drag penalty than the baseline BR wing, which may also be attributed to larger vibration amplitudes in the single-layer wing. Figure 6-17. Experimentally measured design optimality over baseline drag. Figure 6-18. Experimentally measured design optimality over baseline efficiency. The results for optimal efficiency ( Figure 6-18) show substan tial im provements over the three baseline designs for a range of moderate angles: 8 18 The optimization is only conducted at 12 angle of attack; whereas the pr eviously considered optimal designs can be reasonably considered ideal throughout most of the -sweep (up to stall), the conflictive nature of the lift-to-drag ratio is more complex. This can be seen in the numerical data of Figure 5-15, where the baselin e BR, PR, and rigid wings all have the highest L/D for different lift values. It is PAGE 121 121 expected that optimizing at different angles of attack will produce radically different optimal L/D designs, but similar results may be reta ined for the remaining objectives. Of the six aerodynamic objectives considered in this section, wind tunnel testing indicates that two are unmistakably superior to the base lines over a large range of angles of attack (minimum drag and maximum efficiency), and two have similar responses to one or more of the baseline designs for small and moderate angles but are clearly superior for higher angles of attack (minimum lift and pitching moment slopes) One objective (maximum lift) is slightly better at moderate angles (though not beyond the m easured uncertainty), but decidedly inferior during stall, while another objec tive (maximum lift slope) is identical to the baseline for moderate angles, and again inferior during stall. With the exception of these latter two studies, this wind tunnel validation confirms the use of numerical aeroelastic tailoring for realizable improvements to actual MAV wings. This is not to indicate that the latter two st udies have failed: the computed pe rformance of the tailored wings is not always significantly better than the baseline, and may be blurred by experimental errors. The experimental data of these two designs is not significantly better th an the baseline designs, but not measurably worse eith er (for moderate angles). PAGE 122 122 CHAPTER 7 AEROELASTIC TOPOLOGY OPTIMIZATION The conceptual design of a wing skeleton esse ntially represents an aeroelastic topology optim ization problem. Conventional topology optimization is typically concerned with locating the holes within a loaded homogenous structure, by minimizing the compliance [16]. This work details th e location of holes within a carbon fibe r wing shell, holes which will then be covered with a thin, taut, rubber membrane skin. In other words, the wing will be di scretized into a series of panels, wherein each panel can be a carbon fiber laminated shell or an extensible latex rubber skin. Rather than compliance, a series of aerodynamic objective functions can be considered, including L/D, CL, CD, CL Cm etc. While the two wing topologies discussed in th e preceding section (battenand perimeterreinforced wings) have been shown to be effectiv e at load alleviation via streamlining and load augmentation via cambering, respectively, both designs have defici encies. The BR wing experiences membrane inflation from in-betwe en the battens towards the leading edge ( Figure 51), cam bering the wing and contradicting the load a lleviating effects of th e adaptive washout at the trailing edge. Furthermore, the unconstrained trailing edge is only moderately effective at adaptive geometric twist, as the forces in this region are very small ( Figure 5-20). If re-curve is built into the wing section, the forces in this area may push the trailing edge downward, actually increasing the incidence, and thus the loads. The PR wing, being a simpler design, is more effective in its intended purpose (adaptive cambering for increased lift and static stability), but the drastic changes in wing geometry at the carbon fiber/membrane interfaces towards the lead ing and trailing edges of the membrane skin are aerodynamically inefficient. Large membrane in flations are also seen to lead to potentially unacceptable drag penalties as well. All of these deficiencies can be remedied via the tailoring PAGE 123 123 studies considered above, but the greater generality of an aeroelastic topology scheme (due to the larger number of variables) would suggest better potential improve ments in aerodynamic performance. Furthermore, such an undertaki ng can potentially be followed by an aeroelastic tailoring study of the optimal topology for furthe r improvements, as discussed by Krog et al. [115]: topology optimization to lo cate a good design, followed by sizing and shape optim ization. A flexible MAV wing topological optimizati on procedure has some precedence in early micro air vehicle work by Ifju et al. [10], with an array of successfully flight tested designs shown in Figure 7-1. Each of these designs consists of a laminated leading edge, wing tip, and wing root; a series of thin stri ps of carbon fiber are im bedded within the concomitant membrane skin. Both the BR and PR wings are present, al ong with slight variati ons upon those themes. Ifju et al. [10] qualitatively ranks these wing structures b ased upon observations in the field and pilot-reported handling qualities: a crude trial and error process led to the batten-reinforced design as a viable candi date for MAV flight. Figure 7-1. Wing topologies flight tested by Ifju et al. [10]. Several challenges are as sociated with the optim ization procedure considered here. First, a PAGE 124 124 fairly fine structural grid is needed to re solve topologies on the order of those seen in Figure 7-1. The f ine grid will, of course, increase the computational cost associated with solving the set of FEA equations, as the number of variables in the optimization algorithm is proportional to the number of finite elements. The wing is discretized into a set of quadrilaterals, which represent the density variables: 0 or 1. These quadrilate rals are used as panels for the aerodynamic solver, and broken into two triangles for the finite element solver, as shown in Figure 7-2. As in Figure 7-1, the wing topology at the root, leading edge, and wing tip is fixed as carbon fiber, to m aintain some semblance of an aerodynamic shape capable of sustaining lift. The wing topology in the figure is randomly distributed. Figure 7-2. Sample wing topol ogy (left), aerodynamic mesh ( center), and structural mesh (right). Further complications are associat ed with the fact th at these variables are binary integers: 1 if the element is a carbon fiber ply, 0 if th e element is latex membrane. Several binary optimization techniques (genetic algorithms [109], for example) are impractical for the current problem due to the large number of variables, but also due to th e extremely large computational cost associated with each aeroelastic function evaluation. A fairly standard technique for topology optimization problems classifies the density of each element as continuous, rather than binary [16]. Intermediate densities can then be penalized ( implicitly or otherwise) to push the design towards a pure carbon fiber/membrane distribution, with no porous material. PAGE 125 125 The sensitivity of each elements dens ity variable upon the wings aerodynamic performance is required for this gradient-bas ed optimization scheme. As before, the large number of variables and the e xpensive function evaluations precl ude the use of simple finite difference schemes for computation of gradients. An adjoint sensitivity analysis of the coupled aeroelastic system is thus required, as the numbe r of design variables is much larger than the number of objectives/constraints [110]. Further complications ar ise from the fact that second derivatives are also required: im portant MAV aerodynamic performance metrics such as the slope of the lift curve, for example, are sensitivity derivatives that depend upon the characteristics of the aeroelastic system as well. This chapter provides a computational framew ork for computing the adjoint aeroelastic sensitivities of a coupl ed aeroelastic system, as well as interpolation schemes between carbon fiber and membrane finite elements and methods for penalizing intermed iate densities. The dependency of the computed optimal topology up on mesh density, angle of attack, initial topology, and objective function are given, as we ll as the resulting deformation and pressure distributions. The wing designs created via aeroelastic topo logy optimization demonstrate a clear superiority over the baseline BR and PR designs discussed above in terms of load alleviation (former) and augmentation (latter), advantages which are further expounded through wind tunnel testing. Multi-ob jective topology optimization is discussed as well, with the evolution of the optimal wing topology as one travels along the Pareto optimal front. Computational Framework Material Interpolation Topology optimization often minimizes the complia nce of a structure under static loads, with an equality constraint upon th e volume. If the density of each element is allowed to vary continuously, an implicit penalty upon intermediate densities (to push the final structure to a 0-1 PAGE 126 126 material distribution) can be achieved through a nonlinear power law in terpolation. This technique is known as the solid isotropic material with pe nalization method, or SIMP [105]. For the two-m aterial wing considered above (membrane or carbon fiber), the stiffness matrix Ke of each finite element in Figure 7-2 can be computed as: p epmemp1X KKKKK (7-1) where Kp and Km are the plate and membrane elements, re spectively (the latter with zeros placed within rows and columns corresponding to bending degrees of freedom). is a small number used to prevent singularity in the pure membrane element (due to the bending degrees of freedom), and Xe is the density of the element, varying from 0 (membrane) to 1 (carbon fiber). p is the nonlinear penaliz ation power (typically greater than 3). A common criticism of this power law approach is that intermediate densities do not actually exist. This is a particular problem fo r the current application, where each element is either carbon fiber or membrane rubber. The physics of these two elements is completely different, as the carbon fiber is inextensible yet has resistance to bending and twisting, while the opposite is true for the latex. An equal combination of these two (equivale nt to stating that the density within an element is 0.5), while computat ionally conceivable, is not physically possible. The wing topology will not represent a real structure until the density of each element is pushed to 1 (carbon fiber) or 0 (membrane). The power laws effectiveness as an im plicit penalty is predicated upon a volume constraint: intermediate densities are unfavorable, as their stiffness is small compared to their volume [16]. No such volume constraint is utilized here, due to an uncertainty upon what this value should be. Furthermore, for aeronautical a pplica tions it is typically desired to minimize the mass of the wing itself, as discussed by Maute et al. [118]. Regardless, the nonlinear power PAGE 127 127 law of SIMP is still useful for the current application, as demonstrated in Figure 7-3. Both linear and nonlinear m aterial interpolations are given for the lift computation, and the wing topology is altered uniformly. For the linear interpolation (i.e., without SIMP ), the aeroelastic response is a weak function of the density until X becomes very small (~0.001), when the system experiences a very sharp change as X is further decreased to 0. This is a result of the large st iffness imbalance between the carbon fiber laminates and the membrane skin, a nd the fact that lift is a direct function of the wings compliance (the inverse of the weighted sum of the two disparate stiffness matrices in Eq. (7-1)). The inclusion of a non linear penalization power (p = 5) spreads the response evenly between 0 and 1. Aeroelastic topology optimi zation with linear material interpolation experiences convergence difficulties, as the gradie nt-based technique stru ggles with the nearlydisjointed design space; a penalization power of 5 is utilized for the remainder of this work. Figure 7-3. Effect of linear and nonli near material interpolation upon lift. The results from Figure 7-3 suggest a number of othe r potential difficulties with an aeroelastic topology optim ization scheme. First, th e sensitivity of the aer oelastic response to element density is zero for a pure membrane wing (X = 0), as can be inferred from Eq. (7-1). As such, using a pure membrane wing as an initial guess for optimization will not work, as the PAGE 128 128 design wont change. Secondly, two local optima exist in the design space of Figure 7-3, which m ay prevent the gradient-based optimizer from c onverging to a 0-1 materi al distribution. To counteract this problem, an expl icit penalty on intermediate dens ities is added to the objective function, as discussed by Chen and Wu [158]: XN i i1RsinX (7-2) where R is a penalty parameter appropriately sized so as not to overwhelm the aerodynamic performance of the wing topology. This penalty is only added when and if the aeroelastic optimizer has converged upon a design with intermed iate densities, as will be discussed below. Aeroelastic Solver Due to the large number of expected functi on evaluations (~ 200) needed to converge upon an optimal wing topology, and the required aeroela stic sensitivities (computed with an adjoint method), a lower-fidelity aeroelastic model (compared to that utili zed in Chapters 5 and 6) must be used for the current application. An inviscid vortex lattice method (Eq. (4-11)) is coupled to a linear orthotropic plate model a nd a linear stress stiffening membrane model (Eq. (4-4)). The latter module is perfectly valid in predicting membra ne inflation as long as the state of pre-stress is sufficiently large, as seen in Figure 4-5. Furthermore, in-plane s tretching of the laminate is ignored; only out-of-plane displacements (as well as in-plane rotations in the laminate) are computed over the entire wing. The vortex lattice method is reasonably accu rate as well, despite the overwhelming presence of viscous effects within the flow. As seen in Figure 4-11, the lift sl ope is co nsistently under-predicted due to an inability to model the large tip vortices [3], and the drag is underpredicted at low and high angles of attack due to separation of th e lam inar boundary layer [4]. Aeroelas tic coupling is facilitated by consider ing the system as defined by a three field PAGE 129 129 response vector r: T TTTruz (7-3) where u is the solution to the system of finite element equations (composed of both displacements and rotati ons) at each free node, z is the shape of the flexible wing, and is the vector of unknown horseshoe vortex circulat ions. The coupled system of equations G(r) is then: o() KuQ GrzzPu0 CL (7-4) The first row of G is the finite element analysis: K is the stiffness matrix assembled from the elemental matrices in Eq. (7-1), and appropriately reduced based upon fixed boundary conditions along the wing root. Q is an interpolation matrix that converts the circulation of each horseshoe vortex into a pressure, and subsequen tly into the transverse force at each free node. The second row of G is a simple grid regeneration analysis: zo is the original (rigid) wing shape, and P is a second interpolation matrix that conv erts the finite element state vector into displacements at each free and fixed node along the wing. The third row of G is the vortex lattice method. C is an influence matrix depending solely on the wing geometry (computed through the combination of Eqs. (4-11) and (4-12)), and L is a source vector depending on the wings outward normal vectors, the angle of att ack, and the free stream velocity. Convergence of this system can typically be obtained within 25 iterations, and is defi ned when the logarithmic error in the wings lift coefficient is less than -5. One potential shortcoming of this aeroelastic model can be seen in Figure 7-3, where the com puted lift of a wing with no carbon fiber in the design domain (X = 0) is larger than the lift generated by the rigid wing (X = 1). This is due to a combination of membrane cambering towards the leading edge, and a depr ession of the trailing e dge reflex region. In reality, however, PAGE 130 130 the combination of a poorly-constrained trailing edge and unsteady vortex shedding will lead to a large-amplitude flapping vibration, similar to that discussed by Argentina and Mahadevan [62]. W ind tunnel testing of this wing is given in Figure 7-4 at 13 m/s; the cr itical speed of flapping vibration is approxim ately 3 m/s. Figure 7-4. Measured loads of an inad equately reinforced membrane wing, U = 13 m/s. As expected, the measured lift of the membrane wing is significantly less than that measured from the rigid wing in the wind tunnel: the poorly-supported wi ng cannot sustain the flight loads, while the large amplitude vibrations levy a substantial drag penalty. Even a mild amount of trailing edge reinforcement (suc h as that seen in the upper left of Figure 7-1) will prevent this behavior, bu t formulating a constr aint that will push the aeroelastic topology optimizer away from wing designs with a poorly-rein forced trailing edge is difficult, and is not included. This section only serves to highlight one significant shortcoming of the aeroelastic model used here, and to diminish the perc eived optimality of certain wing topologies. Adjoint Sensitivity Analysis As the number of variables in the aeroelas tic system (essentially the density of each element) will always outnumber the number of constraints and objective functions, a sensitivity PAGE 131 131 analysis can be most effectivel y carried out with an adjoint an alysis. The sought-after total derivative of the objective function with respect to each density variable is given through the chain rule: Tdgggd dd r XXrX (7-5) where g is the objective function (a scalar for the single-objective optimization scheme considered here; multi-objective optimization will be discussed below) and r is the aeroelastic state vector discussed above. The term g/ X is the explicit portion of the derivative, while the latter term is the implicit portion th rough dependence on the aeroelastic system [159]. Only aerodynam ic objective functions are considered in th is work: the explicit portion is then zero, unless the intermediate density penalty of Eq. (7-2) is included. The derivative of the aeroelastic state vector with respect to the element densities is found by differentiating the coupled system of Eq. (7-4): d(,) d dd GXrGr 0A0 XXX (7-6) where A is the Jacobian of the aer oelastic system, defined by: G A= r (7-7) Combining Eqs. 7-5 and 7-6 leaves: T -1dggg d G A XXrX (7-8) Using the adjoint, rather than the direct method to solve Eq. (7-8), the adjoint vector is: -Tg aA r (7-9) The system of equations for the adjoint vector does not contain the density of each element PAGE 132 132 (X), and only needs to be solved once. For th e aeroelastic system considered above, the terms that make up the adjoint vector are: dd K0-Q A-PI0 0CzLzC (7-10) T Tg 00S r (7-11) where S is the derivative of the aerodynamic objectiv e function with respect to the vector of horseshoe vortex circulations. For metr ics such as lift and pitching moment, g = ST though more complex expressions exist for drag. Th e sensitivities can then be computed as: Tdggd dd G a XXX (7-12) Only the finite element analysis of the aeroelas tic system contains the element densities, and so this final term can be computed as: d d K u X G 0 X 0 (7-13) Of all of the terms needed to undertake the ad joint sensitivity analysis, only the derivative of the vortex lattice influence matrix C with respect to the wing shape z (a three-dimensional tensor) is computationally intensive, and represents the majority of the cost associated with the gradient calculations at each iteration. In order to solve the linear system of Eq. (7-9), a staggered approach is adapted, rather than solv ing the entire system of (un-symmetric sparse) equations as a whole, as discussed by Maute et al. [110]. Each sub-problem is solved with the sam e algorithm used in the aeroelastic solver (direct sparse solver for the finite element PAGE 133 133 equations, and an iterative Gauss-Seidel solver for the vortex lattice equations), and as such, the computational cost and number of iterations needed for convergence is approximately equal between the aeroelastic solver (Eq. (7-4)) and the adjoint vector solver (Eq. (7-9)). The second derivative of the objective func tion is required if aerodynamic derivative metrics such as CL and Cm are of interest. Two options are available for this computation. The first involves a similar analyti cal approach to the one described above. This would eventually necessitate the extremely difficult computation of A / r, which is seldom done in practice [160]. Finite d ifferences are used here: 2g1gg (( XXX (7-14) The term g/ can be computed using another finite difference, or with the adjoint method described above, substituting the angle of attack for the element densities X. Optimization Procedure In order to ensure the existence of the op tim al wing topologies, a mesh-independent filter is employed along with the nonlinear power penaliza tion. Such a filter acts as a moving average of the gradients throughout the membrane wing, and limits the minimum size of the imbedded carbon fiber structures. Such a tactic shoul d also limit checkerboard patterns (carbon fiber elements connected just at a corner node). The moving average filter modifies the element sensitivity of node i based on the surrounding sensitiv ities within a circular region of radius rmin, as discussed by Bendse and Sigmund [16]: X XN min min i,jj i,j N j1 ij new ii,j j1rdist(i,j)ifdist(i,j)r dg1 dg HXH 0otherwise dX dX XH (7-15) As no constraints are included in the optimi zation (preferring instead the multi-objective PAGE 134 134 approach described below), an unconstrained Fl etcher-Reeves conjugate gradient algorithm [159] is employed. Step size is ke pt constant, at a reasonably sm a ll value to preserve the fidelity of the sensitivity analysis. The upper and lowe r bounds of each design variable (1 and 0) are preserved by restricting the step size such that no density variable can leave the design space, forced to lie on the border instead. In order to increase the chances of locating a global optimum (rather than a local optimum), each optimization is run with thr ee distinct initial designs: Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1. A pure membrane wing (Xo = 0) cannot be considered for the reasons discussed above. Six objective functions are considered: ma ximum lift, minimum drag, maximum L/D, maximum CL minimum CL and minimum Cm Flight speed is kept constant at 13 m/s, but both 3 and 12 angles of a ttack are considered, with a of 1 for finite differences. Both the reflex airfoil seen in Figure 5-16 and a singly-curved airf oil are used, though aspect ratio, planform and peak camber are unchanged. The stiffness of the carbon fiber laminates is as computed by Figure 4-3, and the pre-stress of the membra ne is fixed in both the chordwise and spanwise direction s at 7 N/m. No correction is applied to the free trailing edge, as such a computation would render the pre-stress in this location very small, leading to unbounded behavior of the linear membrane model. The circular radius rmin for the mesh-independent filter is fixed at 4% of the chord. Single-Objective Optimization A typical convergence history of the aeroelastic topology optimizer can be seen in Figure 7-5, for a reflex wing at 3 ang le of attack, with a maximum L/D objective function. The initial guess is an intermediate density of 0.5. Within 4 iterations, the optimizer has removed all of the carbon fiber adjacent to the root of the wing, w ith the exception of the region located at threequarters of the chord, which corresponds to the infl ection point of the reflex airfoil. The material PAGE 135 135 towards the leading edge and at the wing tip is also removed. Further ite rations see topological changes characterized by intersec ting threads of membrane material that grow across the surface, leaving behind islands of carbon fiber. These stru ctures arent connected to the laminate wing, but are imbedded within the membrane skin. Figure 7-5. Convergence hi story for maximizing L/D, = 3, reflex wing. These results indicate two fundamental differences between the designs in Figure 7-1 and those com puted via aeroelastic t opology optimization. The first is the presence of islands; these designs can be built, but the process is significantly more complicated than with a monolithic wing skeleton. Such structures could be avoided with a manufacturability constraint/objective function (suc h as discussed by Lyu and Saitu [161]), but the l ogistics of such a m etric (as above, with the trai ling edge reinforcement constraint) are difficult to formulate. Furthermore, the aeroelastic advantages of freefloating laminate structures are significant, as will be discussed below. A second difference is the fact that the designs of Figure 7-1 are com posed entirely from thin strips of carbon fiber embedded within the membrane, while the PAGE 136 136 topology optimization is apt to utilize tw o-dimensional laminate structures. After 112 iterations in Figure 7-5, the optimization has largely converged (with only m inimal further improvements in L/D), but some material with intermediate densities remains towards the leading edge of the wing. Many t echniques exist for effec tively interpreting gray level topologies [162]; the explicit penalty of Eq. (7-2) is used here. Su rprisingly, the L/D sees a f urther increase with the additi on of this penalty, contrary to the conflict between performance and 0-1 convergence reported by Chen and Wu [158]. The explicit penalty does not significantly alter the topology, but merely forces all of the design variables to their lim its, as intended. The final wing skeleton has thre e trailing edge battens (one of which is connected to a triangular structure towards the center of the membra ne skin), and a fourth batten oriented at 45 to the flow direction. The structure s hows some similarities to a wing design in Figure 7-1 (third row, first colum n), and appears to be a topolog ical combination of a BR and a PR wing, with both battens and membrane inflation towards the leading edge. The optimized topology increases the L/D by 9.5% over the initial design and (perhaps more relevant, as the initial intermediate density design does not techni cally exist) by 10.2% over the rigid wing. The affect of mesh density is given in Figure 7-6, for a reflex wi ng at 12 angle of attack, with L/D maximization as the objective function. The 30x30 grid, for example, indicates that 900 vortex panels (and 1800 finite elements) cover each semi-wing. As the leading edge, root, and wing tip of each wing are fixed as carbon fiber, 480 density design variables are left for the topology optimization. One obvious sign of adequate convergence is the efficiency of the rigid wing, with only a 0.44% difference between that computed on the two finer grids. The three optimal wing topologies are similar, with three distinct carbon fi ber structures imbedded within the membrane skin: two extend to the trailing ed ge and the third resides towards the leading PAGE 137 137 edge. While the 20x20 grid is certainly too coar se to adequately resolve the geometries of interest, the topology computed on the 30x30 grid is very similar to that computed on the 40x40 grid. The computational cost of each optimization iteration upon th e coarser grid is 5 times less than that seen for a 40x40 grid, and will be used for the remainder of this work. Figure 7-6. Affect of mesh de nsity upon optimal L/D topology, = 12, reflex wing. The affect of the initial starting design is given in Figure 7-7, for a refl ex wing at 12 angle of attack, with drag minimization as the objective function. As mentioned above, Xo = 1 (carbon fiber wing), Xo = 0.5, and Xo = 0.1 are all considered. The three final optimal topologies are very different, indicating a larg e dependency upon the initial guess and no guarantee that a global optimum has been located. Nevertheless, the indicated improvements in drag are promising, with a potential 6.7% decr ease from the rigid wing. As expect ed, the denser the initial topology, the denser the final optimized topology. All three wing topologies utilize some form of adaptive washout for load and drag alleviation. The structures must be flexible enough to generate sufficient nose-down rotation of each wing section, but not so flexible that the membrane areas of the wing will inflate and camber, increasing the forces. The wing structure in the center of Figure 7-7 (with Xo = 0.5) strikes the best compromise between the two defo rmations, and provides th e lowest drag. When Xo = 1, the structure is too stiff, relying upon a membrane hinge between the carbon fiber wing PAGE 138 138 and root. When Xo = 0.1, the optimizer is unable to fill in enough space with laminates to prevent membrane inflation. Of the three designs, this is the least tractable from a manufacturing point of view as well. Figure 7-7. Affect of ini tial design upon the optimal CD topology, = 12, reflex wing. The dependency of the optimal topology (max imum lift) upon both angle of attack and airfoil shape are given in Figure 7-8, for both a reflex (left tw o plots) and a cambered wing (right two plots). For the wing with tr ailing edge reflex, the optimal li ft design looks similar to that found in Figure 7-5: trailing edge batt ens that extend no farther up th e wing than the half-chord, a spanwise m ember that coincides with the in flection point of the airfoil, and unconstrained membrane skin towards the leading edge, where the forces are largest. The optimizer has realized that it can maximize lift by both cambering the wing through inflation at the leading edge, and forcing the trailing edge battens downward for wash-in. This latter deformation is only possible due to the reflex (negative camber) in this area, included to offset the nose-down pitching moment of the remainder of the flying wing, and thus allow for removal of a horizontal stabilizer due to size restrictions. Increasing the angle of attack from 3 to 12 shows no significant differe nce in the wing topology, slightly increasing the length of the largest batten. At the lower angle of attack, up to 22% increase in lift is indicated through topology optimization. PAGE 139 139 Figure 7-8. Affect of angle of attack and airfoi l upon the optimal CL topology. For the cambered wing (singly-curved airfoil, right two plots of Figure 7-8), the lift over the rigid wing is, as expected, m uch larger than found in the reflex wings but adequate stability becomes critical. With the removal of the negati vely-cambered portion of the airfoil, most of the forces generated over this wing will be positive, and the topology optimizer can no longer gain additional lift via wash-in. Imbedding batten stru ctures in the trailing edge will now result in washout, surely decreasing the lift. As such, th e optimizer produces a trailing edge member that outlines the planform and connects to the root (sim ilar to the perimeter-reinforced wing designs), restraining the motion of the trailing edge and inducing an aerodynamic twist. Unlike the PR wing, this trailing edge rein forcement does not extend continuously from the root to the tip, instead ending at 65% of the semi-span. This is then followed by a trailing edge batten that extends into the membrane skin similar to the designs seen for the reflex wing in Figure 7-8. Why such a confi guration should be preferred ove r the PR wing design for lift enhancement will be discussed below. As before, increasing the angle of attack has little bearing on the optimal topology, again in creasing the size of the trailing edge batten. A potential increase in lift by 15% over the rigid wing is indicated at the lower angle of attack. Similar results are given in Figure 7-9, with L/D maximi zation as the topology design m etric. Presumably due to the conflictive nature of the ratio, the wing topology that maximizes L/D is a strong function of angle of attack. For the reflex wing at lower angles, the optimal PAGE 140 140 design resembles topologies used above for lift enhancement ( Figure 7-8), while at 12 the design is closer in topology to one with m inimum drag ( Figure 7-7). Increas ing lif t is more important to L/D at lower angles, while decreasing drag becomes key at larger angles. The drag is very small at low angles of attack (technically zero for this inviscid form ulation, if not for the inclusion of a constant CDo), and insensitive to changes via aeroelasticity. This concept is less true for th e cambered wing (right two plots of Figure 7-9), where designs at both 3 and 12 angle of attack utiliz e a structure with trailing edge adaptive washout. At the lower angle, the topology optimizer leaves a large triangular structur e at the trailing edge (connected to neither the root nor the wing tip), and the leadi ng edge is filled in with carbon fiber. At the higher angle of attack, four batt en-like structures are placed within the membrane skin, oriented parallel to the flow, one of which connect s to the wing tip. Potential improvements are generally smaller than those seen above, though a 10% increase in L/D is available for the cambered wing at 12. Figure 7-9. Affect of angle of attack and airfoil upon the optimal L/D topology. Wing displacements and pressure distributions are given for select wing designs in Figure 7-10, for a reflex wing at 12 angle of attack. Corresponding data along the spanwise section 2y/b = 0.58 is given in Figure 7-11. As the wing is mode led with no thic kness in the vortex lattice method, distinct upper and lower pressure distributions are not available, only differential terms. Five topologies are discussed, be ginning with a pure carbon fiber wing. Lift- PAGE 141 141 augmentation designs are represented by a base line PR wing and the topology optimized for maximum lift. Lift-alleviation designs are re presented by a baseline BR wing and the topology optimized for minimum lift slope. Figure 7-10. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for baseline and optimal topology designs, = 12, reflex wing. The differential pressure distri bution over the rigid wing is larg ely similar to that computed with the Navier-Stokes solver in Figure 5-18 and Figure 5-19: leading edge suction due to flow stagnation, pressure recovery (and p eak lift) over the camber, and negative forces over the reflex portion of the wing. As expected, the inviscid so lver misses the low-pressure cells at the wingtip (from the vortex swirling system [3]), and the plateau in the pressu re d istribution, indicative of a separation bubble [27]. This aerodynamic loading causes a moderate w ash-in of the carbon fiber wing (0.1), resulting in a comput ed lift coefficient of 0.604. Computed deformation of the PR wing is likewise similar to that found above (Figure 5-5), though the defor mations are smaller, within the range of validity of the linear finite element solver. The sudden changes in wing geometry at the membrane/carbon fiber interfaces lead to PAGE 142 142 sharp downward forces at the lead ing and trailing edges, the latter of which exacerbates the effect of the airfoil reflex. Despite this, the membrane inflation increases the camber of the wing and thus the lift, by 6.5% over the rigid wing. Figure 7-11. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology designs, = 12, reflex wing. As discussed above, several disparate deforma tion mechanisms contribute to the high lift of the MAV design located by the aeroelastic topology optimizer (middle column, Figure 7-10). First, the m embrane inflation towards the leading edge increases lift via cambering, similar to the PR wing (the pressure distributions over the two wing structures are identical through x/c = 0.25). The main trailing edge batten structur e is then depressed downward along the trailing edge (due to the reflex) for wash-in, while the forward portion of th is structure is pushed upwards. This structure essentially swivels about the inflection point of the wings airfoil, a deformation which is able to further increase the size of the membrane cambering, and is only possible because the laminate is free-floating within the membrane skin. It can also be seen (from the left side of Figure 7-11 in particular) that the local bending/twisting of this batten PAGE 143 143 structure is minimal: the deformation along this structure is largely linear down the wing. The intersection of this linear trend with the curved inflated membrane shape produces a cusp in the airfoil. The small radius of cu rvature forces very large velociti es, resulting in the lift spike at 46% of the chord. This combination of wash-in and cambering leads to a design which out-performs the lift of the PR wing by 5.6%, but the former effect is troubling. The wash-in essentially removes the reflex from the airfoil (as does the aerodynamic tw ist of the PR wing), an attribute originally added to mitigate the nose-down pitching moment. This fact leads to two important ideas. First, thorough optimization of a single design metric is ill-advised for micro air vehicle design, as other aspects of the flight performance will surely degrade. Its inclusion here is only meant to emphasize the relationship between aeroelastic de formation and flight performance, and show the capabilities of the topology optimization. A better appro ach is the multi-objective scheme discussed below. Secondly, if the design goal is a single-minded maximization of lift, a reflex airfoil is a poor choice compared to a singly-curved airfoil, a shape which the topology optimizer strives to emulate through aeroelastic deformations. Furthe rmore, if the design metric is an aerodynamic force or moment, passive shape adaptations need not be used at all: simply compute the optimal wing shape from the bottom row of Figure 7-10, and build a similar rigid wing. Mass restrictions prevents such a st rategy in traditional aircraft design (though a sim ilar idea can be seen in the jig-shape approach [163], where wing shape is optim ized, followed by identification of the internal structur e which allows for deformation into this shape), but two layers of carbon fiber can adequately hold the inte nded shape without a stringent we ight penalty. However, if the design metric is an aerodynamic derivative (gust rejection or longitudinal static stability, for PAGE 144 144 example), membrane structures must be us ed, as these metrics depend on passive shape adaptation with sudden changes in freestream, a ngle of attack, or control surface deflection. Referring now to the load-allev iating MAV wing structures of Figure 7-10 and Figure 711, the deform ation of the BR wing is relatively small, allowing for just 0.1 of adaptive washout. As discussed above, the BR wing is very sensitive to pre-tensions in the span direction ( Figure 6-1); the structure is too stiff. Less than a 2% drop in lift from th e rigid wing is obtained, and the pressure distributi ons of the two wings in Figure 7-11 are very similar. What load alleviation the BR wing does provide seem s to be due to the membrane inflation from between the leading edge of the battens, and the conc omitant flow deceleration over the tangent discontinuity, rather than the adaptive washout at the tr ailing edge. The load alleviating design located by the topology optimizer (right column, Figure 7-10) is sign ificantly more successful. By filling th e design space with patches of disconnected carbon fiber structures (dominated by a long batten whic h extends the length of the membrane skin, but is not connected to the wings laminate leading edge), the MAV wing is very flexible, but none of the membrane portions of the wing are large enough to camber the wing via inflation. Wing deformation is the same magnitude as that seen in the PR-type wings, but the motion is located at the trailing edge for adaptive was hout, and lift is decr eased by 5%. The local deformation within the membrane between the leading edge and the long batten structure is subs tantial, and the flow deceleration over this point sees a furthe r loss in lift, as with the BR wing. Similar results are given in Figure 7-12 and Figure 7-13, for a cambered wing at 12 angle of attack. T he three baseline wings are again sh own (carbon fiber wing, PR, and BR), as well as the designs located by the topology optimization to maximize lift and minimize lift slope. As the forces are generally larger for the cambered airfo il, the deformations have increased to 5% of the PAGE 145 145 root chord. The negative forces at the trailing edge of the airfoil are likewise absent. As before, the PR membrane wing effectively increases th e lift over its carbon fi ber counterpart through adaptive cambering, along with aerodynamic penalties from the shape discontinuities at the leading and trailing edge of the membrane skin. Figure 7-12. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for baseline and optimal topology designs, = 12, cambered wing. There is an appreciable amount of upward deformation of the PR wings trailing edge carbon fiber strip, leading to washout of each flexib le wing section, degrading the lift. As such, the aeroelastic topology op timizer can maximize lift ( Figure 7-12, middle column) by adding more m aterial to this strip and negating the moti on of the trailing edge. As discussed above, this strip does not continue unbroke n to the wing tip, but ends at 65% of the semispan. The remaining membrane trailing edge is filled with a free-floating carbon fi ber batten. Such a configuration can (theoretically) improve the lift in several ways, similar to the trailing edge structure used for lift optimization in Figure 7-10. Placing a flexible m embrane skin between tw o rigid supports produces a trade-off: the PAGE 146 146 cambering via inflation increases lift, but this metr ic is degraded by the sh arp discontinuities in the airfoil shape. Towards the inner portion of the MAV wing, this trade-off is favorable for lift. Towards the wingtip however (either due to the cha nges in chord or in pressure) this is no longer true, and the topology optimizer has realized that overall lift ca n be increased by allowing this portion of the trailing edge to washout, there by avoiding the negative pr essures seen elsewhere along the trailing edge. Figure 7-13. Deformations and pressures along 2y/b = 0.58 for baseline and optimal topology designs, = 12, cambered wing. The forward portion of this batten structure also produces a cusp in the wing geometry, forcing a very strong low pressure spike over the upper portion of the airfoil, further increasing the lift, as before. Due to the inviscid formul ation, further grid resolution around this cusp will cause the spike to grow larger, as the velocity around the small ra dius approaches infinity. The presence of viscosity will attenuate the speed of the flow, and thus both the magnitude of the low pressure spike and its beneficial effect upon lift. The aeroelastic topology optimizer predicts a 3.5% increase in lift over the PR wing, a nd 12.5% increase over the rigid wing, though the PAGE 147 147 veracity of the former beneficial comparison requires a viscous flow solver to ascertain the actual height of the low-pressure spike at x/c = 0.68. The batten-reinforced design of Figure 7-12 is substantially m ore effective with the cambered wing, than with the reflex wing. As discussed above, reflex in the wing pushes the trailing edge down, limiting the ability of the batt ens to washout for load reduction. This can also be seen by comparing the airfoil shapes between Figure 7-13 and Figure 7-11: the cambered wing shows a continuous increase in the deform ati on from leading to trailing edge, while most of the deformation in the reflex wing is at the flex ible membrane/carbon fiber interface. Aft of this point, deformation is relatively constant to the trailing edge. The 1.6 of washout in the cambered BR wing decreases the load throughout most of the wing and decreases the lift by 8.5% (compared to the rigid wing), but, as before, the loadalleviating design located by the to pology optimizer (right column, Figure 7-12) is superior. Sim ilar to above, the design utilizes a series of disconnected carbon fibe r structures, oriented parallel to the flow, and extending to the trai ling edge. The structures are spaced far enough apart to allow for some local membrane inflatio n, but this cambering only increases the loads towards the trailing edge. The discontinuous wing surface forces a number of high-pressure spikes on the upper surface, notably at x/c = 0. 2 and 0.6. This, in combination with the substantial adaptive washout at the trailing edge decreases the lift by 13. 6% over the rigid wing and by 5.6% over the BR wing. Three of the wing topologies discussed above (minimum CL minimum drag, and minimum pitching moment slope, all optimized for a reflex wing at 3 angle of attack) are built and tested in the closed loop wing tunnel, as seen in Figure 7-14. Though the aeroelastic model relies on a sizable state of pre-stress in the m emb rane skin to remain bounded, all three of the PAGE 148 148 wings are constructed with a slack membrane. This is to ensure similarity between the three wings (pre-stress is very difficult to control), and also to compare the force and moment data to the baseline membrane data acquired above ( Figure 5-12 Figure 5-15). Figure 7-14. Wing topology optimized for minimum CL built and tested in the wind tunnel. Results are given in Figure 7-15, for a longitudinal -sweep b etween 0 and 30. All three structures located by the topol ogy optimizer show marked improvements over the baseline experimental data, validating the use of a low fidelity aeroelastic model (vortex lattice model coupled to a linear membrane solver) as a su rrogate for computationally-intensive nonlinear models. With the exception of very low (where de formations are small) and very high angles of attack (where the wing has stalled), the optimized designs consistently outperform the baselines. As discussed above, this is not expected to be tr ue for L/D, where design st rategies vary strongly with incidence ( Figure 7-9). It should also be noted that the three optim ized designs in Figure 7-15 provide shallower lift slopes, less drag, and steeper pitching m oment slopes, respectively, than the experimental data gathered from the designs utilizing aeroelastic tailoring ( Figure 6-15 Figure 6-18). This confirm s the idea that topology optimization ca n out-perform tailoring of the baseline MAV wings, as the former has a larger number of variab les to work with. The two techniques need not be mutually exclusive: having located suitable wi ng topologies, the designs can be subjected to a PAGE 149 149 tailoring study for further bene fit to the flight performance. Figure 7-15. Experimentally measured forces and moments for baseline and optimal topology designs, reflex wing. Multi-Objective Optimization The need to simultaneously consider more th an one design metric for aeroelastic topology optimization of MAV wings is demonstrated above: optimizing for lift prompts the algorithm to remove the reflex, by depressing the flexible tr ailing edge. The downward forces provided by the reflex offset the nose-down pitching moment of the remainder of wing, and are therefore essential for stability. Design and optimization with multiple performance criteria can be done my optimizing one variable with constraints upon the others (as disc ussed by Maute et al. [118] for aeroelastic topology optim ization). For MAV design however, the formulation and bounds of these constraints are uncertain, and the method does not provide a clear pi cture of the inherent trade-off between variables. The current work minimizes a convex combination of two objective functions (as discussed by Chen and Wu [158] for topology optimization). Successiv e optimizations with different relative weighting betw een the two metrics can fill out the Pareto optimal front. The PAGE 150 150 computational cost of such an undertaking is la rge, and adequate locati on of the front is not ensured for non-convex problems (such as seen in Figure 6-14). The objective function is now: 11,min 22,min 1,max1,min 2,max2,minffff g1 ffff (7-16) where is a weighting parameter that varies between 0 and 1, and f1 and f2 are the two objective functions of interest. These functions are pr operly normalized, with the minimum and maximum bounds computed from the single-object ive optimizations (optimizing with set as 0 or 1). Eq. (7-16) is cast as a minimizati on problem, and the sign of f1 and f2 is set accordingly. As before, the objective function can be augmented with th e explicit penalty of Eq. (7-2) as needed. Typical convergence history results are given in Figure 7-16, for simultaneous m aximization of L/D and minimization of the lift slope. The weighting parameter is set to 0.5, for an equal convex combination of the tw o variables. The values given for CL (~ 0.4) are smaller than experimentally measured trends (~ 0.5, from Table 5-1), as the inviscid solver is unable to p redict the vortex lift from the tip vortex swirling system [27]. Beginning with an inte rmediate density (Xo = 0.5), the optimizer is able to decrease the convex combination (g) from 0.7 to 0.3, using similar techniques seen ab ove. All of the carbon fiber material adjacent to the root, leading edge, and wingtip is removed. Intersecting streams of membrane material grow across the wing, leaving behind disc onnected carbon fiber structures. The lift-to-drag ratio monotoni cally converges after 25 itera tions, while the lift slope requires 70 iterations to converg e to a minimum value. An explicit penalty on intermediate densities is employed at the 80 iteration mark, pr oviding a moderate decrease in the combination objective function. The lift-to-d rag ratio is improved as well th rough the penalty, though the lift slope suffers. As before, the penalty only serves to force the density variables to 0 or 1, and does not significantly alter the wing topology. PAGE 151 151 Figure 7-16. Convergence history fo r maximizing L/D and minimizing CL = 0.5, = 3, reflex wing. The multi-objective results of Figure 7-16 can be directly co mpared to th e single-objective results of Figure 7-5, where only L/D must be improved. For the latter, L /D can be increased to 4.17, with the inclusion of trailing edge batt ens for adaptive wash-in, and an unconstrained membrane skin towards the leading edge for cambe ring via inflation. This is a load-augmenting design, and as such the lift slope is very high: 0.040. In order to strike an adequate compromise between the two designs, the multi-objective optimizer leaves the trailing edge battens, but fills the membrane skin at the leading edge with a disconnected carbon fiber st ructure. The L/D of this design obviously degrades (4.05), but the lif t slope is much shallower (0.038), as desired. The Pareto front for this same trade-off (maximum L/D and minimum CL ) is given in PAGE 152 152 Figure 7-17, along with the performance of the 20 baseline MAV wing designs ( Figure 7-1), and the design located by the single-objectiv e topology optim izer to maximize CL All results are for a reflex wing at 3 angle of attack. Focu sing first on the baseline wings, the BR and PR wings represent the extremes of the group in te rms of lift slope, as expected. The homogenous carbon fiber wing has the lowest L/D (implying that for a reflex wing at this flight condition, any aeroelastic deformation will improve efficiency regardless of the type), while a MAV design with 2 trailing edge battens as the largest L/D. Figure 7-17. Trade-off between efficiency and lift slope, = 3, reflex wing. The aeroelastic topology optimization produces a set of designs th at significantly outperform the baselines, in terms of individually -considered metrics (maximum and minimum lift slope, maximum L/D), and multiple objectives: all of the baselines are removed from the computed Pareto front. The optimized designs la y consistency closer to the fictional utopia point as well which for Figure 7-17 is at (4.18, 0.0366). The en tirety of the Pareto front is not PAGE 153 153 convex, but the topology optimizer is still able to ade quately compute it. The data points are not evenly spaced either, with = 0.4 and 0.2 both very close to the solution with optimal L/D ( = 0). This would suggest that despite the normalizing efforts, maximizing L/D carries greater weight than minimizing the lift slope, an imba lance which may be remedied through nonlinear weighting [111]. The results of Figure 7-16 also indicate that usi ng an exp licit penalty to force the design to a 0-1 density dist ribution favors L/D, but not CL In terms of the two metrics in Figure 7-17, none of the designs along the Pareto optimal front are technically superior: th ey are non-dom inated, in that no other design exis ts within the data set that out-performs another design in bo th metrics. Other performance indices, not included in the optimization, can then be used to select an adequate design. For micro air vehicle applications, payload, flight duration, or agility/control metrics can be used, as discussed by Torres [3]. Realistic knowledge of the low-fidelity aeroelastic m odels limitations (the perceived superiority of an unconstrained membrane wing in Figure 7-3 is destroyed by large nonlinear flapping vibrations [62], for example), or manufacturability [161] may also be used to select a design. It should also be noted that at higher ang les of attack, the trade-off between high efficiency and low lift slopes doe s not exist. As discussed ( Figure 7-9), increasing the incidence prom otes an aeroelastic structure with streamlini ng to improve L/D, a deformation that will also decrease the lift slope. Wing displacements and pressure distributions for selected wings along the Pareto front of Figure 7-17 are given in Figure 7-18, for a reflex wing at 3 angle of attack. Corresponding data along the spanwise section 2y/b = 0.58 is given in Figure 7-19. When = 1 (sing le-objective optimization to minimize the lift slope), the aeroe lastic topology optimizer locates a design with several disconnected structures imbedded with in the membrane, including a long batten that PAGE 154 154 extends the length of the membrane skin. The wing is flexible enough to adaptively washout, but the remaining patches of membrane skin ar e not large enough to inflate and camber. Figure 7-18. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for designs that trade-off between L/D and CL = 3, reflex wing. Gradually adding weight to the L/D design metr ic removes the structures from the leading edge of the membrane skin, leavi ng batten-like structures at the trailing edge of the wing. The former transition allows the membrane to inflat e and camber the wing, wh ile the latter provides wash-in through depression of the trailing edge. The cambering membrane inflation does not grow monotonically with decreasing but the trailing edge deform ation does: from 0.25 of washout to 0.75 of wash-in. The size of the depressed trailing edge portion also grows in size. Decreasing shifts the lift penalty (pressure spike on the upper surface) forward towards the membrane/carbon fiber interface, and the lift spik e (due to the surface geometry cusp at the leading edge of the batten stru ctures) aft-ward. However, th e design that maximizes L/D ( = 0) has no spike, with a smooth pressure and displace ment profile aft of the lift penalty towards the leading edge. This may be indicative of the de trimental effect the airfoil cusp has on drag. PAGE 155 155 Figure 7-19. Deformations and pressures along 2y /b = 0.58 for designs that trade-off between L/D and CL = 3, reflex wing. The trade-off between the drag and longitudina l static stability of a membrane MAV wing is very important: the latter is typically improved through large membrane inflations. The resulting tangent discontinuities in the wing surf ace produce pressure spikes oriented axially, and the exaggerated shape prompts the flow to separate above and below the membrane [14]. The trade-off is given in Figure 7-20 for a reflex wing at 12, for both the 20 baseline designs and the Pareto front located with t opology optim ization. Compared with the data seen in Figure 7-17, the base line designs at this higher angle of attack fail to adequately fill the design space; their performance generally falls with in a band. The streamlining of the BR wing provides the lowest drag (of the baselines), but doesnt significan tly out-perform the homogenous carbon fiber wing. As expected, the PR wing has the largest static stability margin of the baselines, but the drag penalty is large (and probably under-predict ed by the inviscid flow solver). The topology optimizer is able to locate a design with the same drag penalty, but a steeper pitching moment slope: by 5.6% over the PR wing. Th e baselines designs, in general, lie closer to the Pareto front PAGE 156 156 than seen in Figure 7-17, but the optimized designs are still superior in terms of Pareto optimality and individual m etrics. The optimal drag desi gn (3.8% less than the BR wing) begins with two carbon fiber structures imbedded within the membra ne skin, one of which is a long batten that extends the length of the design domain. Figure 7-20. Trade-off between drag and pitching moment slope, = 12, reflex wing. By adding weight to the st atic stability metric (Cm ), this long batten breaks in two pieces; the foreword section shrinks into a slender batten imbedded in the leading edge of the membrane skin. The aft-ward section gradually accumulate s along the trailing edge, merges with the root, and forms the trailing edge support. As discu ssed above, this reinforcement does not connect monolithically to the wingtip; this space is filled with a trailing edge batten. The superiority of this design is confirmed by the wind tunnel data of Figure 7-15. The Pareto optimal front of Figure 7-20 shows a more pronounced convexity than seen in Figure 7-17, though the data points are still not evenly spa ced with PAGE 157 157 Similar data is given in Figure 7-21, for the trade-off between maximum lift and minimum lift slope, for a cam bered wing (no reflex) at 12 angl e of attack. Such a trade-off is of interest because minimizing the lift slope of a membrane MAV wing, while an effective method for delaying the onset of stall or rejecting a sudden wi nd gust, typically decreases the pre-stall lift in steady flight as well; a potentially unacceptable c onsequence. Certain aeroelastic deformations, such as a passive wing de-cambering, would provi de a wing with higher li ft (than the baseline carbon fiber wing, for example), but a shallower lift slope. Figure 7-21. Trade-off betw een lift and lift slope, = 12, cambered wing. Such a motion is unusual for low aspect ratio membrane structures however: none of the baseline designs have both larg er lift and a smaller lift slope than the carbon fiber wing. The correlation between CL and CL within the set of baseline desi gns is very strong, and all the designs fall very close to a si ngle line, clustered in three gr oups. Any baseline design with adaptive washout (free trailing edge) has lift slopes between 0.035 and 0.037, any overly-stiff PAGE 158 158 design with battens oriented pe rpendicular to the flow (or the carbon fiber wing) has a slope between 0.038 and 0.039, and the PR wing has a lift slope of 0.041. The strong data correlation is in sh arp contrast to the results of Figure 7-17 for the reflex wing, where the baseline structures are well-d is tributed through the design space. This emphasizes the large role that the doubly-curved airf oil can play in producing many different types of aeroelastic deformation, providing greater freedom to the designer and better compromise designs. Despite this, the magnitude of the variability is higher for the cambered wing, as the forces are generally larger: CL can be varied by 14.5% for the reflex wing in Figure 7-17, but by 26.4% for the cam bered wing in Figure 7-21. These numbers can be increased further with the use of nonlinear m embrane stru ctures, but deformations must be kept at a moderate level to preserve the fidelity of the linear finite element model in the current work. As wing structures with high lift and shallow lift slopes are rare the set of baseline designs lies close to the Pareto front in Figure 7-21. None are superior however, in terms of individual m etrics or Pareto optimality. The designs located by the topology optimizer to maximize lift and maximize lift slope are almost identical, though disp arate designs can be obtained with a reflex wing, as noted above. The PR wing is very effective for cambered wings at higher angles of attack, and lies close to these two optimums. Th e slight convexity in the Pareto front produces two designs with the sought-aft er higher lift and lower lift slope than the homogenous carbon fiber wing. The topology highlighted in Figure 7-21 increases the lift coefficient from 0.842 to 0.876 and decreases the lif t slope from 0.038 to 0.036, and is found from an equal weighting of the two metrics ( = 0.5). Wing displacements and pressure distributions for selected wings along the Pareto front of Figure 7-21 are given in Figure 7-22, for a cambered wing at 12 angle of attack. Corresponding PAGE 159 159 data along the spanwise sect ion 2y/b = 0.58 is given in Figure 7-23. Shallow lift slopes are provided with a series of disconnected batten stru ctures oriented parallel to the flow. As a weight for high lift is added to the objective function, a large carbon fiber region grows at the trailing edge, but is connected to either the root or the wing tip. This allows for both washout and m embrane cambering, and produces the MAV design with higher lift and shallower lift slopes than the carbon fiber wing ( = 0.5). Further decrease in flattens the chord of the trailing edge structure and removes the disjointed battens at the leading edge, to maximize lift. Figure 7-22. Normalized out-of-plane displacement s (top) and differential pressure coefficients (bottom) for designs that trade-off between CL and CL = 12, cambered wing. The locus of aeroelastic deform ation clearly shifts from the trailing edge to the mid-chord of the wing as the structures produce higher lift. Washout monotonically decreases with (from 3 to 0.1 of wash-i n). Membrane deformations are largest when = 0.5, though the design that maximizes lift shows the largest change in camber, owing to the significant adaptive washout of the former, as discussed. Similarly, the ae rodynamic penalty at th e leading edge of the PAGE 160 160 membrane/carbon fiber interface is largest with the compromise design. The severity of the surface cusp (and the concomitant lift spike) increases with decreasing emphasizing its usefulness as a lift-augmentation device. As discussed above, the severi ty of this spike is certainly over-predicted by the inviscid flow solver, though similar trends are seen using NavierStokes solvers for wings with tangent discontinuities Figure 5-21. Figure 7-23. Deformations and pressures along 2y /b = 0.58 for designs that trade-off between CL and CL = 12, cambered wing. PAGE 161 161 CHAPTER 8 CONCLUSIONS AND FUTURE WORK The results given in this work detail a co m prehensive research effort to understand and exploit the static aeroelasticity of membrane mi cro air vehicle wings. The flow structures of such wings are exceedingly complex, characte rized by low Reynolds numbers (flow separation, laminar-turbulent transition, r eattachment, vortex shedding, vort ex pairing), low aspect ratios (strong tip vortex swirling, low pr essure wing tip cells), and unstable interactions between the two (vortex destabilization for b ilateral asymmetry). The wings structural mechanics are also difficult to predict: a topologically-complex orth otropic wing shell is covered with a thin extensible latex skin, a membrane with an inherently nonlinear response. Aeroelastic fixed membrane wi ng topologies can be broadly di vided into two categories: load-alleviating, and load-augmenting. The form er can use streamlining to reduce the drag, or adaptive washout for gust rejection, delayed sta ll, or attenuated maneuver loads. The latter increases the loads via adaptive cambering or wash-i n, for improved lift and static stability; the wing may also be more response to pull-up maneuvers, etc. Wing topology is given by a distinct combination of stiff laminate composite members and a thin extensible rubber membrane sheet, similar to the skeletal structure of a bird wing, or the venation patterns of insect wings. This work discusses aeroelastic analysis and optimization in three phases. First, given a set of wing topologies (a batten-reinforced design for adaptive geometric twist, a perimeterreinforced design for adaptive aerodynamic twis t, and a homogenous laminate wing), how does the membrane inflation affect the complex flow st ructures over the wing? Secondly, how can the various sizing and strength variab les incorporated within the wing structures be tuned to improve flight performance, in terms of both individual metrics and compromise functions? Third, can these baseline wing topologies be improved upon? How does the distribution of laminate shells PAGE 162 162 throughout a membrane skin aff ect the aeroelastic response? No model currently exists that can accurate ly predict such aeroelasticity (the threedimensional transition is the biggest numerical hurdle), and so the current work utilizes a series of low-fidelity aeroelastic models for efficien t movement through the design space: vortex lattice methods and laminar Navier-Stokes solvers are coupled to linear and nonlinear structural solvers, respectively (detailed in Chapter 4). Due to th e lower-fidelity nature of the models (despite which, the computational cost of this coupled ae roelastic simulation is ve ry large), experimental model validation is required. Su ch characterization is conducted in a low speed closed loop wind tunnel. Aerodynamic forces and moments are measured using a strain gage sting balance with an estimated resolution of 0.01 N. Structural displa cement and strain measurements are made with a visual image correlation system; a calibrated camera system is m ounted over the test section, as discussed in Chapter 3. Chapter 5 provides a detailed analysis of th e flow structures, wing deformation, and aerodynamic loads of a series of baseline membrane MAV wings. At small angles of attack, the low Reynolds number flow beneath a MAV wing sepa rates across the leading edge camber, the flow over the upper surface is largel y attached, and the tip-vortex swirling system is weak. The opposite is true has the incidence is increased : the bubble on the upper surface grows, eventually leading to stall. The lift curves of the low aspe ct ratio wings are typically shallow, with a large stalling angle. Low pressure cells deposited on the upper surface of the wing tip by the vortex swirling grow with angle of attack, adding nonlinearities to the lift and moment trends. The structural deformation of a batten-reinfo rced wing has two main trends: the forces towards the leading edge are very large, and induce membrane infl ation in-between the battens. This increases the camber over the wing, and thus the lift. A second trend comes from the free PAGE 163 163 trailing edge of the BR wing, which deflects upw ard for a nose-down twist, decreasing the wing lift. These two effects tend to offset for lower angles of attack, and the aerodynamics follow the rigid wings very closely. At higher angles the adaptive washout dominates, decreasing the incidence of a wing section by as much as 5 and decreasing the slope of the wings lift curve. Outside of the promise such a wing shows for gust rejection and benevolent stall, the data also indicates that the streamlining decreases drag. The deformation of a perimeter-reinforced wing is characterized by adaptive aerodynamic twist: the membrane skin inflates, constrained at the leading and traili ng edges by the stiff carbon fiber perimeter. Lift, drag, and pitching moment s are consistently stronger than measured from the rigid and BR wings, as a result of the camberi ng motion. The slope of the pitching moment curve is considerably steeper, providing much-need ed longitudinal static st ability to a wing with severe space and weight constrai nts. The large drag penalty of the wing is partly due to a pressure spike at the tangent discontinuity betw een the inflated membrane and the carbon fiber, and partly due to the greater amount of separate d flow over the PR wing. Interactions between the separated longitudinal flow and the wing tip vortices are clearly visible in the PR wing, possibly indicating a greater propensity for rolling instabilities. The stretching of the membrane skin in the PR wing is more two-dimensiona l without the restrictiv e presence of battens. It is shown in Chapter 6, both numerically and experimentally, that unconventional aeroelastic tailoring can be us ed to improve MAV wing performance. The chordwise and spanwise membrane pre-tension, number of plain weave carbon fiber layers, laminate orientation, and batten thickness are all considered, with the first three variables identified as critical through a series of one-factor-at-a-time tests. In creasing stiffness is seen to tend aerodynamic behavior towards a rigid wing, thought many local optima exist and can be PAGE 164 164 exploited. A comprehensive numerical review of the design space is provided with a full factorial designed experiment of th e three aforementioned variables. This data is then used to optimize six aerodynamic variables, as well as compromises between each. The six designs resulting from the single-objective optimizations are built and tested in the wind tunnel: five show improvements over the baseline designs, one has a similar response. While the flexible wing structures have been shown to effectively alter the flow fields over a MAV wing, aeroelastic topology optimization (C hapter 7) can be used to improve on the shortcomings of the previously-considered base line designs. Results are superior to those computed via tailoring, as the number of variables is much larger: the wing is discretized into a series of panels, each of which can be membra ne or carbon fiber laminate. The computational cost is severe: hundreds of iterati ons are expected for convergence, and a sensitivity analysis of the coupled aeroelastic syst em must be conducted. The optimization is able to identify a seri es of interesting designs, emphasizing the relationships between flight c ondition, airfoil, design metric and wing topology. For load alleviation, the algorithm fills the membrane skin with a number of disconnected laminate structures. The structure is flexible enough to washout at the trailing edge, but the patches of exposed membrane skin are not large enough to in flate and camber the wing. Such a design has less drag and a shallower lift curve than the batte n-reinforced wing. For load augmentation, the topology optimizer utilizes a combination of cam bering, wash-in, and wing surface geometry cusps to increase the lift over th e perimeter-reinforced wing. As a wing design optimized for a single metric is of minor usefulness, the topo logy optimizer is expanded to minimize a convex combination of two metrics for computation of the Pareto front. Three such designs are built and tested in the wind tunnel, c onfirming the computed superior ity over the baseline wings. PAGE 165 165 Several future aeroelastic optimi zation studies are of interest. First, it is desired to upgrade the model fidelity used in the topology optim ization described above. In order to limit computational cost, the work uses several linea r modules: a vortex lattice solver and a linear stress-stiffening membrane solver, computed on a relatively coarse topology grid. Such a model is unable to capture several important nonlinearit ies, including flow separation and tip vortex formation. This can be remedied by using an unsteady Navier-Stokes solver coupled to a nonlinear membrane structural dynamics solver, increasing the computati onal cost by several orders of magnitude. The large number of variables (~1000) require s the use of a gradient-based optimizer; the higher-fidelity models will increas e the complexities involved in the sensitivity analysis of the coupled aeroelastic system as well. Of particular interest is gust response: how the membrane wing responds to a sinusoidal wind cycle, where it is desired to minimize the overall response for smoother flight. Objective functions may be the change in lift, integrated ov er the gust cycle. A second interest is the wing topology that delays the stall of the fixed wing. Conventional optimization formulations for this problem are difficult, as the stall angle is not a direct output from the aeroelastic system, but the angle at which the slope of the lift curve become s negative. The optimizer will have to compute the lift at a set number of (large) angles of a ttack, and interpolate betw een the data points to estimate the stalling angle. Secondly, these aeroelastic topology optimiza tion techniques will be extended to flapping micro air vehicle wings. The structure of these wi ngs is very similar to the fixed wings discussed above (thin membrane skins reinforced with lami nate plies), and so the two-material model is appropriate. As with the gust cycle, lift and th rust will have to be computed over an entire flapping cycle, and then integrated to produce a s calar objective function. Furthermore, lift and PAGE 166 166 thrust will conflict: thrust relies on wing twist via deformation for th rust generation, while lift is dependent upon the leading edge vortex, which can be disrupted by excessive deformations. This requires successive optimizations of a convex combination of the two weighted metrics to fill out the trade-off curve (assuming that this Pareto front is convex). The optimal design can then be selected from this front based upon metrics not considered in the formal optimization: trim requirements, manufacturability etc. The flow structures that develop over flapping wing systems are very complicated, uns teady vortex driven flows. Navier-Stokes solvers can adequately handle these phenomena, but the computational cost may be prohibitive. Topology optimization of flapping wings may requi re lower-fidelity aerodynamic methods for effective navigation through the design space. Finally, the aeroelastic topology optimization of both the fixed and flapping wings can be followed by a tailoring study for additional improvements to the flight performance. This is a standard optimization process: topology optimizati on, interpretation of th e results to form an engineering design, followed by sizi ng and shape optimization (or in this case, tailoring). Both laminate thickness/orientation and membrane pre-tension can be used, as above. 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PAGE 179 179 BIOGRAPHICAL SKETCH Bret Kennedy Stanford was born in Rich m ond, Virginia on September 30, 1981, though his grandmother claims it was on September 29. School was never really an option for young Bret, forced by his parents at an ear ly age to join the circus instead He was taught to read, write, and juggle by a kindly group of clowns, despite his extreme terror of anything with big floppy shoes, a phobia which continues un abated to this day. Bret wa s reunited with his parents two years later, an act which was prompted by a re cent increase in the Child Tax Credit. Several brush-ins with the law led to the Stanford familys expulsion from Virginia, escorted to the North Carolina border by a group of uns ympathetic state troopers. The family subsequently relocated to Tampa, Florida in the fall of 1988, though Brets grandmother claims it was in the summer. Brets time in Tampa was mostly spent selli ng hand-carved limestone trinkets and jewelry to tourists. At the age of 17, he was rejected from most of the universities along the eastern seaboard, who were collectively unimpressed with his artesian and entertainment backgrounds. A clerical error granted him acceptance to the Univers ity of Florida. He arrived in Gainesville in the fall of 1999 (a date his grandmother genera lly agrees upon) with the intent of studying French post-modern theatre. Nine convoluted ye ars later he received hi s doctorate in aerospace engineering. Upon graduation, he plans on throwing all of his newfound knowledge, books, and lab journals into the River, in order to start a Beach Boys cover band. He hopes that his advisor will handle all of the journal review replies as th ey come back from the editors, so the research will not have gone to waste. |