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PAGE 1 TUNABLE HOLDER FOR COMMERCIALLY AVAILABLE BORING BARS By LONNIE A. HOUCK III A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 1 PAGE 2 2009 Lonnie A. Houck III 2 PAGE 3 To all of my immediate family members: Mom, Dad, Dianne, Lance, and especially my wife, Ashley. Without their motivation and unwavering support I honestly can say I would not be here today. 3 PAGE 4 ACKNOWLEDGMENTS I thank my advisor, Professor Tony Schmitz; his guidance and personality have left a permanent mark on my life and professional career. I would also like to thank my committee, Professors John Schueller and Gloria Wiens, the students of the Machine Tool Research Center, the National Science Foundation for partial financial support (grant number CMMI0238019), and Kennametal for providing the boring bars used in this study. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF FIGURES.........................................................................................................................7 ABSTRACT.....................................................................................................................................9 CHAPTER 1 INTRODUCTION..................................................................................................................11 Chatter.....................................................................................................................................11 Current Methods and Limitations for Dealing with Chatter...................................................12 New Method of a Tunable Holder and Mass Tuning.............................................................13 2 DESIGN OVERVIEW...........................................................................................................14 Damped Dynamic Absorber...................................................................................................14 Beam Modeling and Receptance Coupling Techniques.........................................................14 Analytical Models...................................................................................................................17 Iteration Process......................................................................................................................18 Analytical Design Results.......................................................................................................18 Sensitivity of Holder Design..................................................................................................19 Finite Element Model.............................................................................................................20 Flexure/ Holder Designs.........................................................................................................20 Finite Element Analysis Design Iterations.............................................................................21 3 PROTOTYPE, EXPERIMENTAL SETUP AND RESULTS...............................................29 Manufacturing Procedures......................................................................................................29 Component A: Flexure....................................................................................................29 Component B: Brass Sleeve............................................................................................30 Component C: Steel Boring Bar......................................................................................30 Assembly................................................................................................................................30 Testing Procedure and Results................................................................................................30 4 COMPARISON OF FEA, AND PROTOTYPE MASS TUNING TRENDS........................36 Mass Tuning Introduction.......................................................................................................36 FE Analysis.............................................................................................................................36 Prototype Measurements........................................................................................................37 5 CONCLUSIONS AND INDUSTRY APPLICATIONS........................................................41 Conclusions.............................................................................................................................41 5 PAGE 6 Industry Ideas..........................................................................................................................41 APPENDIX A BEAM RECEPTANCE MODELING....................................................................................44 B MATLAB CODE....................................................................................................................46 Boring Bar Design..................................................................................................................46 FreeFree Response of Boring Bar.........................................................................................53 Program to couple to freefree beams together.......................................................................56 Program to Read Impact Test Data and Determine Modal Response....................................59 Code for Sensitivity Analysis.................................................................................................59 LIST OF REFERENCES...............................................................................................................64 BIOGRAPHICAL SKETCH.........................................................................................................65 6 PAGE 7 LIST OF FIGURES Figure Page 21 Example 2DOF system......................................................................................................23 22 Assembled/component systems.........................................................................................23 23 Analytical fixedfree FRF for 0.625 inch diameter boring bar..........................................23 24 Analytical FRF of boring bar coupled to holder................................................................24 25 Amplitude ratio for assembly.............................................................................................24 26 The FRFs for clampedfree boring bar (denoted rigid) and holderbar assembly (denoted new holder) with stiffness and damping factors of 20........................................25 27 The FRFs for clampedfree boring bar (denoted rigid) and holderbar assembly.............25 28 Sensitivity study.................................................................................................................26 29 Reduction in amplitude for boring barholder assembly relative to fixedfree boring bar for range of holder natural frequencies........................................................................26 210 Three holder designs: leaf, notch, and circular notch........................................................27 211 Finite element harmonic analysis of the fixedfree boring bar..........................................27 212 Finite element harmonic analysis of holder.......................................................................28 213 Finite element harmonic analysis of assembly..................................................................28 31 Complete model and prototype..........................................................................................32 32 Flexure with dovetail.........................................................................................................32 33 Brass round........................................................................................................................33 34 Boring bar..........................................................................................................................33 35 Measured FRF of the boring bar and assembly.................................................................34 36 Test setup for measurements..............................................................................................34 37 Tool post and assembly......................................................................................................35 38 The FRF of assembly clamped to the tool post..................................................................35 41 Base and sleeve sliding scheme.........................................................................................38 7 PAGE 8 42 Finite element models of sleeve position in mass tuning trend analysis...........................38 43 Finite element results of sleeve position in mass tuning trend analysis. Note the change in scales for the amplitude axis. ............................................................................39 44 Prototype mass tuning trends results..................................................................................40 51 Industrial boring bar holder conceptual representation.....................................................43 8 PAGE 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the degree of Master of Science TUNABLE HOLDER FOR COMMERCIALLY AVAILABLE BORING BARS By Lonnie A. Houck III May 2009 Chair: Tony L. Schmitz Major: Mechanical Engineering During metal cutting operations, vibratory motion between a cutting tool and work piece can lead to nonbeneficial cutting performance. Such vibrations can cause the cutting tool, work piece, and/or machine to become damaged. Selfexcited vibrations, or chatter, between the cutting tool and work piece can cause poor surface finish, tool breakage, and other unwanted effects. When chatter does occur, the machining parameters must be changed and, as a result, productivity may be adversely affected. One example of tools that may encounter excessive vibration is boring bars, which are typically used to fabricate deep holes. A primary difficulty in their use is that, because the holes tend to be deep and narrow, the boring bars must be long and have small diameters. Therefore, during machining, the variable cutting force causes the tool to deflect and leave a wavy surface behind. When the cutting edge encounters this wavy surface in the next revolution, additional forces and deflections may be caused which can lead to chatter. Several methods for reducing boring bar vibration are currently used including, for example, internal vibration absorbers. Here, we describe a new method to reduce tool vibrations by providing a flexible holder with dynamics tuned to match the boring bar dynamics. The flexible holder supports the boring bar and acts as a dynamic absorber for the boring bar. The 9 PAGE 10 flexible holder natural frequency is matched to the clamped natural frequency of the tool, thereby reducing the amplitude of vibration at the free (cutting) end of the bar. In this paper we present both an analytical solution, which applies EulerBernoulli beam theory, combined with receptance coupling techniques and a finite element model for predicting the assembly dynamics. A holderboring bar assembly is designed and frequency response measurements of the boring bar alone are compared to the measured response of a prototype holderboring bar assembly. The dynamic stiffness of the holderboring bar assembly is compared to the stiffness of the cantilever boring bar alone. Stiffness improvements up to 68% are observed for the holderboring bar assembly. A new modal mass tuning system is implemented that does not require changing the tool characteristics, such as overhang length, to match the holder and clamped bar natural frequencies. Instead, the modal mass effect is realized by adjusting the position of a mass attached to the tool and enables the tool dynamics to be tuned to the holder dynamics. The overall goal of the design is to use a single holder for a set of varying length and diameter boring bars. The holder can then be quickly and efficiently tuned for use (through the modal mass effect) for the current boring bar with predetermined mass positions. 10 PAGE 11 CHAPTER 1 INTRODUCTION In this section the phenomenon of chatter in machining processes is explained. Current methods and tools used to help reduce chatter in boring bars are outlined and limitations of each presented. A new method is described and advantages over current methods given. Chatter During a metal cutting operation, any vibratory motion between a cutting tool and workpiece may lead to nonbeneficial cutting performances. Furthermore, such vibration may cause the cutting tool or the machine tool to become damaged. Self excited vibrations, frequently called "chatter," between the cutting element of a machine tool and the surface of the workpiece cause poor surface finish, tool breakage and other unwanted effects that have long plagued machining operations. Such vibrations arise especially when the tool is long and flexible, which generally leads to larger deflections. When chatter does occur the machining parameters must be changed and, as a result, productivity may be adversely affected. When using even the stiffest and most advanced machine tool design, vibration still exists. For example, a long cantilevered boring bar with a single cutting edge at its free end will, by its interaction with the workpiece, tend to vibrate. This vibration may be considered to be selfexcited because its amplitude depends on both the current vibration and the vibration (as imprinted on the workpiece surface) in the previous cutting pass. Such selfexcited vibration occurs near the natural frequency of the cantilevered tool. As noted, one example of tools that may encounter issues due to excessive vibration is boring bars. Boring bars are used to fabricate deep holes. A primary difficulty in their use is that, because the holes tend to be deep and narrow, the boring bars must be long and have small diameters. Therefore, during machining, the variable cutting force causes the tool to deflect and 11 PAGE 12 leave a wavy surface behind. When the cutting edge encounters this wavy surface in the next revolution, additional forces and deflections may be caused which can lead to chatter, or unstable machining. The results of chatter are poor surface finish and hole accuracy, large forces and deflections, and potential damage to the tool, workpiece, and/or machining center. Current Methods and Limitations for Dealing with Chatter To reduce these vibrations, various methods have been employed. Simple methods include reducing the material removal rate (by lowering the depth of cut) or decreasing the length of the tool. Reducing the material removal rate interferes with production. Changing the length of the tool restricts the depth of the hole that can be fabricated. A common machinist rule of thumb is to keep the length to diameter ratio (L:D) at or below 6 to 1. This limitation can prevent the accurate fabrication of small slender holes. Another method to reduce vibration in boring bars includes mounting upon or within the bar a dynamic vibration absorber, which is typically comprised of a cylindrical mass of a high density material supported on rubber bushings. When optimally tuned, the mass oscillates in response to vibration produced in the boring bar to cancel out vibration. The absorber may be tuned to accommodate the boring bar for the speed at which the workpiece or boring bar is rotating, the length of the boring bar and the type of cutting tool connected at the end of the bar. Such an adjustment is made by longitudinally forcing pressure plates at opposing ends of the cylindrical mass, which serve to compress the rubber bushings against the mass. This simultaneously shifts the position of the mass and alters the stiffness of the rubber bushings to change the dynamics of the absorber boring bar assembly. However, even with such a design available, each time the boring bar is to be used under different conditions, it must be separately tuned. Additionally, the dynamic absorbers is most commonly placed inside the boring bar, i.e., the boring bar is hollowed and the springmass12 PAGE 13 damper system is added inside the boring bar structure. This places limits on the boring bar diameter, which must be large enough to contain the dynamic absorber, and increases the boring bar cost. New Method of a Tunable Holder and Mass Tuning This thesis describes a new method of reducing vibrations for a cutting tool by using a flexible holder. The holder and tool dynamics are matched at the design stage which eliminates the need for complex instrumentation to tune the assembly and does not require modification of the tool. Since there is no tool modification, this method is not limited by the size requirements for dynamic absorber inserts. The holder incorporates a new method referred to here as mass tuning. In this approach, the modal mass of the holder is adjusted, thus changing the assembly dynamic response. Since the tool length remains unchanged and only the holders characteristics are altered, this method is not limited by the 6 to 1 L:D ratio. 13 PAGE 14 CHAPTER 2 DESIGN OVERVIEW In this chapter, the design procedure used to model the tunable holder for boring bars is described. The widely used dynamic absorber effect is applied in an analytical design method using beam theory and receptance coupling. A finite element (FE) analysis approach is also used. Justification for design and material selections are provided. Damped Dynamic Absorber If a force with an excitation frequency close to the natural frequency of a system is present the system may experience large vibrations, referred to as resonance. The addition of a properly designed second system, or dynamic absorber, can alter the characteristics of the original system to change the resonant frequency and reduce the vibration amplitude at the prescribed forcing frequency. The dynamic absorber effect results from an interaction between modes associated with the individual substructures, e.g. the holder and tool in the holdertool assembly. The dynamic absorber moves the original resonant peak of the single degreeoffreedom (SDOF) system, but introduces two new peaks in the 2DOF system, split about the original SDOF resonant frequency. This effect is well known and can be found in many vibrations textbooks [1]. Beam Modeling and Receptance Coupling Techniques In receptance coupling, experimental, analytical or numerical representations of direct and cross frequency response functions (FRFs) for the individual components are used to predict the final assemblys dynamic response at any spatial coordinate selected for component measurements/models [26]. In this method, unlike modal coupling, for example, FRFs are required only at the coordinate of interest (the tool point for metal cutting) and any connection coordinates; the number of modeled structural modes in each component does not define the 14 PAGE 15 number of required measurement locations (to obtain square modal matrices) and no matrix inversions are necessary (only vector manipulations are required). As shown in Figure 21, two SDOF springmassdamper systems, A and B, are to be connected through the linear spring element, kc, to form the new 2DOF assembly C. The receptance matrix, G(), for the assembled system will now be derived using the receptance coupling method [2]. The 2x2 receptance matrix (for the 2DOF system) will be calculated by columns. We will determine the first column by applying a virtual force, F1, to coordinate X1. Figure 22 displays the assembled and component systems with F1 applied to the assembled system. For simplicity, the springmassdamper systems are represented by continuous bodies (which may, in general, contain several degrees of freedom). Considering the (unassembled) substructures in Figure 22, the displacements at the two coordinates can be written as shown in Equations 21. The notation H refers to the spatial receptance matrices of the individual components before assembly. The equilibrium condition for the components is given in Equations 22. The compatibility conditions are shown in Equations 23. 22221111fHxfHx (21) 121Fff and (22) 211fFf 11xX and 22xX ckfxx212 (23 Substitution of Equations 21 and 22 into 23 yields Equations 24. Substituting the result shown in 24 into the equilibrium condition gives 25. These are expressions for the forces acting on the individual components. 15 PAGE 16 1111221121FHkHHfc (24) 11112211111FHkHHfc (25) The first column of the receptance matrix is now determined by substitution into the appropriate displacement/force relationships. Equations 26 and 27 give expressions for the G11() and G21() receptance terms. 1112211111111111HkHHHHFXGc (26) 11122112212211HkHHHFXGc (27) The second column of the receptance matrix is found by applying a virtual force, F2, to coordinate X2 of the assembly. The displacements for this case are the same as those given in Equations 21. The new equilibrium and compatibility equations are shown in Equations 28 and 29, respectively. Substitution and combining operations similar to those for the first column give expressions for G12() and G22(). 221Fff and (28) 122fFf 11xX and 22xX ckfxx121 (29) 22122111121121HkHHHFXGc (210) 2212211222222221HkHHHHFXGc (211) 16 PAGE 17 Analytical Models The closedform, EulerBernoulli beam receptances described by Bishop and Johnson [2] (see Appendix A) were implemented to describe an ISO A10SCLPR2 NE4 boring bar with a minimum of a 6:1 L:D ratio. This high L:D ratio bar was selected since the focus of this work is the improvement of the dynamic stiffness for these inherently low stiffness situations. The material properties and dimensions were defined and simulations completed in the Matlab programming environment (See Appendix B). The 0.625 inch (15.9 mm) diameter was chosen because this is the smallest diameter for commercially available tunable boring bars (with dynamic absorbers located inside the bar). Figure 23 shows the analytical FRF for the fixedfree (cantilever) beam model representing our boring bar. Using the peak picking method [7] a stiffness value of mN51035. 6and a damping ratio of were determined. Next, a fixedfree beam model of the boring bar holder was defined with stiffness 20 times greater than the boring bar, but with the same natural frequency. The holder model was then coupled to the freefree model of the boring bar using the receptance coupling approach. The FRF of the combined beam and bar is shown in Figure 24. 41050.6 Figure 24 shows the first bending mode of the boring bar is shifted downwards from 306 Hz to 287 Hz with a 74% reduction in amplitude. A second mode is also observed, but shifted upwards by the same amount to 324 Hz with the same amplitude. In the bandwidth shown, the system goes from one bending mode to two bending modes split about the first mode due to the addition of the holder. The increased dynamic stiffness of the coupled system is counterintuitive (i.e., adding a flexible element increases the assembly stiffness), but is similar in nature to the wellknown dynamic absorber effect [8]. 17 PAGE 18 Iteration Process The role of the holder stiffness and damping characteristics on the system are shown in Figure 25. The top graph shows the ratio of the maximum holderbar assembly FRF magnitude to the maximum fixedfree boring bar magnitude as a function of a stiffness factor, barholderkkFactor where ki is the modal stiffness value, while damping is held constant. The bottom is the reverse; damping is varied while stiffness remains constant. In this case, barholderccFactor where ci is the modal damping value. In both instances, the bar and holder natural frequencies were matched for all factor values. A decrease in amplitude is observed for an increase in the stiffness/damping factors. However, since the decrease is approximately logarithmic, the percent change varies inversely with an increase in the factor. The percent change is less than one for both stiffness and damping factors of 10. This leads to a design goal of a holder with a first natural frequency of 306 Hz (see Fig. 23) and a minimum stiffness value of 10 times the stiffness of the bar, or N/m, and a 10 times damping increase, or 65103561010356..45061010506.. 310 Analytical Design Results A model for a system with stiffness and damping factors of 20 is shown in Figure 26. A nearly complete attenuation of the fundamental bending mode from the fixedfree (or rigidly clamped) boring bar is observed. This shows that, in the limit, as the stiffness and damping factors approach infinity, the amplitude of the resulting FRF approaches zero. In actual systems, increasing the area moment of inertia for the bending axis can be used to increase stiffness. However, increased damping is difficult to realize. Typical damping ratios for solid boring bars 18 PAGE 19 are less than one percent. Therefore, an easily producible holder could exhibit a stiffness factor of over 100 (for long, slender boring bars), but damping factors near unity. Figure 27 shows an example with a holder stiffness factor of 100 and a damping factor of one. These results show a 47.8 percent reduction in amplitude from the boring bar alone. Sensitivity of Holder Design The sensitivity of the reduction in amplitude for mismatches in the bar and holder natural frequencies was determined analytically. In the previously presented analytical models the natural frequency of the bar and the holder were equal. Since producing a holder with a natural frequency that matches the boring bars natural frequency exactly is difficult, this sensitivity analysis was completed in order to provide an initial assessment of the feasibility of and accuracy required for prototype manufacturing. For this analysis, the natural frequency for the holder modeled analytically in Figure 27 (with a stiffness of 100 times the bar and the same damping value) was varied between 70% and 130% of the boring bars fixedfree natural frequency (306 Hz). The individual cases are shown in Figure 28, where each dotted line represents a different holder; the range of natural frequencies are 214.2 Hz to 397.8 Hz in steps of 5.75 Hz. The heavy solid lines represent the nominal natural frequencies. Each holder was analytically coupled with the boring bar. The maximum amplitude of the assembly was determined for each combination and compared to the fixedfree boring bar. The amplitude reduction was then normalized to unity and the process was repeated for three different damping values. The results are summarized in Figure 29. In this figure, a nearly 50% loss in the desired amplitude reduction is seen relative to the ideal conditions represented in Figure 27. A high sensitivity is shown as a +/10% variation in natural frequency (i.e., holder/bar between 0.9 and 1.1 in Figure 29) leads to approximately a 5.5 times increase in frequency response amplitude 19 PAGE 20 versus the ideal design. If the design is not within +/20%, almost no benefit is achieved. The reader may note, however, that the assembly can conveniently be tuned to achieve the matched frequency conditions by adjusting the boring bar overhang length if required. Finite Element Model Finite element (FE) modeling was also used to aid in the design and analysis of the holder. This enabled the convenient analysis of complicated geometries, which is more challenging when applying the analytical beam models that assume constant crosssections. All models were developed using Autodesk Inventor 10 and FE models were imported into ANSYS Workbench 10 software package for analysis. Workbench subroutines were used to automatically generate the mesh for each part. Mesh refinement was introduced around critical areas. Solid threedimensional (3D) elements were used and no cross section approximation or beam simplification were applied. This differs from the twodimensional (2D) beam theory used in the analytical design and enables a comparison and validation of the 2D approximations. Boundary conditions were implemented as appropriate. Each model was then checked for convergence and a final simulation completed. Flexure/ Holder Designs Three holder types were initially explored in order to control the natural frequency and stiffness of the holder: leaf, notch, and circular notch (see Figure 210). The first two designs require large dimensions in order to obtain large stiffness values. The complex shapes also require nontraditional manufacturing processes, such as wire electric discharge machining, which can increase manufacturing cost. The circular notch was selected because it enables high stiffness values, while retaining a simple design and enabling ease of manufacture. Once the geometry was chosen, the next step was to select a material. 6061T6 aluminum was chosen due to its good machinability and high thermal expansion coefficient; a thermal 20 PAGE 21 shrink fit was used to rigidly connect the holder to the boring bar. A brass sleeve was used to tune the natural frequency of the holder to the boring bar. Brass alloy 360 was chosen for it high density and good machinability. Because the natural frequency of the holder must be approximately equal to the frequency of the bar, but the holder has a higher stiffness, it requires significant mass. The use of brass kept the overall dimensions of the holder reasonable as compared to the use of aluminum for the entire assembly. Finite Element Analysis Design Iterations The ISO A10SCLPR2 NE4 boring bar was modeled and then a harmonic analysis was carried out. Figure 211 shows the results of the ANSYS model for fixedfree boundary conditions. A first natural frequency of 419 Hz was observed. The natural frequency and amplitude of the FE results differed substantially from the analytical model. This is due to: 1) limitations of EulerBernoulli beam theory (shear effects are neglected); and 2) the analytical model assumption of a constant cross section. The FE model overcomes these limitations and it, therefore, more accurate. Mass is removed at the tip where the cutting surface is located as well as hole through the entire length of the bar to allow coolant oil to pass through. Also, since boring bars are designed to be used in tool holders a flat is machined on the cylinder so that set screws may be used to clamp the bar to the tool post. All these factors reduce mass which drives the natural frequency up as shown in the FE model. Since the FE method required an input force for the harmonic analysis, a 1 N force was applied to the tip. This resulted in an amplitude of N/m. Next, several holder design iterations were completed while varying notch size, diameter, and overhang until the FE results 510472. 21 PAGE 22 returned a value close to the 419 Hz reported for the boring bar. The fixedfree holder model, including the cylindrical brass mass (clamped around the holder), is shown in Figure 212. Finally, the two models were rigidly coupled (to approximate the thermal shrink fit). The coupled system showed the split in frequencies about the 419 Hz boring bar natural frequency to 295 Hz and 533 Hz with a reduction in amplitude of 74% to m/N. The results are displayed in Figure 213. 610326. The graph in Figure 213 shows a slight asymmetry in the two modes. This is because the fundamental bending mode natural frequencies of the bar and holder are not exactly the same: 417 Hz for the holder compared to 419 Hz for the (clampedfree) boring bar. If the two systems are not perfectly matched, then, depending on the system characteristics, one mode dominates. This effect was exploited as discussed in Chapter 4, Mass Tuning. 22 PAGE 23 Figure 21. Example 2DOF system. Figure 22. Assembled/component systems. Figure 23. Analytical fixedfree FRF for 0.625 inch diameter boring bar. 23 PAGE 24 Figure 24. Analytical FRF of boring bar coupled to holder. B A Figure 25. Amplitude ratio for assembly. A) As a function of the stiffness factor. B) As a function of the damping factor. 24 PAGE 25 Figure 26. The FRFs for clampedfree boring bar (denoted rigid) and holderbar assembly (denoted new holder) with stiffness and damping factors of 20. Figure 27. The FRFs for clampedfree boring bar (denoted rigid) and holderbar assembly (denoted new holder) with stiffness factor of 100 and damping factor of 1. 25 PAGE 26 B A Figure 28. Sensitivity study. A) Boring bar dynamics. B) Various holder dynamics. The heavy solid lines represent the nominal natural frequencies, while the dotted lines represent the various nonideal holder responses. Figure 29. Reduction in amplitude for boring barholder assembly relative to fixedfree boring bar for range of holder natural frequencies. 26 PAGE 27 Figure 210. Three holder designs: leaf, notch, and circular notch. A B Figure 211. Finite element harmonic analysis of the fixedfree boring bar. A) Model. B) Magnitude versus frequency. 27 PAGE 28 Figure 212. Finite element harmonic analysis of holder. A B Figure 213. Finite element harmonic analysis of assembly. A) Model. B) Magnitude versus frequency. 28 PAGE 29 CHAPTER 3 PROTOTYPE, EXPERIMENTAL SETUP AND RESULTS In this chapter, the manufacturing techniques used to create the prototype are described. The individual components were tested first and then the complete assembly was fabricated and tested. The experimental setup and testing procedures are detailed. Frequency response function (FRF) results are presented. Manufacturing Procedures Justification for material selection is given in Chapter 2. Figure 31 shows the complete model and the assembled prototype with each component labeled. In Figure 31, component A is the aluminum flexurebased holder (a dovetail notch was added so that it could be fixed to a traditional tool post). Component B is the round brass mass that was used for tuning the flexure natural frequency. Component C is the ISO A10SCLPR2 boring bar. Component A: Flexure The flexure was fabricated using a combination of traditional manufacturing processes. An aluminum round was first turned on a lathe in order to obtain the 2 inch diameter selected in the design stage. Next, the heat shrink fit hole was roughed out with standard jobber bits and finished with a 0.6235 inch reamer, providing a 0.0015 inch interference. The circular notches were then machined on a Mikron UCP600 Vario 5axis computer numerically controlled (CNC) milling machine using a standard convex radius cutter. Preformatted circular interpolation subroutines available in the Heidenhain controller (conversational programming format) were used to create the notches. Finally, the flexure was transferred to a manual 3axis milling machine and the dovetail was cut using a standard 60 deg dovetail cutter. Figure 32 shows the final model of the flexure 29 PAGE 30 Component B: Brass Sleeve The brass sleeve was manufactured on the 5axis Mikron milling machine. First, the available subroutines were used to machine out the circular pocket. Next, the 5axis capability of the milling machine was used to rotate the round and machine the rectangular pockets where the 832 socket head cap screws were to be inserted in order to clamp the round to the flexure. This provided a nominally rigid connection while enabling easy change of mass position so that the holder flexure could be properly tuned to the desired natural frequency. The screws holes were predrilled using the 5axis machine and then hand tapped. Finally, a band saw was used to cut a slot partially through the part so that it could be clamped to the aluminum holder. See Figure 33 for the modal and picture of the brass round. Component C: Steel Boring Bar Figure 34 shows the boring bar and the corresponding solid model. This model was used for design purposes. Assembly The aluminum flexure was placed in an oven and allowed to reach a uniform temperature of 300 deg C. The steel boring bar was then inserted into the hole and the assembly was air cooled. This shrink fit procedure provided a nominally rigid connection between the bar and holder. The brass round was then positioned around the flexure and secured in place using the cap screws. Testing Procedure and Results Before the bar was inserted into the flexure, initial FRF tests were performed to verify the design natural frequency. These initial tests were completed by clamping the flexure in a vise to avoid potential complications of the dovetail connection. The boring bar frequency response was also tested separately by clamping it in a large vise. After the bar was shrink fit in the flexure, 30 PAGE 31 test were performed on the assembly (again using a vise to mimic a rigid ground connection). Figure 35 shows the measured FRF of the boring bar and the complete assembly (both held in the same vice). Figure 36 displays a photograph of the test setup. A SDOF response for the boring bar alone is observed in Figure 35 (within the bandwidth of interest). Once the bar is clamped in the holder, the desired 2DOF response with reduced amplitude and increased dynamic stiffness is seen. The results in Figure 35 show a 52.6 % reduction in amplitude for the holder system as compared to the boring bar alone. A second mode with a much smaller amplitude is also seen in the boring bar measurement. This is explained by the asymmetry in the boring bar and imperfect tap test that produced a cross coupling effect. Next a standard lathe tool post holder was fixed in the vice and the assembly was secured to the dove tail; this provided a more realistic test of actual clamping conditions. Figure 37 shows the tool post and the flexurebrass roundboring bar assembly fixed on the tool post. Figure 38 shows the FRF of the assembly when mounted on the tool post (note that the scale is the same as Figure 37). The same 2DOF behavior and an even larger reduction in amplitude is observed. The increased amplitude reduction of 68.4% can possibly be explained by the larger Columbic damping in the dovetail connection. Additional modes are observed as well. These are introduced by tool post holder itself and highlights the need to consider base characteristics in the design for commercial use. 31 PAGE 32 1 A B 2 3 3 1 2 Figure 31. Complete model and prototype. A) Model. B) Prototype. The individual parts are labeled: 1flexible holder, 2brass sleeve, and 3boring bar. Figure 32. Flexure with dovetail. 32 PAGE 33 A B Figure 33. Brass round. A) Solid model. B) Machined prototype. A B Figure 34. Boring bar. A) Model. B) Actual bar. 33 PAGE 34 Figure 35. Measured FRF of the boring bar and assembly. Figure 36. Test setup for measurements. 34 PAGE 35 A B Figure 37: Tool post and assembly. A) Tool post. B) Assembly on tool post. Figure 38. The FRF of assembly clamped to the tool post. 35 PAGE 36 CHAPTER 4 COMPARISON OF FEA, AND PROTOTYPE MASS TUNING TRENDS In this chapter, the mass tuning technique is presented and the advantages over length adjustment are given. FE analysis is applied to investigate mass tuning and the results are compared to prototype measurements. Mass Tuning Introduction Currently, the boring bar length must be shortened or its diameter increased to improve stiffness. This places limitations on the size and depth of holes that can be produced. The new method presented here takes advantage of the coupled system dynamic characteristics. Instead of altering the tool, the base characteristics are changed to yield a coupled system dynamic response that provides acceptable dynamic stiffness. For the dynamic absorber effect exploited here to be effective, the base holder and clamped tool natural frequencies must be similar. Rather than adjusting the tool length or holder length to realize the matched frequencies, we have implemented the mass tuning approach. We modify the modal mass, and not the actual mass, of the base. This enables a quick and easy change of the base system characteristics for each tool change. This is achieved by sliding the brass sleeve along the axis of the base as shown in Figure 41. As the sleeve is moved toward the clamped end the modal mass of the system is reduced and vice versa. FE Analysis The sleeve positions and results for the mass tuning trend from the FE analysis are provided in Figures 42 and 43. In first step, when the modal mass is the lowest (sleeve is closest to the clamped end), the flexure frequency is highest and the system acts approximately as a SDOF system. The mass is shifted away from the clamped end toward the free end in steps 23 and the flexure natural frequency approaches the boring bar natural frequency. The system is 36 PAGE 37 thus tuned and a 2DOF system with a reduced amplitude is observed. As the modal mass is increased beyond the tuned position, step 4, the system once again begins approaching a SDOF system (the holder clamped frequency is now lower than the boring bar clamped frequency). Prototype Measurements Figure 44 shows the results from nine positions of the brass sleeve, each increasing in distance from the clamped end (19) measured on the prototype using the same procedure described in Chapter 3. The trend shown in the FE analysis is again observed. When the sleeve is at a position closest to the clamped end, step 1, the coupled system acts as a SDOF cantilever. As the brass sleeve is shifted away and the system is tuned a 2DOF system is seen. Once the mass is shifted beyond the tuned position, steps 79, the system once again approaches a SDOF system. 37 PAGE 38 Figure 41. Base and sleeve sliding scheme. Figure 42. Finite element models of sleeve position in mass tuning trend analysis. 38 PAGE 39 Figure 43. Finite element results of sleeve position in mass tuning trend analysis. Note the change in scales for the amplitude axis. 39 PAGE 40 Figure 44. Prototype mass tuning trends results. 40 PAGE 41 CHAPTER 5 CONCLUSIONS AND INDUSTRY APPLICATIONS In this chapter conclusions are given. Concepts and ideas for industry application are also presented. Conclusions A flexible tool holder which acts as a dynamic absorber for a boring bar was studied. By introducing flexibility into the holder (using a notched flexure geometry) and matching its fundamental natural frequency to the first cantilever natural frequency of the boring bar, the holder effectively served as a dynamic absorber for the boring bar. An analytical approach was used to select the nominal holder response for an ISO A10SCLPR2 NE4 boring bar with a 0.625 in (15.9 mm) diameter. Finite element models were used to refine the design. Components were machined and frequency response function measurements were completed. Results for the analytical and finite element analysis were then compared to the manufactured prototype. The dynamic stiffness of the holderboring bar assembly was compared to the stiffness of the cantilever boring bar alone. Stiffness improvements up to 68% were observed for the holderboring bar assembly Therefore, the addition of the flexible holder can significantly increase the dynamic stiffness of boring bar which can increase the material removal rate (MMR) without the limitations of the methods currently in practice. Industry Ideas In order for this design to be applied in an industry environment it must be capable of incorporating different sizes, lengths and diameters, of standard boring bars. The concept is to provide one flexible holder with a set of boring bars of varying lengths and diameters. An integral mass slider would be designed that has predetermined positions for the mass. This would enable tuning of the holder for each individual bar. The change would only take seconds 41 PAGE 42 and current design limitations would be eliminated. A further improvement would be to remove the connection with the tool post holder. Instead, the holder itself would be fixed to the lathe in the same way the traditional tool post holder is done. This would avoid the need to consider base characteristics for each individual machine in general since the assembly should be much less stiff than the lathe. A conceptual representation is shown in Figure 51. 42 PAGE 43 Figure 51. Industrial boring bar holder conceptual representation. 43 PAGE 44 APPENDIX A BEAM RECEPTANCE MODELING Bishop and Johnson [2] showed that the displacement and rotationtoforce and moment receptances for uniform EulerBernoulli beams could be represented by simple closedform expressions. For a cylindrical freefree beam with coordinates j and k identified at each end, the frequencydependent direct and cross receptances are given by: 3351FiEIFhhkkjj 3381FiEIFhhkjjk (A1) 3211FiEIFllkkjj 32101FiEIFllkjjk (A2) 3211FiEIFnnkkjj 32101FiEIFnnkjjk (A3) 361FiEIFppkkjj 371FiEIFppkjjk (A4) where E is the elastic modulus, I is the 2nd area moment of inertia, is the frequencyindependent damping coefficient, and: LiEIm124 (A5) LsinhLsinF 1 13 LcoshLcosF LcoshLsinLsinhLcosF 5 LcoshLsinLsinhLcosF 6 LsinhLsinF 7 LsinhLsinF 8 LcoshLcosF 10 (A6) 44 PAGE 45 In Equations (A5), the cylindrical beam mass is given by 422Lddmio where do is the outer diameter, di is the inner diameter (set equal to zero if the beam is not hollow), L is the length, and is the density; the cylinders 2nd area moment of inertia is 6444ioddI ; and is the frequency (in rad/s). 45 PAGE 46 APPENDIX B MATLAB CODE Boring Bar Design % % boring_bar_design.m %% Lonnie A. Houck III (6/10/2004) (Modified) % % T. Schmitz (4/28/03) % % clc % clear all % close all % % % Clampedfree beam (Boring bar alone to determine correct natural frequency and baseline stiffness and damping) % d = 15.875e3; % tool diameter, m % L = 6*d; % overhang, m % density = 7800; % density, kg/m^3 % m = density*pi*d^2/4*L; % mass, kg % E = 2e11; % highspeed steel modulus, N/m^2 % I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 % eta = 0.0015; % structural damping factor for base (estimated) % EI = E*I*(1 + i*eta); % complex structural stiffness % % w = (1:0.1:2e3)'*2*pi; % Lw = ((w.*w)*m/(L*EI)).^(1/4); % lambda(omega) % LwL = Lw*L; % lambda(omega) beam length % EILw1 = EI*Lw; % complex structural stiffness beam length % EILw2 = EI*Lw.^2; % complex structural stiffness beam length^2 % EILw3 = EI*Lw.^3; % complex structural stiffness beam length^3 % % F1 = sin(LwL).*sinh(LwL); % F4 = cos(LwL).*cosh(LwL) + 1; % F5 = cos(LwL).*sinh(LwL) sin(LwL).*cosh(LwL); % F6 = cos(LwL).*sinh(LwL) + sin(LwL).*cosh(LwL); % % H11 = F5./(EILw3.*F4); % % % Fit to clampedfree response % wnx1 = 1239.55*2*pi; % zetax1 = eta/2; % kx1 = 2.23e6; % fit = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); 46 PAGE 47 % % figure(1) % subplot(211) % plot(w/2/pi, real(H11), 'r', w/2/pi, real(fit), 'b:') % legend('Clampedfree', 'Fit') % ylabel('Real (m/N)') % subplot(212) % plot(w/2/pi, imag(H11), 'r', w/2/pi, imag(fit), 'b:') % ylabel('Imag (m/N)') % xlabel('Frequency (Hz)') % clear all %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all clc % Redefine rigidly clamped boring bar for comparison to holder base + freefree boring bar f = (1:0.1:2e3)'; w = f*2*pi; d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 eta = 0.0015; % structural damping factor for base (estimated) EI = E*I*(1 + i*eta); % complex structural stiffness Lw = ((w.*w)*m/(L*EI)).^(1/4); % lambda(omega) LwL = Lw*L; % lambda(omega) beam length EILw1 = EI*Lw; % complex structural stiffness beam length EILw2 = EI*Lw.^2; % complex structural stiffness beam length^2 EILw3 = EI*Lw.^3; % complex structural stiffness beam length^3 F1 = sin(LwL).*sinh(LwL); F4 = cos(LwL).*cosh(LwL) + 1; F5 = cos(LwL).*sinh(LwL) sin(LwL).*cosh(LwL); F6 = cos(LwL).*sinh(LwL) + sin(LwL).*cosh(LwL); H33 = F5./(EILw3.*F4); 47 PAGE 48 %Design Parameters damping_factor = 20; stiffness_factor = 20; max_amp_stiff = zeros(stiffness_factor,1); max_amp_damp = zeros(damping_factor,1); for cnt=1:stiffness_factor; % Define holder damping_factor = 3; stiffness_factor = cnt; wnx1 = 305.6*2*pi; %1239.55 zetax1 = eta/2*damping_factor; kx1 = 2.23e6*stiffness_factor; h33 = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); l33 = zeros(1, length(w)); % Assume all other receptances are zero n33 = zeros(1, length(w)); p33 = zeros(1, length(w)); h44 = zeros(1, length(w)); l44 = zeros(1, length(w)); n44 = zeros(1, length(w)); p44 = zeros(1, length(w)); h34 = zeros(1, length(w)); l34 = zeros(1, length(w)); n34 = zeros(1, length(w)); p34 = zeros(1, length(w)); h43 = zeros(1, length(w)); l43 = zeros(1, length(w)); n43 = zeros(1, length(w)); p43 = zeros(1, length(w)); % Define freefree boring bar d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 eta = 0.0015; % structural damping factor EI = E*I*(1 + i*eta); % complex structural stiffness % Determine freefree receptances 48 PAGE 49 [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); % Couple base and boring bar [ha11, la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); % % Couple two freefree subcomponents % [row col] = size(f); % for n = 1:row % R11 = [h11(n) l11(n); n11(n) p11(n)]; % R12 = [h12(n) l12(n); n12(n) p12(n)]; % R21 = [h21(n) l21(n); n21(n) p21(n)]; % R22 = [h22(n) l22(n); n22(n) p22(n)]; % % R33 = [h33(n) l33(n); n33(n) p33(n)]; % % G11 = R11 R12*(R22+R33)^1*R21; % ha11(n) = G11(1,1); % end max_amp_stiff(cnt) = abs(min(imag(ha11))); cnt end H33_1 = H33; ha11_1 = ha11; figure(1) subplot(211) plot(f, real(H33_1), 'r', f, real(ha11_1), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') legend('Rigid', 'New holder','Location','NorthWest') ylabel('Real (m/N)','FontSize',12, 'Fontweight', 'bold') axis([000 400 1.4e3 1.4e3]) subplot(212) plot(f, imag(H33_1), 'r', f, imag(ha11_1), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') axis([000 400 3.5e003 .0005]) tic 49 PAGE 50 for cnt_d=1:20; % Define holder damping_factor = cnt_d; stiffness_factor = 11; wnx1 = 305.6*2*pi; %1239.55 zetax1 = eta/2*damping_factor; kx1 = 2.23e6*stiffness_factor; h33 = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); l33 = zeros(1, length(w)); % Assume all other receptances are zero n33 = zeros(1, length(w)); p33 = zeros(1, length(w)); h44 = zeros(1, length(w)); l44 = zeros(1, length(w)); n44 = zeros(1, length(w)); p44 = zeros(1, length(w)); h34 = zeros(1, length(w)); l34 = zeros(1, length(w)); n34 = zeros(1, length(w)); p34 = zeros(1, length(w)); h43 = zeros(1, length(w)); l43 = zeros(1, length(w)); n43 = zeros(1, length(w)); p43 = zeros(1, length(w)); % Define freefree boring bar d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 eta = 0.0015; % structural damping factor EI = E*I*(1 + i*eta); % complex structural stiffness % Determine freefree receptances [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); % Couple base and boring bar [ha11, la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); 50 PAGE 51 % % Couple two freefree subcomponents % [row col] = size(f); % for n = 1:row % R11 = [h11(n) l11(n); n11(n) p11(n)]; % R12 = [h12(n) l12(n); n12(n) p12(n)]; % R21 = [h21(n) l21(n); n21(n) p21(n)]; % R22 = [h22(n) l22(n); n22(n) p22(n)]; % % R33 = [h33(n) l33(n); n33(n) p33(n)]; % % G11 = R11 R12*(R22+R33)^1*R21; % ha11(n) = G11(1,1); % end max_amp_damp(cnt_d) = abs(min(imag(ha11))); cnt_d end H33_2 = H33; ha11_2 = ha11; cnt = 1:20; cnt_d = 1:20; figure(2) subplot(211) plot(f, real(H33_2), 'r', f, real(ha11_2), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') legend('Rigid', 'New holder','Location','NorthWest') ylabel('Real (m/N)','FontSize',12, 'Fontweight', 'bold') axis([000 400 1.4e3 1.4e3]) subplot(212) plot(f, imag(H33_2), 'r', f, imag(ha11_2), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') axis([000 400 3.5e003 .0005]) figure(3) subplot(211) plot(cnt, max_amp_stiff, 'r') set(gca,'FontSize',12, 'Fontweight', 'bold') legend('Varying Stiffness') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') subplot(212) 51 PAGE 52 plot(cnt_d, max_amp_damp, 'b') set(gca,'FontSize',12, 'Fontweight', 'bold') legend('Varying Damping') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlabel('Factor','FontSize',12, 'Fontweight', 'bold') %Our Holder % Define holder damping_factor = 1; stiffness_factor = 100; wnx1 = 305.6*2*pi; %1239.55 zetax1 = eta/2*damping_factor; kx1 = 2.23e6*stiffness_factor; h33 = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); l33 = zeros(1, length(w)); % Assume all other receptances are zero n33 = zeros(1, length(w)); p33 = zeros(1, length(w)); h44 = zeros(1, length(w)); l44 = zeros(1, length(w)); n44 = zeros(1, length(w)); p44 = zeros(1, length(w)); h34 = zeros(1, length(w)); l34 = zeros(1, length(w)); n34 = zeros(1, length(w)); p34 = zeros(1, length(w)); h43 = zeros(1, length(w)); l43 = zeros(1, length(w)); n43 = zeros(1, length(w)); p43 = zeros(1, length(w)); % Define freefree boring bar d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 eta = 0.0015; % structural damping factor EI = E*I*(1 + i*eta); % complex structural stiffness % Determine freefree receptances [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); % Couple base and boring bar 52 PAGE 53 [ha11, la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); figure(4) subplot(211) plot(f, real(H33), 'r', f, real(ha11), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') legend('Rigid', 'New holder','Location','NorthWest') ylabel('Real (m/N)','FontSize',12, 'Fontweight', 'bold') axis([200 400 1.4e3 1.4e3]) subplot(212) plot(f, imag(H33), 'r', f, imag(ha11), 'b:') set(gca,'FontSize',12, 'Fontweight', 'bold') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') axis([200 400 3.5e003 .0005]) toc FreeFree Response of Boring Bar % bishop_free_free.m % T. Schmitz, G.S. Duncan (12/13/03) % Function to calculate freefree receptances % Coordinate system: 1 is Bishop's 'l' end, 2 is '0' end; by convention 1 will be the right end, 2 the left end function [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, length, m); w = f*2*pi; % frequency, rad/s lambda = (w.^2*m/(length*EI)).^0.25; lambda_length = lambda*length; F1 = sin(lambda_length).*sinh(lambda_length); F2 = cos(lambda_length).*cosh(lambda_length); F3 = cos(lambda_length).*cosh(lambda_length) 1; F4 = cos(lambda_length).*cosh(lambda_length) + 1; F5 = cos(lambda_length).*sinh(lambda_length) sin(lambda_length).*cosh(lambda_length); F6 = cos(lambda_length).*sinh(lambda_length) + sin(lambda_length).*cosh(lambda_length); F7 = sin(lambda_length) + sinh(lambda_length); 53 PAGE 54 F8 = sin(lambda_length) sinh(lambda_length); F9 = cos(lambda_length) + cosh(lambda_length); F10 = cos(lambda_length) cosh(lambda_length); EIlam = EI*lambda; EIlam2 = EI*lambda.^2; EIlam3 = EI*lambda.^3; h11 = F5./(EIlam3.*F3); l11 = F1./(EIlam2.*F3); n11 = l11; p11 = F6./(EIlam.*F3); h22 = F5./(EIlam3.*F3); l22 = F1./(EIlam2.*F3); n22 = l22; p22 = F6./(EIlam.*F3); h12 = F8./(EIlam3.*F3); l12 = F10./(EIlam2.*F3); n12 = F10./(EIlam2.*F3); p12 = F7./(EIlam.*F3); h21 = h12; l21 = n12; n21 = l12; p21 = p12; % figure(1) % subplot(211) % plot(f, real(h22)) % title('h22 and h11') % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(h22)) % title('h22 and h11') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) % % figure(2) % subplot(211) % plot(f,real(n22),f,real(n11)) % title('n22/l22 and n11/l11') 54 PAGE 55 % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % legend('n22/l22','n11/l11') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(n22),f,imag(n11)) % title('n22/l22 and n11/l11') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) % legend('n22/l22','n11/l11') % % figure(3) % subplot(211) % plot(f,real(h12)) % title('h12/h21') % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(h12)) % title('h12/h21') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) % % figure(4) % subplot(211) % plot(f,real(n12),f,real(l12)) % title('n12/l21 and l12/n21') % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % legend('n12/l21','l12/n21') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(n12),f,imag(l12)) % title('n12/l21 and l12/n21') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) % legend('n12/l21','l12/n21') % % figure(5) % subplot(211) % plot(f,real(p22)) % title('p22/p11') 55 PAGE 56 % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(p22)) % title('p22/p11') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) % % figure(6) % subplot(211) % plot(f,real(p12)) % title('p12/p21') % xlabel('Frequency (Hz)') % ylabel('Real (m/N)') % xlim([50 max(f)]) % subplot(212) % plot(f,imag(p12)) % title('p12/p21') % xlabel('Frequency (Hz)') % ylabel('Imaginary (m/N)') % xlim([50 max(f)]) Program to Couple to FreeFree Beams Together function [h11, l11, n11, p11, h44, l44, n44, p44, h14, l14, n14, p14, h41, l41, n41, p41] = couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); % Couple two freefree subcomponents [N col] = size(f); h11=reshape(h11,1,1,N); l11=reshape(l11,1,1,N); n11=reshape(n11,1,1,N); p11=reshape(p11,1,1,N); h12=reshape(h12,1,1,N); l12=reshape(l12,1,1,N); n12=reshape(n12,1,1,N); p12=reshape(p12,1,1,N); h21=reshape(h21,1,1,N); 56 PAGE 57 l21=reshape(l21,1,1,N); n21=reshape(n21,1,1,N); p21=reshape(p21,1,1,N); h22=reshape(h22,1,1,N); l22=reshape(l22,1,1,N); n22=reshape(n22,1,1,N); p22=reshape(p22,1,1,N); h33=reshape(h33,1,1,N); l33=reshape(l33,1,1,N); n33=reshape(n33,1,1,N); p33=reshape(p33,1,1,N); h34=reshape(h34,1,1,N); l34=reshape(l34,1,1,N); n34=reshape(n34,1,1,N); p34=reshape(p34,1,1,N); h43=reshape(h43,1,1,N); l43=reshape(l43,1,1,N); n43=reshape(n43,1,1,N); p43=reshape(p43,1,1,N); h44=reshape(h44,1,1,N); l44=reshape(l44,1,1,N); n44=reshape(n44,1,1,N); p44=reshape(p44,1,1,N); RS11 = [h11 l11; n11 p11]; RS12 = [h12 l12; n12 p12]; RS21 = [h21 l21; n21 p21]; RS22 = [h22 l22; n22 p22]; RS33 = [h33 l33; n33 p33]; RS34 = [h34 l34; n34 p34]; RS43 = [h43 l43; n43 p43]; RS44 = [h44 l44; n44 p44]; R11=zeros(2,2,N); R41=zeros(2,2,N); R14=zeros(2,2,N); R44=zeros(2,2,N); for n=1:N R11(:,:,n) = RS11(:,:,n) RS12(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS21(:,:,n)); 57 PAGE 58 R41(:,:,n) = RS43(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS21(:,:,n)); R14(:,:,n) = RS12(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS34(:,:,n)); R44(:,:,n) = RS44(:,:,n) RS43(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS34(:,:,n)); end h11 = R11(1,1,:); l11 = R11(1,2,:); n11 = R11(2,1,:); p11 = R11(2,2,:); h14 = R14(1,1,:); l14 = R14(1,2,:); n14 = R14(2,1,:); p14 = R14(2,2,:); h41 = R41(1,1,:); l41 = R41(1,2,:); n41 = R41(2,1,:); p41 = R41(2,2,:); h44 = R44(1,1,:); l44 = R44(1,2,:); n44 = R44(2,1,:); p44 = R44(2,2,:); h11=reshape(h11,1,N); l11=reshape(l11,1,N); n11=reshape(n11,1,N); p11=reshape(p11,1,N); h14=reshape(h14,1,N); l14=reshape(l14,1,N); n14=reshape(n14,1,N); p14=reshape(p14,1,N); h41=reshape(h41,1,N); l41=reshape(l41,1,N); n41=reshape(n41,1,N); p41=reshape(p41,1,N); h44=reshape(h44,1,N); l44=reshape(l44,1,N); n44=reshape(n44,1,N); p44=reshape(p44,1,N); 58 PAGE 59 Program to Read Impact Test Data and Determine Modal Response % Lonnie Houck % 5/18/06 % Reads TXF Files used in Boring Bar application and plots results % Needs file txfnew.m clear all close all clc % Read in File Names and transform in Txf subroutine [f AAA] = Txfnew('bar'); % Eliminate values below 100 hertz index = find(f > 100); f = f(index); AAA = AAA(index); HAA = AAA; figure(1) subplot(211) plot(f,real(HAA)) set(gca,'FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') ylabel('Real (m/N)','FontSize',12, 'Fontweight', 'bold') axis([100 1000 2.5e4 2.5e4]) subplot(212) plot(f,imag(HAA)) set(gca,'FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') axis([100 1000 4e004 .0001]) Code for Sensitivity Analysis %%Lonnie Houck %MATLAB Code for Sensitivity Analysis 59 PAGE 60 clear all close all clc %Define Paramters f = (1:0.005:2e3)'; %Hz w = f*2*pi; %rad/s eta=.0015; % structural damping factor for base (estimated) %% Define Fixed Free Holder damping_factor = 1; stiffness_factor = 100; wnx1 = 305.6*2*pi; %1239.55 zetax1 = eta/2*damping_factor; kx1 = 2.23e6*stiffness_factor; h33 = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); l33 = zeros(1, length(w)); % Assume all other receptances are zero n33 = zeros(1, length(w)); p33 = zeros(1, length(w)); h44 = zeros(1, length(w)); l44 = zeros(1, length(w)); n44 = zeros(1, length(w)); p44 = zeros(1, length(w)); h34 = zeros(1, length(w)); l34 = zeros(1, length(w)); n34 = zeros(1, length(w)); p34 = zeros(1, length(w)); h43 = zeros(1, length(w)); l43 = zeros(1, length(w)); n43 = zeros(1, length(w)); p43 = zeros(1, length(w)); %% Define Fixed Free Boring Bar d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 EI = E*I*(1 + i*eta); % complex structural stiffness Lw = ((w.*w)*m/(L*EI)).^(1/4); % lambda(omega) LwL = Lw*L; % lambda(omega) beam length EILw1 = EI*Lw; % complex structural stiffness beam length EILw2 = EI*Lw.^2; % complex structural stiffness beam length^2 EILw3 = EI*Lw.^3; % complex structural stiffness beam length^3 F1 = sin(LwL).*sinh(LwL); F4 = cos(LwL).*cosh(LwL) + 1; F5 = cos(LwL).*sinh(LwL) sin(LwL).*cosh(LwL); F6 = cos(LwL).*sinh(LwL) + sin(LwL).*cosh(LwL); H33 = F5./(EILw3.*F4); 60 PAGE 61 %%Plot The 2 Fixed Free Responses figure(1) subplot(211) plot(f, imag(H33), 'r','LineWidth',3) set(gca,'FontSize',12, 'Fontweight', 'bold') legend('FixedFree Boring Bar', 'Location','NorthEast') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlim([200 400]) subplot(212) plot(f, imag(h33), 'b','LineWidth',3) set(gca,'FontSize',12, 'Fontweight', 'bold') legend('FixedFree Holder', 'Location','NorthEast') ylabel('Imag (m/N)','FontSize',12, 'Fontweight', 'bold') xlabel('Frequency (Hz)','FontSize',12, 'Fontweight', 'bold') xlim([150 450]) hold on %% Vary Holder Between +/30%*wnbar low_f = 0.7*305.6; high_f = 1.3*305.6; f_compare = linspace(low_f,high_f,33); h33 = zeros(length(w),length(f_compare)); for l = 1:length(f_compare) damping_factor = 1; stiffness_factor = 100; wnx1 = f_compare(l)*2*pi; %1239.55 zetax1 = eta/2*damping_factor; kx1 = 2.23e6*stiffness_factor; h33(:,l) = (wnx1^2/kx1)./(wnx1^2 w.^2 + i*2*zetax1*wnx1.*w); l33 = zeros(1, length(w)); % Assume all other receptances are zeron33 = zeros(1, length(w)); p33 = zeros(1, length(w)); h44 = zeros(1, length(w)); l44 = zeros(1, length(w)); n44 = zeros(1, length(w)); p44 = zeros(1, length(w)); h34 = zeros(1, length(w)); l34 = zeros(1, length(w)); n34 = zeros(1, length(w)); p34 = zeros(1, length(w)); h43 = zeros(1, length(w)); l43 = zeros(1, length(w)); n43 = zeros(1, length(w)); p43 = zeros(1, length(w)); end subplot(212) plot(f, imag(h33), 'b:') hold on %% Couple base and boring bar % Redefine freefree boring bar d = 15.875e3; % tool diameter, m L = .1917; % overhang, m density = 7800; % density, kg/m^3 61 PAGE 62 m = density*pi*d^2/4*L; % mass, kg E = 2e11; % highspeed steel modulus, N/m^2 I = pi/64*(d)^4; % 2nd area moment of inertia, m^4 eta = 0.0015; % structural damping factor EI = E*I*(1 + i*eta); % complex structural stiffness ha11 = zeros(length(w),length(f_compare)); for l = 1:length(f_compare) % Determine freefree receptances [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); [ha11(:,l), la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = Couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33(:,l), l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); end for k=1:length(f_compare) max_response1(k) = max(imag(ha11(:,k))); end eta=.0020; EI = E*I*(1 + i*eta); % complex structural stiffness for l = 1:length(f_compare) % Determine freefree receptances [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); [ha11(:,l), la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = Couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33(:,l), l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); end for k=1:length(f_compare) max_response2(k) = max(imag(ha11(:,k))); end eta=.0025; EI = E*I*(1 + i*eta); % complex structural stiffness for l = 1:length(f_compare) % Determine freefree receptances [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = bishop_free_free(f, EI, L, m); [ha11(:,l), la11, na11, pa11, ha44, la44, na44, pa44, ha14, la14, na14, pa14, ha41, la41, na41, pa41] = Couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33(:,l), l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); end for k=1:length(f_compare) max_response3(k) = max(imag(ha11(:,k))); 62 PAGE 63 end max_bar = ones(length(f_compare),1); max_bar = max(imag(H33))*max_bar; figure(2) %mesh(f_compare/305.6,f,imag(ha11)) plot(f_compare/305.6, 1(max_response1/max(imag(H33))),f_compare/305.6, 1(max_response2/max(imag(H33))),f_compare/305.6, 1(max_response3/max(imag(H33)))); %plot(f_compare/305.6,max_bar,'r','LineWidth',3) 63 PAGE 64 LIST OF REFERENCES S. Rao, Mechanical Vibrations, 4th Ed., Pearson Education, 2004. R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, Cambridge, 1960. J. Ferreira, D. Ewins, Nonlinear receptance coupling approach based on describing functions, in: Proceedings of the 14th International Modal Analysis Conference, Dearborn, MI, 10341040, 1995. J. Ferreira, Transfer Report on Research Dynamic Response Analysis of Structures with Nonlinear Components, Internal ReportDynamics Section, Imperial College, London, UK, 1996. Y. Ren, C. Beards, A generalized receptance coupling technique, in: Proceedings of the 11th International Modal Analysis Conference, Kissimmee, FL, 868871, 1993. Klosterman, W. McClelland, I. Sherlock, Dynamic simulation of complex systems utilizing experimental and analytical techniques, ASME 75WA/Aero9 (1977) 114. D. Ewins, Modal Testing: Theory and Practice, Research Studies Press, Ltd., Somerset, England, 1995. G.S. Duncan, M.F. Tummond, T.L. Schmitz, An investigation of the dynamic absorber effect in highspeed machining, International Journal of Machine Tools & Manufacture 45 (2005) 497507. 64 PAGE 65 BIOGRAPHICAL SKETCH Lonnie A Houck III was born in the small town of Perry, FL, about 90 miles north of Gainesville. Upon graduating from Taylor County High School, Lonnie enrolled in the undergraduate mechanical engineering program at the University of Florida. As a junior, he began research in the Machine Tool Research Center (MTRC). He graduated with a Bachelor of Science degree in 2006 and continued his graduate education in the MTRC. He received his Master of Science degree in 2009 from the University of Florida. Currently, Lonnie is working for Alstom Power as the technical authority of all structures and casings for the industrial gas turbine division. 65 