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Page i Dedication Page ii Acknowledgement Page iii Table of Contents Page iv Page v Page vi List of Tables Page vii List of Figures Page viii Page ix Page x Key to abbreviations Page xi Abstract Page xii Page xiii 1. Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 2. Genetic algorithms: Theory and application Page 9 Page 10 Page 11 Page 12 Page 13 3. Finite element model refinement using genetic algorithms Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 4. Modal test excitation and sensor placement: Current techniques Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 5. Modal test excitation and sensor placement: New techniques Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 6. Premodal test planning algorithm application: NASA eightbay truss Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 7. Premodal test planning algorithm application: Microprecision interferometer truss Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 8. Premodal test planning application: Car body Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 9. Conclusions and future work Page 105 Page 106 Page 107 References Page 108 Page 109 Page 110 Page 111 Page 112 Biographical sketch Page 113 Page 114 Page 115 Page 116 

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VIBRATION TESTING BY DESIGN: EXCITATION AND SENSOR PLACEMENT USING GENETIC ALGORITHMS By CINNAMON BUCKELS LARSON 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 1996 To Jim and Audrey ACKNOWLEDGEMENTS I would like to sincerely thank my advisor, Dr. David Zimmerman, for all of the support, advice, and knowledge he has given to me. He has worked hard on my behalf, obtaining funding for my support and providing me with several research opportunities. He gave me the guidance to learn and the room to grow. I will forever be indebted to him. I would like to thank my husband, Jim, his parents, and my daughter, Audrey. Without their love and support I never would have made it. I would like to thank my sisters, Beth, Kim, Cynthia, and Erin, my brother Laing, and my best friend Leslie for their neverending encouragement and love. I am truly blessed. I would like thank my dear friends Mohamed Kaouk and William Leath for their advice, support, and companionship through graduate school. I would also like to thank the entire staff and faculty of the Aerospace Engineering, Mechanics, and Engineering Science department. Specifically, I would like to acknowledge my committee, Dr. Norman FitzCoy, Dr. Daniel Drucker, Dr. Marc Hoit, Dr. Peter Ifju, and Dr. Bavani Sankar for their advice. I would like to acknowledge Sandia National Laboratories for their financial support and for the research opportunities they have given me. Specifically, I would like to thank Ed Marek, Clay Fulcher, and Scott Klenke. I would also like to acknowledge General Motors for providing data for my studies. I would like to thank the Florida/NASA Space Grant Consortium whose financial support made my graduate studies possible. TABLE OF CONTENTS page ACKNOWLEDGEMENTS ........................................... iii LIST OF TABLES ................................................ vii LIST OF FIGURES ............................................... viii KEY TO ABBREVIATIONS ......................................... xi ABSTRACT ....................................................... xii CHAPTERS 1 INTRODUCTION ...................................... ...... 1 1.1 Finite Element Model Refinement ............................. 2 1.2 Modal Testing: Sensor and Actuator Placement .................. 5 1.3 Current Study Objective .................................... 7 2 GENETIC ALGORITHMS: THEORY AND APPLICATION .......... 9 3 FINITE ELEMENT MODEL REFINEMENT USING GENETIC ALGORITHMS ..................................... 14 3.1 Introduction ........................................... 14 3.2 Model Refinement Problem Formulation ....................... 14 3.3 Genetic Algorithm Application ............................... 15 3.4 Numerical Example: Six Bay Truss FEM ...................... 17 3.4.1 M odel Refinement .................................... 18 3.4.2 Effect of Noise .................. ..................... 20 3.5 Conclusions ........................................... 22 4 MODAL TEST EXCITATION AND SENSOR PLACEMENT: CURRENT TECHNIQUES ..................................... 25 4.1 Introduction ...................................... 4.2 Effective Independence ............................. 4.3 Kinetic Energy .................................... 4.4 Eigenvector Product .............................. 4.5 Driving Point Residue .............................. 5 MODAL TEST EXCITATION AND SENSOR PLACEMENT: NEW TECHNIQUES .................................. 5.1 Introduction ................ 5.2 Mode Indicator Function ........ 5.2.1 Excitation Placement ...... 5.2.2 Sensor Placement ........ 5.3 Observability and Controllability 5.3.1 Excitation Placement ...... 5.3.2 Sensor Placement ......... 6 PREMODAL TEST PLANNING ALGORITHM APPLICATION: NASA EIGHTBAY TRUSS .............................. .... 49 6.1 Introduction ...................... 6.2 NASA EightBay Test Bed .......... 6.2.1 Excitation Placement .......... 6.2.2 Sensor Placement ............. 6.2.3 Results: Random Sensor Location 6.3 Computational Efficiency ........... 6.4 Conclusion ..................... 7 PREMODAL TEST PLANNING ALGORITHM APPLICATION: MICROPRECISION INTERFEROMETER TRUSS ................. 7.1 Introduction ................... .................... 7.2 MicroPrecision Interferometer Test Bed ....................... 7.3 Excitation Placement ...................................... 7.4 Sensor Placement ......................................... 7.4.1 Unconstrained Sensor Placement ........................ ....... 25 ....... 25 ....... 29 ....... 31 ....... 3 1 ....... 35 7.4.2 Triaxially Constrained Sensor Placement .................. 76 7.4.3 Unconstrained vs. TriaxiallyConstrained Sensor Sets ........ 80 7.5 Effect of Model Error ....................................... 80 7.5.1 Excitation Placement with Model Error ................... 82 7.5.2 Sensor Placement with Model Error ...................... 83 7.6 Computational Cost ....................................... 88 7.7 Conclusions ............................... ............. 89 8 PREMODAL TEST PLANNING APPLICATION: CAR BODY ................................... ............ 5 95 8.1 Introduction ..................................... ....... 95 8.2 Excitation Placement ....................................... 95 8.3 Sensor Placement ................... ........ ........... 102 9 CONCLUSIONS AND FUTURE WORK .......................... 105 REFERENCES ....................................... ........... 108 BIOGRAPHICAL SKETCH .......................................... 112 LIST OF TABLES Table pa 5.1 GMIF Design Variable Description .................................. 38 6.1 EightBay Truss Frequencies and Mode Description .................... 50 6.2 Percent Difference in FE and Identified Frequencies .................... 57 6.3 Total Floating Point Calculations for Each Placement Technique .......... 60 7.1 Reduced MPI FEM Frequencies Compared with MPI Modal Test Frequencies .................................... ...... 64 7.2 Original, GMIF, and GCON Excitation Locations MIF Values ............ 66 7.3 Difference Between Pre and PostCorrupted Model Frequencies and Mode Shapes ..................................... 82 7.4 Number of Sensor or Triax Sets That Change W hen M odel Error Is Added ...................................... 87 7.5 Floating Point Calculations for MPI Sensor and Excitation Placement ...... 89 8.1 Car Body Excitation Location and Orientation ........................ 96 8.2 Mode Indicator Function Values for Various Excitation Placements ........ 97 8.3 Controllability Angles (in degrees) for Excitation Placements ............. 101 8.4 Excitation Placement Techniques Floating Point Calculations ............. 101 8.5 Triaxial Sensor Placement Techniques Floating Point Calculations ......... 104 LIST OF FIGURES Figure page 1.1 Finite Element Modeling ........................................ 2 1.2 Finite Element Model Refinement ................................. 3 1.3 M odal Testing ........................................... ..... 6 2.1 Genetic Algorithms as Robust Problem Solvers ....................... 9 2.2 Coding of a Four Design Variable Problem .......................... 11 2.3 CrossOver Examples ........................................... 12 2.4 Genetic Algorithm Flow Chart .................................... 13 3.1 Six Bay Truss with 25 DOFs ..................................... 17 3.2 Generational Data, Measured Modes with No Noise ................... 19 3.3 Generational Eigensolution Data, Measured Modes with No Noise ....... 21 3.4 FRF after 0, 5, 10, and 20 Generations, Measured Modes with No Noise ... 21 3.5 Generational Data, Measured Modes with 15% Noise .................. 23 3.6 Generational Eigensolution Data, Measured Modes with 15% Noise ...... 23 3.7 FRF After 20 Generations, Measured Modes with 15% Noise ........... 24 4.1 Typical Driving Point Residue (NASA 8Bay Truss) ................... 34 5.1 Typical M IF Plot ................. ...... .... ...... ............ 37 5.2 Excitation Selection by GM IF ..................................... 38 5.3 State Space Variable Description .................................. 42 5.4 Controllability and Observability ................................. 44 6.1 NASA 8Bay Truss ............................................. 50 6.2 EightBay Excitation Locations ........................... ........ 51 6.3 EightBay Excitation Locations Frequency Response of Time Domain Data ............................................. 53 6.4 EightBay Excitation Placement CrossOrthogonality of Identified and FEM Modes 1 to 5 .................................. 53 6.5 EightBay Sensor Locations ...................................... 55 6.6 EightBay Sensor Placement CrossOrthogonality of Identified and FEM Modes 1 to 5 ......................... ......... 56 6.7 EightBay CrossOrthogonalities of Five Techniques Compared to 300 Random Sensor Sets ................. ...... ....... .... ....... ... 59 7.1 MPI Structure .................... .......... .... ............. 62 7.2 Excitation Placement on MPI Structure ............................. 63 7.3 Typical Frequency Response for MPI Structure ....................... 65 7.4 Comparison of Selected Excitation and Random Excitation MIF Values ... 68 7.5 Comparison of Selected Excitation and Random Excitation Controllability Angles .......................................... 69 7.6 CrossOrthogonality Between FE Modes and Identified Modes .......... 70 7.7 Unconstrained MPI Sensor Sets ................................... 72 7.8 CrossOrthogonality Between MPI FE and Identified Modes, 18 Unconstrained Sensors ...................................... 74 7.9 Triaxially Constrained MPI Sensor Sets ................ ....... . 77 7.10 CrossOrthogonality Between MPI FE and Identified Modes, 6 Triaxially Constrained Sensors .................................. 78 7.11 Model Error Added to MPI FEM ................................. 81 7.12 True vs. Corrupted MPI Mode Shapes .............. ............. .. 81 7.13 GMIF Derived Excitation Locations ................................ 83 7.14 GCON Derived Excitation Locations ............................... 84 7.15 Model Error Effect on Unconstrained MPI Sensor Sets ................. 85 7.16 Model Error Effect on Triaxially Constrained MPI Sensor Sets ........... 86 7.17 CrossOrthogonality Between MPI Identified Modes Using Corrupted Model and Uncorrupted FEM Modes (18 Unconstrained Sensors) .............. 91 7.18 CrossOrthogonality Between MPI Identified Modes Using Corrupted Model and Uncorrupted FEM Modes (6 Triaxially Constrained Sensors) ... 93 8.1 Car Body Shaker Locations ....................................... 96 8.2 MIF Values for Excitation Placements Compared to 500 Randomly Located 3Point Excitations ...................................... 99 8.3 Controllability Values for Excitation Placements Compared to 500 Randomly Located 3Point Excitations .................. .......... 100 8.4 Triaxially Located Car Body Sensor Sets ............... ............. 102 KEY TO ABBREVIATIONS AKE ..... ARS ...... DOF ...... DPR ...... El ........ ERA ...... EVP ..... FEM ..... FRF ...... GA ....... GCON .... GMIF ..... GMRA .... KE . ..... MIF ...... MPI ...... OBS ...... average kinetic energy average random sample degree of freedom drivingpoint residue effective independence eigensystem realization algorithm eigenvector product finite element model frequency response function genetic algorithm genetic controllability algorithm genetic mode indicator function algorithm genetic model refinement algorithm kinetic energy mode indicator function microprecision interferometer observability Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy VIBRATION TESTING BY DESIGN: EXCITATION AND SENSOR PLACEMENT USING GENETIC ALGORITHMS By Cinnamon Buckels Larson May 1996 Chairperson: Dr. David C. Zimmerman Major Department: Aerospace Engineering, Mechanics and Engineering Science This dissertation is an investigation of the use of genetic algorithms for the purposes of finite element model refinement and premodal test planning. The objective of a model refinement technique is to use information, about a structure, obtained during a vibration test to update the analytical model. The product of this process is an updated model of a structure which possesses dynamic properties closer to the dynamics obtained from the modal test of the structure. A genetic algorithm is used to vary finite element structural parameters to obtain an updated model with measured modal properties. Although one purpose of a modal test is to use the information to update finite element models, the information obtained may be used for other purposes such as damage assessment, critical loads and frequency determination, and vibration control design. The type of information to be realized from a vibration test may well govern how and where the structure is excited and observed. The principal purpose of the current work is to explore the subject of premodal test planning for excitation and sensor placement. An overview of several existing sensor and excitation placement techniques is presented as a platform for the current study. The sensor placement techniques include effective independence, kinetic energy, and eigenvector product and the excitation placement techniques include eigenvector product, kinetic energy, and drivingpoint residue. Two new sensor and two new excitation placement techniques are developed using normal mode indicator functions, and the concept of modal controllability and observability along with genetic algorithms. The new and existing techniques are compared using three finite element models: the NASA eightbay truss, the Jet Propulsion Laboratory MicroPrecision Interferometer test bed, and a car body. CHAPTER 1 INTRODUCTION The area of structural engineering encompasses the design, manufacture, and test of a wide variety of systems. These basic steps are all used in the design of any structure, whether it is a household appliance, an automobile, a bridge, an aircraft, or a spacecraft. In the past, systems were overengineered and overbuilt resulting in an increase in the time and material required to build them. Often times the steps in the engineering process were repeated several times until the designed system performed satisfactorily. With the advent of the computer, tools have been and are continuing to be developed which enable the structural engineer to improve on each step of the engineering process. These tools not only help to limit the time, material, and cost that it takes to complete the engineering steps but they also help to limit the repetition of these steps. Structural computer modelling and vibration testing are two tools that have been developed with the aid of computers. Knowledge about material properties and structural dynamics may be used to create computer models of a system, which in turn may be used to predict dynamic performance and limitations. Once the system has been built, modal testing may be used to gain a greater insight into the dynamics of the structure and to update the computer model. An updated computer model may be used as a health monitoring tool for the structure after it goes into use. 1.1 Finite Element Model Refinement One of the most common modelling techniques is finite element modelling (FEM) which can be used to represent the continuous medium of a structure as a connection of finite elements. This enables a system with distributed mass, damping, and stiffness properties to be represented as a lumped parameter system with discrete mass damping and stiffness properties (Figure 1.1). In other words, an infinite degree of freedom system is represented as a finite degree of freedom system. continuous  medium I I FINITE ELEMENT MODEL Figure 1.1 Finite Element Modeling A wide variety of finite element modelling software is available to the designer. The basic steps involved in the modelling process are as follows: 1. Divide continuum into a finite number of elements. 2. Select node points where equilibrium conditions are enforced. 3. Determine element types and properties. elemental type (rod, plate, etc.) and location elemental displacement, stiffness, stressstrain, nodepoint lodes 4. Assemble elemental matrices (mass,stiffness, and damping). 5. Develop equilibrium equations for node point location. lumped parameters \__________ 6. Create global mass damping and stiffness matrices from which frequencies and mode shapes are calculated. The resulting FEM may be used to evaluate the efficacy of a design before it is built. Critical loads, resonant frequencies, and mode shapes may be predicted using the FEM and appropriate changes to the design based on these values may be made. Once the structure has been built, the FEM may be used to predict the performance of the structure under working conditions as well as serve as a damage assessment tool. However, while the original FEM is a predicted representation of a particular structure, the dynamic performance of the FEM very rarely matches the performance of the asbuilt structure. In order to correlate the dynamics of the structure to those of the FEM, the model must undergo a refinement process. The basic steps of FEM refinement are shown in Figure 1.2. The dynamic properties of the FEM are compared to the dynamic properties extracted from a vibration test of the structure. The resulting information is used to refine the model so that the modal properties of the FEM agree with modal properties from the vibration test. Figure 1.2 Finite Element Model Refinement Some of the earliest work done in model refinement was proposed by Rodden (1967), who explored using modal test data to generate analytical mass and stiffness properties of the structure being tested. The early work of Rodden has broadened into the modern FEM refinement techniques. Algorithms used to address the FEM refinement can be broadly classified as falling into one of four different approaches: optimalmatrix updates, sensitivity methods, eigenstructure assignment techniques, and minimumrank perturbation methods. Survey papers providing an overview of these techniques are provided in papers by Ibrahim and Saafan, 1987; Heylen and Sas, 1987; and Zimmerman and Kaouk, 1992. In the optimal matrix update formulation, perturbation matrices for the mass, stiffness, and/or damping matrices are determined which minimize a given cost function subject to various constraints. Typical constraints may include satisfaction of the eigenproblem for all measured modes, definiteness of the updated property matrices and preservation of the original sparsity pattern of the property matrices. Baruch and Bar Itzhack (1978) worked on an optimal update of the global stiffness matrix with a cost function that minimized the Frobenius norm of the perturbation matrix. Their work was expanded to look at updating mass, damping, and stiffness matrices (Berman and Nagy, 1983; Fuh et al., 1984; Hanagudet al., 1984). Kabe (1985) and Kammer (1987) expanded on this work further by looking at matrix updating while preserving the sparsity pattern of the original FE global matrices. Sensitivity methods for model refinement make use of sensitivity derivatives of modal parameters with respect to physical design variables (Martinez et al., 1991) or with respect to matrix element variables (Matzen, 1987). When varying physical parameters, the updated model is consistent within the original FE program framework. A variety of derivatives and optimization techniques have been used (Collins et al., 1974; Chen and Garba, 1980; Adelman and Haftka, 1986; Creamer and Hendricks, 1987; Flanigan, 1991). In the current work a physical parameter update technique using a genetic algorithm is developed. Inman and Minas (1990) proposed designing pseudocontrollers to be applied to the FEM in an iterative fashion resulting in a match between measured and FE modal properties. These controllers were then translated into matrix updates. These techniques, known as controlbased eigenstructure assignment techniques, are based on work done in eigenstructure control (Andry et al., 1983). Zimmerman and Widengren (1990) proposed a noniterative eigenstructure assignment formulation using an algebraic Riccati equation. Finally, the development of a minimum rank update theory has been recently proposed as a computationally attractive approach for model refinement and damage detection (Zimmerman and Kaouk, 1992). The update to each property matrix is of minimum rank and is equal to the number of experimentally measured modes which the modified model is to match. 1.2 Modal Testing: Sensor and Actuator Placement Regardless of the method used to perform FEM refinement, a modal test must be performed on a structure or its components in order to obtain the experimental information to correlate with the analytical information contained in the FEM. Finite element model refinement is only one use for modal data. Modal analysis is also a tool for damage assessment and force reconstruction. Several issues may govern the use of modal tests for these purposes. The final use of modal test data governs the pretest planning associated with modal testing. The placement of actuators for structural excitation purposes and the placement of sensors for structural response observations may well depend on whether the data will be used for modal parameter estimation, mode orthogonality for FEM correlation, identification of uncertain parameters in FEMs, structural health monitoring, or force reconstruction. The science of modal testing is thoroughly discussed in D.J Ewins' book Modal Testing: Theory and Practice. The basic steps involved in a modal test are discussed here for completeness and are pictured in Figure 1.3. The structure being tested must be dynamically excited and the response of the structure to this input must be measured. The excitation may be accomplished using an impact hammer, a shaker, or a release from an initial structural displacement. The response of the structure is generally measured using piezoelectric accelerometers, which are mounted on the structure in various locations. The force and response signals are sent to a processor or analyzer, after being filtered and amplified, from which a frequency response function (FRF) of the structure is obtained. The modal properties (mode shapes, damping ratios, and frequencies) may be obtained from the analysis of the FRF. FORCE TRANSDUCER shaker impact hammer conditioning   amplifiers frequencies structure mode shapes RESPONSE TRANSDUCER: signal processor triax single DOF Figure 1.3 Modal Testing Modal testing of a structure can be a costly venture in terms of time and money. Premodal test planning can be essential in saving cost by determining ahead of time the appropriate transducers and analyzers to use for the job at hand. Another aspect of pretest planning, the one associated with the current study, is the optimal selection of the location of the force and response transducers. It is desired to obtain the most information at the least cost, which involves minimizing the number of transducers to be used. The selection of the transducers will also depend on what the modal data will be used for (i.e., model refinement or damage detection). A majority of the sensor placement research that has been done may be broadly classified into two areas, system identification and optimal control. Yu and Seinfield (1973), Le Pourhiet and Le Letty (1978), Omatu et al. (1978), Sawaragi et al. (1978), and Qureshi et al. (1980) have all done work in the area of sensor placement for system identification. Shah and Udwadia (1978) and Udwadia and Garba (1985) have done work in sensor placement for structural parametric identification. Kammer (1991) approached the system identification sensor placement problem for the purpose of FEM validation. Goodson and Polis (1978) researched the selection of sensors for optimal structural control. Juang and Rodriguez (1979) looked at sensor placement for identification and control purposes. Information about the FEM has been used by modal test designers to place sensors and actuators on a structure for the purpose of modal testing. Finite element DOFs which have high kinetic energy are good choices for sensor or actuator placement because more information may be extracted about and more energy may be input to the structure at these points (Kammer, 1991; Flanigan and Hunt, 1993; Lim, 1991). Jarvis (1991) proposes using FE mode shape products to find sensor and actuator locations for modal testing. Kientzy et al. (1989) used driving point residues or modal participation factors to determine modal test excitation locations. Mode indicator functions have been used by the modal test engineers to tune modes during a modal test (Hunt et al., 1984). 1.3 Current Study Objective Proper modal test planning is needed in order to obtain the largest amount of information about a structure relative to the task of the data at the smallest cost. The refinement of finite element models is a task which benefits from modal test planning. The objective of the current studies is to explore the areas of finite element model refinement and premodal test planning. Specifically, the use of the optimization technique, genetic algorithms (GAs), in these two areas will be examined. The genetic algorithm is an optimization tool which has been developed in the past 20 years. An overview of the theory and applications of the GA is given in Chapter 2. A structuralparameter update modelrefinement algorithm is developed using a genetic algorithm in Chapter 3 and is applied using a FEM of a twodimensional truss structure. The topics of premodal test planning actuator and sensor placement are examined in Chapters 4 and 5. In Chapter 4 several techniques which have been developed in existing literature are outlined. The excitation placement techniques of kinetic energy eigenvector product, and driving point residue and the sensor placement techniques of effective independence, kinetic energy, and eigenvector product are reviewed. In an effort to improve on the current sensor and actuator placement technologies, two new actuator and sensor placement techniques are developed in Chapter 5. The normal mode indicator function (MIF) of a FEM is used with a GA to optimally find excitation locations. In addition the MIF is used as a tool to locate sensors. The second sensor and actuator placement algorithms use a degree of controllability and observability calculated using the FEM information. The effectiveness of the current sensor and actuator placement techniques and those developed in Chapter 5 are explored using several structural FEMs in Chapters 6,7, and 8. In Chapter 6 NASA's EightBay truss, in Chapter 7 the microprecision interferometer truss, and in Chapter 8 a FEM of a General Motors car body are used as examples to explore the effectiveness of all of the sensor and actuator placement algorithms for the purpose of premodal test planning. Concluding remarks and a discussion of future work in the areas outlined above are given in Chapter 9. CHAPTER 2 GENETIC ALGORITHMS: THEORY AND APPLICATION The motivation behind the development of GAs is that they are robust problem solvers for a wide class of problems, as depicted in Figure 2.1. However, it should be noted that they are not as efficient as nonlinear optimization techniques over the class of problems which are ideally suited for nonlinear optimization; namely continuous design variables with a continuous differentiable unimodal design space. Genetic Algorithms have the capability to solve continuous, discrete and a combination of continuous and discrete optimization problems. Nonlinear Optimization Genetic Algorithms Random Walk Problem Class Figure 2.1 Genetic Algorithms as Robust Problem Solvers Genetic algorithms are an optimization method which is based on Charles Darwin's survival of the fittest theories (Holland, 1975). The basic concept of the GA is that a population of designs is allowed to evolve over a period of time. The most fit members of that population are most likely to survive thus enabling their genetic code (or design information) to be passed down to future generations. More than just the information contained in the initial population may be passed down to future generations. As in nature as the evolutionary process progresses, mutations may occur in the offspring which may or may not result in more fit population members. Ideally, this evolutionary process will result in a population of members more fit than the original initial population. Genetic Algorithms may therefore be described as a directed random search or as a compromise between determinism and chance. Genetic algorithms are radically different from the more traditional design optimization techniques. Genetic algorithms work with a coding of the design variables, as opposed to working with the design variables directly. The search is conducted from a population of designs (i.e., from a large number of points in the design space), unlike the traditional algorithms which search from a single design point. The GA requires only objective function information, as opposed to gradient or other auxiliary information, which is usually required in other optimization techniques. The GA is based on probabilistic transition rules, as opposed to deterministic rules. There are five main operations in a basic GA: coding, evaluation, selection, crossover and mutation. Coding is the process in which each design variable is coded as a qbit binary number. Discrete variables would each be assigned a unique binary string. A continuous design variable Bi is approximated by 2q discrete numbers between lower and upper bounds for the design variable, B = Bn + inary# (B B n) (2.1) i imin 2q 1 imax imin where Bimin and Bimax are the lower and upper bounds on the ith continuous design variable and binary# is an integer number between zero and 2q 1. The continuous to discrete coding is like that of an analog to digital converter used in control systems. A population member is obtained by concatenating all design variables to obtain a single string of ones and zeros. Thus, a population member contains all information to completely specify the total design. For example, consider a design which has three continuous variables B1, B2 and B3 represented by 5bit, 6bit, and 4bit numbers and a discrete design variable B4 which can take on four different values. An example of a population of members containing this design information is pictured in Figure 2.2. A population is defined to be a grouping of npop members, where npop is the number of members in the population. Design Variables 5 bit 6 bit 4 bit 2 bit population continuous continuous continuous discrete member member B1 B2 B3 B4 1 10111 010001 0011' 11 2 0 001 1 0 1 1 1 0 1 0 1 101 1 0 npop 1 0 1 0 1 10 00 00 11 0 1 Figure 2.2 Coding of a Four Design Variable Problem Evaluation is the process of assigning a fitness measure to each member of the current population. The fitness measure is typically chosen to be related to the objective function which is to be minimized or maximized. No gradient or auxiliary information is used; only the value of the fitness function is needed. Therefore, GAs are less likely than traditional "hill climbing" algorithms to become "trapped" at a local minima or maxima. Additionally, because no gradient information is required, the design space is allowed to be discontinuous. Selection is the operation of choosing members of the current generation to produce the prodigy of the next generation. Selection is biased toward the most fit members of the population. Therefore, designs which are better as viewed from the fitness function, and therefore the objective function, are more likely to be chosen as parents. Crossover is the process in which design information is transferred to the prodigy from the parents. Crossover amounts to a swapping of various strings of ones and zeroes between the two parents to obtain two children. Two possible types of crossover are illustrated in Figure 2.3. Point Crossover Pattern Crossover parent 1 1 0 1000 1 1 010001 parent 2 1 1 1 0 1 0 0 1 10 0 1 0 0 swap: point swap x x x pattern child 1 0 10o 1 0 0 1 1 1 0 1 0 0 child 1 1 1:0 0 0 1 1 0 0 0 0 0 1 Figure 2.3 CrossOver Examples Mutation is a low probability random operation that may perturb the design represented by the prodigy. The mutation operator is used to retain design information over the entire domain of the design space during the evolutionary process. Holland (1975) developed the concept of schema, which for a very simple GA implementation explains why GAs work. Schema are a similarity template defined by O's, l's and x's, where x's are the don't care symbol. Thus, for a design coded with a total of 8bits, one schema would be 10xxxxxx. All designs which have a 1 in the most significant bit and a 0 in the 2nd most significant bit would be said to contain this schema. Holland's Schema Convergence Theorem states that under certain combinations of selection, crossover, and mutation, the expected number of schema H at generation k+1, n(H,k+l), is given as fitH n(H,k + 1) = (1 e) n(H, k) (2.2) tavg where E is a number much less than 1 and fitH and fitavg are the average fitness of all designs containing schema H and of the population as a whole, respectively. Thus, if those designs which contain schema H have on the whole a higher average fitness than the overall general population, the expected number of schema H in the next generation will be greater than or equal to the number of schema H in the current generation. The proof is valid only for a specific combination of selection, crossover and mutation. It should be noted that a general proof for more complex GAs has not been developed. However, there exists a wide body of literature which demonstrates the power and capability of advanced GAs (Schaffer 1989, Grefenstette 1987). A flow chart summarizing the GA process is shown in Figure 2.4. CODING Create Initial Population EVALUATION Evalute Population Fitness (objective function calculation) SSELECTION Apply Selection Criteria NO (which members reproduce?) F CROSSOVER and MUTATION evaluate create new members stopping New Design criteria Figure 2.4 Genetic Algorithm Flow Chart CHAPTER 3 FINITE ELEMENT MODEL REFINEMENT USING GENETIC ALGORITHMS 3.1 Introduction As discussed in Chapter 1, an important tool in the design of engineering structures is the finite element model (FEM). Recall that FEM refinement techniques may be classified as optimal matrix updates, sensitivity updates, eigenstructure assignment updates, and minimumrank perturbation updates. Sensitivity methods use sensitivity derivatives of modal parameters with respect to physical design variables or with respect to matrix element variables. These derivatives are used in order to determine what changes to make to the physical parameters or elemental matrices of the FEM in order to obtain in a refined model with measured modal properties. A GAdriven model refinement technique is developed to update FE structural parameters to provide an updated model with the measured modal characteristics. This model refinement technique is illustrated using a numerical example. 3.2 Model Refinement Problem Formulation For a given undamped structure it is assumed that an nDOF FEM is developed and results in the secondorder linear differential equation of motion, Mx + Kx = 0 (3.2.1) where M and K are the original analytic (nxn) mass and stiffness matrices, and x is the (nxl) position vector. The overdots represent differentiation with respect to time. The eigenvalue problem associated with Eq. (3.2.1) can be written as ?2Mr + Kr = 0 (3.2.2) where Xr is the rth eigenvalue and r is the rth mass orthogonal eigenvector of the original analytical system. It is assumed that the original analytic model of Eq. (3.2.1) does not satisfy the eigenvalues (Xmr) and the mass orthogonal eigenvectors (Omr) of the experimentally measured system, X2mrMm(r + Kmmr = 0 (3.2.3) where Mm and Km are the experimentally derived mass and stiffness matrices of the structure. Therefore, a discrepancy between the original analytic and measured modal information will result in an eigenvalue problem of the form 2,[M + AM(p)]r + [K + AK(p)]!kmr = 0 (3.2.4) where AM(p) and AK(p) are perturbation matrices which are functions of the structural parameters vector p. These perturbation matrices represent the mismatch between the original analytic mass and stiffness matrices and the experimentally derived mass and stiffness matrices. In order to develop an updated analytical model which is in agreement with measured modal data, a structural parameters vector p must be found which satisfies Eq. (3.2.4) for all measured modes. In the following sections, a model refinement technique is developed which employes the use of a GA to find the structural parameters vector which will result in an updated FEM whose modal properties match the measured modal properties of the asbuilt structure. 3.3 Genetic Algorithm Application As explained in Chapter 1, a FEM is a lumped parameter representation of a continuous structure. Information about the geometry and material of a structure are used to estimate its properties and to create the FEM. Mass, density, Young's modulus, crosssectional area, and moment of inertia are some example of the properties which are used to develop a FEM. Since these properties are estimated for the structure, they may be perturbed to give an updated FEM with the same modal properties as experimentally measured from the true structure. A Genetic Model Refinement Algorithm (GMRA) is developed which uses a GA to search for updated structural parameters which will result in an updated FEM with corresponding measured modal properties. An outline of the steps of GMRA, which follow those for a GA given in Chapter 2, follows. Coding. The design variables used in GMRA are continuous design variables which represent the structural parameters to be changed. Limits may be set on the amount of perturbation allowed for each design variable or structural parameter. This enables the user to allow small perturbations to variables about which they are certain, such as crosssectional area or moment of inertia and larger perturbations to variables about which they are less certain, such as density or Young's modulus. Evaluation. The most fit members of a population are those which minimize a chosen objective function. An objective function has been formulated which states, the most fit structural parameters vector p is one which results in an updated analytical model which gives the smallest value for the objective function obj = xur + m1 Qu 1mr hr (3.3.1) r= 1 r=1 where Xur and (ur are the rth eigenvalue and eigenvector of the updated analytical model. The first summation of Eq. (3.3.1) provides for the minimization between measured eigenvalues and updated analytic eigenvalues. As the updated eigenvalue approaches the measured eigenvalue the first summation approaches zero. The absolute value of the difference in each measured and updated eigenvalue is divided by the corresponding measured eigenvalue to insure that each frequency contributes equally to the objective function. The second summation of Eq. (3.3.1) is the 2norm between the difference in measured and updated eigenvectors and provides for the minimization between measured and updated mode shapes. As the updated eigenvector approaches the measured eigenvector the second summation approaches zero. The variables w, and hr are weights which can be changed in order to emphasize agreement between specific measured and updated eigenvalues/eigenvectors. By changing the weights, emphasis can be placed either on updated eigenvalue or updated eigenvector agreement with measured data. 3.4 Numerical Example: Six Bay Truss FEM A finite element model of a six bay truss with 25DOFs was developed to test GMRA. A picture of the truss is given in Figure 3.1. A= llcm2 E = 7.03x109 kg/m2 p = 2685 kg/m3 l = 0.75 m Figure 3.1 Six Bay Truss With 25 DOFs In Figure 3.1, A is the cross sectional area, E is the modulus of elasticity, p is the density of all of the members, and I is the length of each bay. It is assumed that the analytic model is incorrect and needs to be changed in order to facilitate agreement in analytic and measured modal properties. To obtain experimental modal information, the FEM of the six bay truss is altered and the "experimental" modal information is calculated. It is assumed that the dimension of the measured eigenvector is the same as that of the analytic eigenvector. This can be accomplished using an eigenvector expansion algorithm (Berman and Nagy, 1971; Smith and Beatie,1990; Zimmerman and Smith, 1992). An alternate formulation would be to let the vector norm calculation of Eq. (3.3.1) take place over only those components of the eigenvector that are measured, therefore, eliminating any error that may be introduced by expanding the measured eigenvectors. 3.4.1 Model Refinement As a first example, it is assumed that all of the structural parameters are known to be correct except for E, the modulus of elasticity. In addition, the properties of all of the diagonal members, all of the horizontal members, and all of the vertical members are linked. GMRA is used to find the updated structural parameters vector, Pu, which minimizes Eq. (3.3.1), u = {Ed Eh Ev} (3.4.1) The components of pu (Ed, Eh, and Ev ) are the moduli of elasticity of the diagonal, horizontal, and vertical truss members. The original analytic model has the same modulus of elasticity for all members, which will be referred to as Enom. The resulting analytical parameters vector is PA = Enom Enom Enom} (3.4.2) The value for Enom is 7.03x109 kg/m2. The structural parameters vector which is used to generate the experimentally measured model is pm = {0.95Enom 0.90Enom 0.92Enom} (3.4.3) The genetic algorithm is instructed to search for three design variables (components of the structural parameters vector) which are in the range of 0.7Enom to 1.3Enom. An initial population representing the horizontal, vertical, and diagonal moduli of elasticity is randomly generated within the limits of 0.7Enom to 1.3Enom. A member is added to this initial population which represents the original analytic model parameters vector. Since the original population is randomly generated, there is a good possibility that one of those initial population members will be more fit than the member representing the original analytic model. Therefore, the improvement in the updated eigensolution would in part be due to a random search. Even though this random chance is a benefit of genetics, in order to show the true improvement to the original analytic model by the genetic algorithm, an initial 19 population is chosen with all members less fit than the member representing the original analytic model. In order to facilitate frequency matching, the weights of Eq. (3.3.1) are set to emphasize minimization of the eigenvalue portion of the cost function. Five measured modes are supplied and GMRA is instructed to run for twenty generations (Figure 3.2). Minimum and Average Objective Functions Diagonal Truss Members 0 1.2 o0 S0.8 0 ^  0.8 0  I~ 0. 0 10 20 0 10 20 Generation Generation Horizontal Truss Members Vertical Truss Members S 1.2 1.2 o 0 0 0 0.8 0.8 0 10 20 0 10 20 Generation Generation Figure 3.2 Generational Data, Measured Modes with No Noise The top left graph of Figure 3.2 shows the average fitness (dashed line) and maximum fitness (solid line) of the current population at each generation. As the generations increase the average and maximum fitness improves, which corresponds to a decrease in the value of the objective function. After the first generation there is substantial improvement in the updated objective function over the original analytic objective function value. Since the members of the population which are randomly generated are all less fit than the original analytical model member, the decrease in objective function over the first generation is due to the children of the initial population. Since it is the goal to minimize the objective function, this is a desirable trend. Also as the generation increases, the average fitness approaches the maximum fitness. This is due to the fact that as generations evolve, the overall population tends toward the most fit member. After a certain number of generations, the diversity in the population members decreases. The other three graphs of Figure 3.2 show how the actual design variables modulii of elasticity) are varying over the generations. The straight lines on these graphs correspond to the "experimental" diagonal, horizontal, and vertical moduli of elasticity which were used to generate the measured modes. It is seen that within the first few generations, the parameters have quickly converged near their "experimental" values. In later generations it is seen that Ev, the vertical member's elastic modulus, varies widely. Physically, this is due to the fact the lower modes are fairly insensitive to the stiffness of the vertical members. The generational trends of the eigensolution of the updated model are shown in Figure 3.3. The top left graph is the norm of the difference in the measured and updated eigenvectors. It can be seen that over some of the generations the value of this norm increases instead of decreasing. This is due to the fact that the eigenvalue portion of the cost function is weighted more heavily in order to insure modal frequency agreement. It can be seen that an increase in the norm in Figure 3.3 corresponds to the overall updated eigenvalues moving closer to the measured eigenvalues resulting in a decrease in the value of the objective function. A comparison of the first three frequencies of the updated model with respect to the measured frequencies over 20 generations is shown in the other three graphs of Figure 3.3. The FRFs of the pregenetics model and the FRFs of the postgenetic models after 0, 5, 10, and 20 generations are pictured in Figure 3.4. An immediate improvement in the FRF of the system can be seen after five generations. 3.4.2 Effect of Noise The example presented would be expected to behave differently when there is noise present in the measured modal information. To simulate noise in the measured data, 5% and 15% random noise were added to the measured eigenvectors. It is assumed that the modal frequencies are measured accurately. As in the case with no noise, an initial population is Norm (MeasuredUpdated) Eigenvectors 3  2 0 0 10 20 30 Generation ona Frequency: Measured(..) Upoate 67 66 65 64 63 rst Frequency: Measured(..) Updated() Thi 124 122 120 118 1 10 20 Generation rd Frequency: Measured(..) Updatec 0 10 20 0 10 20 Generation Generation Figure 3.3 Generational Eigensolution Data, Measured Modes with No Noise 105 . 108 o 1011 Frequency (Hz) 10s 10~ ai 0 100 200 34 Frequency (Hz) After 20 Generations 0 100 200 300 0 100 200 300 Frequency (Hz) Frequency (Hz) (...) Updated Model () Measured Model Figure 3.4 FRF After 0, 5, 10, and 20 Generations, Measured Modes with No Noise 105 0 108 gio 105 S108 Cr: d() generated with one member which represents the original analytic model and with other members randomly generated which have cost function values greater than that for the original analytic model member. This is to show how the cost function is minimized due to genetics and not just due to a random selection of a more fit design. GMRA was run using 5% and 15% noise with 5 measured modes. The generational results for 15% noise are pictured in Figures 3.5 and 3.6, and the FRF is pictured in Figure 3.7. Graphically a similar trend was observed for 5% noise. An immediate improvement in the cost function can be seen after 5 generations, as shown in the top left graph of Figure 3.5. The generational updated eigensolution data of Figure 3.6 shows a similar trend to that of the example with no noise. The norm of the difference in measured eigenvectors with 15% noise and the updated eigenvectors is given in the top left graph of Figure 3.6. The straight dotted line in this figure corresponds to the norm of the difference in measured eigenvectors with and without noise. The FRF pictured in Figure 3.7 shows how the model which was updated using noisy data compares with measured model without noise and with the original analytic model. Because the cost function was weighted to be more heavily affected by the frequencies, the effect that noisy modes may have had on the update was minimized. 3.5 Conclusions One of the benefits of using a GA to search for an updated parameters vector is that the search is conducted from several points in the design space whereas conventional gradient sensitivity methods conduct the search from a single point. This helps enable GMRA to avoid getting stuck in a local minimum in addition to completing the search faster. Based on the evaluation of the data of this example, GMRA was successful in identifying an updated parameters vector which resulted in an updated FEM with measured modal properties. One, draw back to GMRA is that it requires an eigensolution of the FEM in order to calculate the objective function of Eq. (3.3.1). For large FEMs this objective function evaluation is a computationally expensive calculation, and would need to be redesigned to make GMRA a feasible model refinement tool. Minimum and Averaee Obiective Functions 10 20 Generation Horizontal Truss Members Diaeonal Truss Members 10 20 Generation Vertical Truss Members 0 10 20 0 10 20 Generation Generation Figure 3.5 Generational Data, Measured Modes with 15% Noise Norrm (MeasuredUpdated) Eigenvectors st Frequency: Measured(..) Updated() 34 33 32 0 10 20 30 Generation Scond Frequency: Measured(..) Updated() 0 Thi 124 122 120 118 10 20 Generation rd Frequency: Measured(..) Update< l__ 0 10 20 0 10 20 Generation Generation Figure 3.6 Generational Eigensolution Data, Measured Modes with 15% Noise 105 106 107 Q. 10S 109 1010 Analytic Model(), Measured Model No Noise(), and Updated Model (...) 101 1 I1I 0 50 100 150 200 250 300 Frequency (Hz) Figure 3.7 FRF After 20 Generations, Measured Modes with 15% Noise CHAPTER 4 MODAL TEST EXCITATION AND SENSOR PLACEMENT: CURRENT TECHNIQUES 4.1 Introduction In the current literature various techniques for excitation and sensor selection for modal testing exist. These techniques vary in computational complexity, cost, and accuracy. Several of these techniques were explored in the current study as a basis for comparison for the excitation and sensor placement techniques developed in the next chapter. An overview of the excitation placement techniques of kinetic energy, driving point residues, and eigenvector products and of the sensor placement techniques of effective independence, kinetic energy, and eigenvector product is given in the following sections. 4.2 Effective Independence Effective independence (El) is a technique developed to place sensors for the purpose of obtaining structural information for FEM verification for large space structures (Kammer, 1991). It follows from the work done by Shah and Udwadia (1978) and Udwadia and Garba (1985). The sensor locations are chosen such that the trace and determinant of the Fisher information matrix (corresponding to the target modal partitions) are maximized and the condition number minimized. By maximizing the determinant of the Fisher information matrix, the covariance matrix of the estimate error would be minimized, thus giving the best estimate of the structural response. A reduced sensor set is obtained in an iterative fashion from an initial candidate set by removing sensors from those DOFs (i.e., removing rows from the Fisher information matrix) which contribute least to the linear independence of the target modes. In order to perform test analysis mode shape correlation using a crossorthogonality criterion, the measured modes obtained during the modal test must be linearly independent. A summary of the derivation given in Kammer's paper (1991) follows. The output of the sensors can be expressed as the product between the FEM target mode matrix partitioned to a candidate set of sensors, Os, and the modal coordinates q Us = Osq + W2 = H + I2 (4.2.1) with Gaussian white noise Yo2 added. It is assumed that the FEM mode shapes are linearly independent. The sensors are sampled and an estimate of the state of the system is calculated as q = TI: s] sus (4.2.2) In order to obtain the best estimate of the state of the structure, the covariance matrix of the estimate error must be a minimum. The covariance matrix is given by P E[(q )(q q)T] = 2] (4.2.3) Assuming that the sensors measure displacement (acceleration may also be considered), the covariance matrix may be rewritten as S4TT21 P= ~~~0 s =Q1 (4.2.4) where Q is the Fisher information matrix and can be rewritten as Q = s = 2o (4.2.5) 0 0 Ao will now be referred to as the Fisher information matrix. In order to minimize the covariance error P, Q must be maximized; therefore Ao must be maximized. Kammer (1991) states that the determinant of the Fisher information matrix for the best linear estimate is a maximum for all linear unbiased estimators. Therefore, one wishes to maximize the determinant of the Fisher information matrix. From Eq. (4.2.5), the Fisher information matrix is calculated to be the product of the transpose of the target mode matrix times the target mode matrix, Ao = ITs (4.2.6) The first step is to calculate the eigenvalues XA and eigenvectors yA of the Fisher information matrix. Since it is assumed that the original FEM mode shapes are linearly independent, Ao will be positive definite, the eigenvalues will be real and positive and the eigenvectors will be orthonormal. The next step is to form the matrix product, G = ['FsWA] [sIA] (4.2.7) where & is an element by element multiplication. The columns of G sum up to be the eigenvalues of A0. Next the G matrix is scaled by the inverse of the eigenvalues of A0, FE= G kA1 (4.2.8) The effective independence vector is then calculated by summing the rows of the FE matrix, n FE1j j=1 n n FE2j ED= 1. (4.2.9) n Z FE(nDOF)j j=l where n is the number of target modes. The ith term in the ED vector is hypothesized to be the contribution of the ith sensor to the linear independence of the FEM modes. A value of 1.0 in the ED vector corresponds to a DOF that is essential to the linear independence of the target modes (i.e., that DOF must be retained as a measurement location). The DOF which contributes least to the linear independence (i.e., lowest ED value) is removed from the FEM target mode matrix. The Fisher information matrix Ao, the G and FE matrices, and effective independence vector ED are recalculated and the next sensor location is deleted from the target set. This iterative process is performed until the desired number of sensors remains. The minimum number of sensors required for identification corresponds to the number of target modes supplied. The previously described technique chooses single DOFs to place sensors. Often times a modal tester uses triaxial sensors instead of single DOF sensors. Assume the FEM being used to place sensors has 3 DOF per node. The El algorithm is modified to choose 3 DOFs at a time (corresponding to a node point) which contributed least to the linear independence of the target modes were eliminated over each iteration. The El value for each node is calculated as a sum of the El of each DOF of that node. The ED values for the 3 DOFs at each node are summed as, ED(1) + ED(2) + ED(3) EDti = ED(4) + ED(5) + ED(6) (4.2.10) ED(s 2) + ED(s 1) + ED(S) The 3 DOFs which contributes least to the linear independence (i.e., lowest EDtriax value) are removed from the FEM target mode matrix. However, if 1 of the DOFs for a particular node had an EI value of 1.0, meaning that that DOF was essential to the linear independence of the target modes, that node point would be retained, regardless of the ranking of its node point El sum rating compared to the other node points. The Fisher information matrix Ao, the G and FE matrices, and effective independence vector EDtriax are recalculated and the next triax sensor location is deleted from the target set. This iterative process is performed until the desired number of sensors remains. It is suggested that to increase computational efficiency for large FEM, the original FEM should be reduced down to a candidate set of measurement locations larger than the number of sensors to be placed before performing the effective independence calculations. One suggested technique for this reduction is modal kinetic energy, which is discussed in the next section. 4.3 Kinetic Energy The use of kinetic energy for optimal sensor placement as well as target mode identification has been discussed in several papers (Salama et al., 1987; Kammer, 1991). The modal kinetic energy is calculated using the FEM mass matrix and target modes. The kinetic energy of the ith DOF of the jth mode is given as nDOF KEij = ij Mik kj (4.3.1) k=l where nij is the ijth entry of the FEM modal matrix 4, Mik is the ikth entry of the FEM mass matrix M, and nDOF is the total number of DOFs of the mass and modal matrices. The kinetic energy matrix, KE, can be expressed as the matrix product DOF1 KE = (D M = DOF2 (4.3.2) SKEDOFn where @ denotes an element by element multiplication of the matrix D and the matrix resulting from the product of M and 0. The rows of the KE matrix correspond to the DOFs of the model and the columns correspond to the modes of the FEM. Locations for actuation or sensing are chosen as those DOFs with a maximum value of kinetic energy for a given mode. For example, assume the 4 contains FEM modes 1 through 10 for a given structure and one wishes to sense or excite the third mode. The DOF (row) with a maximum kinetic energy value for the third mode or column of the KE matrix would be selected. It is assumed that by placing the sensors at points of maximum kinetic energy, the sensors will have the maximum observability of the structural parameters of interest. If the modal test designer wishes to place triaxially constrained sensors, then a KE matrix may be calculated by summing the rows of Eq. (4.3.2) corresponding to DOFs for each node. Then the node points with maximum KE over the modes of interest may be chosen as locations for actuators or sensors. KEDOF1 + KEDOF2 + KEDOF3 SKEDOF + KEDOF2 + KDOF3 (4.3.3) KEtriax (4.3.3) KEDOFn2 + KEDOFn1 + KEDOFn The kinetic energy objective function precludes placing any sensors or actuators at nodal points since there is no motion and zero kinetic energy at these points (i.e., the Q entry would be zero resulting in a zero product). This could be a limiting factor in the pretest planning. To combat this problem, sensors can also be placed using maximum average kinetic energy (AKE) technique. A sensor is placed at a DOF with a maximum average kinetic energy over a range of modes of interest. In using an average kinetic energy, a DOF is not necessarily excluded if it is a node point of a particular mode. The average kinetic energy vector is calculated as N SKE1k k=l N KE2k AKE = k= 1 /N (4.3.4) N SKE(ndof)k k=1 where N is the number of modes in the mode shape matrix 0 (i.e., the number of columns of the KE matrix). The sensor or actuator locations are found by finding the DOFs of the maximum average kinetic energies. Triaxially constrained sensors may be placed by taking the sum of the average kinetic energy for the DOFs for each node and choosing the nodes with maximum average kinetic energy. In addition, it should be noted that the mass weighting inherent to the kinetic energy and average kinetic energy approaches causes the sensor or excitation placement to become dependent on the finite element discretization of the structure. There is an inherent bias against the placement of sensors in the areas of the structure in which a fine mesh size (and thus small mass) is used. 4.4 Eigenvector Product This technique uses modal products from the reduced FEM eigenvectors to identify possible locations for sensors or excitation. By choosing a frequency range of interest and the corresponding FEM eigenvectors (or modes) in that range, the eigenvector product is calculated as EVP = 21 (0 2 0 . N (4.4.1) where ( represents an element by element multiplication of the mode shape vectors 4. The ith entry of the EVP is given as EVPi = (i1)i2i3 .iN (4.4.2) This product is calculated for all candidate DOF sensor or actuator locations. A maximum value of this product corresponds to a candidate location of reference or excitation (Jarvis, 1991). This technique also precludes the placement of sensors at nodal points which result in zero eigenvector products. If this presents a problem for a given test case, the eigenvector product can be replaced equivalently by an absolute value eigenvector sum, over the FE target modes of interest. The eigenvector product may be used to place triaxially constrained sensors by summing up the entries of Eq. (4.4.2) which correspond to a particular node point. The node points with the maximum eigenvector product sum are then chosen as points of reference. 4.5 Driving Point Residue A FEM can be used to identify the best locations and directions for exciting a structure by an evaluation of driving point residues (DPRs) or modal participation factors (Kientzy et al, 1989). A DPR is a measure of how much a particular mode is excited at a particular DOE The point and direction of excitation are chosen where the DPRs are maximized (to excite a given mode) or minimized (to avoid exciting a given mode). An equation of motion in Laplace domain for a structure may be written as [Ms2 + Cs + K]X(s) = F(s) or B(s)X(s) = F(s) (4.5.1) where M, C, and K are the (nxn) mass, damping and stiffness matrices, s is the complex Laplace variable (s = ( + ico), and F(s) is the transformed excitation forces. Equation (4.5.1) may be solved for the transformed displacement responses, X(s), X(s) = H(s)F(s) where H(s) = B(s)1 (4.5.2) and H(s) is referred to as the transfer matrix. The system transfer matrix for a structure with damping can be expressed in the form H(s) = + (4.5.3) k=lSk S k where Rk and Rk* are the modal residues and Xk and Xk* are the complex conjugate pairs of eigenvalues of the transfer matrix. The residues can be written in terms of the mode shapes Ok as kN Tk H(s)= s Ak kX sT A 1 (4.5.4) k= S k Sk where Ak is the mode shape scaling constant. For a structure which is lightly damped, the following two inequalities are true: Ok < (Wk and Imaginaryl{k} < Real{(k} for k = 1 to N (4.5.5) When these conditions are imposed, the mode shape scaling constants can be written in the form A = for k = 1 to N (4.5.6) Ak mk(Ok and the residues become Rk(a, b) = [k(a)k(b)] for k=1 to N (4.5.7) (mkwk) where Rk(a,b) is the residue between DOF a and DOF b, Ok(a) is the kth mode shape component at DOF a, are scaled such that they are mass orthonormal, (i.e., TM4=I, where columns are the mode shapes 4k (for k=1 to N), M is the FEM mass matrix, and I is the identity matrix) then the residues (in terms of the displacements) may be written as Rk(a,a) k(a) for k=1 to N (4.5.8) or equivalently in terms of acceleration Rk(a, a) = k(a)2wk for k=1 to N (4.5.9) The easiest way to evaluate several residues at once is to display them graphically. The DPRs that were calculated for the NASA 8bay truss are shown in Figure 4.1. The DPRs are graphed in order of weighted average residue in order to discriminate against zero DPRs. The weighted average residue is calculated as waDPR = average DPR x minimum DPR (4.5.10) Each vertical line on the graph represents the range of DPRs from maximum to minimum over all the modes of interest for a single candidate DOF The highest weighted average which is the best driving point is displayed first. The residues in the top graph are the square root of the sum of the squares of the residues in the x, y, and z direction plotted on a log scale. The bottom graphs are the residues for the x, y, and z direction plotted on unit normalized linear scales. The top graph is used to choose the node at which to place the excitation device. The bottom graphs are used to find the x, y, or z direction of the excitation. In order to insure that an excitation location will give uniform participation of as many target modes as possible, it is desired to find a high average residue for a given DOF as well as Weighted Average dprs co "ttff" t t tt tI" I : c 4 cn 0 0 2 3 4 1 5 8 6 7 13161514191820171110129 21242322252827262933130 2 node 0 5 10 15 20 25 30 node X,Y,Z dof dprs 3 I I I z "02 a, y N 0 5 10 15 20 25 30 node Figure 4.1 Typical Driving Point Residue (NASA 8Bay Truss) a small residue range over all the modes of interest. For this example the highest weighted average DPRs are at nodes 2, 3, 4, and 1 as seen in the top portion of Figure 4.1. The bottom portion of this figure shows that the optimal directions for excitation at these node points would be in the x and/or z direction, because the larger residues are in these directions. CHAPTER 5 MODAL TEST EXCITATION AND SENSOR PLACEMENT: NEW TECHNIQUES 5.1 Introduction In an effort to improve on the existing sensor and excitation placement techniques, two new sensor placement techniques and two new excitation techniques are developed in the current work. The first excitation and sensor placement techniques are based on the FEM normal Mode Indicator Function (MIF) calculation. The second excitation and sensor placement techniques are based on the observability and controllability calculations of the modes of the FEM. The effectiveness of these techniques, along with the techniques discussed in Chapter 4, will be explored in subsequent chapters using several different structural testbeds. 5.2 Mode Indicator Function The mode indicator function (MIF) was first developed to detect the presence of real normal modes in sine dwell modal testing (Hunt et al, 1994 and Williams et al, 1985). This function also serves as a useful metric for pretest analysis. While it is somewhat useful for assessing the efficacy of sensor layout, its true utility lies in assessing the effectiveness of a particular input in exciting the system modes. The first step in calculating the MIF is the calculation of an acceleration frequency response function using the FEM mode shapes and frequencies, Hk i k (5.2.1) = msr(w (02 + j2Sr(0) r= 1 Mr(( r where m number of modes in frequency range of interest or rth mode Okr force input point k of the rth mode Oir response point i of the rth mode o discrete frequency at which to calculate Hik Or frequency of the rth mode Sr viscous damping ratio of rth mode msr modal mass of the rth mode Next, the normal MIF is calculated using Hik as ([Real(Hik(o)) x H(ik()) MIF((o) = i= (5.2.2) (kH,k(W)2) i=1 where L is the total number of response points. The MIF is nearly 1.0 except near a normal mode, at which point it drops off considerably since the frequency response becomes mostly imaginary at that point (i.e., Real(Hik(o)) is very small). A plot of a typical MIF is given in Figure 5.1. 5.2.1 Excitation Placement In pretest planning, an excitation is desired which exhibits a sharp drop in the MIF at each mode of interest, indicating that the mode is well excited. The Genetic Mode Indicator Function (GMIF) excitation selection algorithm uses a genetic algorithm (Holland, 1975) to find excitation locations and their orientations on a structure to optimally excite a given mode or range of modes. The success of the excitation is based on the MIFs of the chosen excitation locations. If more than one excitation is sought then a MIF must be calculated for each. A single excitation need not exhibit a sharp MIF drop for all modes as long as the union of the MIFs for all of the excitation sources exhibits a large drop for each target mode. Two algorithms have been developed. The first is an unconstrained version which searches for 0 10 20 30 40 50 60 70 frequency (Hz) Figure 5.1 Typical MIF Plot node point excitation locations with forces being applied in any direction at the node points. The second algorithm is a constrained version in that the direction of the excitation is constrained to be 0, 30, 45, 60, or 90 degrees in each x, y, and z plane. The constrained algorithm was developed to provide an improvement in algorithm speed by reducing the number of search points in the design space. In addition, the attachment of the excitation hardware on the structure during the modal test would be easier if the angles of orientation are limited. An outline of the GMIF algorithms follows. Coding. The GA chooses an initial population of node points and directions for excitation location. The node points and the directions are referred to as design variables. The design variables are represented differently for the constrained and unconstrained versions of the GMIF algorithm. In both the constrained and unconstrained cases the node point locations of the excitations are treated as discrete design variables. Discrete design variables represent a finite number of variables to search over, and for this application they represent all of the node points in a FEM that are being considered as possible excitation locations. The direction design variables are two angles in spherical coordinates, a and 3, which are used to calculate the direction of the force as seen in Figure 5.2. For the z F force P *(r,La,) x = r cosa sinp y = r sina sin3 r z= r cospl node F = cosasinP3i + sinasinlj + cospk x Figure 5.2 Excitation Selection by GMIF constrained algorithm the orientation of the excitation is considered discrete in the sense that there are a finite number of angles (i.e., 0, 30, 45, 60, or 90 degrees) from which the GA selects a and p. For the unconstrained case, the angles of orientation are considered to be continuous in that the GA searches over all possible angles. For both cases the force is assigned a unit magnitude in order to only evaluate the angle of orientation of the force and not the magnitude. Table 5.1 presents a list of variables used in the GMIF selection algorithm. Table 5.1 GMIF Design Variable Description Unconstrained Constrained Design Variable Type Design Variable Type node discrete node discrete a (any angle) continuous a (0,30,45,60,90 degrees) discrete P (any angle) continuous p (0,30,45,60,90 degrees) discrete Evaluation. The next step in the GA is to evaluate the fitness of each population member or excitation. The fitness of a member is based on the calculation of the MIF corresponding to each force that makes up a single member. All of the MIFs for a single member are assembled into a MIF matrix, miffl(Wo) miffl(W2) . miffl(omm Force 1 MIFm= if.(W1) miff(W2) . mif2(Om+ Force 2 (5.2.3) miffni()m oif(2) mf 2 m.ifn(o m Force nf 1st 2nd mth natural frequencies of interest Next the minimum of each column of MIFm is taken to find the maximum dropoff values of the union of the MIFs of each force resulting in a minimum MIF vector, column MIF = minimum (MIFm) (5.2.4) The objective function is calculated as a weighted sum of the elements of MIFv, m Jobj = wMIFvi (5.2.5) i=l The weights may be used to emphasize the dropoff values of particular modes. The objective function of Eq. (5.2.5) is designed to find excitation sources which exhibit sharp MIF dropoffs for as many modes as possible. Selection, crossover, and mutation. Once the fitness of the initial population is established the population is allowed to evolve over a fixed number of generations. The information contained in the initial population is crossed over between members and sent to the next generations. Members of new generations which are more fit than the previous generation (i.e., have better dropoff values) replace the less fit members in the evolving population. Mutations that occur in the population allow for the population to remain diverse during the evolutionary process, keeping the design search space open. 5.2.2 Sensor Placement Once an excitation source has been selected, the MIF corresponding to the chosen excitation source may be used along with a GA to locate a sensor set. First, the FRF matrix is calculated for the FEM under consideration using the chosen excitations. When the MIF is calculated to evaluate an excitation source, all DOFs of the mode shape matrix are used to calculate the frequency response matrix, H. When the MIF is used to evaluate a sensor placement, only the sensor candidate DOF or three DOFs in case of a triax sensor set, is used to calculate the frequency response matrix, H. A MIF must be calculated for each force for a candidate sensor and the minimum MIF value for each mode is taken. The MIF values for the target modes for the ith DOF are taken as the minimum MIF values for all of the forces in an excitation set, MIFi(forcel) column MIFi(force2)  MIFji = minimum (5.2.6) MIFi(forcenf)  The MIF vector of Eq. (5.2.6) is calculated for all candidate sensor DOFs. A weighted sum of the MIF values for each DOF is made and assembled into the MIF vector, Z wMIFi MIFv = wMIF2 (5.2.7) I wMIFn where w is an (lxm) weight vector used to emphasize MIF dropoff values. The variable n is the total number of candidate sensor DOFs for unconstrained sensor placement or the total number of candidate sensor nodes for triaxiallyconstrained sensor placement. Once MIFv has been calculated, the node or dof with minimum MIFv sum is retained as the first sensor. The MIFv vector is recalculated using all remaining DOFs plus the single sensor chosen, and the next dof or node is chosen with minimum MIFy value. This iterative process is performed until the desired number of sensors is chosen. 5.3 Observability and Controllability Consider the set of discrete linear secondorder differential equations of motion corresponding to a particular nDOF FEM of a structure, Mx(t) + Dx(t) + Kx(t) = Bu(t) (5.3.1) y(t) = Cx(t) (5.3.2) where M, D, and K are the (nxn) mass, damping, and stiffness matrices, x(t) is the (nxl) displacement vector, and u(t) is the (nxl) input function of the system. The over dots represent differentiation with respect to time. By choosing z(t x(t)] (5.3.3) z(t) = [x(t)J Eq. (5.3.1) can be rewritten in state space form as z(t) = MK M1D z(t) + M ] (5.3.4) or equivalently z(t) = Az(t) + Bu(t) (5.3.5) where A is the (2nx2n) state matrix, B is the (2nxo) input influence matrix, and u(t) is the (ox 1) input function vector. The output of the system defined by Eq. (5.3.2) may be rewritten as y(t) = Cz(t) where C = [C 0] (5.3.6) The vector y(t) is the (lxl) system output, and C is the (lx2n) output influence matrix. Figure 5.3 is a pictorial representation of the matrices and vectors of a state space system of equations and describes the purpose of each. An important consideration in the control of the system described by Eq. (5.3.4) is if the system is controllable and observable. Another consideration is the observability and controllability of the modes of the system defined by Eq. (5.3.4). Several techniques for calculating the observability and controllability of modes have been explored. One of the most common tests for controllability and observability is the PopovBelevitchHautus (PBH) test (Kailath, 1980). For the purpose of vibration control, it is most common to 42 Input Space / Output Space u(t) y (t0 map B map C how and where energy what information is is injected into system State Space extracted from system x(t) Smap A how system transforms and dissipates energy Figure 5.3 State Space Variable Description overstep the conversion of Eq. (5.3.1) into state space form and to calculate the observability and controllability directly from Eq. (5.3.1). The PBH eigenvector test for a secondorder system (Laub and Arnold, 1984) states that given the system defined in Eqs. (5.3.1) and (5.3.2): 1. The ith mode will not be controllable from the jth input if and only if there exist a left eigenvector qi such that qi[X2M + XiD + K] = 0T (5.3.7) qTbj = 0T (5.3.8) 2. The ith mode will not be observable from the kth output if and only if there exist a right eigenvector pi such that [XM + XiD + K]pi = 0 (5.3.9) cki = 0 (5.3.10) where bj is the jth column vector of the input influence matrix, B, and gk is the kth column vector of the output influence matrix, C. This evaluation of controllability and observability tells whether or not the modes are completely observable or controllable; it does not address the issue of degree of observability and controllability. The issue of degree of controllability and observability is explored in a paper by Hamdan and Nayfeh (1989). In this work the matrices QTB and CP are used to evaluate the degree of controllability and observability of the modes of a system. The matrix QT is the transpose of the matrix whose columns are the m left eigenvectors of Eq. (5.3.7) and B is the output influence matrix whose columns are the o output influence vectors. Ti qT I I QTB = b2 q b b2 . b (5.3.11) n I I I The matrix C is the output influence matrix whose I rows contain the output influence vectors and P is the matrix whose columns are the m right eigenvectors of Eq. (5.3.9). c I  CP = 2 2. . (5.3.12) The (mxo) matrix QTB contains information about the controllability of the modes and the (lxm) matrix CP contains information about the observability of the modes. If the ijth entry of QTB is 0 then the ith mode is uncontrollable from the jth input. Similarly, if the kith entry of CP is 0 then the ith mode is unobservable from the kth output. If the ijth entry of the controllability matrix is nonzero, then what information may be gained about the degree of controllability of the ith mode from the jth input? The ijth element of the QTB matrix is the vector dot product of qi and bj. If the two subspaces spanned by these vectors are parallel then the ith mode is completely controllable from the jth input, and if the two subspaces are orthogonal then the ith mode is completely uncontrollable from the jth input. If the two subspaces are neither orthogonal or parallel then the angle between the two is an indication of the degree of controllability of the ith mode from the jth input. This relationship is illustrated in Figure 5.4 and the magnitude of the vector dot product is, (5.3.13) A similar argument may be made for the observability of the ith mode from the kth output using the magnitude of the vector dot product, IckPil = II Pi I cos (ki (5.3.14) The angle 0ij is a direct measure of the degree of controllability of the ith mode and )ki is a direct measure of the degree of observability of ith mode. The degree of controllability and observability decrease as ij and Oki go from 0 to t/2 as shown in Figure 5.4. CONTROLLABILITY OBSERVABILITY bj Qi ck Pi completely controllable completely observable completely controllable completely observable Oij = 7c/2 complete uncontrollable complete3 uncontrollable Oki = 7C/2 completely uncontrollable Figure 5.4 Controllability and Observability The above argument has been made from a dynamic controls perspective. The same argument may be used to gain information about actuator and sensor placement during modal tests of a structure. Using the FEM of a particular structure the degree of IqbjI = 1 q ill bj II cos Oi controllability of a modal test excitation layout may be used to optimally select an excitation location. Similarly, the degree of observability of a modal test sensor layout may be used to select a sensor configuration which will result in an increase in the amount of modal information obtained. 5.3.1 Excitation Placement The degree of controllability based on the calculation of the angle between the subspace spanned by a mode shape of the system and the subspace spanned by the input influence vectors of the matrix of Eq. (5.3.1) is used to evaluate how effective the input u(t) may be in controlling the modes of the system. Consider that Eq. (5.3.1) is the equation of motion for a particular structure and that the right hand side, Bu(t), is the force that will be applied to excite the structure for modal testing. In order to gain the most information from the modal test, an excitation location which will excite a chosen range of target modes well is required. The measure of modal controllability is an indication of how great an effect a particular input, bj, may have on the mode shapes of the system. An input with a higher degree of controllability over a mode will be more successful in exciting that mode than an input with a lower degree of controllability over that mode. Therefore, it is proposed that the angle Oij of Eq. (5.3.13) may be used as a measure of how successful the input excitation bj will be in exciting mode qi. Since there are an infinite number of possible input influence vector values, an optimization technique is needed to search for an input influence vector which maximizes the controllability of the target modes. A genetic algorithm is employed as the optimization tool for this purpose. Coding. The coding of the Genetic Controllability (GCON) Algorithm is identical to the coding of the GMIF algorithm. One design variable represents the node point locations of the forces, the other design variables represent the angle orientations of the forces in spherical coordinates as described in Figure 5.2. The difference between the GMIF and the GCON algorithms is in the fitness evaluation of the population. Evaluation. The fitness of each population member is based on the calculation of the controllability vector. The location and orientations of each force in a population member is used to calculate an input influence vector. The j th force of a member is used to calculate a portion of the input influence vector as, cos ctf sin Pfj' b = sin sin ifj (5.3.15) cosf, (J The unit magnitude vector bfj is calculated for all j forces of a member and assembled into the global input influence vector. The global input influence vector, b, is initially an n DOF vector of zeros. Once the individual force unit input influences are calculated, they are placed in the global input influence vector, b, at the DOFs of each corresponding force node point, '0 bfl 0 b = f2 (5.3.16) bf3 0 hfnf 0 where nf is the total number of excitation devices represented in a population member. Since the magnitude of the big's affect the controllability of the system, each bfj is scaled to unit magnitude so as to compare the directions of the forces as apposed to the magnitudes. The unit input influence vector is used in conjunction with the lefthand eigenvectors of Eq. (5.3.7) to calculate the (mx1) degree of controllability vector from Equation 5.3.13, S= 0 2 (5.3.17) 0m The ith entry of the degree of controllability vector is Scos qTb (5.3.18) Oi = COS1  II 9i II The entries of the (mx1) controllability vector, 0, represents the controllability of each of the m modes of the system from the locations and directions defined by b. The algorithm is designed to find excitation location which exhibit the highest degree of controllability for the modes of interest. Therefore, the most fit members of a population are forces which minimize the entries of the vector Q. Doing so minimizes the angle between the input vector subspace and the mode shape subspaces thus increasing the amount of controllability and the amount of power input into the modes. The objective function is calculated as a weighted sum of the entries of 9, m Jobj = i0i (5.3.19) i=l The weight may be used to emphasize the controllability of particular modes over other modes. Selection, crossover, and mutation. The population is allowed to evolve over a fixed number of generations as in the GMIF algorithm. The most fit members are those that minimize the objective function of Eq. (5.3.19). 5.3.2 Sensor Placement The degree of observability of the modes of the system in Eq. (5.3.1) is based on the calculation of the angle between the modes of the system and the output influence matrix. When performing a modal test of structure, it is not likely that all DOFs in the FEM will be instrumented during the test due to cost constraints. In order to get the most information about the modes of the system, a reduced sensor set which has the greatest degree of target mode observability should be chosen. Therefore, the angle 0 of Eq. (5.3.14) will be used as a 48 measure of how successful a sensor configuration is in measuring a group of chosen target modes. There are a finite number of DOFs or sensor possibilities represented in a FEM, therefore, an optimization technique is not needed. In order to evaluate each DOF location individually, the output influence matrix, C, is set equal to an (nxn) identity matrix. Therefore, the observability of the kth DOF of the ith mode is obtained from Equation 5.3.15 as ki = kil (5.3.20) II Pi II where Pki is the kith entry of the right eigenvector matrix of Eq. (5.3.12) and pi is the ith column of the right eigenvector matrix, P. If this value is calculated for all candidate DOFs and all target modes the resulting (nxm) observability matrix, Il 12 m (P21 (22. * 2m ( = (5.3.21) 4nl Sn2 . nm The rows of the observability matrix represent DOFs and the columns represent the modes. Once the observability matrix has been calculated, the DOFs for sensor location must be evaluated. The observability column corresponding to each mode is sorted, and the DOFs with the minimum 0 values (i.e., greatest observability) for each mode are selected as sensor locations. CHAPTER 6 PREMODAL TEST PLANNING ALGORITHM APPLICATION: NASA EIGHTBAY TRUSS 6.1 Introduction The NASA 8bay truss is used to compare the techniques discussed in Chapters 4 and 5 in placing sensors and actuators for modal testing and system identification purposes. The kinetic energy, average kinetic energy, eigenvector product, driving point residue, and controllability techniques are used to place three excitation devices on the truss. A numerical simulation is performed to evaluate the effectiveness of each technique to excite the first five target modes of the structure. A crossorthogonality check between the identified and finite element modes is performed in addition to a frequency match comparison. Effective independence, kinetic energy, average kinetic energy, eigenvector product, and observability techniques are used to place sensors on the 8bay truss, in order to best identify the first five modes of vibration. The structural response of the truss is numerically simulated and measured at those DOFs corresponding to the sensor locations obtained using the various techniques. The eigensystem realization algorithm (ERA) is then used to evaluate the effectiveness of each sensor set with respect to modal parameter identification (Juang and Pappa 1985). A set of three hundred random sensor locations are compared to the five sensor location techniques. The cost effectiveness of each of the excitation and sensor selection techniques is evaluated. 6.2 NASA EightBay Test Bed The NASA eightbay truss, pictured in Figure 6.1, is modeled with 96 DOFs and is considered to be lightly damped. Using the FE mass and stiffness matrices supplied by NASA, the FE mode shapes and frequencies were calculated. When the true modal tests were performed on the truss, it was assumed that the first five modes were successfully identified (Kashangaki, 1992). Table 6.1 list the first five frequencies and mode descriptions. 29 25 21 13 17 18 22 7 3 2 14 28 9 f 10^ 19 /'24 1 ~ 1 20 7 1 12 16 y x 3 8 Figure 6.1 NASA 8Bay Truss Table 6.1 EightBay Truss Frequencies and Mode Description Mode Frequency (Hz) Description 1 13.925 1st yx bending 2 14.441 1st yz bending 3 46.745 1St torsional 4 66.007 2nd yx bending 5 71.142 2nd yz bending 6.2.1 Excitation Placement Kinetic energy, average kinetic energy, eigenvector product, driving point residue, and controllability techniques are each used to place three excitation devices on the 8bay truss to best excite the first five modes of the structure. The excitation locations for the five techniques are pictured in Figure 6.2. The kinetic energy technique placed two excitation y Z Kinetic Energy z Average Kinetic Energy and Controllability Eigenvector x Product S, x Driving Point x x Y Residue z z Figure 6.2 EightBay Excitation Locations sources towards the cantilevered end of the truss and one towards the center of the truss. The other four techniques clustered all of the excitation sources at the cantilevered end of the truss. It is interesting to note that the kinetic energy technique put all excitation sources in the zdirection. The remaining four techniques placed excitation sources in both the x and z directions, in addition to clustering two of the sources at a single node. The average kinetic energy and controllability techniques placed the excitation devices in the same location as seen in Figure 6.2. In a true modal test, the two excitation sources which were placed at a single node could be combined into one excitation source at that particular node in the xzdirection, thus reducing the number of excitation sources needed. The kinetic energy placement could not result in this option. Once the excitation sources have been determined using the five techniques, the truss's response to an impulse at the chosen excitation locations is numerically simulated for the first five modes and 5% noise is added to the response data. The response, measured at all 96 DOFs to fully evaluate the effectiveness of the each excitation placement, is sampled for a length of 2 seconds at 200 Hz. Five percent noise is added to the response which is sent to ERA for system identification. A comparison of the measured frequencies and crossorthogonalities of the identified and FE models is calculated for each excitation placement. The five excitation placement techniques were all successful in exciting the structure at the target frequencies based on the comparison of the original FE and identified frequencies and mode shape. All of the techniques had excellent matching between identified and FEM frequencies with differences much lower than the industry standard of 5% (Flanigan and Hunt, 1993). A Frequency Response Function (FRF) is plotted for each of the excitation devices in Figure 6.3. The crossorthogonalities between identified and FEM mode shapes for the five excitation placement techniques are pictured in Figure 6.4. All offdiagonal terms for each of the excitation sets are less than or equal to 0.02, which is well within the industry standard of < 0.02 offdiagonals for primary modes (Flanigan and Hunt, 1993). Of the five techniques, the crossorthogonality of kinetic energy was worst although it was well within the acceptable range of offdiagonal values. The crossorthogonality of average kinetic energy, controllability, eigenvector product, and driving point residue techniques had similar values; all of the offdiagonal elements for each of the techniques were : 0.01. Recall that kinetic energy placed all excitation sources in the zdirection at three separate nodes, and that the other three location techniques placed excitation sources in both the x and z directions and collocated two sources at one node. The similar placement configurations at the cantilevered tip of the truss resulted in similar crossorthogonalities. Based on the frequency matching and crossorthogonality between identified and FEM frequencies and mode shapes, and on the FRFs, each of the five excitation location 0 lo 20 30 410 0 w o 70 80 0 10 20 30 40 0 60 70 80 Frequency(Hz) Frequency(Hz) S Eigenvector Product 1 Driving Point Residue 10to' 10 10d 10o 10 10 Iio 0 10 20 30 4 50 860 70 80 0 10 28 3 40 50 0 70 0 Frequency(Hz) Frequency(Hz) Figure 6.3 EightBay Excitation Location Frequency Response of Time Domain Data Kinetic Enereg Average Kinetic Energy and Controllability Mode'j  Mode Figure 6.4 EightBay Excitation Placement Crossorthogonality of Identified and FEM Modes 1 to 5 techniques identified an acceptable three point excitation location for exciting the first 5 modes of the 8bay truss. 6.2.2 Sensor Placement Using the FE modes and frequencies, the five sensor placement techniques, effective independence, kinetic energy, average kinetic energy, eigenvector product, and observability were assigned the task of best identifying the first 5 modes of vibration by placing 15 sensing devices on the truss. The five sensor set configurations are pictured in Figure 6.5. Each of the techniques clustered the sensors in two locations on the truss at the cantilevered end and at the midspan. Effective independence, average kinetic energy, and eigenvector product techniques collocated fourteen of the fifteen sensors at seven node points. The kinetic energy technique collocated twelve of the fifteen sensors at six node points and observability collocated eight sensors at four node points. From a cost standpoint, the collocation of as many sensors as possible is desired. None of the five techniques placed sensors in the y DOF This is to be expected since the first five modes do not include significant motion in the ydirection. The simulated response, with 5% noise added, obtained using the excitation configuration determined from the average kinetic energy technique was used to test the sensor sets. The exact same response was used to test each sensor configuration, by taking from the 96 DOF response only those DOFs corresponding to the sensor locations to be evaluated. The response data of each sensor set were sent to ERA for frequency and mode shape identification. The excitation placement found using the average kinetic energy technique is used to excite the structure to test the various sensor locations. Each of the five sensor placement techniques measured the target frequencies well as can be seen from the percent difference in the FE and identified frequencies given in Table 6.2. The 96 DOF FEM mass matrix was reduced to a 15 DOF mass matrix using exact reduction and the crossorthogonality between y Z Effective Independence Kinetic Energy Average Kinetic Energy Eigenvector Product Observability xz T xz Figure 6.5 EightBay Sensor Locations identified and FEM modes 1 through 5 was calculated using the reduced mass matrix. The crossorthogonality for each of the sensor sets is given in Figure 6.6. For each of the five cases the crossorthogonality terms were within acceptable limits. The offdiagonal terms corresponding to the primary modes were all 4 0.02. Effective independence, kinetic energy, and eigenvector product techniques resulted in similar crossorthogonalities (all offdiagonals are 4 0.01). Observability technique gave the worst crossorthogonality results of the five techniques even though the offdiagonal elements remained within acceptable values. The improved performance of the effective independence, kinetic energy, observability, and eigenvector product techniques over the average kinetic energy technique can clearly be seen in the next section when the five techniques are compared to the random sensor sets. Kinetic Energy Average Kinetic Energy Eigenvector Product Observability  1.00 a 0.02 m 0.01 m<0.01 1.00 0.02 0.00 Figure 6.6 EightBay Sensor Placement Crossorthogonality of Identified and FEM Modes 1 to 5 Table 6.2 Percent Difference in FE and Identified Frequencies MODE El KE AKE EVP CON 1 0.042 0.052 0.066 0.042 0.17 2 0.018 0.004 0.012 0.018 0.19 3 0.141 0.036 0.013 0.142 0.30 4 1.140 0.032 0.455 1.140 0.79 5 0.021 0.008 0.037 0.021 0.15 6.2.3 Results: Random Sensor Location Three hundred random sets of 15 sensors each were generated and evaluated in order to assess the level of increased performance of the various sensor placement algorithms against pure chance. The same time domain response with 5% noise used in the previous section was partitioned to the random sensor configurations. For the time domain data of each sensor set, ERA is used to identify the first five frequencies and mode shapes. The crossorthogonalities and frequency differences between the identified and FEM modes and frequencies were calculated. Figure 6.7 is a comparison of crossorthogonalities for the 300 random sensor sets and for the five sensor placement techniques (EI,KE, AKE, EVP, and OBS). The bar portion of each graph corresponds to each random sensor set value and the straight lines correspond to the five evaluated sensor configurations (El, KE, AKE, EVP, and OBS) and to the average value of all the random sensor sets (ARS). The top graph of Figure 6.7 is a plot of the maximum offdiagonal elements of the crossorthogonality matrix, the center graph is the average offdiagonal of the crossorthogonality matrix, and the bottom graph is the two norm of the crossorthogonality matrix minus the identity matrix. In general, the average random sensor set was within the acceptable limits on frequency matching and crossorthogonality. This is due to the fact that a large number of sensors were placed on the truss and approximately 1/3 of the trusses node points would be instrumented by the random sensor sets. Statistically, the random sets would have good chances of capturing pertinent modal information. Of the five techniques evaluated, kinetic energy, effective independence, eigenvector product, and observability gave better results than 97% of the random sets as can be seen in Figure 6.7. The maximum offdiagonal and the average offdiagonal of the crossorthogonality matrices of the three techniques are less than those for the average random set. In addition the two norm of the difference between the crossorthogonality and the identity matrix for the three techniques is lower than that of the average random set. However, the average kinetic energy gave results similar to the average random configuration, and showed little to no improvement over the purely random placement of fifteen sensors. The maximum offdiagonal of the AKE set was larger than that for the average random set and the average offdiagonal and two norm were approximately equal to those of the average random set. 6.3 Computational Efficiency For this particular example, the results for each of the excitation and sensor placement techniques are all relatively comparable. The targeted modes and frequencies are excited by all of the excitation placements and properly identified by all of the sensor sets evaluated. This is illustrated by the acceptable differences in FEM and identified frequencies and crossorthogonality values. The agreement between all the techniques can be partially contributed to the fact that the "the modal test" was a numerical simulation. The differences in the results for excitation evaluation and identification may be greater for a true modal test. However, an important issue that must be considered when using the discussed techniques for excitation and sensor location is the computational cost of each evaluation versus the accuracy of the modal test results. As can be seen from Table 6.3, the most efficient technique for excitation and sensor location is the eigenvector product technique. It may well be that on a more complicated example, the more computationally efficient techniques may result in modal test configurations which give less accurate modal information than the more complex placement techniques. A larger system model with more DOFs may make the CrossOrthogonality: Maximum OffDiagonal CrossOrthogonalitv: Average OffDiagonal Iw i 0 50 100 150 200 250 3 Two Norm: CrossOrthogonality Identity 0.08 I 1.1,I AKE ARS OBS EVP El KE ARS AKE EVP El OBS KE 00 ARS AKE B OBS a EVP KE 0 50 100 150 200 250 300 Figure 6.7 EightBay CrossOrthogonalities of Five Techniques Compared to 300 Random Sensor Sets Z 0.01 00.006 J l 2.)rt 0.07 0.06 Z 0.05 (q 0.04 "' ! 4111 tradeoffs between the computational cost of placement and the accuracy of the modal identification more apparent. Table 6.3 Total Floating Point Calculations for Each Placement Technique Technique Total flop count Placement El 812,000 sensors KE 92,700 sensors & excitation AKE 93,200 sensors & excitation EVP 480 sensor & excitation DPR 7,600 excitation CON 7,600 excitation OBS 7,600 sensor 6.4 Conclusion Based on the evaluation of the numerical simulation, each of the five excitation techniques successfully placed three excitation sources on the structure which would excite the first five modes of vibration. The sensor placement techniques of effective independence, kinetic energy, eigenvector product, and observability found sensor locations which showed better frequency matching and crossorthogonality than 97% of the random sensor sets. The sensor set obtained using average kinetic energy showed no improvement in crossorthogonality or frequency matching over those of the random sensor sets. Based on the similar results of the placement techniques for sensors and actuators, a more complex structure will now be used to compare the techniques discussed in this chapter as well as other techniques outlined in Chapter 5. CHAPTER 7 PREMODAL TEST PLANNING ALGORITHM APPLICATION: MICROPRECISION INTERFEROMETER TRUSS 7.1 Introduction A comparative study of several premodal test planning techniques is presented using the Jet Propulsion Laboratories' MicroPrecision Interferometer (MPI) testbed. Mode indicator functions calculated using a reduced FEM of the structure and degrees of target mode controllability are used in conjunction with genetic algorithms to find location and orientation of two excitation sources in order to optimally excite a chosen range of target modes during a modal test. Effective independence, kinetic energy, eigenvector product, observability, and mode indicator function techniques are used to place a combination of sensors on the structure for the purpose of modal identification. The sensors are placed in two ways: independent sensor placement and triaxially constrained placement. A numerical simulation of the response of the structure is used to evaluate the effectiveness of each of the placement techniques to identify the target modal parameters of the structure. The effect of FEM error on the various placement techniques is evaluated. 7.2 MicroPrecision Interferometer Test Bed The MPI, shown pictorially in Figure 7.1, is a tested that has been built in order to study structural control systems in the development of space interferometers. Modal tests were performed on the MPI structure by two independent groups (Sandia National Laboratories and the Jet Propulsion Laboratories (RedHorse et al., 1993; Carne et al., 1993; LevineWest et al., 1994). left extending boom z x right extending boom Figure 7.1 MPI Structure The FEM used to evaluate the placement techniques in the current work was obtained from Sandia National Laboratories (RedHorse et al., 1993). The model used is a 240 DOF Guyanreduced FEM which has been updated using the data obtained from the modal test of the structure. The 240 DOFs correspond to three DOFs (x,y,z) at each of the 80 node balls. The frequencies from the Guyanreduced FEM corresponding to the first 12 nonrigidbody modes are given in Table 7.1 and are compared to actual frequencies obtained during the modal test. 7.3 Excitation Placement During the original modal test of the MPI structure, two excitation sources were used as pictured in the top portion of Figure 7.2. The lower portion of this figure is the excitation configurations that were obtained by optimizing the MIFs of the FEM using a GA (GMIF) and by optimizing the modal controllability of the FEM using a GA (GCON). Both the original and the GMIF excitation locations have an exciter on the two extending booms although they are oriented differently. The GMIF setup moves the excitation of the right extending boom to an interior point in comparison to the original configuration. The GCON technique placed an exciter at the midpoint of the left extending boom and an exciter at the top of the main boom. Figure 7.3 gives typical frequency responses for the two excitations of the three techniques shown in Figure 7.2; the responses are measured at the sensor location shown in Figure 7.2 in the ydirections. The straight line corresponds to the first force located by each technique and the dotted line corresponds to the second. Original node 79 0.000 x 0.707 y 0.707 z node 41 0.707 x 0.000 y 0.707 z GMIF derived excitation node 77 0.5225 x 0.8323 y 0.1538 z z Xy x node 19 0.2471 x 0.9618 y 0.1178 z GCON derived excitation node 6 sensor 0.8660 x 0.5000 y 0.0000 z node 67 0.000 x 0.000 y 1.000 z Figure 7.2 Excitation Placement on MPI Structure Table 7.1 Reduced MPI FEM Frequencies Compared with MPI Modal Test Frequencies Mode Frequency (Hz) Frequency (Hz) FEM Modal Test 1 7.82 7.75 2 11.66 11.65 3 12.75 12.67 4 29.52 29.36 5 34.45 34.06 6 37.76 37.34 7 42.81 42.25 8 47.30 46.04 9 51.14 50.69 10 52.36 53.00 11 55.41 56.82 12 61.40 60.04 The excitation devices placed by the GMIF algorithm were selected to minimize an objective function which was dependent on the MIF of each of the two excitation locations. The MIF will be nearly 1.0 except near normal modes, at which point it drops off considerably. This dropoff indicates that the mode is well excited. Therefore, it is desirable to find two excitation sources (location and orientation) which exhibit a sharp drop at all of the normal frequencies. The GMIF objective function was designed to find an excitation sources) which exhibits sharp drop offs at normal frequencies as discussed in Chapter 5. The GCON algorithm, as discussed in Chapter 5, was designed to find excitation sources which minimize the angles between the input influence vector subspace and the subspaces spanned by the modes of the system thus maximizing the controllability of the modes of the structure. By choosing excitations with maximum FE modal controllability, the amount of energy being imparted to the FE modes of the system by the excitation is theoretically maximized. In order to compare the three excitation sources, the MIF drop off values for the first twelve modes of each of the excitations is calculated and shown in Table 7.2. The sharpest dropoff value for each of the modes is highlighted in bold in the table. Original Modal Test Excitation frequency (Hz) GMIF Excitation frequency (Hz) node 41 node 79 Snode 19  node 77 node 6 node 67 frequency (Hz) Figure 7.3 Typical Frequency Response for MPI Structure Table 7.2 Original, GMIF, and GCON Excitation Locations MIF Values Original GMIF GCON MODE 41 79 19 77 6 67 1 0.0014 0.0006 0.0003 0.0106 0.0345 0.0129 2 0.0063 0.2601 0.0105 0.0048 0.0095 0.0079 3 0.0394 0.0022 0.4106 0.0428 0.0211 0.0256 4 0.0854 0.2017 0.0228 0.0706 0.4023 0.0287 5 0.0236 0.0420 0.0836 0.0184 0.0174 0.0408 6 0.4636 0.0609 0.0393 0.6855 0.2495 0.0862 7 0.0484 0.0489 0.0911 0.7126 0.6753 0.0372 8 0.0755 0.3263 0.3318 0.0644 0.0996 0.1947 9 0.1537 0.7790 0.7521 0.0824 0.0550 0.5615 10 0.7228 0.8957 0.1674 0.8050 0.7625 0.2260 11 0.4077 0.0407 0.0826 0.2747 0.4372 0.0747 12 0.0724 0.8088 0.4408 0.0591 0.0703 0.2013 min MIF 2 / 12 7 / 12 3 / 12 The GMIF excitations exhibit a sharper dropoff than the original excitations' MIFs for 10 of the 12 target modes. The GCON exhibit a sharper dropoff than the original excitation for 6 of the 12 modes. Comparing all three techniques, the GMIF had the most minimum dropoff values at 7 followed by the GCON technique at 3, and the original excitation at 2. This is not a surprise since the GMIF is designed specifically to find excitation sources which exhibit the greatest dropoffs. An improvement in the dropoff of the GMIF excitation over the original excitation can especially be seen for the tenth mode. It is interesting to note that even though the GMIF has the best overall MIF dropoff values, the minimum dropoff values of the GCON excitations are well within acceptable levels. The highest minimum value for the GCON technique is 0.25 for the fifth mode. However, in the next chapter it will be shown that an excitation with good controllability values does not necessarily have acceptable MIF values. In order to evaluate the performance of the genetic algorithm for excitation placement, a set of 500 random 2point excitations is generated. The number of random excitations, 500, was chosen because the GMIF algorithm evaluated approximately 400 population members in the search for the chosen GMIF excitation. The MIF values for each of the random excitations are calculated, sorted and graphed in Figure 7.4 and the controllability angles for the random excitations are calculated, sorted, and graphed in Figure 7.5. The top graph of Figure 7.4 shows the maximum MIF value for the original, the GMIF derived, and the GCON derived excitations superimposed on the graph of 500 random designs in order to compare the values. The top portion shows the GMIF excitation has a smaller maximum MIF than 97% of the random population. The bottom portion of Figure 7.4 is a graph of the genetic MIF excitation placement algorithm objective function values of the random and selected excitations. An optimal excitation according to the GMIF objective function is one which has as small as possible maximum MIF drop off value. The bottom portion of the figure shows that the GMIF excitation outperformed 100% of the random population. This shows that the genetic algorithm was successful in finding a good excitation based on the objective function designed in a more computationally efficient manner than an exhaustive search. The same evaluation was performed for the controllability angle calculations as pictured in Figure 7.5. The controllability excitation placement algorithm objective function is designed to find excitations which have a minimum controllability angle sum over the target modes to be excited. Even though the original and GMIF excitations have a smaller maximum controllability angle as seen in the middle graph of Figure 7.5, the GCON excitation has a better angle sum as seen in the bottom graph. In fact, the GCON excitation and the original excitations outperform 100% of the randomly generated excitations. Numerical simulations of the MPI structural response to simultaneous impulses applied at the two excitation locations were calculated within the MATLAB environment. Five percent noise was added to the simulated time responses of the structure. These time responses were used along with the Eigensystem Realization Algorithm (ERA) to identify EXCITATION Maximum MIF 100 150 200 250 300 350 400 4E GMIF Objective Function S I I SI I i I 50 100 150 200 250 300 350 400 450 500 Random Excitations random original GCON GMIF random original GCON GMIF Figure 7.4 Comparison of Selected Excitation and Random Excitation MIF Values the twelve target mode shapes and frequencies (Juang and Pappa, 1985). The evaluation of the success of ERA to identify the frequencies and mode shapes was based on a frequency percent difference comparison between identified and FE frequencies and on a crossorthogonality check between FE and identified mode shapes using an exactly reduced mass matrix. The reduction is exact in the sense that the frequencies and mode shapes of the reduced system match exactly their counterparts in the unreduced model (O'Callahan et al., 1989). When ERA was used to identify system mode shapes and frequencies, it missed the fifth and tenth frequencies and mode shapes when the original 41/79 DOF excitation locations were used to numerically simulate structural excitation. To illustrate this, the crossorthogonality between FEM and ERA identified mode shapes was calculated, and is 2 $1.5 1 0.5 0  Minimum Controllability Angle O U I  I  I  I  I  I  I  I  85 75 70 0 50 100 150 200 250 300 350 400 450 500 90 Maximum Controllability Angle 89 88  87 86 I uou '1060 0 0 50 100 150 200 250 300 350 oo GCON Objective Function 400 450 500 o 01020 <1000 98C1 I I j 980' II i I 0 50 100 150 200 250 300 350 400 450 500 Random Excitations random GMIF original GCON Figure 7.5 Comparison of Selected Excitation and Random Excitation Controllability Angles pictured in Figure 7.6. For this example, the 240 DOF simulated response was partitioned to the sensor configuration obtained using the El technique as discussed in the following section. These poor crossorthogonality results are corroborated by the frequency response function shown in Figure 7.3 in which poor excitation can be seen for modes 5 and 10. The GMIF and GCON derived excitations resulted in successful ERA identification of all 12 mode shapes and frequencies of the original FEM.  lfl fe\ EXCITATION random GMIF original GCON random GCON GMIF original 1.00 0.50 0.25 modes [] 1.00 Effective Independence <0.25 Unconstrained Sensor Set E <0.02 Figure 7.6 CrossOrthogonality Between FE Modes and Identified Modes 7.4 Sensor Placement Six sensor selection techniques, effective independence, kinetic energy, average kinetic energy, eigenvector product, observability, and MIF were used to place sensors on the MPI structure. Most of these techniques were previously evaluated for sensor placement using the NASA eightbay testbed in Chapter 6. In that study, the five techniques of effective independence, kinetic energy, average kinetic energy, eigenvector product, and observability, performed equally well. This could be due to two reasons: (i) the structure lacked significant dynamic complexity required to distinguish between the methods or (ii) the methods were actually so similar that they led to similar results regardless of structural dynamics. One purpose of this study is to again evaluate the five techniques on a more complex dynamic system in addition to evaluating the efficacy of using the MIF for sensor placement. The second purpose is to investigate the suitability of the techniques when the sensors are constrained to be placed in a triaxial fashion. Eighteen sensors were placed in two different studies using the six techniques in order to best identify the 12 target FEM mode shapes and frequencies. First, the techniques were used to choose 18 of the 240 DOFs as sensor locations. In the second study, the techniques were constrained to choose 18 triaxially constrained sensors (i.e., 6 triaxsensor sets). The excitations selected using the GMIF discussed in the previous section were used to excite the MPI structure numerically in order to test the various sensor configurations. 7.4.1 Unconstrained Sensor Placement The first placement study evaluated the six techniques' placement of 18 sensors on the MPI structure at any of the 240 DOFs (x,y,z of the 80 node balls). The first 12 flexible modes of vibration were chosen as the target modes for each technique. The locations of the sensors obtained using each of the techniques are pictured in Figure 7.7. All of the techniques evaluated placed a majority of the 18 sensors at the ends of the three booms. In addition all of the techniques except MIF placed sensors in the two DOFs for each boom which exhibited the greatest range of motion (i.e., xy for the primary boom, xz for the extending right boom and yz for the extending left boom). The EVP technique clustered all 18 sensors at the boom tips, and the AKE techniques clustered 17 of the 18 sensors at the boom tips, with one sensor being placed near the midspan of an extending boom. The KE technique placed 15 sensors at the boom tips with 3 sensors near the midspan of the two extending booms. The El technique placed 13 sensors at the boom tips and at least one sensor near the midspan of the main and extending booms. The observability technique placed 17 of the 18 sensors at the boom tips and 1 sensor at the midspan of an extending boom. The MIF technique resulted in the most unusual sensor configuration with several sensor being placed in the zdirection on the main boom. The zdirection is not the primary direction of motion for this portion of the truss. Of the twelve target modes shapes, modes 2 through 11 exhibit a bending mode similar to that of secondmodecantileveredbeam bending in at least one of the main or extending booms. Secondmode bending is clearly exhibited by the two extending booms for all target modes and is exhibited by the main boom for some of the twelve target modes. The sensor configurations chosen by the El and KE techniques are particularly suited to capture this secondbending mode shape due to their placement of some sensors at midspans of the three booms. The FEM of the MPI structure was used with MATLAB to simulate a time response of the structure to an impact applied at the GMIF chosen excitation locations for all 240 DOFs. Five percent uncorrelated noise was added to the time response. The noisy response was then partitioned to each of the six sensor sets and was sent to ERA for mode shape and frequency identification. All the techniques resulted in percent frequency difference between FEM and identified frequencies of much less than 1% which is well within industry accepted standards (Flanigan and Hunt, 1993). Crossorthogonalities between FEM and identified mode shapes were calculated for each of the techniques and are pictured in Figure 7.8. In order to calculate the crossorthogonalities, the 240 DOF FE mass matrix was reduced to 12 DOFs using exact reduction. For this size model, exact reduction was computationally acceptable, therefore, it was used to get the best crossorthogonality comparison. All of the offdiagonal elements of the crossorthogonality matrix for the EI technique are within the industry accepted standards of <0.02 for primary modes (Flanigan and Hunt, 1993). This can be seen graphically in Figure 7.8. For the KE, AKE, and MIF techniques, almost all of the offdiagonal elements are <0.02. Some of the entries are between 0.02 and 0.1 which is within the industry standard for secondary modes (<0.1). The OBS technique resulted in the most modes having crossorthogonalities <0.10 which is acceptable for secondary modes. The 8th mode was not successfully identified by the OBS sensor set. The crossorthogonality for the EVP technique was poor for all target modes. The crossorthogonality of the EVP technique was evaluated with no noise, 1%, 2%, and 5% noise added to the time response. Of the four time responses evaluated, only the response with no noise gave acceptable crossorthogonality values. Based on these calculation, the EVP technique was unsuccessful in finding an acceptable sensor set. Effective Independence Kinetic Energy Average Kinetic Energy Observability z y x MIF *z My Ox Figure 7.7 Unconstrained MPI Sensor Sets Eigenvector Product a) Effective Independence 1.00 <0.10 <0.02 b) Kinetic Energy 1.00 <0.10 <0.02 Figure 7.8 CrossOrthogonality Between MPI FE and Identified Modes 18 Unconstrained Sensors 1.00 0.10 0.02 1.00 0.10 0.02 1.00 0.10 0.02 d) Eigenvector Product 1.00L <0.75 <0.50 <0.25 <0.02 e) Observability J 1.00 < <0.50 * <0.25 * <0.10 * <0.02 e) MIF 1.00 <0.10 <0.02 Figure 7.8 continued 1.00 0.75 0.50 0.25 0.02 1.00 0.50 0.25 0.10 0.02 1.00 0.10 0.02 7.4.2 Triaxially Constrained Sensor Placement The six sensor placement techniques were modified to place 6 triaxially constrained sensor sets (18 total sensors) at any of the 80 node balls of the MPI structure. The resulting 6 triaxsensor sets placed using the six placement techniques are pictured in Figure 7.9. The EI, KE, and AKE techniques grouped two sensor sets at or near the end of each boom, the OBS technique put one, two, and three sensors at the end of each boom, and the EVP technique placed three triaxsets at the ends of only two of the booms. It should be noted that if the EVP placement task were extended to placing 7 triaxsets, the seventh set would be placed at the end of the main boom using the EVP technique. The MIF sensor placement technique put two sensors at the end of the right boom, one sensor at the end of the left boom, one sensor at the base of the left boom, and two sensors on the main boom. As in the case of the unconstrained set, the MIF technique did not place any sensors on the tip of the main boom. The time response of the MPI structure excited by the GMIF actuator locations was partitioned to those DOFs corresponding to the six triax sensor locations chosen by the six placement techniques. The partitioned numerical data with noise added was sent to ERA in order to evaluate the effectiveness of each of the triaxsensor sets in identifying the system mode shapes and frequencies. All the techniques resulted in percent frequency difference between FEM and identified frequencies of much less than 1% which is well within industry accepted standards. The crossorthogonality calculations between the FEM target modes and the ERA identified modes were performed using an exactly reduced mass matrix as in the previous section, and are shown in Figure 7.10. All of the offdiagonal crossorthogonality values for the KE techniques, shown in Figure 7.10, were within the industry standard of <0.02 for offdiagonal elements for primary modes. The EI, AKE, and MIF techniques resulted in crossorthogonalities which were within this standard for most of the modes, but which were slightly above the offdiagonal standard for a few modes. The OBS technique resulted in acceptable crossorthogonalities for most modes and acceptable secondary crossorthogonalities for modes 7 through 10. The EVP technique resulted in poor offdiagonal crossorthogonality values for all modes. Effective Independence Eigenvector Product z x Energy MTF * triaxial sensor group Figure 7.9 Triaxially Constrained MPI Sensor Sets a) Effective Independence 1.00 0.10 0.02 E 1.00 modest N <0.10 3 2 3 4 E <0.02 1 b) Kinetic Energy 1.00 0.10 0.02 2 1 1.1 * <0.10 modes 6 * <0.02 c) Average Kinetic Energy 1.00 0.10 0.02 El 1.00 S<0.10 modes * <0.02 Figure 7.10 CrossOrthogonality Between MPI FE Modes and Identified Modes 6 Triaxially Constrained Sensors d) Eigenvector Product 1.00 <0.75 <0.50 <0.25 <0.02 e) Observability 1.00 <0.10 <0.02 Figure 7.10 continued 1.00 0.75 0.50 0.25 0.20 1.00 0.10 0.02 1.00 0.10 0.02 7.4.3 Unconstrained vs. TriaxiallyConstrained Sensor Sets Based on the crossorthogonalities and frequency differences between FE and identified mode shapes and frequencies the EI, KE, AKE, and MIF techniques located sensor sets for the unconstrained and constrained examples which were successful in identifying the target mode set. Only a few of the crossorthogonalities were slightly above acceptable primary mode values, and all values were within secondary mode standards. For the unconstrained sensor set, the El sensor set resulted in identified modes with the best crossorthogonality with the FE target modes. However, for the triaxiallyconstrained example, the KE sensor set resulted in identified modes with the best crossorthogonality with FE target modes. In both constrained and unconstrained cases the EVP technique resulted in identified modes with poor crossorthogonalities with FE mode shapes. However, the EVP and OBS triaxially constrained sensor sets showed an improvement over the unconstrained sets. This is unusual because as a rule, the constrained sets do not perform as well as the unconstrained sets. 7.5 Effect of Model Error In order to investigate the effect that model error has on the various placement techniques, error was added to the original Guyanreduced FEM of the MPI structure as seen in Figure 7.11. Specifically, 1/3 of the struts' crosssectional areas were decreased by 20%, 1/3 of the struts' crosssectional areas were increased by 20%, and the remaining 1/3 of the struts were unchanged. The resulting differences in precorrupted and postcorrupted model frequencies and mode shapes are listed in Table 7.3. The second column represents the percent differences in the frequencies of the two models. The third column represents the root mean squared (RMS) values of the absolute differences in the mode shapes of the two models. The differences between the pre and postcorrupted model mode shapes are shown pictorially in Figure 7.12. The true modes are plotted along the horizontal axis and the corrupted modes are plotted along the vertical axis. 81 AA = change in crosssectional area 20% AA 0% AA +20% AA Figure 7.11 Model Error Added to MPI FEM 0.2 2 0.1 0 0.1 0.2 mode I 0 0. 2 0.1 0 0.1 0.2 mode 5 .02 0.1 0 0.1 0.2 0 01..2 0.1 0 0.1 0.2  2 0.1 0 0.1 0.2 0 mode 0:2 0.1 0 0.1 0.2 0, S.2 0.1 0 0.1 0.2 0.2 0.1 0 0.1 0.2 6.2 0.1 0 0.1 0.2 .01 2 0.1 0 0.1 0.2 Figure 7.12 True vs. Corrupted MPI Mode Shapes Table 7.3 Difference Between Pre and PostCorrupted Model Frequencies and Mode Shapes MODE Frequency % difference Mode Shape RMS values 1 3.15 0.90 e3 2 3.23 5.20 e3 3 2.01 5.20 e3 4 3.27 4.70 e3 5 1.27 5.60 e3 6 1.27 10.9 e3 7 1.76 13.9 e3 8 0.45 11.8 e3 9 2.23 9.00 e3 10 3.47 10.4 e 3 11 0.12 14.7 e3 12 1.96 14.1 e3 7.5.1 Excitation Placement with Model Error Once error was introduced into the MPI FEM, the GMIF and GCON excitation placement techniques using the corrupted FEM were run. The GMIF derived excitations for the uncorrupted and corrupted models are pictured in Figure 7.13, and the GCON derived excitations for the uncorrupted and corrupted models are pictured in Figure 7.14. The GMIF excitation placement changed slightly when the model error was added; only node 77 switched to node 76 when model error was added. The GCON excitation moved the shaker from the mid span to the tip of the left boom. The directions for all of the exciters were changed when model error was added. In order to evaluate the excitations obtained using the corrupted FEM, the time response of the MPI structure to impacts at the new excitations was numerically simulated using the original uncorrupted model. This time response was then partitioned to the uncorrupted unconstrained El sensor set of section 7.4.1 and was sent to ERA for identification. As in the case of the uncorrupted model excitations, the target frequencies and mode shapes were node 19 0.2471 x node 77 0.9618 y 0.5225 x 0.1178z 0.8323 y 0.1538 z Uncorrupted FEM sensor z  y ~node 19 x 0.5036 x 0.6800 y 0.5329 z node 76 0.5890 x 0.5714 y 0.5714 z Corrupted FEM Figure 7.13 GMIF Derived Excitation Locations successfully identified based on percent difference and crossorthogonality calculations. Based on these results, the error added to the FEM had little to no effect on the excitation placement configurations' success in exciting the uncorrupted target mode shapes of the structure for both the GMIF and GCON excitation placement techniques. 7.5.2 Sensor Placement with Model Error Both the unconstrained and triaxially constrained sensor placement problems were evaluated after error was added to the FEM using the six placement techniques previously discussed. The changes in sensor set configurations for the unconstrained and constrained sets are shown pictorially in Figure 7.15 and Figure 7.16. The original sensors placed using the uncorrupted FEM are represented by the boxes. Any sensors that were removed from the original sensor set after model error was introduced are represented by circles and any node 6 0.8660 x 0.5000 y 0.0000 z node 67 0.000 x 0.000 y 1.000 z Uncorrupted FEM z sensor node 53 0.5000 x ) Y 0.8660 y 0.0000 z node 79 0.4330 x 0.2500 y 0.8660 z Corrupted FEM Figure 7.14 GCON Derived Excitation Locations sensors that were added to the original set after model error was introduced are represented by triangles. The total numbers of sensors that changed for the unconstrained and triaxially constrained sensor sets after model error was introduced are listed in Table 7.4. The general distribution of the sensors was mostly maintained after model error was added for all of the unconstrained sensor sets except for the MIF sensor set. For the constrained sensor sets, five of the six placement techniques resulted in a changed sensor set after model error was added. The El technique moved one triaxset from the main boom tip to midboom. The KE technique moved one triaxset from the left extending boom tip to midextendingboom, and the EVP technique moved a triaxset from the left extending boom to the main boom. The AKE technique resulted in no sensor change after model error was added. The OBS and MIF 85 m Y l] X Effective Independence Eigenvector Product Z x Kinetic Energy Average Kinetic Energy Observability z MIF y 0 original sensor set O removed using model error A added using model error Figure 7.15 Model Error Effect on Unconstrained MPI Sensor Sets Effective Independence Kinetic Energy Average Kinetic Energy z _y Observability MIF * original triaxial sensors A added using model error removed using model error Figure 7.16 Model Error Effect on Constrained MPI Sensor Sets Eigenvector Product technique changed over half the triaxially constrained sensors. For both the unconstrained and triaxially constrained cases, the MIF sensor placement technique was particularly sensitive to FE model error. The original uncorrupted FEM response to the GMIF derived excitation was used to evaluate the new sensor sets obtained with the corrupted FEM. The time response discussed in the previous section was partitioned to the new sensor configurations and ERA was used to identify mode shapes and frequencies. Both the unconstrained and constrained sensor sets obtained using the corrupted FEM were successful in identifying the target frequencies within 1%, for all six techniques evaluated. The resulting crossorthogonalities between identified (using error sensor sets) and original FEM mode shapes were calculated and are pictured in Fig. 7.17 and Fig. 7.18. Table 7.4 Number of sensors or triax sets that change when model error is added MPI Sensor Set El KE AKE EVP OBS MIF unconstrained 2 of 18 2 of 18 1 of 18 6 of 18 2 of 18 16 of 18 constrained 1 of 6 1 of 6 0 of 6 2 of 6 3 of 6 4 of 6 For the unconstrained sensor sets, the EI, KE, AKE, and MIF techniques resulted in generally acceptable crossorthogonality values for the twelve target modes shown in Fig. 7.17. Only a few offdiagonal entries of the crossorthogonalities resulting from these sensor configuration were above the acceptable limit of <0.02 for primary modes, but were still within the acceptable limit of <0.10 for secondary modes. The error added to the model in this example had little effect on the placement techniques' success in identifying sensor configurations which resulted in successful modal information identification. The EVP and OBS techniques resulted in poor crossorthogonalities as was the case when the uncorrupted model was used. For the triaxially constrained sensor configuration, the model error did not greatly affect the uncorrupted crossorthogonality results for the EI, KE, AKE, and MIF techniques, as 