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Ii};- Einliczv} can-zyb «$14.15!.(5vl‘ S nil...)- ....:.(,...n: :1 .. . . 2.. . ¢‘ .4 LIBRARY l Michigan State . l U i‘li'velsiiy This is to certify that the dissertation entitled ORTHOGONAL DECOMPOSITION METHODS FOR MODAL ANALYSIS presented by UMAR FAROOQ has been accepted towards fulfillment of the requirements for the PhD. degree in Mechanical Engineering Major Professdr’s Signature 12/18/2009 Date MSU is an Affirmative Action/Equal Opportunity Employer PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5’08 KIProj/AocsPreleIRCIDateDueindd “r“ 33:!“ _~ 4“ s... _._. ‘ ORTHOGONAL DECOMPOSITION METHODS FOR MODAL ANALYSIS By Urnar Farooq A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Mechanical Engineering 2009 ABSTRACT ORTHOGONAL DECOMPOSITION METHODS FOR MODAL ANALYSIS By Umar Farooq The output-only class of experimental modal analysis has recently gained tremen- dous popularity. The distinct advantage of using these methods is the measurement of responses only, without the need of input measurements. On the other hand, the proper orthogonal decomposition (POD) for some time, has been recognized as a standard empirical modal analysis scheme. This work is an attempt to apply some generalizations of POD to tie the gap between output—only modal analysis and de- composition methods. The state-variable modal decomposition (SVMD) is a generalization of POD method developed in this work. It enables frequency, damping, and mode—shape esti- mations from free-response ensemble data for generally damped linear multi-degree— of-freedom systems. The method extracts modal parameter information from an eigenvalue problem that is formed by correlations of state-variable ensembles. N u- merical simulations are presented to show the method’s efficacy for a broad range of damping levels present in the systems. The smooth orthogonal method (SOD) is another generalization of the POD method that has shown good parameter estimation characteristics from free responses in the past. Extension of the SOD and SVMD decomposition methods for randomly excited systems is also addressed in this work. This work shows that the SOD and SVMD eigenvalue problems formulated from white noise induced response data can be tied to the unforced structural eigenvalue problem, and thus can be used for modal parameter estimation. Numerical simulations for lightly damped vibratory system are explored. Simulations show that for light damping, satisfactory results are obtained for estimating both system frequencies and modal vectors. Experimental investigation of the SVMD method for the free response case is con- ducted in order to help understand practical considerations regarding the method. Using various experimental setups, issues related to optimal data—length and filter settings are explored. Modal coordinates are evaluated, together with modal vec- tors/ frequencies, to qualitatively eliminate the spurious modes obtained during de- composition. The decomposition quality is quantified next by means of two new tools. The spectrum assurance and the spectrum difference criteria are first evaluated on nu- merical examples. These methods are then investigated in real life experiments to quantify the quality of modal coordinates and identification results. We observe that both tools used together help establish the true system modes and eliminate spurious modes. The effect of time record length used in the identification process is studied via numerical examples. Lastly, a comparison of decomposition based methods is made with the common time-domain identification schemes using numerical simulations as well as actual ex- periments. While the comparison highlights the strengths and weaknesses of the compared methods, it also shows that the orthogonal decomposition based methods are, in essence, equally competitive with the time-domain methods. Dedicated to To the greatest Gift of God to the world, Prophet Muhammad (SAWW) To my parents, for their unconditional and imperishable love, My parents Sahibzada Ishtiaq Hussain and Mrs. Shaheen Ishtiaq My nieces Huda and Warda iv ACKNOWLEDGMENTS This work would not have been possible had my advisor Dr. Brian Feeny not inspired me to do so. I would like to extend my deepest and fondest gratitude to him for his support, encouragement and tutelage through thick and thin during this journey. My guidance committee deserves special appreciation. I am indebted to Dr. Hassan Khalil, Dr. Alan Haddow, and Dr. Steve Shaw for their time, help and useful sugges- tions to bring improvements in this work. I would like to thank my late grandfathers Sahibzada Irshad Hussain and Sahibzada Mushtaq Hussain, and my siblings Yasir Waqas and Huma Ishtiaq for their unflagging support and continuous prayers that has helped me survived through the hard times. I would also like to thank my friends and colleagues for their endless support during my years at Michigan State University. I value the friendship of Shahin Nudehi, Elliot Motato, Jeff Rhoads, Jimmy Issa, Nick Miller and Brian Olson and cherish the technical discussions with them. A word of thank for Dr. Gokcek who instilled in me the importance of writing technical papers. I appreciate his guidance in that matter. I would also like to thank Dr. David Chelidze for introducing the beautiful SOD scheme to me that was motivation of this work. Last but not the least, I would like to thank our graduate secretary Aida Montalvo, and my room mates M. Vaqas Ali and Farooq Rauf for their help and support. TABLE OF CONTENTS LIST OF TABLES ................................. x LIST OF FIGURES ................................ xii 1 Introduction ' ................................. 1 1.1 Background ................................ 1 1.1.1 Analytical modal analysis .................... 2 1.1.2 Experimental modal analysis ................... 3 1.1.3 Classification of signals ..................... 3 1.2 Output-only modal analysis ....................... 4 1.3 Proper orthogonal decomposition and output-only time domain methods 5 1.4 Preview of this thesis ........................... 7 1.5 Contributions of this work ........................ 10 2 State Variable Modal Decomposition of Free Responses ............. 13 2.1 Introduction ................................ 13 2.2 Nonsymmetric state-variable modal decomposition ........... 14 2.2.1 State-variable linear vibration model .............. 14 2.2.2 Decomposition strategy ...................... 15 2.3 Numerical examples ............................ 17 2.3.1 Proportionally damped example . . . . . . . . . . . . . . . . . 18 2.3.2 Generally damped example ................... 19 2.3.3 Step size selection ......................... 20 2.4 Robustness under noise .......................... 21 2.5 Summary ................................. 25 3 Random Response .............................. 27 3.1 Introduction ................................ 27 3.2 Smooth orthogonal decomposition .................... 28 3.2.1 Smooth orthogonal decomposition and modal analysis for free vibration .............................. 28 vi 3.2.2 Smooth orthogonal decomposition for systems under random excitation ............................. 30 3.2.2.1 Smooth orthogonal decomposition and random excita- tion ........................... 30 3.2.2.2 White noise ....................... 31 3.3 Numerical example ............................ 32 3.4 State-variable modal decomposition for systems under random excitation 38 3.4.1 State—variable modal decomposition and random excitation . . 38 3.4.2 Numerical example ........................ 40 3.5 Summary ................................. 42 Experimental Investigation .......................... 44 4.1 Introduction ................................ 44 4.1.1 Assumptions ............................ 45 4.2 A single-degree-of-freedom experiment ................. 47 4.3 Clamped-free beam experiment ..................... 50 4.3.1 Filtering effects .......................... 54 4.3.2 Identification results ....................... 55 4.3.3 Modal coordinates- ........................ 58 4.4 Limitations on the number of sensors and active modes ........ 64 4.4.1 Sensitivity analysis ........................ 64 4.4.2 Numerical example ........................ 67 4.4.3 A free-free linear beam experiment ............... 68 4.4.3.1 Identification by SVMD ................ 70 4.5 Summary ................................. 73 Quantitative Assessment of Identification ................... 75 5.1 Use of modal coordinates to evaluate decomposition results ...... 75 5.2 Quantitative assessment of modal coordinate F RF ........... 77 5.2.1 Introduction ............................ 77 5.2.2 The spectrum assurance criterion (SAC) ............ 77 5.2.3 Spectrum difference (SD) ..................... 78 5.3 Numerical simulations .......................... 79 vii 5.3.1 Correct frequency and damping estimates in both F RF 8, with noise contamination in one of the FRFS ............. 80 5.3.2 Correct frequency in both FRFs, with incorrect damping in the tested FRF .............. _ .............. 82 5.3.3 Correct damping in both FRFS, with incorrect frequency in the tested FRF ............................ 84 5.3.4 Lightly damped vs. heavily damped systems .......... 86 5.3.5 Summary of trends ........................ 88 5.4 Experimental data ............................ 90 5.4.1 Large sampling N=12,000 .................... 92 5.4.2 Smaller sampling N=1000 .................... 95 5.4.3 Comments and summary ..................... 99 5.5 Time record length ............................ 100 5.5.1 Results ............................... 101 5.5.1.1 High damping ...................... 101 5.5.1.2 Low damping ...................... 103 5.5.2 Summary of trends ........................ 104 5.6 Summary ................................. 106 Comparison with Other Identification Methods ................. 1 10 6.1 Introduction ................................ 110 6.2 Time—domain modal identification methods ............... 111 6.2.1 Ibrahim time domain method .................. 111 6.2.2 Eigensystem realization algorithm ................ 114 6.2.2.1 A note on obtaining Markov parameters in ERA . . 116 6.3 Numerical simulations .......................... 117 6.3.1 Free response: A three DOF discrete system example ..... 117 6.3.1.1 Noise effects ....................... 119 6.3.2 Forced response: A sixteen DOF system example ....... 120 6.3.2.1 Noise effects ....................... 121 6.4 Clamped-free cantilever beam experiment ................ 123 6.4.1 Identification results ....................... 125 6.5 Summary ................................. 130 viii 7 Conclusions ................................. 134 7.1 Synopsis .................................. 134 7.1.1 Free response ........................... 134 7.1.2 Random response ......................... 135 7.1.3 Experiments ............................ 136 7.1.4 Quantitative evaluation ...................... 136 7.1.5 Comparison to other methods .................. 137 7.2 Recommendations for use of the decomposition methods ....... 138 7.3 Future work ................................ 139 7.3.1 Damping estimation for random excitation ........... 140 7.3.2 Continuous systems ........................ 140 7.3.3 Nonlinear systems ......................... 141 BIBLIOGRAPHY ................................. 144 ix 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 LIST OF TABLES System frequencies estimated from smooth orthogonal decomposition (SOD) with white noise forcing compared against structural eigen fre- quencies in Eq. 3.3 ............................ Estimated SOD modes compared against system modes using the modal assurance criterion ........................ Percentage error computation (SOD) for all modes .......... System frequencies estimated from state variable decomposition (SVMD) with white noise forcing compared against structural eigen frequencies in Eq. (3.3) .......................... System damping ratios estimated from the state-variable decomposi- tion (SVMD) with white noise forcing compared against structural damping ratios in Eq. (3.3) ........................ Estimated modal parameters from the decomposition method com- pared against theoretical and classical experimental methods ..... Estimated modal parameters from the decomposition method for a linear beam free response compared against the undamped structural eigen frequencies and the FF T for data consisting of first N = 12, 000 sample values ................................ Experimental system frequencies (Hz.) estimated from SVMD for a linear beam free response compared against the undamped structural eigen frequencies and the F FT values for data consisting of first N = 1000 sample points. ........................... Experimental modal damping estimated from the SVMD method for the linear beam free response for first N = 1000 sample values. Only the first three damping estimates are experimentally estimated by “pure mode” log decrement method .................... Modal parameters estimation from reduced number of sensors com- pared against the theoretical eigenvalue problem for a three DOF dis- crete system with 8 bit quantization noise. ............... 33 34 37 41 42 49 56 57 68 4.6 5.1 6.1 6.2 6.3 6.4 6.5 Modal frequency estimation for an experimental free-free beam using a number of sensors that is less than active frequencies compared against the theoretical undamped values. .................... Estimated modal parameters from the decomposition method for a linear beam free response for first N = 12, 000 sample values ...... Estimated modal parameters from various decomposition methods com- pared against the theoretical eigenvalue problem for a 3 DOF discrete system example .............................. Modal parameters estimation from various decomposition methods compared against the theoretical eigenvalue problem for a 3 DOF dis— crete system with 8—bit quantization noise. ............... System frequencies estimated from decomposition methods with white noise forcing compared against structural eigen frequencies ...... Experimental system frequencies (Hz.) estimated from various decom- position methods for a linear beam free response compared against the undamped structural eigen frequencies ................. Experimental modal damping estimated from various decomposition methods for the linear beam free response. Only first three damping estimates are computed by using dominantly single mode log decrement method .................................. xi 70 92 118 119 122 126 2.1 2.2 2.3 3.1 3.2 3.3 3.4 LIST OF FIGURES The acceleration of mass one, computed from applying the finite differ- ence matrix D to the displacement ensemble with noise. Upper is the underdamped example, and lower is the overdamped example. Images in this dissertation are presented in color ................ The damping factor estimate for first mode for various levels of noise. The solid line (—) shows system damping while dashed line (- -) presents the estimated damping. The horizontal axis indicates the value b representing the uniform noise in the range :l:1/2b. ...... The modal frequency estimate for first mode for various levels of noise. The solid line (—) shows modal frequency while dashed line (- -) presents the estimated frequency. The horizontal axis indicates the value b representing the uniform noise in the range :l:1/2b. ...... The mass-spring—damper model. The dashpots are figurative to repre- sent the presence of damping, and do not accurately correspond to the example damping matrix. ........................ A comparison between mode shapes from the structural eigenvalue problem (solid lines) and those estimated from the smooth orthogonal decomposition (‘0’ symbols). The sensor locations refer to the mass indices. The location 0 represents the wall attachment. The modes are in ascending order by modal frequency. ................. The frequency estimates improve as N gets large. The solid line (—) represents the true frequency (rad/sec) and (—*) are the estimated frequencies (rad/sec). The percent error in fre- quency estimation computed at N = 104 for modes 1 to 8 are 1.2%, 0.17%, 0.12%, 0.02%, 0.11%, 0.22%, 0.01% and 0.31% respectively also shown in Table 1. .......................... The maximum singular values of XXT/N (m2), VVT/N (m/s)2 and XFT/N (N-m). The dashed line (—0) represents values of XXT/N, the dashed line (—*) represents the values of VVT/N and the line (—*) represents values of XFT/N. We can see by N = 104 samples, XFT/N << (XXT/N, VVT/N) and is perhaps negligible for this and larger values of N ............................. xii 22 24 24 32 35 36 37 3.5 3.6 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 The undamped natural frequency estimate starts to deteriorate for high damping. The solid line (—) represents the true frequency (rad/sec) and (———*) is the estimated frequency (rad/ sec). Here samples size is N = 105 ................................. The MAC values comparison for SVMD for random excitation. MAC value of 1 shows vectors match up. Modes are not in any particular sequence .................................. The experimental setup for study by Rhoads et. a1 .......... The impulse response of SDOF beam experiment for study by Rhoads et. al .................................... The truncated sample points to obtain a “clean” signal for analysis of the study by Rhoads et. al ....................... The F FT log plot of the SDOF beam experiment ........... The experimental setup of a 16 DOF clamped-free beam ....... The beam acceleration snapshots of the acceleration ensemble of a 16 DOF clamped-free beam with its mean removed ............. The linear FFT plot of a 16 DOF clamped-free beam. Only first six modes are shown .............................. The frequency response of the second order high—pass filter for the 16 DOF beam experiment ........................... SVMD is compared for the first beam mode against the theoretical mode. SVMD is shown with + mark, theoretical mode is shown with the solid line —— ............................... SVMD is compared for the second beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line ~——— ............................ SVMD is compared for the third beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line — ............................ xiii 38 41 47 49 50 51 51 52 53 54 08 59 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 O" p—a 5.2 SVMD is compared for the fourth beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line —— ............................ 60 SVMD is compared for the fifth beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line — ............................ 61 SVMD modal coordinates are shown for the first 5 beam modes. Modal coordinates indicate the quality of the decomposition, and can be used to distinguish between computational and actual modes. ....... 62 SVMD modal coordinates are shown for the third beam mode. Modal coordinates can be used to distinguish between computational and ac- tual modes. Actual mode is shown in dotted (- -), whereas spurious is shown in solid line (—). ......................... 63 The FFT of an acceleration time history channel of the experimental free-free beam. Eight active frequencies are visible within the Nyquist range. ................................... 69 Mode shapes obtained by the SVMD of the experimental free-free beam. 71 First four state-variable modal coordinates plot of the experimental free-free beam. .............................. 72 F FT of the first state-variable modal coordinate of the experimental free—free beam. .............................. 72 The low noise F RF plot with 6 = 19 * 7} representing strength of uniform random noise signal amplitude. Here 19 value is 1. The dotted line represents FRF plot with no noise 19 = 0 and is shown for comparison purpose only. ............................... 81 The high noise F RF plot with 6 = 19 * 7) representing strength of uni- form random noise signal amplitude. Here 19 value is 4. The dotted line represents FRF plot with no noise 29, = 0 and is shown for comparison purpose only. ............................... 82 xiv 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 The SAC estimates for frequency identification for various levels of noise. The solid line (—) shows the amplitude SAC while dashed line (- -) presents the amplitude of complex version of the SAC. The horizontal axis indicates the value 1‘) representing the uniform random noise coefficient. When noise level increases, the estimates deteriorate significantly ................................. The spectrum difference estimates for frequency identification for vari- ous levels of noise. The horizontal axis indicates the value 19 represent- ing the uniform random noise coefficient. When noise level increases, the estimates deteriorate .......................... The SAC values for various estimates of identified damping ratios. The solid line (—) shows the amplitude SAC while dashed line (- —) presents the amplitude of complex version of SAC. The damping estimate error ratio Cg/(l ranges from -100 % at 0.001 to +100 % at 2 with (1 being 0.08 at 1. ................................. The spectrum difference values for various estimates of identified damp- ing ratios. The damping estimate error ratio (2 / C1 ranges from -100 % at 0.001 to +100 % at 2 with (1 being 0.08 at 1. .......... The SAC values for various estimates of identified frequency values. The solid line (—) shows the amplitude SAC while dashed line (- -) presents the amplitude of the complex version of SAC. The frequency estimate error ratio w2 /w1 ranges from -100 % at 0.0 to +200 % at 3. The spectrum difference values for various estimates of identified fre- quency values. The frequency estimate error ratio (.02 /w1 ranges from -100 % at 0.0 to +200 % at 3. ...................... The amplitude SAC values for various estimates of identified damping values. The solid line (—) shows the amplitude SAC while dashed line (- -) presents the amplitude of the complex version of SAC. The reference system damping ratio is chosen as (1 =0.02 ......... The spectrum difference values for various estimates of identified damp- ing ratios. The reference system damping ratio is chosen as (1 =0.02 The SAC values for various estimates of identified damping values. The solid line (—) shows the amplitude SAC while dashed line (- -) presents the amplitude of the complex version of SAC. The reference system damping ratio is chosen as (1 =0.2 ............... XV 83 84 85 86 87 88 89 89 90 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 The spectrum difference (SD) values for various estimates of identified damping ratios. The reference system damping ratio is chosen as (1 20.2 .................................... 91 The amplitudes of complex SAC values computed for each FRF gen- erated from the SVMD estimates compared against F RFs from modal coordinates. ............................... 93 The r = 1- SD values computed for each FRF generated from the SVMD estimates compared against FRFs from modal coordinates. For ease of visualization, values of SD subtracted from 1 are plotted instead. 93 The amplitudes of complex SAC values computed for each FRF gen- erated from the FFT estimates compared against FRFs from modal coordinates. The estimates are ordered to match sequence by SVMD for ease of comparison. .......................... 94 The r = 1- SD values computed for each FRF generated from the FF T estimates compared against FRFs from modal coordinates. The estimates are ordered to match the sequence of SVMD. For ease of visualization, values of SD subtracted from 1 are plotted instead. . . . 95 The amplitude of complex SAC values computed for each FRF gener- ated from the SVMD estimates compared against F RF 8 from modal coordinates. ................................ 96 The r=1-SD values computed for each FRF generated from the SVMD estimates compared against F RFs from modal coordinates. The esti- mates are not ordered in any particular sequence. For ease of visual- ization, values of SD subtracted from 1 are plotted instead ....... 97 The amplitude of complex SAC values computed for each F RF gen- erated from the FF T estimates compared against F RF 3 from modal coordinates. The estimates are ordered to match sequence by SVMD for ease of comparison. .......................... 98 The r=1-SD values computed for each FRF generated from the FF T estimates compared against FRFs from modal coordinates. The esti— mates are ordered to match the sequence of SVMD. For ease of visual- ization, values of SD subtracted from 1 are plotted instead ....... 98 xvi 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 6.1 The percent error in estimation of frequency parameters against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level is 10 bits. .................................... The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level is 10 bits. .................................... The percent error in estimation of frequency parameters against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The maximum error appears in the first mode. The quantization noise level is 6 bits. ........... The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period and the maximum error appears in the first mode. The quantization noise level employed is 6 bits. . . . . The percent error in estimation of frequency parameters against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 6 bits. .................................. The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 6 bits. .................................. The percent error in estimation of frequency parameters against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 10 bits. ................................. The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 10 bits. ................................. The mass-spring-damper model. The dashpots are figurative to repre— sent the presence of damping, and do not accurately correspond to the example damping matrix. ........................ 102 103 104 105 106 107 108 109 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 7.1 7.2 The MAC values for all SVMD modes. MAC value of 1 shows vectors match up. Modes are not in any particular sequence .......... 123 The MAC values for all sixteen modes for SOD. MAC value of 1 shows vectors match up. Modes are not in any particular sequence ..... 124 The MAC values for all ITD modes. MAC value of 1 shows vectors match up. Modes are not in any particular sequence .......... 124 The MAC values for all mass-weighted POD modes. MAC value of 1 shows vectors match up. Modes are not in any particular sequence . . 125 Identification methods are compared for the first beam mode. POD is shown with dotted line - -, SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —. ................ 127 Identification methods are compared for the second beam mode. POD is shown with dotted line - -, SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —. ............. 128 Identification methods are compared for the third beam mode. SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —— .................................... 129 Identification methods are compared for the fourth beam mode. SOD is shown with crosses X, SVMD with circles o, and ITD with the solid line — .................................... 130 Identification methods are compared for the fifth beam mode. SOD is shown with crosses x, SVMD with circles o, 'and ITD with the solid line — .................................... 131 The log plot of POVs shown. The plot shows about 4 significant modes. 131 SOD and POD mode shape identification for a higher mode with the system excited in a way to get a dominant 3rd order beam mode re- sponse. SOD is shown with the solid line —, and POD is shown with a dotted line - - ............................... 132 The modal coordinates plot for (.122 and their phase space are plotted for 10 second simulation time. ..................... 142 The modal coordinates plot for (122 and their phase space are plotted for simulation time of 400 seconds. ................... 143 xviii Introduction 1.1 Background odal analysis is a procedure that describes the inherent dy— namic behavior of a structure in terms of its dynamic proper- ties, namely the natural frequencies, damping ratios and mode shape characteristics. Modal analysis is viewed as an efficient tool geared towards understanding and optimizing structural characteristics for a large class of noise and vibration applications, thereby resulting in, for example, lighter, stronger, and safer structures, and superior performances. Simply put, modal analysis is the study of the natural vibration cl'iaracteristics of structures [1]. Modal analysis applications cover a broad range of objectives [2] including: 0 Identification and assessment of vibrations phenomena 0 Validation, correction and updating of the analytical dynamical models. 0 Development of experimentally based dynamic models. 0 Estimation of the dynamic behavior of both individual components and/or as- sembled structures. 0 Design of systems such that resonances are avoided. 0 Optimization of the dynamic properties of structures (mass, stiffness, damping) and model integration with other dynamic areas, such as acoustics and fatigue. 0 Estimation of applied forces from the measured responses. 0 Prediction of the responses due to complex and/or multiple excitations. 0 Structural health monitoring, damage detection and structural integrity assess- ment. 0 Prediction of the effect of structural modifications, for instance, the inclusion of damping in finite element models and updating of finite element models. 1.1.1 finalytical modal analysis Two kinds of modal analysis techniques exist: analytical and experimental. When per- formed analytically, modal analysis utilizes the mathematical model of a structure’s dynamic behavior to compute the modal parameters. The mathematical model is either a set of differential equations obtained by classical principles such as Newton’s laws, Lagrange’s equations, or numerical techniques such as finite element models (FEMS), that could be used to derive an eigenvalue problem. This eigenvalue prob- lem produces a set of mode shapes each with an associated natural frequency and a modal damping value. The mode shapes can be used to define a coordinate change that decouples the system into individual oscillators with characteristic modal frequen- cies and damping ratios. These modal parameters provide a complete description of the structure’s dynamic behavior. To perform this kind of analysis, complete infor- mation of the system, for example, mass, damping, stiffness, and forcing functions, is assumed to be available. 1.1.2 Experimental modal analysis Unlike analytical modal analysis, experimental modal analysis is used to determine the modal parameters by actual testing. There are two ways of performing experi- mental modal analysis: Classical or traditional modal analysis, and operational or output-only modal analysis. In classical modal analysis, a set of frequency response functions (or impulse response functions) is computed, from a set of measured input forces and output responses of a structure [3—6], that are then processed to determine the modal parameters. In the output-only modal analysis, as the name indicates, only the output responses are measured and the excitation and operating forces are unmea- sured inputs. There are various output-only approaches; but typically, an eigenvalue problem is constructed that produces modal parameter estimates strictly based on the obtained data. Next, we will briefly note the types of signals utilized in the experimental modal analysis. 1.1.3 Classification of signals All signals can be broadly classified into either stationary or non-stationary categories. A stationary signal is one whose distributions remain constant in their statistical parameters (for instance amplitude distribution and standard deviation) over time (or space) [7,8]. Another definition of stationary signals (in wide—sense) implies a signal whose power spectrum does not change with time [9]. Stationarity implies that the ensemble will look just the same if the time origin is shifted. Stationary signals are further divided into deterministic and random signals. Ran- dom signals are unpredictable in their instantaneous frequency content and/ or their amplitude level, but their statistical characteristics are still relatively uniform over time. Examples of random signals are rain falling on a roof, noise from waterfalls, jet engine noise, turbulence in pump flow patterns and cavitation, vibration of ship hulls due to flow of water along the hull, and the noise from an air-conditioning system [9—11] All signals that are not stationary category can be considered non-stationary. For non-stationary and / or nonlinear signals, promising methods have been developed for structural analysis, such as Hilbert-Huang transform [12—16] and wavelet analysis [17—26]. The interested reader is referred to these works for understanding these methods. The discussion contained in this work is limited to stationary random signals, and transient and decaying signals that may result from short duration non-stationary phe- nomena with clearly defined beginnings and ends. This would for instance comprise impulse and white noise forcing functions. 1.2 Output-only modal analysis The output-only (or response-only or operational) modal analysis has gained popu- larity over recent years (see for example [27—36]). Advantages of output-only analysis over traditional modal analysis are the following. I) In many real life applications, the nature of input forcing prevents its measurement (for instance earthquake excitation, wind excitations, or traffic loads on structures) and output-only analysis eliminates the need to measure inputs. II) The construction of complex frequency response func- tions or transfer matrix functions in traditional experimental modal analysis requires an experienced engineer to correlate various response rows (or columns) to correctly identify the system modes, and is cumbersome when the modes are not well separated. III) Contrary to traditional modal analysis, in many cases output-only analysis can eliminate the need of stand-alone testing for the structure at various locations (or components). The output-only methods can be either time or frequency based. A non-exhaustive list of time domain based output-only methods includes, the Ibrahim time domain method [27, 37—41], a series of methods based on Prony’s method [3,42], such as the complex exponential method [28, 43]; its single-input, multiple—output extension called the least square complex exponential method [29] and its multi-input, multi- output extension called the polyreference complex exponential method [30], the eigen- system realization algorithm [31, 44, 45], the independent component analysis [46,47], and the stochastic subspace identification methods [32]. Frequency based output- only methods include the orthogonal polynomial methods [33,34], the complex mode indicator function [35], and the recent frequency domain decomposition [36]. For a more detailed survey of modal parameter identification methods, reader is referred to the findings of [43,48-51] that also include historical references such as [42] and [52—54]. 1.3 Proper orthogonal decomposition and output-only time domain methods The proper orthogonal decomposition (POD) produces modes that optimize the sig- nal energy distribution in a set of measured time series. POD can be classified as an output-only method (at least for structural applications) since it requires only the time histories of the coordinate’s outputs without the need of measuring the inputs. POD has emerged as a standard tool that has been applied to turbulence, structures, and many other types of systems. POD has been used, for example, to characterize spatial coherence in turbulence and structures [55—61], to evaluate the dimensions of the dynamics [57—60, 62], to detect modal interactions [63,64], to produce empir- ical modes for reduced order models [65—71], and in system identification [72—75]. The POD is similar to Karhunen-Loeve decomposition [76, 77], or more appropriately Kosambi-Karhunen-Loeve decomposition [78], principal components analysis [79], and singular value decomposition [80]. All of these tools have been compared for struc- tural applications [81]. The similar biorthogonal decomposition has also been applied to fluids and plasmas [82,83]. POD is easily applied to discrete systems, or systems with discrete measurements 113,-(tj), where 2' = 1,...,M are the sensor indices, and j = 1,...,N are the time sample indices. An ensemble matrix is formed as XT = [x1, . . . ,xM], where the vectors x,- have elements of time samples Ii(tj)' So each row of X is a sampled history of one sensor on the structure. Then the matrix R = XXT/N is formed. If the means of the signals are all zero, then R is a covariance matrix. Otherwise, it is referred to as a correlation matrix, and the method can still be applied. The eigenvectors of R are the proper orthogonal modes (POMS), and the eigenvalues are the proper orthogonal values (POVs). If the mass distribution is uniform, the POD produces the normal modes of a structure [8487], including 2-D structures [88]. If the mass distribution is not uniform, but known, or if the stiffness matrix is known, the POD can be weighted to produce the normal modes [84]. POD can be used for modal analysis if the damping is light, and if the system is either impulse excited or randomly excited [89]. The prOper orthogonal modes (POMS) then converge to the linear normal modes, and the proper orthogonal values (POVs) provide the mean squared values of the modal coordinates. Proper orthogonal modal coordinates can be used to obtain information on frequencies and damping ratios. Another POD based decomposition technique is the complex orthogonal decom- position (COD) [90]. Here, the ensemble X of oscillatory signals is expanded into the complex domain to form an ensemble Z of complex analytic signals. This is done non- uniquely [91] by the Fast Fourier Transform (F FT) [92] or Hilbert transform [91,93]. Then a complex Hermitian correlation matrix R = ZZT/N is formed. (A complex Hermitian matrix was also formed in the frequency domain for modal analysis [94].) The complex Hermitian matrix R has real eigenvalues (COVs) and generally complex eigenvectors (COMS). The COVs represent mean squared amplitudes of complex modal oscillations. The COMS can be interpreted as complex modes, representing generally a mix of traveling and standing wave behaviors, which can be extracted and quantified. As such, the COD can characterize the modal content of systems with mixed standing and traveling wave behavior. This renders that COD is also applica- ble for structural analysis of non-stationary signals. Initial indications are that the complex modes can be efficiently extracted. Much like POD for general systems, the COMS will not produce complex linear normal modes except under special circum- stances. But the ongoing work suggests that the COD provides an optimized and systematic characterization of generally oscillatory behavior. A recent addition to the time domain output-only family is smooth orthogonal decomposition (SOD) [95] that has shown good results for modal analysis of lightly damped structures in free response case. The SOD method, described in more detail later, is also viewed as a variant of the proper orthogonal decomposition method being evaluated for structural modal analysis. The SOD method is applicable in blind source separation [96], fatigue damage identification [97,98], and was recently presented as a generalized modal analysis scheme [99]. However, the method has room for development as will be shown later in section 3.2. 1.4 Preview of this thesis The current work extends the POD method for the modal parameter estimation of structural systems under free response as well as under random excitation, using state- variables [100—102]. The current work takes motivation from the SOD method, and modifies the existing POD method to estimate modal frequencies and damping ratios in addition to the linear normal modes as originally estimated by the POD. Similar to the SOD, the state-variable modal decomposition method (SVMD) falls under the time-domain category and exploits the frequency content “hidden” in the time period considered. The orthogonal decomposition-based method SVMD (and also SOD) carries some advantages. For example, SVMD does not involve the possibility of oversized state matrices (at least in the noise-free case) and their spurious modes (as in the Ibrahim time domain and the schemes based on Prony’s method); the estimation of initial states (for instance in stochastic subspace identification methods); or the construction of generalized block Hankel matrices (as in the eigensystem realization algorithm); or spectral density functions (as in the frequency domain decomposition). SVMD thereby invokes minimum assumptions and is simpler in construction. The main idea of the SVMD method is presented in chapter 2. There, we derive the method from a state variable perspective and formulate the parameter estimation as a nonsymmetric, generalized eigenvalue problem. The data-based eigenvalue prob- lem is related to the generalized eigenvalue problem associated with free-vibration solutions of the state-variable formulation of linear multi-degree—of-freedom systems in that chapter. For linear free-response data, the inverse-transpose of the eigenvector matrix approximate to the state-variable modal eigenvectors, and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state-variable model. As such, the eigenvalues lead to estimates of frequencies and modal damping. The method handles damping ratios ranging from low to “moderately high” values, and the estimation is carried out regardless of whether damping is modal or nonmodal, and whether the modes are real or complex. Numerical simulations are presented for both damping cases. The issue of robustness under noise is also addressed. The resemblance of the method with POD can be easily seen by the eigenvalue problem construction. For vibratory systems subjected to random excitations, modal parameter estima- tion is addressed in Chapter 3. Primarily the extension to the SOD method is shown. This work shows that under certain conditions, the SOD eigenvalue problem formu- lated from white noise induced response data can be tied to the unforced structural eigenvalue problem, and thus can be used for modal parameter estimation. Using out— put response ensembles only, the generalized eigenvalue problem is formed to estimate eigen frequencies and modal vectors for an eight-degree—of-freedom lightly damped vi- bratory system. The estimated frequencies are compared against system frequencies obtained from structural eigenvalue problem and the estimated modal vectors are checked using the modal assurance criterion. Numerical simulations are presented for lightly damped structures whose output has been corrupted with noise. Simulations show that for light damping, satisfactory results are obtained for estimating both the system frequencies and the modal vectors. Then, formulation of SVMD for random response is briefly addressed. The same example is used for SVMD eigenvalue prob- lem and the results Show very similar identification values for system frequencies and mode shapes. An experimental investigation for the free response cases is conducted in chapter 4. This gives insight into practical considerations and the issues that may arise if the decomposition method is to be implemented in the real world. Modal analysis is performed for both single and multi-degree—of—freedom systems and two experiments are reported. In the single—degree—of-freedom case a beam is sensed with a strain gauge whereas in the multi-degree-of-freedom cases, accelerometers are employed. In that way, ensemble matrices are observed to be obtained in multiple ways with no impact on the results. The effect of time-record length on the detection of modal parameters is explored. Modal coordinates are computed. Together with the modal parameters and mode shapes, modal coordinates could prove to be an effective tool to efficiently separate actual modes from the spurious ones. Analysis done in this chapter also underscores the advantages and limitations of using the state variable modal decomposition method by evaluating yet another multi-degree-of—freedom system with issues surrounding filtering and reduced observability for such systems. Two new tools are developed in chapter 5 to quantitatively assess the decompo- sition results. Numerical studies are conducted first to explore the strengths and sensitivity of these tools. Employing frequency response functions, these tools pro— vide a quantitative means of evaluating the modal coordinates and the identification results. Then, by using these tools, the actual and fictitious modes are separated in the experiments with a greater degree of authority than the previous qualitative assessments. It is natural to compare the decomposition methods with other time domain modal identification schemes. This comparison is conducted in chapter 6. The selection of time domain methods is based on their impact, historical significance and/ or their relevance to the current experimental modal analysis. Methods are compared both using numerical simulations (with and without presence of noise in the system), and using actual experimental data. The same data set is used by each method for effective comparison. Two numerical simulations for free response and random response cases are conducted. The same multi-degrees-of-freedom experiment explored in chapter 4 is evaluated for comparison. These studies help in understanding the strength and weaknesses of various modal identification methods. Lastly, we conclude in chapter 7 with some observations and potential directions to future work. Extension of the decomposition method to nonlinear systems and / or the use of nonlinear normal modes is addressed. 1.5 Contributions of this work This work is an effort to tie up the current gap between various time domain methods and the orthogonal decomposition methods (like POD) for modal parameter estima— 10 tion. We summarize the contributions of this work as below. 0 The state-variable modal decomposition method is a novel identification scheme that ties a data—based non-symmetric eigenvalue problem to the state—space form of the ordinary differential equations for multi-degree—of-freedom structural systems. It is a response-only method that enables frequency, damping ratios and mode shape estimates for free vibration problems. The method is applicable for both modal and nonmodal damping cases without the need for measuring the forcing inputs. The method produces good modal parameters estimates in low noise cases. 0 Modal parameter identification was extended to the random response case for both smooth orthogonal decomposition and the state-variable decomposition method. In numerical simulations, for the systems subjected to random excita- tion, the decomposition based methods SOD and SVMD, showed good identifi- cation characteristics for lightly damped systems. Identification was still good when the response was contaminated with random noise at the output. 0 The SVMD method was evaluated for free vibration experimental tests. Both single-degree and multi-degree—of-freedom systems were explored. By testing under conditions that may deviate from ideal theoretical assumptions, insight into the importance of data-length for analysis and filtering effects was gained. Modal coordinates for SVMD together with the SVMD modal vectors were seen as a valuable way of detecting system modes from “fictitious” modes and deter- mining the quality of the decomposition. Multiple experimental tests verified that SVMD is a viable method for modal parameter identification. 0 A quantitative analysis of the parameter estimates that are obtained by the decomposition method was achieved by formulating two new tools. To gain confidence in decomposition results, to eliminate the spurious modes and to 11 reinforce the visual feel obtained through the modal coordinates, the spectrum assurance and the spectrum difference criteria were developed. These criteria were evaluated first through numerical studies. Certain benchmark cases were numerically simulated and sensitivity to the system parameters was established. Then, we applied these tools to our experimental results and gained further insight on eliminating spurious modes. The effect of the data record length in the identification by the decomposition method of a system corrupted with noise was also evaluated via a numerical study. High and low damping cases were explored and a trend was established in determining the correct data length for a given test run. A comparison of decomposition based methods SVMD / SOD with other popular output-only modal analysis methods was performed. The methods were com- pared by conducting tests on free and forced response numerical simulations and the actual experimental tests. The same set of data was used for each scheme to maintain consistency in the comparisons. In this way, the strengths and the weaknesses of each scheme were matched. 12 State ‘Variable Modal Decomposition offree Responses 2.1 Introduction r r . 1 his chapter presents a method of decomposing free—response ensem- ble data into modal components, while enabling frequency, damping, and mode—shape estimation for generally damped linear multi-degree- of—freedom systems. The basic idea is a generalization of proper orthogonal decompo- sition (POD) [557757], with similarity also to smooth orthogonal decomposition (SOD) [95] and complex orthogonal decomposition (COD) [90]. Below, the application of these methods to modal analysis is reviewed, and the niche of the new method is staged. The approach presented here is a generalized eigen decomposition based on state- variable ensembles. We will tie this decomposition strategy to state-variable models of generally damped linear vibration systems to reveal a method of obtaining frequency, damping, and mode shape information, without access to input measurements. The method will be illustrated with numerical examples. 2.2 Nonsymmetric state-variable modal decomposition In this section, we summarize the state-variable modeling of vibration systems. The decomposition equation is presented, and then tied to the state-variable model. 2.2.1 State-variatile linear vifiration model The state-variable model of linear vibration systems is used on systems with nonpro- portional (non-Rayleigh [103]) and nonmodal (non-Caughey [104]) damping to obtain damped vibration modes [105, 106]. The equations of motion for free vibrations are Mx+Cx+Kx= 0, (2.1) where x is an n x 1 array of displacements, M, C, and K, are the n x n mass, damping, and stiffness matrices, and the dots indicate time derivatives. Then defining a 2n x 1 T state vector y = [xT,xT], and introducing the equation Mx — Mx = 0, yields unforced equations of motion of the form Ay+By=0, (2.2) where 0 M —M 0 .._[M C], B] 0 K]. (2.3. A and B are 271. x 211 and symmetric, but are neither positive nor negative definite. Assuming a response of the form y = eat?! the eigenvalue problem aAQ + B? = 0 (2.4) in general yields complex eigenvalues a and eigenvectors Q, with Q = [vT,wT]T, where n x 1 vector partition v corresponds to characteristic shapes of velocity states, and partition w represents characteristic shapes in displacement (complex modes). By the construction of y, v = aw. The vectors 9 are orthogonal with respect to matrices A and B. The latter does not imply that the vectors w are orthogonal with respect to M and K. In fact, they are not. 14 2.2.2 Decomposition strategy The decomposition strategy is based on the free—response state-variable ensemble Y = [VT,XT]T, where X is a displacement ensemble, and V = XDT z X is an approximate velocity ensemble, like the SOD [95]. An example of an (N — 2nd) x N matrix of centered finite differences, with 77d = 1 for half the span of the finite difference, is —l 0 1 0 0 1 0 —1 O 1 0 D=2Kt 3 E 1 (2-5) 0 —1 0 1 where At represents the sampling time. Thus V is n x (N — 2nd), and so the first and last "at columns of X are dropped so that Y has compatible partitions. We then take the derivative W = YDT x Y, this time using an (N —4nd) x (N — 2nd) difference matrix D. The first and last nd time samples of Y are then dropped so that the dimensions of Y and W are both 271. x N — 4nd. We form a correlation matrix R = YYT/ (N — 4nd) and a nonsymmetric matrix N = YWT / (N — 4nd). The 2n x 271 eigenvalue problem is then (1R1,_C’ = N1. (2.6) Rewriting this eigenvalue problem, aYYTt_/1 = YWTQ. Making use of Eq. (2.2), w z Y = —A‘1BY, and we have aYYT_ z —YYTBTA—Ty. (2.7) We expect YYT to be invertible. This is true if all displacement measurements are ' independent and if N — 4nd > n. As such, (“(1 z ~—BTA_T:. In matrix form 11A z —BTA_T\II, (2.8) 15 where A is a diagonal matrix of eigenvalues. Taking the inverse-transpose, e-TA-l s —B-1A\p—T, whence BxII-T z —A'I"TA, and hence —A‘1B\II‘T z 111—TA. (2.9) Letting U = ‘II-T, the data eigenvalue problem leads to —A—1BU z UA, (2.10) which is a generalized eigenvalue problem with matrices A and B, the solution of which determines the unknowns U and A. The matrix form of the structural eigen- value problem of Eq. (2.4) is —A—lB = or, (2.11) a generalized eigenvalue problem with the same matrices A and B, the solution of which determines the unknowns ¢I> and I‘. The eigenvalue problems of Eqs. (2.10) and (2.11) have the same solution (within the modal normalization constants), indicating that (P 2 U = ‘P—T and I‘ = A. (The same logic is seen in SOD analysis of linear vibration systems [95].) ‘II and A are 211. x 211 matrices, corresponding to 2n eigenvectors and 271. eigenval- ues, for an n—degree-of—freedom system. If the eigenvectors are complex, they come in conjugate pairs. That is, a conjugate pair of eigenvectors and eigenvalues represents one mode. If the eigen solution is real, an eigenvector characterizes a response con- figuration decaying at a rate contained in the corresponding eigenvalue. If, in fact, the damping is “modal” (Caughey) [104], there will be 71 independent displacement partitions v among the 271 eigenvectors, which correspond to the n more familiar synchronous normal modes. Thus, we expect the eigenvalues of Eq. (2.6) to approximate the state-variable eigenvalues, containing information about damping and frequency. The inverse of the modal matrix from Eq. (2.6) resembles the complex linear normal modal matrix of 16 the state—variable system Eq. (2.4), and contains velocity and displacement partitions. The only approximation in the method is in X % XDT and Y % YDT. Hence we expect reasonable estimations when noise is limited and the step size is sufficiently small compared to characteristic time scales. 2.3 Numerical examples For comparison, we use the example shown in [84] for POD and [95] for SOD. we will treat underdamped and overdamped examples for both proportional and generalized damping. The mass and stiffness matrices are 2 0 0 2 —1 0 M: 0 1 0 ,K= —1 2 —1 (2.12) 0 0 1 0 —1 1 and the initial conditions are x(0) = (1, 0, 0)T, and v(0) = (0,0, 0)T. The undamped natural frequencies of the system are col = 0.4209, 1.122 = 1.000, and 1113 = 1.6801. In each simulation, the time responses for each mass displacement and velocity were numerically computed. Here, we used a step size At = 0.004977 and a sample size N = 3000. This time record closely matches one period of the lowest-frequency mode. We generated the state variable response from y(t) = (I’qoeat where q0 = TAy(0). To emulate an experiment, we kept the displacement ensemble X from the displace- ment partition of the ensemble generated from sampling y(t). We approximated X m V = XDT, and built the state-variable data ensemble Y. The ensemble W = YDT (independent from displacement modal vector u) was then formed and the endpoints were truncated to obtain compatible matrix dimensions. The eigenvalue problem Eq. (2.6) was formulated using Y and W. 17 2.3 .1 froportionally damped example In this example we chose the damping matrix to be proportional to just the mass matrix, such that C = CM. First, we consider an underdamped example with c = 0.5. The system eigenvalues are complex, of the form *Cjo i iwj 1 — CJZ. These values are 011 = 511 = 0.25 + 0.33862, 03 = (r4 = 0.25 + 0.96821, (15 = 5‘6 = 0.25 + 1.66142. The modal damping factors are then (1 = 0.5940, (2 = 0.2500 and C3 = 0.1488. The lowest-frequency mode has a damped frequency of ”d1 = 0.3386. Our time record specified above spans 81% of the associated oscillations period. From the decomposition eigenvalues, estimates of the damping factors and natural frequencies are identical to that of system to more than four signifi- cant figures. For proportional damping three out of six of the displacement partitions of the state—variable modal vectors are independent. The correspond- ing independent displacement eigenvectors were normalized to real form of unit length as 11'1”: (0.3602,0.5928,0.7204)T, u; = (—0.7071,0.0,0.7071)T and 11;; = (0.2338, —0.8524,0.4676)T. All agree with the linear normal modes (LNMS) to 4th decimal place. Now we consider an overdamped example with c = 2.5, leading to two real and four complex structural eigenvalues. The complex eigenvalues are 011 = 511 = 1.25 + 1.12271, whereas the real pairs are (2.427, 0.073) and (2.000, 0.500), which have modal damping factors of (1 = 2.9701,(2 = 1.2500 and (3 = 0.7440. The decomposition method also yielded one complex and two real eigenvalues from which damping factors and natural frequencies are identical to those of the structural problem. Also, the corresponding independent displacement eigenvectors, normalized to real form of unit length agreed with the linear normal modes (LN Ms) to the fourth decimal place. 18 2.3 .2 Generally damped example We will consider underdamped and overdamped examples. For the underdamped example the damping matrix was 1.2 —0.9 0 C: -—0.9 1.2 -0.3 (2.13) 0 ——0.3 0.3 Then KM‘IC 75 CM’IK, rendering the system to be nonmodally damped [104]. In the case of nonmodal damping, the definition of a modal damping factor is slightly ill posed since we trying to estimate the modal damping from the system that is non- modally damped. Besides, the system cannot be decoupled into individual oscillators of the form ('1',- + 2c,w,q, + 5,211,- : 0 (2.14) Nonetheless, the frequency of damped oscillation is the imaginary part of the eigenvalue. The eigenvalues obtained from the structural-model eigenvalue problem Eq. (2.4) are complex, and are a1 = 612 = 0.7620 + 1.419Oi,a3 = 514 = 0.2474 + 1.0061, a5 = 615 = 0.0405 + 0.42182. Using the magnitude of the eigenvalue as “01-”, and comparing the real part to “(J-112]” produces numbers for C}, and 1123-. In this case (”1 = 0.0957, 52 = 0.2388, and 53 = 0.4731, and 1:), = 0.4238132 = 1.0360, and 1.713 = 1.6107. The modal decomposition method eigenvalues are found to be nearly identi— cal to the system structural eigenvalues, and the “damping factors” and “natu- ral frequencies” are identical to fourth decimal place. All unit length normal- ized displacement partitions of the corresponding complex eigenvectors, w? = erg“: (-0.3674 + 0.03171,—0.5901 — 0.01291,—0.7197 — 0.005611)T, wg‘ = w}: (0.6517 + 0.13581, 0.0499 — 0.15391, -0.7908 + 0.10221)? wg‘ = wg‘ = (—0.3526 + 0.1499i,0.9224 —— 0.04312, —0.3469 —- 0.26702)T, agreed with the LNMs to the fourth 19 decimal place. Now a highly overdamped example is considered. The damping matrix was chosen as 6.0 —4.5 0 C: —4.5 7.5 -—3.0 (2.15) 0 —3.0 3.0 The structural modal eigenvalue problem (2.4) produces four real and two complex eigenvalues. The complex eigenvalues are 011 = (11 = 0.1858 + 0.41652, whereas the real pairs are (9.8853,0.2432) and (2.6180,0.3820). The real pairs were matched to produce the correct undamped modal frequency estimates but in reality could pose problems if “true” frequency information is unknown. The “damping factors” are 61 = 0.4073, 62 = 1.5000 and 63 = 3.2665 and corresponding “modal frequencies” are 521 = 0.4561,®2 = 1.000, and 1123 = 1.5504. The decomposition method also produces four real and two complex eigenvalues that are nearly identical to the structural eigenvalues. The displacement partitions of the six eigenvectors are w? = w; = (0.4717 — 0.08292,0.5784 + 0.00542,0.6728 + 0.05341)T , wg‘ = (0.5365,0, —0.8439)T , wf = (0.1989,0, —0.9800)T , wg‘ = (—0.2792,0.8869, —0.3680)T, and wg = (0.1098, —0.7683, —0.6305)T, which agreed with the LNMS to the 4th decimal place. 2.3 .3 Step size selection The differentiation step size is a parameter of the method. If the step size is too large, it results in erroneous parameter estimates and gets worse in over damped cases. If the step size is too small, it results in noise amplification. This implies that the sampling resolution needs to be chosen appropriately to fully capture the features of the transient time response, and in turn to accurately estimate the system parameters, particularly for very highly damped (or very fast) systems. For choosing the sample time, important features of the response are the characteristic settling time, 7'1 j = 4/Cjwj, and the oscillation period, 1’2]- : 2n/wdj. It was concluded by 20 numerous simulations that any step size that lies within 5% of the minimum (over j) of 7'1 j and 'rgj would give reasonable resolution and hence serves our purpose. For data obtained through experiments, the settling time criterion might be good enough to approximate the step size. Time records should be appropriately chosen with regard to the maximum of T1 j and 723-. This information may not be available a-prz'ori, but provides a rough rule of thumb for the required time resolution. 2.4 Robustness under noise To emulate a 10—bit quantization noise in digitization [91], we set up random fluctua- tions. The MATLAB function “rand” was used to generate an ensemble of uniformly distributed random numbers between :l:2_10:1:0, where (1:0 is the maximum initial condition. ‘Rand’ emulates white noise. We verified that the spectrum of our ran- dom data was uniformly distributed. The root mean squared value was very close to 2’101r0/ \/3 The random ensemble was added to the displacement matrix X. We then took numerical derivatives of the new X to build V and subsequently our ensem- ble matrices. All the above mentioned cases were studied under noisy data with a step size of 0.0049767 and N = 3000. To reduce noise amplification from differentiation over short time steps, we used a differentiation step size of 32 (nd = 16). This value was selected after some trials on step sizes and was found to be better than others. Examples of finite-difference mass accelerations with noise are shown in Fig. (2.1) for the proportionally underdamped and overdamped examples. The figure also shows how much time record was used in all examples. For the proportionally damped, underdamped example of c = 0.5, the er- rors in the estimated damping factors and natural frequencies were 0.82%, 0.57% and 0.17%, and 0.09%, 0.05% and 0.01%, respectively. The corresponding com- plex independent displacement eigenvectors normalized to real form agreed with 21 0.5 a l! .v i .2 0- l _ § 1 2 Q’ s ,3 4151 - -1 f l -1.5‘ 1 7 1 1 1 1, ,, , 0 500 1000 1500 2000 2500 3000 Sample Points 1.] I [‘1‘]. l [ |' . l l , [ l l 1 1‘ 1‘ a . 1 "l l . . -2 l] l : ‘l 8 , 2 Q) 8 < 0' 500 1000 1500 200077 2500 473000 Sample Points Figure 2.1: The acceleration of mass one, computed from applying the finite differ- ence matrix D to the displacement ensemble with noise. Upper is the underdamped example, and lower is the overdamped example. Images in this dissertation are presented in color the LNMS to the 4th decimal place. For the overdamped case of c = 2.5, two eigenpairs were overdamped and one was underdamped. We were able to get 22 good estimates of the underdamped eigenvalues and also the slow part of the sys- tem eigenvalues. The underdamped system frequency and damping had 1.8% and 0.56% errors respectively. The slow eigenvalues were 0.0752 and 0.4903, with cor- responding vectors (0.3576,0.5925,0.7218)T and (0.7021,—0.0067, —0.7120)T that are comparable against state-variable model eigenvalues (0.073 and 0.500) and the vectors (0.3602,0.5928,0.7204)T and (0.7071,0.0,~—0.7071)T respectively. For the state-variable model, the slow and fast eigen-pairs correspond to the same eigen- vector and thus are easy to visualize, the decomposition estimates the vector as- sociated with the slow eigenvalues within reasonable accuracy. The identified eigenvectors associated with the fast eigenvalues were (0.3930,0.5925,0.7033)T and (0.7312,0.1141,—0.6725)T that may still be reasonable. But the fast eigenvalues, even with faster sampling and a larger differentiation step size, were corrupted by noise and gave erroneous results. This means we were unable to get good estimates of C j and 023- for the overdamped system. From the slow eigenvalues though, we can estimate modal time constants and given the system frequencies (e.g. by FEM), we can still estimate damping ratios with reasonable accuracy. In the generally damped examples, for the underdamped case of Eq. (2.13), the errors in the estimated damping factors and natural frequencies were 0.52%, 0.04% and 0.67%, and 0.04%, 0.11% and 0.13%, respectively. The displacement partitions of the six complex eigenvectors agreed with the LNMS to the 3rd decimal place. For overdamped case of Eq. (2.15), the same interpretation about fast and slow eigenvalues holds as described in case of proportionally damped overdamped case. Lastly, we consider the effect of increasing the noise level. Here the system is chosen to be modally under damped with C = 0.5M. We used a differentiation step size of 32 (nd 2 16). As the noise bit level was incremented (doubling the noise level), the error typically increased, until at five bits, the error in the worst mode (highest damping factor) was 55% Fig. (2.2) and Fig. (2.3). 23 .0 on U 4,. *3 0 5— _ .§ I l ‘."\. fl . "K‘ E” 0.4“ ‘~.\ 'E X s Q 03— ~ ] 55% Error"""’/T»°‘ 0. l 1 L . 1 5 0 9 8 7 6 5 Increased Noise (Bits) Figure 2.2: The damping factor estimate for first mode for various levels of noise. The solid line (——) shows system damping while dashed line (- -) presents the estimated damping. The horizontal acris indicates the value b representing the uniform noise in the range :I:1/2b. 0.55, 25% Erro r--/"”' I 0.5[ ,-’ “ Frequency estimate o-o-c-I-I-I-n- "l- I-I-.-'- . I-.- '.’ ‘~ e -~'-o fi" ~.~.-.'u 10 9 8 ‘7 6 e 5 Increased Noise (Bits) Figure 2.3: The modal frequency estimate for first mode for various levels of noise. The solid line (——) shows modal frequency while dashed line (- -) presents the estimated frequency. The horizontal axis indicates the value 1) representing the uniform noise in the range $1 / 2b. 24 That some frequency estimates improved with noise is probably random happen- stance. We briefly looked into varying the number of data. The results were similar with N = 3730 (a full response cycle of the lower mode damped response), and the results deteriorated, primarily for the lowest mode, when N was increased to 9000. The suspected reason is that, with the modal damping factor of 0.5940, the modal response was settled for most of the time record, and therefore the lowest modal data was dominated by the noise. In fact, one of the better results came with N = 1865, representing a half cycle of lower mode response. For further lower values of N , results deteriorated. It seems that the time record should include at least a half cycle of the mode to be identified, but also for modes with significant damping, the time record should not be too long. Observations with various step sizes were also made. When noise is present, longer differentiation step sizes are beneficial at larger noise levels. However, the differentiation step size promotes error if it is a “large” fraction of the response period of the mode to be identified. These are simple observations from a few numerical experiments. More is explained when we conduct the experiments in next chapters. These examples suggest that the decomposition method is robust under low levels of noise that might be seen in experiments. In multiple simulations, we found that examples with damping factors up to around C = 0.8 were successful. With higher damping we could estimate slow time constants but not C]- and raj. In many exper- iments, accelerometers are used so displacement signals will be less prone to high frequency noise since accelerations are integrated to obtain X and V. 2.5 Summary We have proposed a method for extracting modal information from free responses of generally damped linear multi—degree—of—freedom systems. A generalized nonsymmet- 25 ric eigenvalue problem was posed, involving “correlations” of state-variable ensembles. The resulting eigenvalues and eigenvectors can be complex in general. The eigenval- ues hold information of damping and frequency, while the eigenvectors indicate the mode shapes associated with the state-variable model. The method is applicable to free multi-modal responses with small or “large” damping, and with modal or nonmodal damping in noise-free cases (even for over- damped systems). In the noisy cases, however, we were successfully able to identify frequencies and damping factors when damping factors had values up to C = 0.8. To obtain modal parameters under these conditions, the system input need not be measured; only the response is measured. We applied this scheme to several numerical examples, for which accurate es- timates of modal parameters (frequency, damping factor or decay rate, and mode shapes) were possible regardless of the damping value. The sampling rate should be sufficiently smaller than the shortest characteristic time interval, defined by decay time or oscillation period. For nonmodally damped systems, the eigenvalues can ac- curately quantify the decay rates, damped frequencies, and mode shapes, including complex modes. In the presence of noise, large damping factors could not be ex- tracted, but the slow decay rates were still estimated. Numerical examples suggest that there is an optimal choice of the differentiation time step, and the time record, for extracting modal frequencies and damping in the presence of noise. 26 Random Response 3.1 Introduction I andom excitation is one of the five common types of excitations that f is applied to the test structures for modal analysis (the others being ” EL transient( i(mpact, step or chirp), stepped-sinusoidal, slow sine sweeps, and periodic). A great deal of literature exists on experimental modal analysis for randomly excited systems. Such analysis requires deriving FRFS from random exci- tations (see for example [4,6]). Here we focus on the modal analysis of randomly excited systems, where only the response needs to be measured and the inputs do not need to be measured. We will primarily focus on the SOD analysis for systems under random excitation. We will begin by reviewing the SOD method for free response and will extend the thory to the random response. Next, we will present a numerical example for a lightly damped, linear multi-degree—of-freedom system. Then, we will Show that the SVMD analysis of random response holds quintessential Similarity with the SOD, that will be augmented with the same numerical example. 27 3.2 Smooth orthogonal decomposition 3.2.1 Smootli [orthogonal decomposition and modal analysis for fiee vibra- tion The “smooth orthogonal decomposition” (SOD) [95] can be applied to lightly damped symmetric vibration systems with inhomogeneous mass distributions to find struc- tural modes. First, an n x N ensemble matrix X of displacements is obtained from N time samples of n displacement signals. Then an ensemble V m X of velocities is formed. This can be done by finite difference through a matrix D, such that V = XDT where X is an ensemble of displacements (in structures case). Next the velocity covariance matrix S = VVT/Nv is formed, where NU is the number of velocity samples. If the finite difference covers two adjacent samples, such that vi(tj) = :ci(tj+1) — ;r,-(tj), then NU = N — 1, if vi(tJ-) = sci(tj+1) —— $,(tj_1), then NU = N — 2, and so on. Keeping the displacement data that correspond to the calcu- lated velocity data, the ensemble X is pared down to the same dimensions as V. The matrix R = XXT/Nv is then formed, representing a covariance matrix if the mean of the displacement data is zero. Then the SOD is based on a generalized eigenvalue problem cast as AR? 2 St_’1. (3.1) For a free multi-modal response with light damping, the eigenvalues A approximate the frequencies squared, and the inverse—transpose of the modal matrix '1' approximates the linear modal matrix. To see this, consider first the symmetric undamped vibration system of the form Mi + Kx = 0, (3.2) which is associated with the eigenvalue problem —w2MQ + K59 = 0, which in matrix form is K<1> .—_ MA, (3.3) 28 where A is a diagonal matrix of eigenvalues and the columns of are the eigenvectors. The eigenvalues and eigenvectors provide the modal frequencies and the mode shapes. The SOD eigenvalue problem Eq. (3.1) can be written as AXXTE = VVTE, or Axng = XDTDXT} (3.4) Close examination [95] shows that DTDXT z -—XT. If the system damping is negligible, then from the symmetric vibration model of Eq. (3.2), we would find —XT = XTKM—l. Hence, Eq. (3.4) becomes AXXTQ = XXTKM‘lw. In matrix form, xxTrA = xxTKM-lql. (3.5) Assuming XXT is invertible (n modes are active and NU 2 n), we have \IIA = KM_1\I’. Taking the inverse transpose and noting symmetry, \IV’I'PA—1 = K-1M\II_T, and hence erv—T = MW’TA. (3.6) Comparing equations (3.6) and (3.3), the eigenvalue problem of SOD has reduced to the eigenvalue problem Eq. (3.3) of the undamped vibration system. The SOD modal matrix is thus related to the structural linear modal matrix as = \II'T. Chelidze and Zhou [95] derived this relationship starting with an optimization representation of the eigenvalue problem. SOD is applicable for symmetric, but otherwise general, mass and stiffness distribu— tions. SOD directly produces estimates of the modal frequencies from the eigenvalue problem. Insight to modal participation is not directly obtained, but can come from analysis of the modal coordinates, dependent on how modal vectors are normalized. Limitations of SOD are that SOD is restricted to lightly damped systems, and it has not been justified or studied for random excitatirms. Also, sufficient numbers of sensed displacements are needed. 29 3.2.2 Smooth orthogonal decomposition for systems under random excita- tion Previously, proper orthogonal decomposition for modal analysis was justified for ran- dom excitation [89]. Here the SOD will be justified for white noise excitation. 3.2.2.1 Smooth orthogonal decomposition and random excitation Consider the symmetric vibration system neglecting damping, M)": + Kx = f(t), (3.7) where f (t) is a random excitation. In terms of the sampled ensemble matrices, MX + KX = F, with F representing the ensemble matrix of the sampled f (t), and therefore DTDXT z —XT = XTKM‘1 — FTM_1. Hence, from the matrix form of (3.4) 1 T 1 .T —1 1 T —1 — = —XX KM \II — — . . NXX ‘IIA N NXF M \II (3 8) The elements in the matrix 7%7XXT represent cross correlations (with zero delay) between responses, and are expected to be nonzero. The elements in the matrix L = 1%,){FT represent cross correlations (with zero delay) between responses and random inputs. In other words, the elements Lij are the means of the products 513,-(t) fj(t). If their expected values are zero, then this term can be neglected, and the decomposition eigenvalue problem would then converge, as N gets large, to 1 T _1 T -—1 Nxx \PA— Nxx KM \11, (3.9) which is the same as Eq. (3.5), and thus reduces to the undamped structural eigenvalue problem if XXT is invertible. Under this condition, the SOD, even with random excitation, would produce the modal frequencies and mode shapes of the system. Thus we are interested in conditions for which L -—+ 0 as N gets large. Elements of L have the form 1 N m 00 L,,- = 17 121121 /_OOhu(T)f1(t/.~ — ridrmtn. (3.10) 30 where h,l(t) is an element of the impulse response matrix, between f (t) and x. In this form hil(t) is a linear combination of modal coordinate impulse response functions, each sinusoidal with a modal frequency. Interchanging the order of sums, Z/wh 1%,ng routed E21 ff; h..,-I(T)ijl(r)dr (3.11) where ijl(r) is the cross correlation between the forcing functions associated with coordinates j and 1. 3.2.2.2 White noise Here there are two useful possibilities. One is that the forcing on all coordinates are statistically independent. For example, independent bombardment of each coor— dinate by random turbulence fluctuations might qualify. Then ijl(r) = R§(r)6fl, where R; (r) is the autocorrelation of the jth forcing term. If the forcing functions are modeled as white noise, then Cf 1(r )= yj6(r)5fl, where 6(r) is the Dirac delta function, and 63-1 is the Kronecker delta. Another possibility is that each forcing term is dependent, for example in random base excitation. Then fj(r) = yjf(r), and hence C360) = fl'lefCr). If the forcing function is modeled as white noise, then ijl(r) = 'yj16(r). In either of these mwhite noise cases, we have the form Lz'j = I: 1/_O:Oh ('r)r)'yj[6( =1: hil(0) 73-]. For a typical vibration system, the impulse response function will be such that [121(0): 0, whence Lij — -.0 Thus, for white noise, the response and excitation are uncorrelated, and the matrix form of the SOD eigen— value problem of Eq. (3.4) represents the undamped structural eigenvalue problem of Eq. (3.3) for large N. As such, the SOD should produce estimates of the modal frequencies and mode shapes of the undamped model under white noise excitation. The natural excitation algorithm (referred to as NEXT) [107,108] also arrives at a 31 similar conclusion albeit in a different way. There, it was shown that for a system subjected to uncorrelated white noise inputs, the cross correlation between various outputs would be a sum of complex exponential functions of the same form as the sum of impulse response functions of the original system. Thus, NEXT would ac- commodate using output-only methods for modal parameter identification in case of independent (uncorrelated) white noise forcing. 3.3 Numerical example We simulated the eight-degree-of-freedom linear vibratory system shown in Fig. 3.1. The system observes light modal damping and was excited by white noise applied to the first mass with zero initial conditions. The mass and stiffness matrices are given ’ k] [:T k2 7x2 [:3. k8 l—x: k3 2m -’V\/\/\/— m -’\NV\F m "VVVV m __33._ E -.-.-. ..... + C] C2 C3 Cg Figure 3.1: The mass-spring-damper model. The dashpots are figurative to represent the presence of damping, and do not accurately correspond to the example damping matrix. as ”20000000“ '2—1000000' 01000000 —12—100000 00100000 0—12—10000 M=000100001K= 00—12—1000, 00001000 000—12—100 00000100 0000—12—10 00000010 00000—12-1 _00000001, ,000000—11_ (3.12) ' and the damping matrix is chosen to be C = cM, where c = 0.01. 32 The system was simulated for 1000 seconds (later on we found out this time to be equal to 34 periods of the fundamental mode) with SIMULINK toolbox in Matlab, which uses a fixed-step Dormand Prince (a member of family of Runge—Kutta methods) differential equation solver [109] to evaluate the response of the system. White noise forcing was generated using the Gaussian white noise generator function that generates discrete—time normally distributed random numbers with sampling time step matching the solver step size chosen as 0.1, resulting in generation of 10,000 data points. The forcing was observed to have a mean approaching zero. Both displacement and forcing matrices were saved to the Matlab workspace for further processing. The excitation was applied to the first mass only. In the decomposition, the V ensemble was formed with centered finite differences with a total step of two samples, such that difference matrix D was NU x N, and V = XDT was 71 x NU, where NU = N — 2. From the data decomposition eigenvalues 5 estimates of the natural frequencies are compared to the true modal frequencies in Table 3.1. Table 3.1: System frequencies estimated from smooth orthogonal decomposition (SOD) with white noise forcing compared against structural eigen frequencies in Eq. 3.3 SOD Structural EVP % Error 0.1816 0.1838 1.19 0.5275 0.5266 0.17 0.8133 0.8143 0.12 1.0963 1.0966 0.02 1.3875 1.3859 0.11 1.6449 1.6412 0.22 1.8368 1.8366 0.01 1.9524 1.9586 0.31 The modal assurance criterion [110,111] is a useful tool for testing whether the estimated modes are consistent with the system modes. The normalized inner prod- ucts (squared) between estimated and true modes are seen in Table 3.2. Values of 33 assurance criterion Table 3.2: Estimated SOD modes compared against system modes using the modal Mode 1 2 3 4 5 6 7 8 1 0.9993 0.0001 0.0001 0.0001 0.0000 0.0002 0.0004 0.0003 2 0.0003 0.9998 0.0004 0.0005 0.0001 0.0008 0.0019 0.0012 3 0.0003 0.0000 0.9994 0.0013 0.0002 0.0022 0.0051 0.0035 4 0.0003 0.0002 0.0003 0.9999 0.0005 0.0056 0.0127 0.0094 5 0.0000 0.0001 0.0001 0.0004 1.0000 0.0006 0.0022 0.0011 6 0.0003 0.0007 0.0023 0.0057 0.0005 0.9999 0.0279 0.0136 7 0.0008 0.0017 0.0051 0.0136 0.0018 0.0256 0.9997 0.0307 8 0.0005 0.0008 0.0021 0.0070 0.0012 0.0145 0.0296 0.9999 near unit magnitude indicate modal vectors that nearly line up. For visualization, the modal vectors from smooth orthogonal decomposition and the structural eigenvalue problem are compared in Fig. 3.2. With random excitation, results are expected to converge as N increases. For this example, we increased N, with the time step fixed at 0.1, and plotted the estimated frequencies in Fig. 3.3 (Table 3.3 shows the percent errors). Increasing N improves the frequency estimates. The period of the lowest-frequency mode is about 34 sec- onds. Estimates of this mode converged within about 10,000 samples, or 1000 seconds (about 30 first-mode cycles of random response). Important in the convergence is the relative contributions of matrices 7%,XXT, 7%;VVT and fifXFT. The maximum singular values of these matrices are plotted in Fig. 3.4, indicating that 7%7XFT approaches zero (while the other matrices’ singular values settle to finite values), thereby becoming negligible in Eq. 3.8 for large N. The example problem studied had a maximum damping ratio of C = 0.027 in the system corresponding to fundamental frequency of col = 0.1838. With increasing damping, the results deteriorated as seen in Fig. 3.5, even with increased sample size 34 g 0.5 8 0.5 O 2 2 0< 00 m -0 5 0 2 4 6 8 0 1 1 3rd Mode 3 4th Mode 9 5th Mode E 6th Mode <2 7th Mode é 8th Mode 9? 0 2 4 6 8 0 2 4 a 8 Sensor Locations Sensor Locations Figure 3.2: A comparison between mode shapes from the structural eigenvalue prob- lem {solid lines) and those estimated from the smooth orthogonal decomposition ( ‘o ’ symbols ) The sensor locations refer to the mass indices. The location 0 repre- sents the wall attachment. The modes are in ascending order by modal frequency. N = 105. When the system had a first-mode damping factor of C = 0.8, the error in cor- responding frequency estimation was z 8%. We also see in the Fig. 3.5, that the frequency estimation is very good for the ideal undamped case, for which the theory was developed. 35 0.6 0.55] 2 A 4 6 0.5]. N (Sample P011115) N (Sample Points) 0.9 1.2] 1,1 W is 1 . 102 104 106 N (Sample Points) N (Sample Points) ,, 1.6 £14 ‘ 102 104 106 '10 104 106 N (Sample Points) 4 2\ 1.5 - 0 . 102 104 106 102 104 106 N (Sample Points) N (Sample Points) Figure 3.3: The frequency estimates improve as N gets large. The solid line (-) represents the true frequency (rad/sec) and {— *) are the estimated frequencies {rad/sec). The percent error in frequency estimation computed at N = 104 for modes 1 to 8 are 1.2%,0.17%,0.12%,0.02%,0.11%,0.22%,0.01% and 0.31% re- spectively also shown in Table 1. 36 Table 3.3: Percentage error computation (SOD) for all modes 104 10 N (Sample Points) Sample Points 102 102-5 103 103-5 104 104-5 105 105-5 106 1.11 9.68 6.69 4.95 1.08 1.19 0.70 0.48 0.38 0.00 1.12 2.54 2.14 2.82 0.49 0.17 0.17 0.11 0.05 0.03 1.13 2.60 3.53 1.30 0.12 0.12 0.20 0.02 0.07 0.01 154 6.38 4.51 0.57 0.29 0.02 0.11 0.04 0.00 0.00 135 8.60 0.85 0.21 0.39 0.11 0.08 0.06 0.02 0.00 1.16 12.94 0.13 1.87 0.37 0.22 0.01 0.00 0.01 0.01 157 21.79 1.42 1.45 0.27 0.01 0.05 0.03 0.04 0.00 158 68.04 5.83 2.25 0.39 0.31 0.14 0.07 0.03 0.01 2 7% 1 r ' ' 1 fr 1 v 1 .'~ .’ ’~ 0 ‘. 0‘ '.'~,., “It. ”,15‘ ,’ '~o----o ----- e" " 3 8" E y ”.“s I ' s E 1‘ I '0' “ d 3 ~ I... I... g) "I ’. ~~ "'. * . .U-J " ’0 ~.o 0.5-- "’o’ .1 ~o «‘1' ."‘.”’.. Figure 3.4: The maximum singular values of XXT/N (7112), VVT/N (m/s)2 and XFT/N {N-m). The dashed line (i) represents values of XXT/N, the dashed line (—*) represents the values of VVT/N and the line (—*) represents values of XFT/N. We can see by N = 104 samples, XFT/N << (XXT/N,VVT/N) and is perhaps negligible for this and larger values of N 37 0.2 I 1 I I I I I l 0.18” .-'--"-.---.- .- a a) O -..._.. 9 “O.. (U .h. E 0 450.16" r LIJ 8 $0.14- - C' 93 LL 0.12" 4 0.1 l l l l 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Damping Factor C Figure 3.5: The undamped natural frequency estimate starts to deteriorate for high damping. The solid line (—) represents the true frequency (rad/sec) and (— *) is the estimated frequency (rad/sec). Here samples size is N = 105 3.4 State-variable modal decomposition for systems under random excitation In this section, we explore the SVMD method for random response. We apply the decomposition strategy in the Similar way as described in section (2.2.2), and we will see that there is a remarkable resemblance in the SVMD and the SOD analysis of randomly excited systems which will be fully exploited. 3.4.1 State-variahle modal decomposition and random excitation Consider a MDOF damped vibration system, M5”: + ex + Kx = 1(1), (3.13) 38 where f (t) is a random excitation and the matrices M, C, and K represent the system mass, damping and stiffness terms in the usual manner. T We define a state vector y = [53, xT], and again use the equation Mx—Mx = 0, to obtain forced equations of motion of the form Ay + By = F, (3.14) where MC 0K F where F represents the ensemble matrix of the sampled f (t) Thus similar to sec- H” M1 M10, F=l°l tion 2.2.2, we use the state-variable ensemble Y = [VT, XT]T, where V = XDT % X is the approximate velocity ensemble, and D is the matrix of centered finite differ- ences. The vector form of the state-variable eigenvalue problem of Eq. (2.6) is 1 (1ng = NYWTg (3.16) Now, taking the derivative W = YDT x Y, and making use of the Eq. (3.14), W x Y = —A'—1BY + A"1F, we get 1 T ~_1 T T —T) 1 -T —T,, aNYY g... NYY B A E+NYF A g. (3.17) In matrix form, 1 1 1 - NYYTWA = —NYYTBTA_T\II + NYFTA_T\II. (3.18) We are interested in the conditions when YFTA_T‘II z 0. Close examination of this matrix reveals l ’T —T _}_-V T —T NYFA \P_N_X][0 F [A \II 1'OXDTFT _T =— A 1;: 3.19 N 0 XFT ( ) 39 The elements of Eq. (3.19) contain the matrices XFT and XDTFT. Of these matrices, XFT has exactly the same form as the Eq. (3.10) discussed in the section (3.2.2.1) for the SOD forced eigenvalue problem while the other matrix XDTFT is much similar but involves a linear combination of terms in F. This matrix could also be seen as VFT which won’t exploit hv(0), Since hv(0) 75 0. However, by looking at the singular values of VFT/N, it was observed in this example (but not shown) that the singular values of VFT/N have similar strength to those of XFT/N and thus may be approximated to 0 for larger N values. This is not a proof but gives some evidence of success in approximating VFT % 0. Thus, for the white noise cases as detailed in section (3.2.2.2), the response and the excitation are uncorrelated, and the matrix form of the SVMD eigenvalue problem Eq. (3.18) represents the unforced structural eigenvalue problem Eq. (2.11) for large N. Just like SOD, the SVMD should also produce estimates of the modal parameters (frequency and damping), and mode shapes of the MDOF system subjected to white noise excitation. 3 .4 .2 Numerical example To perform modal parameter identification using SVMD for randomly excited systems, we revisit the same example that was used for SOD random analysis. Since the initial conditions, the simulation time interval, the step sizes, the numerical integration scheme, and the set of data points (N = 10,000), are kept exactly the same as the SOD example, the interested reader can draw conclusions about both methods. For the SVMD decomposition, the ensemble matrices Y and W were formed and assembled in the eigenvalue problem. From the data decomposition eigenvalues, estimates of the natural frequencies are compared to the true modal frequencies in Table 3.4. The modal vectors from state-variable modal decomposition and the structural 40 Table 3.4: System frequencies estimated from state variable decomposition (SVMD) with white noise forcing compared against structural eigen frequencies in Eq. (3.3) SVMD Structural EVP % Error 0.1826 0.1838 0.65 0.5292 0.5266 0.49 0.8094 0.8143 0.60 1.0967 1.0966 0.00 1.3822 1.3859 0.27 1.6361 1.6412 0.30 1.8275 1.8366 0.49 1.9453 1.9586 0.67 1 0.9 0.8 0.7 g .1 a 0.6 > Sr.) 0.5 2 0.4 0.3 é 0.2 0.1 I1 0 Figure 3.6: The MAC values comparison for S VMD for random excitation. MAC value of 1 shows vectors match up. Modes are not in any particular sequence eigenvalue problem are compared using the modal assurance criterion in Fig. 3.6. The frequency and mode shape estimates Show the same trend as those seen in the SOD modal identification. Similarly, it was also seen (but not Shown) that the estimates improved when the sample size increased. However, SVMD was unable to produce good damping estimates (not shown) that were expected from the theoretical 41 analysis. This needs further investigation. However, we suspect that this is primarily due to involvement of differentiating of the X ensemble twice to obtain the velocity and acceleration ensembles. This can be seen by looking at the damping values ob- tained by the SVMD decomposition that uses the SIMULINK generated acceleration and velocity ensembles (as opposed to generating velocity and acceleration ensembles by differentiating the displacement ensemble) for N = 104'5 samples, as shown in Table 3.5. Table 3.5: System damping ratios estimated from the state-variable decomposition (SVMD) with white noise forcing compared against structural damping ratios in Eq. (3.3) Damping ratio Structural EVP SVMD (1 0.0272 0.0281 (2 0.0095 0.0101 (3 0.0061 0.0074 (4 0.0046 0.0047 (5 0.0036 0.0038 (6 0.0030 0.0028 (7 0.0027 0.0028 (8 0.0026 0.0027 Further increase of sample points, showed further reduction of the error in damping ratio estimates. Furthermore, system damping ratios up to C = 0.8 were estimated within 5% of accuracy. 3.5 Summary The extension of smooth orthogonal decomposition and the state-variable decompo— sition methods for modal parameter estimation under random excitation has been presented. Analysis suggests that, for undamped systems, if the expected value of the product between response and excitation variables is zero, then the decomposi- tion methods converge to an equivalent representation of the undamped structural 42 eigenvalue problem, and therefore should produce estimated modal frequencies and mode shapes for randomly excited structures. This was justified for white noise exci- tation, and the convergence to the structural eigenvalue problem was demonstrated in a simulation. In the SOD simulation example, random excitations were applied to a linear eight- degree-of-freedom structural system while damping was kept light. It was shown that the mean of the product of displacement matrix and forcing vector approaches zero as sufficiently large number of samples are captured. This in turn means that the effect of forcing in formulating the eigenvalue problem becomes negligible. Therefore, the smooth orthogonal decomposition eigenvalue problem from data of the randomly forced problem in essence becomes representative of the free structural eigenvalue problem. The example problem studied was subjected to light damping. As damping in- creases, the SOD estimation results slowly deteriorated. While this work focused on white—noise excitation, if it can be shown that the mean of product between the response and the forcing approaches zero (in reference to equations (3.8) and (3.10)) for other classes of random excitation, this would broaden the applicability of smooth orthogonal decomposition for randomly excited systems. For random responses, all of the conclusions reached for smooth orthogonal decom- position also apply to the state-variable modal decomposition and thus any of these methods is equally good candidate for modal frequencies and mode shapes estimates of lightly damped system. The SVMD could not produce good damping estimations unless V was known accurately, separately from XDT. In such case, SVMD estima- tions got beyond small damping restrictions. 43 fizz-1‘”! fxpen'mental Investigation, 4.1 Introduction revious work using numerical studies has shown that for low noise cases, the decomposition based methods work very well in estimating modal parameters for free response and in some cases forced response. However, to implement the decomposition methods in the physical applications, ex— perimental studies must be conducted. These types of studies can be helpful in under— standing practical aspects of the identification problem, for instance, proper selection of system model order, issues surrounding number of sensors, linearity, boundary conditions, and effects of filtering. This chapter, thus deals with the experimental verification of the ideas presented in the previous chapters. The experimental studies conducted herein address both the advantages and the limitations of applying the decomposition methods in non-perfect conditions. In this work, three kinds of experimental studies are conducted. First, a struc- tural system modeled as a single—degree-of—freedom (SDOD) system is investigated in section (4.2) and then a clamped-free beam experiment that presents a structural system modeled as a multi-degree—of—freedom (MDOF) is addressed in section (4.3). Modal analysis is performed on both kinds of experiments using the SVMD method 44 [IOU—102]. Practical aspects of data length and effects of filtering are addressed fol- lowed by the use of modal coordinates for qualitatively establishing true or spurious modes. Finally, we investigate an experimental study reported in the reference [112] to highlight the limitations on using the decomposition method. 4.1.1 Assumptions Let, for any arbitrary input f (t), the system output y(’t) be given as y(t) = [+00 h(r)f(t — r)dr (4.1) ——00 In Eq. (4.1), h(r) is a weighting function (impulse response function) defined as the output of the system at any time to a unit impulse applied at a time T = 0. For the response of the systems detailed herein, the experiments assume the following outline. A rigorous treatment of the subject can be found at, for example in [5, 113, 114]. o The system understudy is linear. A system is linear if response characteristics are additive and homogeneous, that is, it obeys the principle of superposition [115419]. This means that doubling or halving the inputs would respectively double or half the outputs at any particular frequency. This also suggests that h.(t) is not dependent on the input [120]. o The system is time invariant, implying that the impulse response function h(r) and its Fourier transform H (w) do not change whether the input is applied now or the input is delayed t seconds from now. That is h(t, 7') = h(r) for ~00 < t < +00 (4.2) In simple terms, the system parameters (mass, stiffness and damping) are con- stant. o The system is considered stable such that every possible bounded input pro- duces a bounded output for systems having all of their poles in the left half 45 plane. This is true if the function Mr) is absolutely integrable [120]. To see this, man ——- Um 12mm — adv s f: |h(r)l|f(t — our (4.3) If the input f (t) is bounded by a finite constant R, that is, |f(t)l S R W (4-4) then, using the Eq. (4.4) in Eq. (4.3), we see that +00 Ile s R / lh(r)ldr (4.5) —CXD Or simply, +00 / |h(r)|dr < 00 (4.6) —00 Similar conclusion for a MIMO case is presented in [117]. The condition in Eq. (4.6) is deemed both necessary and sufficient [121] (reproduced here without proof) rendering that the system is BIBO (bounded input bounded output) stable. For a detailed discussion on stability in linear systems, please see for example [115—118, 121]. o The system should be physically realizable. In other words, h(r) = 0 for 7' < O (4.7) This condition asserts that the system would not respond before the excitation is applied. Thus, Eq. (4.1) can be modified to y(t) : Air—00 h(r)f(t — T) ([7 (4.8) The above conditions are generally true for calculating any F RF 8 and hence are applicable to the output-only methods’ experimentation as well. 46 4.2 A single-degree-of—freedom experiment In this section, experimental data of a study conducted by Rhoads et. al [122] is analyzed using the SVMD [100] method. The beam experiment was conducted in such a way that the system behaves as a single—degree-of—freedom (SDOF) system since the beam was resonated into its fundamental mode, the excitation turned off and the free response examined, and therefore is used for evaluating the SVMD method as a SDOF case. The purpose of the original experiment conducted in Dynamics and Vibrations Research Lab at MSU, was to augment the analysis of parametric amplification in mechanical systems. Modal parameters in the original work were obtained using the classical methods [92] and are used as a benchmark for comparison. Figure 4.1: The experimental setup for study by Rhoads et. al The experimental setup of Rhoads et. al [122] comprised a cantilever spring steel beam of dimensions (190 x 19 x 0.5 mm3) vertically mounted on a electromagnetic 47 shaker (MB Dynamics Phi-500) as shown in Fig. 4.1. The base excitation provided by the shaker was measured by a 3-axis accelerometer (Analog Devices ADXLIOSEM- 3). The beam was sensed with two electrical resistance strain gauges (Vishay Micro measurements division, model EA-I3—120LZ-120) mounted on either sides of the beam that constitute a half-bridge circuit [123]. A strain gauge signal conditioner (Vishay Micro measurements division, Model 2210) was then employed to amplify and balance the signal strain gauge Wheatstone bridge circuit and after filtering the signal at 100 HZ cut-off frequency, the signal was fed to a data acquisition system (TEAC GX-l Integrated Recorder) where the strain signals were recorded. For the analysis contained in this work, the system was excited in a different way from Rhoads et. al [122]. Here, instead of the shaker, the excitation was provided by a finger flick that incurs a free-vibration response [124]. Data were acquired for approximately 12.267 seconds and 63814 data points were obtained as shown in Fig. 4.2. The strain data were then recorded in a comma separated value (CSV) format Excel spreadsheet file and subsequently sent to MAT- LAB for further processing. For analysis purposes, data were truncated down to get 2 10,000 points to obtain a “clean” portion of the signal, since the initial part of the signal was observed to be “choppy”, and the later part contained high frequency noise. Thus, signal samples from N = 10, 000 to N = 20, 000 were used as shown in Fig. 4.3. The experimental frequency obtained by F F T [92] was about 11.49 Hz as seen in Fig. 4.4, with the damping factor obtained using the logarithmic decrement method [125, 126] from a nearly single-mode response being estimated at 3 0.0065. By looking at the Fig. 4.2, we observe that the damping is both viscous and Coulomb. Nonethe- less, we can carry out the identification by SVMD on the data. The frequency value obtained via SVMD method is 11.54 Hz. The computed damping factor from SVMD is z 0.0066. The representative equivalent viscous damping identification is consis- 48 Strain Amplitude 3 l l l l l Sample Points 6 x104 Figure 4.2: The impulse response of SDOF beam experiment for study by Rhoads et. al tent with the log decrement method calculation. The theoretical natural frequency calculated is 11.5 Hz, with mass of the sensor being ignored since it is very small. The associated error values respectively are shown in Table 4.1. Table 4.1: Estimated modal parameters from the decomposition method compared against theoretical and classical experimental methods Parameter Theoretical Classical Experimental Methods SVMD Frequency (Hz) 11.50 11.49 11.54 Damping Ratio Q - 0.0065 0.0066 Error in Frequency 0 0.08% 0.34% Error in Damping - 0% 1.53% 49 Strain Amplitude O -2 0 2000 40100 60100 80100 10000 Sample Points Figure 4.3: The truncated sample points to obtain a “clean” signal for analysis of the study by Rhoads et. al 4.3 Clamped-free beam experiment This section describes the modal parameter estimation of a clamped—free beam using the SVMD method for a free vibration experiment. The system is assumed to follow the assumptions described in section (4.1.1). The experimental setup of the system is described as below. A 941 x 52 x 3 mm3 clamped-free uniform steel beam as shown in Fig. 4.5 was prepared for the experiment conducted in the Dynamics and Vibrations Research Lab at MSU. The beam was sensed with 16 PCB model number 352B10 accelerometers, each of which weighs 0.7 gm with a sensitivity listed in catalogue at about 10 mV/ g (individual sensitivities vary slightly), equally spaced from clamp to the beam tip via beeswax [4]. Beam properties hence are: elastic modulus E = 190 x 109 GPa, density 50 The FFT plot of SDOF beam experiment 2 10 f ‘ 3 E 1 10 - 0 10 . . l . m . 1 . —1 0 1 2 1O 1 10 10 Frequency (Hz.) Figure 4.4: The FFT log plot of the SDOF beam experiment Figure 4.5: The experimental setup of a 16 DOF clamped-free beam p = 7500 Kg/m3, mass per unit length including the sensors mass m = 1.1907 Kg/m (without the sensors mass m = 1.1787 Kg/m but mass of the sensors was accounted 51 for in calculating the theoretical natural frequencies of the system), and moment of inertia I = 1.17 x 10"10 m4. Measurement signals from the accelerometers were then processed via signal conditioning amplifier PCB Model number 481A02, the output of which was then fed to a data acquisition system (TEAC GX—l Integrated Recorder) where the acceleration measurement signals were recorded by passing through a low pass filter and then converted into ASCII.txt format. Further processing of these files was done in Matlab. r" , 150 l l l 50r- Acceleration Ensemble (g) C: -100— -150 . - . . - - 1 o 2 4 6 3 1o 12 14 16 Sensor Locations Figure 4.6: The beam acceleration snapshots of the acceleration ensemble of a 16 DOF clamped-free beam with its mean removed. The data were sampled at a rate of 5 kHz. The data were digitally filtered with the cut-off frequency of the low-pass filter set at 0.4 kHz. This value was set well below the Nyquist frequency to avoid aliasing effects. The beam was excited with an impulse, with the resulting response monitored by the accelerometers. Measurement recording time of approximately six seconds was used with sampling step used as 1 / F s, where F3 52 100- a 50 mm s -100 -150 -200 - l 50 100 200 250 1 50 Frequency (Hz.) Figure 4.7: The linear FFT plot of a 16 DOF clamped-free beam. Only first six modes are shown. is the sampling frequency. The number of samples thus obtained were N = 36. 327. The data were also high-pass filtered to reduce the low frequency noise effects on subsequent integrations of the acceleration signals. As such, the ensemble matrix of acceleration time histories A of size 16 x 36327 was formed. For uniformity, each acceleration signal was then individually calibrated per its accelerometer sensitivity specifications. Using the Matlab routine “cumtrapz”, the velocity V and displacement X ensembles were obtained numerically. All ensembles were then processed to remove the respective means. On one of the acceleration signals. the FFT algorithm was conducted and is plotted in Fig. 4.7. Next, the correlation matrices P 2 YYT and Q = YWT, for the decomposition method SVMD were formed, where Y = [VT, XT]T and W 2 [AT. VT[T that were obtained by numerically integrating the acceleration ensemble. 53 4 .3.1 filtering eflects I N O Magnitude (dB) 4;. O I (I) O L4L4im1l 1 1 111111l 1 141g11ii 1 1911111 ._x (D O 90 Phase(deg) rlmmiil 1 L 111 ill 1 1 L11‘l1l 0.01 0.1 1 10 100 Frequency (Hz) Figure 4.8: The frequency response of the second order high-pass filter for the 16 DOF beam experiment. To remove a low frequency “drift” in the integrated signal ensembles, data were high-pass filtered at a filter frequency of 1.4 Hz. This value was selected at about half of the first natural frequency obtained from the FFT plot Fig. 4.7. Note that the filter frequency was evaluated in the range of 0.5 Hz to 2.0 Hz with no significant change observed in the high frequency estimates, and some improvements seen in the lowest frequency estimate as the filter frequency approached to 0.5 Hz. This suggests that as a limiting case, a maximum of one half of the fundamental frequency by FFT should be employed as the filter frequency. 54 The frequency response of the second order high-pass filter modeled as a SDOF system is shown in Fig. 4.8. The filter amplitude at twice the break frequency is observed to be 0 dB (1 in the linear case), thus implying that application of the filter does not reduce the estimated frequencies amplitudes. The phase shifting effects caused by this filter can be removed by running the filter “backwards”. In that way, the filter becomes a fourth order system. The filter was applied prior to, and after each numerical integration of the signals. Thus, to maintain consistency, the acceleration ensemble was filtered thrice, the ve- locity ensemble twice and the displacement ensemble only once. Cast this way, the filter order became 12 for the ensembles. 4.3 .2 Identification results We kept the first N = 12,000 points to minimize the contribution of the high fre- quency noise dominating the decayed signals in the later part of the ensembles. These reduced sized ensemble matrices Y and W were then assembled into P and Q of the SVMD eigenvalue problem from which the eigenvectors and eigenvalues were extracted. From the acceleration snapshots in Fig. 4.6, we see that the beam acceler- ation had a dominant second mode, implying that the impulse input, or equivalently the initial conditions, had a stronger effect on the second modal coordinate than the rest [105]. The theoretical frequency values and the values from taking the FFT and the log decrement method are tabulated in Table 4.2. The log decrement method is applied to carefully excited, dominantly single-mode excitation responses (this was obtained by constraining a nodal point while plucking the beam). The SVMD method identified modal frequencies and the respective damping ratios as also presented the same table. The error thus in the frequency estimates is 1.41% and 0.30% respectively. Since the high modes are expected to damp out rather quickly, for the same experi- 55 Table 4.2: Estimated modal parameters from the decomposition method for a linear beam free response compared against the undamped structural eigen frequencies and the F F T for data consisting of first N = 12. 000 sample values. Parameter Theoretical Classical Methods SVMD ml 2.74 2.75 2.78 (.12 17.16 17.20 17.18 (1 _ 0.0048 0.0049 (2 - 0.0040 0.0032 mental run, in order to see the higher modes, data were further pared down to the first N = 1000 points with everything else being kept the same. Now the decomposition method was able to estimate the first five modes. The obtained frequency identifica- tion results for the first five modes are shown in the Table 4.3. Damping estimates for the first three modes computed on the carefully excited, dominantly “pure mode” re— sponses by the log decrement method are compared against the identification scheme in the Table 4.4. Table 4.3: Experimental system frequencies ( Hz. ) estimated from SVMD for a linear beam free response compared against the undamped structural eigen frequencies and the FF T values for data consisting of first N = 1000 sample points. Method Theoretical FFT SVMD 021 2.74 2.75 3.01 mg 17.16 17.20 17.18 w3 48.05 47.21 47.39 (.114 94.16 94.28 88.96 (1)5 156.66 155.9 153.78 It is clear from these two (N = 12,000 and N = 1000) data sets that a trade-off exists between the number of modes to be estimated and the accuracy acquired from the estimation. For an estimation of lower modes, a large—time data set is expected to be useful since higher modes generally dissipate quickly, leaving lower modes to dominate most of the recorded response. On the other hand, for the estimation of 56 Table 4.4: Experimental modal damping estimated from the S VMD method for the linear beam free response for first N = 1000 sample values. Only the first three damping estimates are experimentally estimated by “pure mode” log decrement method. Method Log Decrement SVMD (1 0.0048 -0.035 (2 0.0040 0.0040 (3 0.0036 0.018 (4 - 0.0077 C 5 - 0.0017 higher modes, a relatively short-time data set could be utilized with the caveat that “short-time” data set begins to err in the lower modal parameter estimates if the time record is short compared to the lower modal periods. This can be observed in the damping estimate of the first mode that was completely “missed” by the decomposition method even though the frequency estimate for that mode remains reasonable in Table 4.4. Mode shapes are now estimated using the SVMD method using the N = 1000 sample points. The eigenvectors obtained from the method are normalized and are plotted separately for each mode in the figures 4.9, 4.10, 4.11, 4.12, and 4.13. These plots in general, accord with the frequency estimates with a distortion observed in the lowest mode. This distortion is speculated to be due to low frequency limit on the PCB sensor (per product specs, sensor’s working range is 2-10,000 Hz, the lower limit being close enough to the fundamental frequency of the beam). In both the long and short time-record examples, the number of identified modal sets is well less than number, n, of sensors. The remaining identified modes are spurious. The spurious modes are distinct from the estimated true modes and their complex conjugate pairs, and are dominated by noise. It is conceivable that spurious SVMD frequencies could be similar to the true 57 0.6 . I . I 1 0.5 I 0.4 I 0.3 Mode 1 I 0.2 I 0.1 l 0 5 1O 15 Sensor Location Figure 4.9: S VMD is compared for the first beam mode against the theoretical mode. SVMD is shown with + mark, theoretical mode is shown with the solid line -—. frequencies. In this case, the quality of modal coordinates may give a. clue to which modes are true. 4.3.3 Modal coordinates The POD uses proper orthogonal modal coordinates (POCs) to determine the modal frequencies and which modes correspond to which frequencies, as explained in detail in references [84, 85]. W) = Z (110)}: (4.9) 58 0.4 0.3 - 0.1~ Mode2 _O'50 2 4 6 8 10 12 14 16 SensorLocafion Figure 4.10: SVMD is compared for the second beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line —. 0.6 I I I I I I T _0-4 at; L 1 1 1 0 2 4 6 8 1O 12 14 16 SensorLocafion Figure 4.11: SVMD is compared for the third beam mode against the theoretical mode. SVMD is shown with + mark, the theoretical mode is shown with the solid line 7. 59 0.4 T ,J l l I l I 0.2 — 0.1- Mode4 SensorLocafion Figure 4.12: S VMD is compared for the fourth beam mode against the theoretical mode. S VMD is shown with + mark, the theoretical mode is shown with the solid line 7. where qi(t) are the SVMD modal coordinates. In matrix form, X 2 WC), where elements in the ith row of Q are the ith SVMD coordinate histories. Noting that for SVMD \II‘II—1 : I, the SVMD modal coordinates Q (SVMCs) are now expressed. These are calculated from Y = \IIQ (4.10) where columns of \I/ are the inverse—transpose of the eigenvectors obtained by solving SVMD eigenvalue problem (2.6). Thus, modal coordinates Q are simply obtained by Q = \r—lY (4.11) Alternatively, one can use Q = (PTY, where (I) is the non-Hermitian matrix of eigenvector matrix obtained by directly solving the SVMD eigenvalue problem (2.6). 60 0.5 I I I I I I l 01* Mode5 + 1. i- _ .4 1 1 1 1 1 0 0 2 4 6 8 10 12 14 16 SensorLocafion Figure 4.13: SVMD is compared for the fifth beam mode against the theoretical mode. S VMD is shown with + mark, the theoretical mode is shown with the solid line Modal coordinates for the SVMD (SVMCS) are presented in Fig. 4.14. It is shown in the references [84,85], that the POD directly yields modal dominance and that proper orthogonal modal coordinates give information on modal frequencies. While SVMD gets frequency and damping estimates directly, modal coordinates can indicate the quality of decomposition, and the estimates of the corresponding modal frequen- cies can be obtained from each modal coordinate time history. For instance, one simple way would be to divide the sampling frequency by a complete cycle of each modal coordinate history. In this particular example, for the first SVMC, the period is not complete (still SVMD got the first mode with data that was less than half of the modal period), but for the higher modes, it is quite easy to calculate the modal frequency. Another way would be to take an FFT of any individual modal coordinate. 61 As such, frequencies estimated by this way may or may not be as accurate as the ones computed by the SVMD directly, but they do provide a good starting point for fur- ther investigations. It appears from Fig. 4.14, that a small quantity of lower modes (mode 1 or 2) has leaked into the higher—modal coordinate histories thus showing a significant low—frequency perturbation on these signals specially in the mode 3 and mode 4. ‘_ 008* 1 N0.01 O a 0 2 o > 51 (I) -0.01 -0.08 * 0 500 1000 0 500 1000 2E 6 2E-6- (0 ‘1' O O 2 0 E 0 51 -2E—6 m '4E_60 500 1000‘: u0 500 1000 3 2E-6 3 0| 1 1 ‘ ‘ w —2E-6 ‘ -4E-6 0 500 1000 Time History Figure 4.14: S VMD modal coordinates are shown for the first 5 beam modes. Modal coordinates indicate the quality of the decomposition, and can be used to distin- guish between computational and actual modes. With the exception of fourth mode, coordinates are smoothly shaped. A smooth periodic non-noisy modal coordinate history can intuitively indicate an actual mode. For the fourth mode, the latter half of the modal coordinate history is corrupted by noise but the first half is good, so even though borderline, it is still a candidate of an actual mode. Thus, the first half of fourth SVMC was tested, together with 62 Fig. 4.12 and the frequency estimated by SVMD, it was concluded to be an actual system mode. (Later we will see another way of analyzing these candidates) Also, if SVMC3 0.00003 f 4 a s . . . . . 0.00002 L _:;::::1* [(1,111 111 O -0.00002 -0.00003 ~ -0.00004 - - ’0'0000‘0 100 200 300 400 500 600 700 800 900 1000 Time samples Figure 4.15: S VMD modal coordinates are shown for the third beam mode. Modal coordinates can be used to distinguish between computational and actual modes. Actual mode is shown in dotted (- -), whereas spurious is shown in solid line (r)- spurious frequencies are estimated by the SVMD, modal coordinates can “visually” help determine the real modes from fictitious ones. As an example, this is seen in Fig. 4.15 where modal frequencies estimated by the SVMD are close (47.39 and 44.81 Hz), but the modal coordinates clearly indicate the noisy, hence the spurious frequency estimate (shown in the figure as the solid line). 63 4.4 Limitations on the number of sensors and active modes In the theoretical development of the SVMD method (section 2.2), it was implicitly assumed that sufficient number of measurement sensors are always available for ob- taining the time series data for modal parameter estimation. This number at the minimum is greater than or equal to the system frequencies. In real life cases how— ever, structural systems are composed of infinite degrees-of-freedom, and as such only a limited number of sensors can be applied to any system. We have seen that if the number of sensors 71 is greater than the number of active modes, spurious modes are present. Conversely, in a situation, where the number of available sensors would be less than the active (dominant) system modes, the SVMD may run into problems. To see this, we will conduct a sensitivity analysis followed by a numerical simulation and an actual experimental test in the following sections. 4.4.1 Sensitivity analysis If the number of sensors employed for identification is less than active/ dominant fre- quencies present in the system, the identified frequencies may deviate from the actual frequencies. To explore this, in this section we analyze an oscillatory system with a harmonic frequency that is contaminated by a small amplitude noise perturbation frequency with only one sensor being used by the system. We are interested in the conditions under which the SVMD identification of the harmonic signal would pro- duce results with minimum or (in ideal case) no estimate deviations. The analysis is carried out by constructing the SVMD eigenvalue problem by this under—sensed system and by obtaining an approximate solution of system eigenvalues. Consider the harmonic response of a signal containing two frequencies “’1 and 622 such that the amplitude A2 of 1122 is very small as compared to the amplitude A of wl, i.e A2 = 614. Then 64 ;r(t) = Acosw1(t) + €ACOSL1)2(t) Taking the derivative twice, we obtain —w1A sin w1(t) — eAw2 sin w2(t) 15(1) = —a%4cosw1(1) — mag cos w2(t) Forming the Y and W matrices, we get Y W Thus, the key non—symmetric matrices YYT and YWT are obtained as L +2w1w26A2 sin colt sin w2t L—E 4:1:1 4:11 1 tog/42 sin2 wlt +w362A2 sin2 wgt J -w1A2 sin wlt cos colt —wleA2 sin wlt cos wgt —w26A2 sin wgt cos colt 2 cog/12 sin wgt cos wgt] 65 -—w1A sin colt — wgeA sin wgt Acosw1t+ eA cos wzt -—w%Acos wlt — wgeA cos wgt —w1A sin colt — wgeA sin w2t r —w1A2 sin wlt cos wlt —wleA2 sin wlt cos wgt —w26A2 sin wgt cos wlt 2 l —e w2A2 sin wgt cos wgt] I l A2 cos2 colt ] +£2.42 0052 wgt l +2042 cos wlt cos thJ (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) and ' I 3 2 - l ‘ “’1 A sm wlt cos wlt ( 1 col/12 sin2 colt +w§wleA2 sin colt cos w2t ] +w§ezA2 sin2 wgt +wge2A2 sin wgt cos wgt l +2wlw26A2 sin wgt sin wlt] T l +w¥w26A2 sin wgt cos colt] : f 2 2 ’2 l f 2 _ _ l —w1A cos colt —w1A SlIl wltcoswlt —w%eA2 cos wlt cos wgt ] —wleA2 sin wlt cos wgt 2A2 , t t — A2 ' t t —ew1 cos all cos c112 wge 811101.12 cos 1121 1 —w%e2A2 cos2 wgt] l —e2w2A2 sin w2t cos wgt] J (4.18) At . . . . Multiplying both sides by —N— and approx1mat1ng the summations as integrals, we note that all of the trig identities can be solved/approximated. Substituting these values in the SVMD eigenvalue problem and simplifying, we obtain was + saw 2 0 A2 + 62142 (PA (4.19) 0 2 0 (M¥A2 + 110362.42) = ‘ 2 ‘1’ (w%A2 + wgezAz) 0 2 that further simplifies to 0 1 0 1 ND 2 (w%A2 + toge2A2) 0 (I) = (to? + (4)362) (I) (4-20) (A2 +€2A2) (1+62) The identified eigenvalues of this system are . 2 2 2 A2 2 M (4.21, (1 + 62) 66 We observe that if the noise perturbation amplitude is small, that is if e z 0, then the eigenvalues are approximately 1.12% and the SVMD correctly identifies the system frequency. However, if e is significant, the eigenvalues estimation suffers greatly. This would be the case where either the system may have high noise amplitudes, or the system may have many active/dominant modes such that the frequency amplitudes cannot be correlated (i.e A and A2 are independent, as opposed to the assumption made earlier in this analysis where frequency amplitudes were related). Addition of a third frequency perturbation to the original signal approximately (1.0% + (.0362 + wgez) (1 + e2 + 62) from the actual frequency. results in A2 = , that further deviates the SVMD identification This analysis is of a simple system that underscores the repercussions of having limited availability of the sensors and significance of low noise amplitude on the signals when using the SVMD for modal parameter identification. Next, we will present two examples to verify the analysis. 4.4.2 Numerical example In this example, we again simulate the numerical three degree-of—freedom mass-spring- damper example using modal damping (c = 0.01M) shown in section (2.3) with a small difference as explained below. The system uses N = 2000 data points, with time step At = 0.01, is corrupted with 8 bit quantization noise, and uses a differentiation step size of 32 (nd = 16). The response is constructed in the similar way shown in section 2.3 from y(t) = q = q0eat where q0 = (DTAy(0). While constructing the modal amplitudes q(t), we added a perturbation frequency w = 2623, with an amplitude of order 6 = 0.1 to the third frequency signal. This way the system has four frequencies with only three sensors. The estimated parameters are shown in table 4.5. 67 Table 4.5: Modal parameters estimation from reduced number of sensors compared against the theoretical eigenvalue problem for a three DOF discrete system with 8 bit quantization noise. Method System SVMD Q,- Freq. Damp. Freq. Damp. “’1 0.4209 0.0119 0.4208 0.0125 tog 1.0000 0.0050 0.9958 0.0050 023 1.6801 0.0030 1.7844 0.0054 Modal identification of the first two modes is very good. We see that the third mode is “off”. The approximation of Eq. (4.21) estimates the undamped system frequency at (.113 = 1.7049, that was not achieved due to presence of damping and noise in the system. (\Nhen we ran a simulation for undamped case, SVMD estimated (.123 = 1.7483 that is closer to the predicted value). It is clear from this simulation study that the decomposition method at best, is only as good as the number of sensors available to it. 4 .4 .3 fllfiee-free linear beam experiment In the works of references [86, 112], a linear steel beam was explored to gain insight into the POD method. We will use the same beam data to understand the observability issues surrounding the SVMD methodl. 3 uniform steel beam with In the experiment conducted, a 12.7 x 12.7 x 1500 mm free—free boundary conditions was prepared to experimentally compare its normal IIIOdes with the extracted POMS. The beam was horizontally hung with two soft Strings attached at the ceiling and the vibration responses in the horizontal direction Were measured under the impulse hammer impact excitation (at one free end) applied In the same direction. This way, at least eight beam modes were excited. The number of beam modes were controlled by adjusting the Nyquist frequency of measurement \ 1Special thanks to Dr. S. Han for providing experimental data for this study. 68 [112] and by using an anti-aliasing filter. To measure the response of the system, six equally spaced accelerometers were attached along the length of the beam. These accelerometers had varying sensitivities, a discussion of which can be seen in [86]. These also made the structure non-homogeneous since they weighed differently and carried significant mass as compared to the beam (the heaviest sensor was reported to be about 3% of the beam mass). The signals were then simultaneously fed into the B&K 2035 FF T signal analyzer (it uses 2048 sampling points) with 8-channel input module. The time record was used as 1 second, hence the sampling frequency was 2048 Hz. The acceleration time response histories and the power spectral densities of the six beam sensor responses can be seen in [112]. so 1 1 1 . 1 ' ' ‘ ‘ 0_ 1 g —50 — l —100 — I ‘1500 $0 260 300 400 500 600 700 800 900 1000 Frequency (Hz) Figure 4.16: The FFT of an acceleration time history channel of the experimental free-free beam. Eight active frequencies are visible within the Nyquist range. Even though the plot shown in [112], shows seven frequencies, there exists yet another frequency within the Nyquist range (= 1024 Hz) of frequencies that can be 69 observed by applying an F FT over one of the acceleration time histories (6th sensor chosen here but it is an arbitrary choice) as shown in the Fig. 4.16. 4.4.3.1 Identification by SVMD The SVMD is now applied on this free-free beam experiment. For this particular example, acceleration ensemble was integrated twice in SIMULINK to get the velocity and displacement ensembles (it resulted in a slightly better identification this way, instead of using the usual MATLAB numerical integration “cumtrapz” scheme). As noted in section 4.3.1, filter application is important and again, the high-pass filter frequency was selected at 14.5 Hz that is approximately half of the beams’ first natural frequency. Then the SVMD eigenvalue problem was formulated and the frequency estimation results computed that are shown in the Table 4.6. Table 4.6: Modal frequency estimation for an experimental free-free beam using a number of sensors that is less than. active frequencies compared against the theoret- ical undamped values. Method System SVMD fl,- Freq. Freq. wl 29.32 30.19 (.122 80.83 76.54 (.23 158.46 , 158.16 1.14 261.94 248.40 (.115 391.30 - of, 546.53 610.71 Parameter identification is reasonable up to three or may be four modes. Iden- t'ification for the third frequency seems very good but that is probably a random 1lappenstance. SVMD is not able to identify even the fifth and sixth modes (that Were expected if we are using 6 sensors, but since all of the 8 beam modes are aLCtive/dominant, the decomposition suffers in identification as discussed in section 70 0.5 ‘— N a: o a: 13 U 0 O 2 § —0.5 ‘05 1 2 3 4 5 6 1 2 3 4 5 6 0.5 0.5 (O V a) a) 8 Oi 8 0 2 2 —O.‘= % ‘ — .5 "1 2 3 4 5 6 0"1 2 3 4 5 6 Sensor Location Sensor Location Figure 4.17: Mode shapes obtained by the S VMD of the experimental free-free beam. 4.4.1). Mode shapes for the first four modes are shown in Fig. 4.17. Mode shapes do not reveal a smooth shape and are difficult to interpret if used alone. The mode shape for the third frequency also augments our suspicion that the accurate estimate for the third frequency is only random. Perhaps the only consolation for mode shapes is that POD (and SOD) also reveal the same modal content (not shown). SVMD modal c0- ordinates are also shown in Fig. 4.18 and appear to be giving meaningful information for perhaps the first two modes only (Zooming “in” into the time axis of the higher modes did not show any different results). Rather surprisingly, an FFT computation of the first state-variable modal coordinate is able to detect system frequencies very well as shown in Fig. 4.19 in which the first mode is about 70 st above the rest. In fact, these frequencies were precisely the same as identified in reference [112] as the experimental system frequencies. In [112], the authors reported that POD (on 71 SVMC 1 SVMC 2 2E 5 1E5 1 E-5 0 l . -1E-5 -2E 5 —1E 5 O 500 1000 1500 2000 O 500 1000 1500 2000 -5E-6 a: 0 500 1000 1500 2000 ‘F ”0 500 - 100 1 ‘ Time Samples 0 500 2000 Time Samples Figure 4.18: First four state-variable modal coordinates plot of the experimental free— free beam. 200 I I I I -250» - :=—300 - 3." E -350- ~ -400» - A :n 1 L l 1.5—w ”“U 200 400 600 800 1000 1200 Frequency (Hz) Figure 4.19: FFT of the first state-variable modal coordinate of the experimental free—free beam. 72 acceleration signals) was able to get better mode shapes on higher modes where the acceleration signals were more significant. 4.5 Summary We have experimentally evaluated the modal parameter identification of the structural systems using the state-variable modal decomposition method. Two experimental studies are conducted that incorporate identification for the two systems modeled as SDOF and MDOF system, respectively. The identified systems are then matched against the theoretical results. It is possible that our experiments have some deviation from standard assumptions such as linearity, time invariance and ideal boundary conditions, and as such identification can result in significant errors. The decomposition method showed good results for both SDOF and MDOF cases. For the MDOF case, a greater number of sensors was used than the active/ dominant frequencies present in the system and the system frequencies, damping ratios and mode shapes were obtained. We observed that data length can be slightly manipu- lated to either identify greater number of modes or increase accuracy in lower modal identification. The trend observed was that short-time data results in higher modes estimates whereas longer-time data is good for lower modes estimates. Some spurious modes may appear in parameter identification that by using the modal coordinates together with the modal vectors can be visually separated. The observations stated herein were gained by testing multiple experimental test beams. It was also observed that low frequency noise issue can be addressed by an appropriate filter selection. To highlight the limitations of the decomposition method, identification was per- formed on another experimental data from previous works. It was learnt that if number of active / dominant system modes is greater than the number of sensors avail- able, the SVMD method may not be able to accurately identify except maybe a 73 limited number of modal parameters. In that case, modal coordinates and modal vectors could still be used as a valuable tool for parameter identification at least in qualitative sense. 74 Quantitative fllssessment of Identification e have seen modal decomposition methods as a way of estimat- ing system parameters. We have also seen that modal coor— dinates obtained from the decomposition methods shed light on decomposition results in a qualitative manner. We will now focus on developing tools to provide further insight on the quality of identification results. Furthermore, in this chapter, by employing these new tools, we hope to explore some limitations and issues that the decomposition techniques may inherently possess. 5.1 Use of modal coordinates to evaluate decomposition re- sults lVlodal'coordinates are a useful measure to describe the quality of decomposition methods. Works of Feeny [85] and Chelidze [95] for instance. bear testimony to this statement. While modal coordinates, in general, could be very useful in qualitative examination of decomposition effectiveness. there, however lies a caveat in using this measure alone. Interpretation of pictorial presentations involves user‘s judgement that is quite 75 acceptable if the user is experienced and could easily distinguish between actual and spurious results. However, sometimes a “feel” may not be altogether conclusive es- pecially when the user is not fully experienced in correlating the results. In some extreme cases, this may potentially lead to erroneous conclusions. Therefore, there is a need of a quantitative tool applicable to modal coordinates to determine the quality of decomposition. This would be useful in practical situations, where interpretation of the results may be critical. Since our interest mostly is in manipulation of the modal coordinates that are inherently decoupled, we focus on system response of SDOF systems. Typically, we would introduce the frequency response function (F RF). The FRF of a system is approximated by taking the Fourier transform (FT) of the input and output of the impulse response of the system. If we assume that the impulse is nearly ideal, we can estimate the F RF from the response only to an unknown scaling constant. The well known form of a normalized FRF for an ideal mass—spring-damper system as a function of frequencies is 1 : 1— (w/wn)2 + i2Cw/wn Ha) (5.1) Note that, for experiments, the F RF would either be generated from taking the F FT of a modal coordinate that will be termed H, or by using estimated frequency and damping, in that case it will be termed as H0. We will now use this FRF to introduce two criteria that will be helpful in de- termination of the decomposition quality. Assuming that the decomposed modal coordinate behaves as a linear SDOF system, the features of its approximate FRF are to be compared to H (w) based on extracted system parameters can and C. 76 5.2 Quantitative assessment of modal coordinate FRF 5.2.1 Introduction We will propose two tools to quantify modal coordinates quality from the decoru- position. These are termed as the spectrum assurance criterion and the spectrum difference criterion and are explored below. 5.2.2 (1' He spectrum assurance criterion (.SfllC) The spectrum assurance criterion (SAC) bears similarities with the modal assurance criterion (MAC) [111]. While the MAC compares the relative consistency between mode shapes of target modal vector with the ideal modal vector by using inner prod- ucts, the SAC uses inner products to compare frequency response functions (FRFs). The FRF formed from the modal coordinates is matched against an ideal frequency response that is generated from the damping and frequency estimates obtained from the decomposition or by other means such as FFT. By using the FRFS, the spectrum assurance criterion (SAC) is defined as ____ mamas? lfllTlflllfilTlHal where |fl| represents the magnitude of reference frequency response function vector (5.2) _H_ that is obtained from the modal coordinate spectrum sampled in the to space, and I flgl represents the magnitude of constructed (actual or obtained) frequency response function vector & also sampled in the to space. In this sense, the vectors fl and 519 are obtained from Lflw) and fldw) and thus the elements in those vectors are function evaluations at sampled frequencies, i.e, H,- = Hi(w,-) and Hm- = Hanoi). The value lfllTlfil is an inner product between the magnitudes of the FRFS £12 and fi- 77 The SAC is a discrete version of the operation _ <|H(w)l, lam»? SAC ’ «Hon, |H(w)|),Ho(w)>2 ,H> SACIC = I (56) The ratio of inner product values obtained lies between 0 and 1, with 1 being the most matching value. 5.2.3 Spectrum inference (511)) Another measure in parallel to the SAC is the mean of difference between spectrum amplitudes. With this measure, in contrast to the SAC, values approaching zero would indicate that a true mode has been extracted with correct parameter estimates. For each frequency vector of length N that is twice the system bandwidth, the mean value of the difference between obtained and referenced FRFs is presented as (1 7V _ ) W? le-l —aIH0.-I} 78 where 5 represents the normalized area under the curve of the amplitude of the either le(w)Idw f—l. N . le0(w)|dw and H2 - Wi§1{lHll}' The values of SD range from 0 (perfect estimates) to 1 (no match). FRFS H or H0, obtained from a = 5.3 Numerical simulations For the purpose of demonstration, a few case studies on numerically simulated data will be shown first. These numerical examples address systems with combinations such as 1. Correct frequency and correct damping (with and without noise). 2. Correct frequency but incorrect damping. 3. Incorrect frequencies, but correct damping 4. Lightly damped systems vs. heavily damped (peaked vs. flat frequency re— sponse) The spectrum assurance and spectrum difference criteria are now applied to a simulation study that incorporates the cases (1-4) stated above. Two FRFs are gener- ated representing the idealized system and the actual or measured system respectively. For this study the natural frequency of the ideal system is chosen as col = 1 rad/sec and damping ratio is (1 = 0.08 unless specified otherwise. The frequency samples are equally spaced between a frequency range of 0.01 to 3.1 rad /sec resulting in 310 sample points. This frequency range corresponds to twice the bandwidth of the sys- tem (about three times the system frequency). This range was chosen to incorporate contributions of target system and residual effects from other modes, if any. The construction of the measured system (for instance FRFs estimated from modal coordinates) is similar. In the cases to follow, we will vary the parameters of the measured system and address the cases one by one. 79 Note that, in most simulation studies to follow, the range of parameter (frequency or damping) estimates in identification is considerably wider than those estimated in the actual physical experiments. This is done to gain an understanding of the applicability domain of these two criteria and such establish an “acceptable” level for these criteria. The identification in physical experiments is expected to be in a much smaller domain. 5.3.1 Correct frequency and damping estimates in fiat/i ffflfs, wit/i noise contamination in one of tlie 97675 The motivation of this set of tests is to evaluate the effect of noise in a system (this could be measurement, instrumentation or process noise). Results from this analysis will provide the information in a case where the decomposition method would have correctly identified values of frequency and damping ratio. However, the presence of noise in the system could distort the FRF estimated from the modal coordinate time history, and hence the SAC and SD could deteriorate. The complex random noise vector 17 is generated by the MATLAB function “rand” for the already established range of frequencies. The mean of this vector is adjusted to be close to zero with a range of :l:1 :l: 2'. The expected root mean squared (RMS) value of the amplitude of the noise signal a = ([273 which is very close to the root mean squared value obtained from our data. Next, we introduce noise coefficient values 29. The coefficient 19 values range from 0 (no noise) to 4 (high noise). The noise vector is scaled by 19, such that the total noise vector is 6 = i9 * n. The noise vector samples 52' are added to the samples of the frequency response Hi to produce a noise contaminated response function. The SAC and SD computations are thus computed between the noisy and the reference FRFS. Sample low and high noise signal values added to the FRFs are shown in Fig. 5.1 and Fig. 5.2 respectively. 80 FRF plot with noise coefficient value 1 I I I I I Amplitude —I N N 03 01 O 01 C.) l I I l I I _n O l 0 0.5 1 1.5 2 2.5 3 3.5 Frequency (rad/sec) Figure 5.1: The low noise FRF plot with 6 = 19 * 77 representing strength of uniform random noise signal amplitude. Here 19 value is 1. The dotted line represents FRF plot with no noise 1? = 0 and is shown for comparison purpose only. The normal and complex version of the SAC and SD criteria are evaluated by comparing ideal FRF with the noise added FRF and are shown in Fig. 5.3 and Fig. 5.4 respectively. The values shown for both criteria are very good for low noise cases. However, for increased values of noise, both criteria suffer though spectral difference seems to settle down at a fixed value for greater noise content. The SAC value is near 1 for low noise values in Fig. 5.3, while the SD value is near zero in Fig. 5.4. Also note that the complex version of the SAC seems more sensitive in revealing the modal coordinates than the normal SAC. The complex SAC and the SD performances are similar. From the results, we establish that even if the decomposition methods correctly identify modal parameters, the effect of noise in modal coordinates could adversely affect the confidence level achieved in identification. If noise level is kept low however, the identification results would be well synchronized with the quantification assess- 81 FRF plot with noise coefficient value 4 12 I 1 I I l I Amplitude o: 4- I I ’ \ 2— , \ v" ‘\ 0 1 l l T- ---I ------ ‘1 ------ 1- 0 0.5 1 15 2 2.5 3 Frequency Figure 5.2: The high noise FRF plot with 6 = 1) * 7] representing strength of uniform random noise signal amplitude. Here i) value is 4. The dotted line represents FRF plot with no noise 1? = 0 and is shown for comparison purpose only. ment tools. We also conclude that if a modal coordinate is dominated by noise, for example with a spurious mode, the SAC and SD will produce indicative values. 5.3.2 Correct frequency in 50th TR?» wit/i incorrect damping in the tested firm In this section and next, we will explore the cases where one of the parameters has been identified correctly while the other has not. Thus. keeping the reference values constant, we will vary the estimation values of the tested FRFs and evaluate the effect on the SAC and SD criteria for such a change. The results from this study will provide insight on sensitivity of the SAC and SD tools on these individual parameters. 82 Amplitude SAC against increased noise level 1 r I I I I T I 09* I 0.8 I 0.7 0.6 SAC 0.5 - 0.4 - 0.3~ 0.2 * 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Increased Noise Level —> 0.1 ' ‘ Figure 5.3: The SAC estimates for frequency identification for various levels of noise. The solid line (—) shows the amplitude SAC while dashed line (— -) presents the amplitude of complex version of the SAC. The horizontal axis indicates the value 1? representing the uniform random noise coefficient. When noise level in- creases, the estimates deteriorate significantly. The FRF for the measured system is generated by assuming that the decompo- sition method has identified the frequency content correctly but has identified the wrong damping ratios. This, in real life, would generally be the case for high damp- ing values (SVMD works well for low or moderately damped systems) or low damping with high percentage error. The damping estimates are presented as ratios of (2/(1 with C1 = 0.08 being the reference value. The error in estimates vary from —100% to +100% corresponding to ratios of 0 to 2 in the plots shown in Fig. 5.5 and Fig. 5.6 for SAC and spectrum difference estimates respectively. ' It can be inferred from the plots that both the SAC and spectrum difference seem 83 Amplitude SD against increased noise level 0.8 I I I T I I I 0.6 I once 3 0.5- ~ iff 1’ 0.4 _ T pectrum O to W 0.2- - 0.1 T y— l I 0 0.5 1 1 .5 2 2.5 3 3.5 4 Increased Noise Level —> Figure 5.4: The spectrum diflerence estimates for frequency identification for var- ious levels of noise. The horizontal axis indicates the value 1? representing the uniform random noise coefficient. When noise level increases, the estimates dete- riorate. to indicate a close match if the frequency is identified correctly, and have low sensi- tivity to damping errors, unless the damping is greatly under estimated. Nonetheless, both SAC and spectrum difference seem to indicate a close match if the frequency is identified correctly, and have low sensitivity to damping errors. The complex SAC has a similar effect. 5.3.3 Correct damping in hath fflfs, with incorrect frequency in the tested 970' In general, many physical systems have low damping ratios, estimates of which may not always be of interest, so often the frequency estimates take the center stage. This case is similar to the previous case with the difference that now modal decomposi- tion method has correctly identified damping estimates but errs in estimating the frequency estimates. We will explore the SAC and SD tools to observe their efficacy 84 SAC vs C2I§1 with correctly identified 0 SAC O i 1 i 0 0.5 1 1.5 2 CZIC'H Figure 5.5: The SAC values for various estimates of identified damping ratios. The solid line (— ) shows the amplitude SAC while dashed line {- -) presents the ampli- tude of complex version of SAC. The damping estimate error ratio C2/C1 ranges from -100 % at 0. 001 to +100 (70 at 2 with (1 being 0. 08 at 1. in such condition. In the plots of Fig. 5.7 and Fig. 5.8, the SAC and SD values are pictorially pre- sented for various estimates of frequency values where the horizontal axis is non- dimensional frequency ratio of estimated over actual value. The SAC plot Fig. 5.7 that resembles a resonance curve shows the SAC value drop to z 0.5 when the fre- quency estimate is “off” by 25 ‘70. The spectrum difference value in Fig. 5.8 for the 25% difference in estimate is close to 0.35 corresponding to 35% error in SD value. The SD values decrease slightly as the frequency ratio increases near the frequency ratio 3, since the estimated frequency approaches the cutoff frequency at those values. This means for the SD computation in this range, the actual FRF vector |ff_o| is only partially used by the SD scheme. 85 (ll {1} it’ll 1 4Spectrum difference vs C2/C1 with correctly identified (2 _L N I 1 Spectrum difference Figure 5.6: The spectrum difference values for various estimates of identified damp- ing ratios. The damping estimate error ratio (2/(1 ranges from -100 ‘70 at 0. 001 to +100 % at 2 with (1 being 0. 08 at 1. From this study, we see that both the SAC and SD produce indicative values if only damping is identify correctly except for a small range of frequency values. Thus, only if frequency estimate errors are within 15 ‘70, we could expect to see reasonable SAC and SD values. 5.3.4 Lightly damped vs. heavily damped systems In this section, we want to study the behavior of the SAC and SD criteria in the low and high damped cases. The system frequency is assumed to be correctly identified by the decomposition method. We would like to see whether these methods are sensitive to reference damping ratios. Here, we vary the damping ratio to observe SAC and SD values for both flat and peak responses. Two cases of light to high damping are considered. For lightly damped case, the value of (1 was set to 0.02. For the highly damped case, the value of (1 was set to 0.2. For the light damping case, the plots of both SAC and SD are 86 SAC vs (oz/(o1 with correctly identified l; 1 . I I . . 0.8- : ‘. i I I I I 0.6 *’ l ‘| " O ' | < ' | (D l ‘ 0.4 ' " \ l \ r ‘\ 0.2 - ' s . ’ s I s ‘ I ~ ~ - - - - 0 _ — ' l l l l '1'- ' ' ' .- 0 0.5 1 1.5 2 2.5 3 (oz/e)1 Figure 5.7: The SAC values for various estimates of identified frequency values. The solid line (—) shows the amplitude SAC while dashed line (— -) presents the am- plitude of the complex version of SAC. The frequency estimate error ratio Log /w1 ranges from —100 ‘70 at 0.0 to +200 % at 3. presented in Fig. 5.9 and Fig. 5.10 respectively. For the high damping case, the plots of both SAC and SD are presented in Fig. 5.11 and Fig. 5.12 respectively. These plots show that if the frequency content is identified correctly, erroneous damping values will not bear a large significance over both SAC and the SD criteria. Furthermore, the performance of both the SAC and SD is nearly independent of the level of damping in an underdamped reference system. This is fruitful since most of structural system have low damping values and hence for these quantification tools, the frequency estimates carry more importance. 87 Spectrum difference vs (oz/(o1 with correctly identified C 1 I I I I I Spectrum difference 0 0.5 1 1.5 2 2.5 3 Lag/W1 Figure 5.8: The spectrum diflerence values for various estimates of identified fre- quency values. The frequency estimate error ratio (.02 /w1 ranges from -100 % at 0.0 to +200 % at 5’. 5.3.5 Summary of trends We have now explored the SAC and SD criteria for various test cases. Both criteria require generation of the FRFs from system parameters. Based on these numerical studies, we see that these tools could be used for quantitative assessment of identifi- cation results. From the numerical studies presented above, we gather the following information. 1. The complex SAC appears to be more sensitive than the normal SAC amplitude in both frequency and damping estimations and is similar to the SD. 2. The noise and frequency errors are well expressed using both SD and SAC criteria. 88 SAC with correctly identified 9 for light damping SAC 0 0.5 czic, Figure 5.9: The amplitude SAC values for various estimates of identified damping values. The solid line (—) shows the amplitude SAC while dashed line (- -) presents the amplitude of the complex version of SA C. The reference system damp- ing ratio is chosen as (1 =0. 02 0850 damping comparison with correctly identified (2 with light damping . l’ _ 804 C i E0.3- ~ 3 502 8 . 8. ”’01” / 0 I I J 0 0.5 1.5 2 Cle, Figure 5.10: The spectrum diflerence values for various estimates of identified damp- ing ratios. The reference system damping ratio is chosen as (1 =0. 02 89 SAC damping comparison with correctly identified (2 for high damping ‘ I W 1 — 1.5 2 as. Figure 5.11: The SAC values for various estimates of identified damping values. The solid line (——) shows the amplitude SAC while dashed line (— -) presents the amplitude of the complex version of SAC. The reference system damping ratio is chosen as (1 =02 3. Both the SAC and SD are less sensitive to damping errors than other types of errors in underdamped systems. 5.4 Experimental data Next, we will use the established criteria on actual test data and detail an example to distinguish spurious modes from actual modes. The fixed-free steel beam experiment conducted in chapter 4 is used here to eval- uate the efficacy of the SAC and SD criterion for real-life experiments. The reference and actual FRFs are generated as follows. The reference FRF is generated from the estimates of system parameters obtained by the decomposition method. Simi- larly, for comparison another set of reference FRFs is generated from the first five frequency estimates using the FFT of the acceleration time signal. Since we do not have the damping estimates from the F FT, the damping estimates from decomposi- tion method are used instead in generating these FRFs. The actual FRF is obtained 90 SD damping comparison with correctly identified Q with high damping 0.7 I T I | I I I I I Spectrum difference .0 .0 .O .0 .0 N 00 -h 01 O) . I l l .0 —L 0 l L l J l l l l 0 0.2 0.4 0.6 0.8 1/ 1.2 1.4 1.6 1.8 2 z2 C1 Figure 5.12: The spectrum difference (SD) values for various estimates of identified damping ratios. The reference system damping ratio is chosen as (1 :02 from the modal coordinate. From the modal coordinate time history, we select the part that starts at the first upward zero crossing. This corresponds to a zero displace- ment and a positive velocity initial condition. This in turn could be associated with an equivalent impulse response with zero initial conditions proportional to an ideal impulse response, whence we obtain the approximate F RF. Two data sets are used for the SAC and SD comparisons. The first data set is comprised of approximately 12,000 points and the second one has only 1000 data points. As observed in chapter 4, longer sampling improves quality in lower mode estimates whereas shorter sampling improves the higher modes. 91 5.4.1 Large sampling 9512, 000 In this case, the SVMD was clearly able to estimate system parameters of the first two modes. However, it is conceivable that in experiments, some of the spurious modal frequencies could be “close enough” to be mistaken for an actual mode. In this case, for both modes, some of the identified frequency and damping pairs are shown in Table 5.1. The first estimate corresponding to each mode is real while the others are spurious. Table 5.1: Estimated modal parameters from the decomposition method for a linear beam free response for first N = 12,000 sample values. Mode FFT Frequency Damping Frequency 1 2.74 0.0033 2.7896 - 0.1590 3.0831 - 0.0263 2.2690 17.20 0.0035 17.174 - 0.0113 18.406 [\DMh—‘H For each mode, the reference and actual F RF s were generated in the way described above. By computing the SAC and SD values between FRFs from each modal coordi- nate history, and each set of estimated modal parameters, we have two n x n matrices of complex amplitude SAC and SD values with n representing the total number of modes. These are plotted in Figures 5.13 and 5.14. The SD values are subtracted from 1 and thus plotted as r .= 1— SD to aid the visualization by representing positive outcomes with tall bars. The SAC values seem to point to pairs (2,2) estimated by SVMD as the second modal parameter; and (10,10), (12,12), (13,13) and (14,14) estimated by SVMD as the first modal parameter. The SD values seem to point to pairs (2,2) estimated by SVMD as the second modal parameter; and (10,10), (12,13), and (13,13) estimated by SVMD as the first modal parameter. In short, the common candidates are (2.2) for second mode and either (10,10) or (13,13) for the first mode. 92 0.5 ; SAC values 00 SVMD FRF 10 Figure 5.13: The amplitudes of complex SAC values computed for each FRF gener— ated from the S VMD estimates compared against F RF 3 from modal coordinates. 0.5 : " 1-SD value r: 5 10 SVMD FRF 15 7 2o 1 3 Modal Coordinate FRF Figure 5.14: The r = 1- SD values computed for each FRF generated from the S VMD estimates compared against FRFs from modal coordinates. For ease of visualization, values of SD subtracted from 1 are plotted instead. 93 We also generated reference FRFS from the frequency estimates of FFT of the acceleration ensemble and these are shown in Figures 5.15 and 5.16. The F FT based FRFs were ordered to match the SVMD based F RFS to ease the visual comparison. The F FT generated SAC values seem to point to (2,2) identified by FFT as first modal parameter; (13,13) identified by FFT as first modal parameter. The FFT generated SD values show the same trend. 1, 0.8 '- w . g 0.6-. > 2 0.4,. (D , 0.2; ; 0.4 0); o 0.3 15 . 0.2 0.1 5 . U0 15 1 3 Modal Coordinate FRF Figure 5.15: The amplitudes of complex SAC values computed for each FRF gen- erated from the FF T estimates compared against FRFs from modal coordinates. The estimates are ordered to match sequence by S VMD for ease of comparison. It is clear from the above plots that the modal pair of (13,13) that corresponds to the first mode as estimated by SVMD along with the modal pair (2,2) that corresponds to the second mode are the true system modes. This can be seen from all four plots where these values are maximum. Note that, these indicated modes from (2,2) and (13,13) agree with the qualitative results obtained via modal coordinates in section 94 1-SD value r 5 Modal Coordinate FRF Figure 5.16: The r = 1- SD values computed for each FRF generated from the FFT estimates compared against FRFs from modal coordinates. The estimates are ordered to match the sequence of SVMD. For ease of visualization, values of SD subtracted from 1 are plotted instead. 4.3.2. 5.4 .2 Smaller sampling 5M=1000 We turn our attention to a smaller sampling size which thereby results in the esti- mation of higher modes by the SVMD. The same procedure as described above is repeated to generate FRFs from modal coordinates which are then matched against FRFs from SVMD and FFT estimates respectively. The only difference is that for the F FT generated FRFs, only the first five estimates are used since SVMD was able to estimate the first five modes. The complex amplitude SAC comparison of SVMD estimates with modal coordinates is presented in Fig. 5.17. The SAC values seem to point to pairs (1,1) estimated by SVMD as the fifth modal parameter; (2,2) estimated by SVMD as the fourth modal parameter; (5,5) estimated by SVMD as the third modal parameter; (11,10), (11,11), (11,12), and 95 U) ..... G) 2 g 0.5 2 w 0.4 o 0 ;:0.2 5 15 ' i SVMD FRF 10 is U o 5 15 1 3 Modal Coordinate FRF Figure 5.17: The amplitude of complex SAC values computed for each FRF gener- ated from the S VMD estimates compared against F RFs from modal coordinates. (12,13) estimated by SVMD as the second modal parameter; and (10,14,), (11,14), (12,14), (13,14), (14,15), (15,15), and (15,16) estimated by SVMD as the first modal parameter. Similarly, the SD values are computed from SVMD and modal coordinates gener- ated FRFs. Again to aid visualization, values are subtracted from 1 and then shown in Fig. 5.18. The SD values seem to point to pairs (1,1) estimated by SVMD as fifth modal parameter; (2,2) estimated by SVMD as fourth modal parameter; (5,5), and (7,5) estimated by SVMD as third modal parameter; (11,11) estimated by SVMD as the second modal parameter; and (14,15), and (1515) estimated by SVMD as the first modal parameter. For the FRFs generated via FFT, the complex SAC and SD values are calculated with SAC values shown in Fig. 5.19 and SD subtracted from 1 are shown in Fig. 5.20. 96 (n 3 ..... ‘>“ o 0.5 0) J- 0.4 II “ 0.3 0.2 " 15 SVMD FRF 15 7 Ll 20 1 3 Modal Coordinate FRF Figure 5.18: The r=1-SD values computed for each FRF generated from the SVMD estimates compared against FRFs from modal coordinates. The estimates are not ordered in any particular sequence. For ease of visualization, values of SD subtracted from 1 are plotted instead. The SAC values here seem to point to pairs (1,1) estimated by FFT as the fifth modal parameter; (2,2) estimated by FFT as the fourth modal parameter; (5,5) estimated by FFT as the third modal parameter; (11,11), and (11,14) estimated by FFT as the second modal parameter; and (15,14), (15,15), and (15,16) estimated by FFT as the first modal parameter. The SD values in this case seem to point to the pairs (1,1) estimated by F FT as the fifth modal parameter; (2,2) estimated by FFT as the fourth modal parameter; (5,5) estimated by FFT as the third modal parameter; ( 11,11) estimated by FFT as the second modal parameter; (15,15) or may be (15,16) estimated by FFT as the first modal parameter. If we combine all the results, these indicate that the pairs (1,1), (2,2), (5,5), (11,11) and (15,15) are the most common candidates for real modes. Note that the indicated 97 SAC Values 15 ’ 3 5 0 1 Modal Coordinates FRF Figure 5.19: The amplitude of complex SAC values computed for each FRF gener— ated from the FFT estimates compared against FRFs from modal coordinates. The estimates are ordered to match sequence by S VMD for ease of comparison. 0.5 r = 1- SD Values FFT FRF 1o 5 o 20 1 3 Modal Coordinates FRF Figure 5.20: The r=1-SD values computed for each FRF generated from the FFT estimates compared against FRFs from modal coordinates. The estimates are ordered to match the sequence of S VMD. For ease of visualization, values of SD subtracted from 1 are plotted instead. 98 modes from (1,1), (5,5), (11,11) and (15,15) are in complete agreement with the “nice looking” plots visualized in section 4.3.2. Also note that the mode from (2,2) augments the “borderline” mode candidate described in section 4.3.3 and confirms its status as a real mode. 5 .4 .3 Comments and summary The SAC and SD criteria are evaluated for an experimental data set. We found both tools to be quite effective in evaluating the identification results. Both utilize FRFs to generate values that help assess the quality of an extracted mode. This quantitative measure augments the qualitative arguments presented in section 4.3.3. We noticed that the SAC can produce excess candidates but perhaps at the same time more clear candidates. We also see that using SAC together with SD is almost always better and hence recommended. We summarize the observations and findings as follows. 1. The FFT frequency spikes are probably more accurate than the SVMD frequen- cies. 2. The SVMD damping values have to be used for both kinds of F RF generations since the damping values were not obtained from the FFT generated FRFS. This may not adversely affect the purpose of evaluating modal coordinates since the damping errors have a weak effect on both SAC and SD values as shown in section 5.3.2. 3. It is likely that a spurious mode will have a bad modal coordinate and that this will produce a complex amplitude SAC far away from 1 or a SD value far from 0. 99 5.5 Time record length We observed from the experimental data that for short time histories, higher modes were extracted, and for longer time histories, lower modes were extracted. We hy- pothesize that if the data record is too short, low frequency modes may not be fully represented and therefore not accurately extracted, and that if the data record is long, quickly damped modes will become dominated by noise for much of the signal, and will not be accurately extracted. We test these ideas, next, by looking at the effect of time record on modal identifications in a simulation study. This section involves a numerical study on a three-degree—of-freedom system with added noise. We will use the numerical example presented in section 2.3. The system parameters are reproduced here which are 2 0 2 —1 0 M: 0 1 0 ,K= —1 2 —1 (5.7) 0 1 0 —1 1 with the initial conditions x(0) = (1,0,0)T, and v(0) = (0,0,0)T. The undamped natural frequencies of the system are wl = 0.4209, v.12 = 1.000, and £123 = 1.6801. We are interested in exploring a couple of options. We want to study the effect of low and moderately high noise as well as the effect of high and low damping values in this time length study. In this study, the frequency and damping estimates are associated with the same eigenvalue, so an error in estimate of one may effect error in the other. The cases that cover the damping and noise ranges are described as follows. The system damping is chosen to be proportional to mass C 2 CM. We will only test the underdamped systems for this study. Since the damping is proportional to the mass matrix, all modes have the same settling time, T3. To have a constant modal settling time was primarily intended for the ease of analysis. While other system damping values could certainly be explored, in this study we wanted to exclude 100 any possible results misleading us away from the actual subject of interest. Those explorations may be subject of a future study. By choosing two values c = 0.5 and c = 0.01, the range of damping ratios lays between 0.003 (low) to 0.59 (high). The individual damping ratios for each mode will be presented the results section. In accordance with section 2.4, quantization noise was added to the system. The two noise levels were selected to be 6 bits (high) and 10 bits (low). The random noise ensemble was added to the displacement matrix X. We then took numerical derivatives of the new X to build V and subsequently our ensemble matrices. The derivative step size was selected as "d = 16 and A t = 0.01 for all the studies detailed herein. The time period of the lowest mode was equal to 14.93 seconds or 1493 data points, similarly the second lowest mode time period was approximated at 6.28 seconds (628 data points) and the highest at 3.74 seconds (374 data points). The simulation was run for the data record sweeping from one tenth of each mode time period to ten times the modal period. From the decomposition, identification results were computed and stored for each run. Then the difference between identified and actual system parameters was calculated. These error results are plotted for frequency and damping differences against the stated dimensionless time record length normalized for each modal period. 5.5.1 Results 5.5.1.1 High damping For the case where c = 0.5, the system damping ratios are (1 = 0.594, (2 =2 0.25, and C3 = 0.1488. These ratios represent the relatively high values as would be seen in some physical experiments. In this case, the 2 % settling time is T3 = 16 seconds, which is about 1.07, 2.54, and 4.27 modal periods for the first, second, and third 101 I. .' . modes, respectively. First, we employed the 10 bit or low level quantization noise. The error results in both damping and frequency estimates are shown in Fig. 5.21 and Fig. 5.22 respec- tively. 20 I I I I 10 _ Modal Period = 14.93 sec. Settling time = 16 sec ._ o ' ' ‘ ' 9 0 2 4 6 8 10 5 20 l l l l >. 0 Modal Period = 6.28 sec, Settling time = 16 sec ac) 10 1 n c- 0 - - - . a) ”I: 0 2 4 6 8 10 20 . 1 . . 1O Modal Period = 3.74 sec, Settling time = 16 sec L ‘ 0 V'— | 1 J 1 0 2 4 6 8 10 Time record normalized by modal period Figure 5.21: The percent error in estimation of frequency parameters against var- ious lengths of data for a 3 DOF underdamped system. Each. time record is normalized by its modal period. The quantization noise level is 10 bits. Next, we increased the quantization noise to 6 bits and reran the simulation. The error results are shown in Fig. 5.23 and Fig. 5.24. From the plots, a consistent trend can be observed. With a small time record, for all modes the error is huge in both frequency and damping estimates. As the data lengths approach roughly a modal period, the error values are greatly reduced. For the higher—noise case, the error values are minimum at about the modal settling time. In such case, as the time record continues to increase, the noise begins to dominate the response, and hence the correlation signals, and the results start to diverge. Since all modes have the same settling time, the lower modes settle more quickly 102 l l T l Modal Period = 14.93 sec, Settling time = 16 sec _ N O 10 — h 0 l I 1 1 2 0 2 4 6 8 10 a 20 r m I 1 g 10 Modal Period = 6.28 sec, Settling time = 16 sec A '5. A E 0 . 1 . . 3 o 2 4 6 8 1o 20 . . . . 10 _v Modal Period = 3.74 sec, Settling time = 16 sec _ 0 1 . 0 2 8 10 4 6 Time record normalized by modal period Figure 5.22: The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is nor- malized by its modal period. The quantization noise level is 10 bits. peak to peak (higher C), or in other words more quickly with respect to their modal- period normalized time variables. Thus with the modal period normalized time plots, we expect the lower modes to diverge more quickly, as observed in the plots. 5.5.1.2 Low damping For the case where c = 0.01, the system damping ratios are (1 = 0.0119, (2 = 0.005, and C3 = 0.003. These ratios represent the relatively low damping values as would be seen in physical experiments. Here again, all modes have the same settling time, T3. In this case, the 2 ‘76 settling time is T3 = 800 seconds, which is about 53.5, 127.4, and 213.9 modal periods for the first, second, and third modes, respectively. In the Figs. 5.25 and 5.26, 6 bit quantization noise was employed and error values DIOtted. In Figs. 5.27 and 5.28, 10 bit quantization noise was used and the identifica— tion error results plotted. In low damping cases, the results seem to imply that as long as the data length is 103 Modal Period = 14.93 sec, Settling time = 16 sec 1 l l l 0 2 4 6 8 10 20 r T 1 1 Modal Period = 6.28 sec, Settling time = 16 sec _ Frequency error a 0 | l 0 2 4 6 8 10 20 l l T I 10 _ Modal Period = 3.74 sec, Settling time = 16 sec _ o ‘ ‘ ‘ ' 0 2 4 6 8 10 Time record normalized by modal period Figure 5.23: The percent error in estimation of frequency parameters against var- ious lengths of data for a 3 DOF underdamped system. Each. time record is normalized by its modal period. The maximum error appears in the first mode. The quantization noise level is 6 bits. less than the modal settling time and that about two modal period oscillations have been captured, errors associated with parameter estimation would be negligible. 5 .5 -2 Summary of trends We evaluated two cases that incorporate moderately “high” and low damping ratios, Via a. numerical example. The effect of inclusion of 10 bit and 6 bit quantization noise values on the signals was studied. ‘Time record was normalized by the modal period in each case. We summarize the observations and findings as below. ° Two important features are modal time period and modal settling time. When time record is short, that is a small multiple or fraction of the modal time 104 20 l l I 10 ‘ Modal Period = 14.93 sec, Settling time = 16 sec ‘ 0 l l l l 0 2 4 6 8 10 ‘9' 2o . . . 1 5 g) 10 Modal Period = 6.28 sec. Settling time = 16 sec _ 5. I E '5 8 0 it s 0 2 4 6 8 10 20 l l l I 10 ii! Modal Period = 3.74 sec, Settling time = 16 sec 0 tfiV’r L I 1 0 2 4 6 8 10 Time record normalized by modal period Figure 5.24: The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is nor— malized by its modal period and the marimum error appears in the first mode. The quantization. noise level employed is 6 bits. period, estimates are expected to be erroneous. Likewise, when time period is long, that is greater than modal settling time, it also results in estimation errors which are attributed to noise dominance in the correlation signals. 0 For estimating frequencies, we might expect to have at least one modal period, in low—noise simulations we saw good results at about two periods. 0 For damping estimates, log decrement might lead us to think that at least a couple oscillation peaks are required, but our low noise simulations indicate that the data between peaks holds the information in it, too. . Based on simulations, the “optimal” time record length seems between modal 105 20 I a I . 10 _ Modal Period = 14.93 sec, Settling time = 800 sec _ ._ 0 J ' ‘ ' 9 0 2 4 6 8 10 a 20 I I I I > 8 1o — Modal Period = 6.28 sec, Settling time = 800 sec — S s o - ‘ ' ‘ L: 0 2 4 6 8 10 20 l I I r 10 _ Modal Period = 3.74 sec, Settling time = 800 sec _ 0 T I l 0 2 4 6 8 10 Time record normalized bv modal period Figure 5.25: The percent error in estimation of frequency parameters against var- ious lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 6 bits. period and modal settling time, a window where the estimation results have minimum errors. 5 .6 Summary We have evaluated the limitations of the identification quality of the decomposition method in this chapter. We have quantitatively examined the modal coordinates that determine the decomposition quality. Two criteria were developed that help assess the quality of decomposition using frequency response functions. Various cases that emulate non-ideal identification circumstances were numerically studied using the pr Oposed criteria. Furthermore, experimental data were also analyzed using these methods. The criteria were applied on FRFS generated by decomposition methods and Compared with FRFs generated from the FFT identification. The spectrum assurance and the spectrum difference criteria used together prove 106 20 F I j I Modal Period = 14.93 sec, Settling time = 800 sec .3 O I O l‘ l- N 0 fi Modal Period = 6.28 sec, Settling time = 800 sec _ Damping error 3 0 1 l L I 0 2 4 6 8 10 20 . . . s 10 _ Modal Period = 3.74 sec, Settling time = 800 sec _ 00 2 4 6 8 10 Time record normalized by modal period Figure 5.26: The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is nor- malized by its modal period. The quantization noise level employed is 6 bits. to be very useful in evaluating the identification results. These criteria quantify an actual mode when spurious results appear in identification. From both numerical and experimental case studies, we see that these criteria are robust even in presence of non—ideal damping identification. Lastly, we addressed the effect of data length on the identification. This was studied via a numerical simulation. It was observed that identification results are best in a time record window associated with the modal time period and modal settling time. 107 2 ' I I I I Modal Period = 14.93 sec, Settling time = 800 sec N Modal Period = 6.28 sec, Settling time = 800 sec Frequency error 00 2 4 6 8 10 5 I I I I l Modal Period = 3.74 sec, Settling time = 800 sec 00 2 4 6 8 10 Time record normalized by modal period Figure 5.27: The percent error in estimation of frequency parameters against var- ious lengths of data for a 3 DOF underdamped system. Each time record is normalized by its modal period. The quantization noise level employed is 10 bits. 108 I l l I Modal Period = 14.93 sec, Settling time = 800 sec CD —3 N I O N .b O) m _.\ O Modal Period = 6.28 sec, Settling time = 800 sec # 1 2 4 6 8 10 I I I N Modal Period = 3.74 sec. Settling time = 800 sec .A 2 4 6 8 10 Time record normalized by modal period 0 Damping error 0 —I N °E °§ Figure 5.28: The percent error in estimation of damping values against various lengths of data for a 3 DOF underdamped system. Each time record is nor- malized by its modal period. The quantization noise level employed is 10 bits. 109 Comparison with Other Identification Methods 6.1 Introduction n this chapter, comparison between various decomposition methods and selected time domain modal identification techniques is discussed. The list of time domain techniques for modal identification of linear systems is non-exhaustive and is probably well beyond the scope of this work. Consequently, the comparison made here presents a rather small but representative portion of the research in the area of system characterization and identification of structural systems. For a list of other comparison methods see for example the works of [43, 48—-50, 99]. In this work, we would like to compare methods that are either most popular, accurate or practical. Specifically. the comparison of the state—variable modal decom- position (SVMD), the proper orthogonal decomposition (POD), and the smooth or- thogonal decomposition (SOD) is being made with the Ibrahim time domain method (ITD) [27, 37, 40] and the eigensystem realization algorithm (ERA) [31, 44, 45]. These are the commonly applied methods in the output-only area of modal identification. For most cases, the identified system modes will also be matched against the POD generated modes (POMS) for verification of results. This would serve dual purpose as the POD comparison would also be employed for tying up the gap as mentioned 110 earlier in section 1.4. We begin by summarizing the derivations of the ITD and the ERA. The interested reader is referred to the original works for detailed discussions. Here, the idea is to highlight the basis of formulating the eigenvalue problem. Next, we conduct a numerical simulation for a three-degrees-of-freedom (DOF) problem and then another simulation study for a sixteen DOF system. Both systems are then also subjected to quantization noise [91]. For the former case, extracted modes are easily verified by direct comparison with the linear normal modes of the system, whereas for the latter system, identified modes are compared to the linear normal modes via the modal assurance criterion [111]. Next, all of the identification schemes are compared in the experimental clamped-free beam studied in section (4.3). Comparisons are made between methods for frequency and damping estimations. Mode shapes corresponding to respective modal frequencies are also matched and plotted for most schemes. We conclude the chapter by some observations obtained through the experiments. 6.2 Time-domain modal identification methods 6.2.1 Ihrahim time domain method One of the earliest time domain methods for modal analysis, the Ibrahim time domain (ITD) method was introduced in 1973 [27], with subsequent improvements following in later years [37,40]. ITD is a single-input-multiple—output (SIMO) method, generally assuming the solution of an n DOF vibration system Eq. (2.1) to be of the form 271 air) = Z ¢,ke3kt,i=1,...,n (6.1) 1:21 where ¢k is a 2n x 1 vector that forms the normal mode shape and sk = -Cwn 2]: wdj provides the system frequencies and damping ratios for kth element. 111 Expanding Eq. (6.1) for n outputs obtained through n sensors with a sampling time record equaling QriAt, with At being the sampling time step, we get " 1 171(1‘0) 17101) 131(f2n.-1) .1: t 17(i .‘r‘(t _) 2t 0) 2 1) . 2 2n 1 = (6.2) . 177100) 177101) Ini‘2n—1) i P $11 $12 . .. $12” " '- eSltO eSltl .. . esthTI—l -] am an egg" e82’0 e82t1 e32‘2n—1 (1,5711 an2 $712” _ L €52nt0 €32nt1 €32nt2n—l - In short form X = ¢I>eSt (6.3) Here, X is a n x 2n matrix of free response measurements from the structure and is known, (I) is an n x 2n matrix of unknown eigenvectors; and eSt is a matrix containing unknown complex eigenvalues and known response measurement times. ITD requires the time shifting of the displacement ensemble by one sampling step (sampling time steps can be chosen), for example t1 = to + At, to generate a new ensemble, termed the shifted matrix, as 131(1‘1) 431(t2) 171(t2n.) x(t) .r(t) In 2.1 2 2 . 2 2n) 2 (64) . 1'n(t1) $rz.(t2) 3372(t2n) . - $11 $12 (1512.” - - eslto esltl €81t2n“‘1 1 4521 0522 c522n A 652’“ 65‘2“ 882%‘1 _ Q5711 $712 @712” L €32nt0 €52ntl gs2nt2n—l _ or in short from as 112 ““1 I [I A x1, = AeSt (6.5) where 'eSIAt 0 0 0 (2.32“ 0 A: , _ O 0 es2nAt4 The shifted matrix has the same dimensions as X and the same is true for the rest of the matrices in Eq. (6.5) in comparison with Eq. (6.3). Another time shift, this time on X17, would generate the second time shift matrix as x2, = A2eSt (6.6) Stacking Eqs. (6.3)-(6.6) together, we form ] XX ] = ] (EA ] eSt or X = ‘I'est (6.7) IT and l :17- ] = [ £162 ] eSt or X = ‘IIAeSt (6.8) 27' The Eqs. (6.7) and (6.8) are restructured to form the eigenvalue problem as AW = \IIA where A = TOT—1 (6.9) Solution of the eigenvalue problem of Eq. (6.9) yields the system modal parame- ters. System frequencies can be obtained by taking the logarithm of the eigenvalues obtained in A scaled by the sampling time, whereas the mode shapes are obtained from the columns of \II. In essence, ITD requires the user to obtain displacement ensemble for a MDOF system and from that generate the two ensembles X and X‘1 . These ensembles 113 are then structured into the eigenvalue problem of Eq. (6.9), solution of which yields modal parameters. 6.2.2 Eigensystem realization agorithm Deemed as one of the most accurate and practical methods [127] for modal parame- ter estimation, the eigensystem realization algorithm (ERA) was developed in 1985 [31] and then improved in following years [44, 128—131]. ERA is a multi-input-multi— output (MIMO) method that requires constituting a Hankel matrix which contains the Markov parameters. This method utilizes singular value decomposition to extend the Ho—Kalman algorithm [132] for realizing a state-variable system model. After obtaining the system model, the model is then transformed to the modal space for modal parameter estimation. Even though the realized system matrix Ac is enough for frequencies and damping estimates, an additional step is required for the mode shape estimation. Following are the salient steps of the ERA method. Consider the state-variable equations of a finite dimensional, discrete time, linear, time invariant dynamical system as x(k + 1) = Ad x(k) + Bd u(k) (6.10) ya) = Cdx(k’) (6.11) where x is an n-dimensional state vector, u is an m-dimensional control input, and y is a p-dimensional output or measurement vector and It indicates discrete samples. The matrices Ad, Bd, Cd are the state, the input and the output matrices respectively, of the discrete model. These are different from the ones shown in section 2.2. From the free response, functions that provide the system time domain description are termed Markov parameters, given by 114 Y(k) = chg—le (6.12) The ERA method begins by forming a generalized block Hankel matrix of order r x s, where r and s are some “optimally” chosen numbers [99], as Y(A:) Y(k+1) Y(k+s) Y(k+1) Y(k+2) Y(k+s+1) H(k — 1) = : _. -. ; (6.13) LY(k+r) Y(k+r+1) Y(j+s+r)‘ The singular value decomposition on the matrix H(O) yields H(O) -_- UZJVT (6.14) Keeping the “desired” singular values (usually 2n [99]) and truncating the matrices U and V to keep the same first dimensions, the realization triple is obtained as A = 2—1/2UTH(1)V2—1/2, B = Zl/ZVTEm, c = EgUZI/Z (6.15) where E; = [171, 0, . . . ,0] , Egg; : [17nd 0, ...0] Modal frequencies and modal damping ratios can then be obtained from real and imaginary parts of eigenvalues of AC after the transition from discrete to continuous system as Ac 2 [111(A) :t 27ri]/chr,i = \/——1 (6.16) with AT being the sampling interval. To obtain the mode shapes, the following transformation is used = Eguzmr (6.17) Where F is the matrix of eigenvectors of the identified system matrix A. 115 6.2.2.1 A note on obtaining Markov parameters in ERA The ERA requires using Markov parameters to construct the Hankel matrices whose correct order is difficult to obtain. A trial and error based approach employed earlier [44] was improved by a procedure that uses an observer based Kalman filter for Markov estimation (OKID), suggested by [133] with further improvements shown in [134]. However, these procedures are complicated and suffer from many difficulties from the identification perspective, as cited in [135]. These are listed as: (I) The initial conditions used in these procedures are typically unknown. (II) The number of Markov parameters in the model increase with the number of discrete samples k. (III) It is required to keep all the non-zero Markov parameter coefficients in the model and this number can be quite large for a lightly damped system. This makes the algorithm computationally expensive. In present work, for the ERA, the required Hankel matrices were obtained in a different way as follows. Using only one sensor (as opposed to the original ERA) with the time series data acquired by that sensor, the square Hankel matrix was obtained by employing the Matlab function “hankel”. This matrix was then pared down to have only nonzero (nonsingular) entries. It was observed that this resulted in a clear diStinction between the “strong” and the “weak” singular values when SVD was applied to the Hankel matrix H (0), thereby verifying the efficacy of proposed approach. After these steps, the remaining ERA steps shown in section 6.2.2 can be easily followed. We will now refer to this approach as modified ERA. Even with this approach, the modified ERA still remains expensive since it uses all the time series data for that particular sensor and thus it may not be an ideal choice for the case where large data is unavoidable. Further study on the subject is required for improved performance. This may require use of stabilization diagrams as frequently used in the SSI methods [32] for model order determination or selecting appropriate rows and columns of the Hankel matrix. 116 6.3 Numerical simulations Structural system models are now numerically simulated and compared using the mentioned identification schemes. Two example models are used to explore both free and forced response cases. 6.3.1 free response: 54 three @017 discrete system example We begin with the simple three DOF mass—spring—dashpot system example investi- gated in [136] for POD, [95] for SOD and [100] for SVMD, and also reported in section 2.3 for free response analysis. The system mass and stiffness parameters are given in Eq. (2.12) and are reproduced here as 2 0 O 2 —1 0 M: 0 1 O ,K= —l 2 —1 (6.18) 0 0 1 0 —1 1 with the reported initial conditions x(0) = (1,0,0)T, and v(0) = (0,0,0)T. The undamped natural frequencies of the system are col = 0.4209, tag 2 1.000, and w3 = 1.6801. In each simulation, the time responses for each mass displacement and velocity were numerically computed. Here, we used a. step size At = 0.01 and a sample size N = 2000. The derivative step size was "d = 1, and for ITD r = 0.01. The system damping was selected to be mass proportional as C = cM where c was chosen as 0.01. This results in lightly damped modes shown in Table 6.1. All of the identification schemes are seen to match up with the system eigenval- ues to the 4th decimal place except SOD. The SOD is reasonably good but fails to perfectly identify the system frequencies. Nonetheless it is expected to improve if a larger data set is used for identification purposes. (It was verified that the largest error in identifying the system frequencies lay within 3 2% for N = 10,000 data points). 117 Table 6.1: Estimated modal parameters from various decomposition methods com- pared against the theoretical eigenvalue problem for a 3 DOF discrete system exam- ple Method System ITD SVMD ERA SOD Qi Freq. Damp. Freq. Damp. Freq. Damp. Freq. Damp. Freq. Damp. col 0.4209 0.0119 0.4209 0.0119 0.4209 0.0119 0.4209 0.0119 0.4595 - (4)2 1.0000 0.0050 1.0000 0.0050 1.0000 0.0050 1.0000 0.0050 0.9869 — w3 1.6801 0.0030 1.6801 0.0030 1.6801 0.0030 1.6801 0.0030 1.6654 - The normalized values of the displacement partitions of the three indepen- dent eigenvectors are w? = (0.3602,0.5928,0.7204)T, w; = (—0.7071,0,0.7071)T, wg= (0.2338,-0.8524,0.4676)T. For ITD and SVMD, the mode shapes The SOD resulted (0.6980,0.0036,—0.7161)T, W; = matched up to the 4th decimal place with the LNMS. in w? = (0.6859,0.6165,0.3867)T, w; = (0.2036, —0.8348,0.5114)T. However, it was verified that the modes produced from the SOD were orthogonal as \IITR\II = I, where R = XXT (see section 3.2). Assuming that the mass distribution was known, the mass weighted POD [136] was formulated by using R = RM as I T I — (I) = —A(I> NXX N (6.19) where (I) is a matrix of eigenvectors that in this case would correspond to LNMS and A is the matrix of eigenvalues that corresponds to mean signal energy distribu- tion in the usual POD manner. results in w? = (0.3470,0.6034,0.7179)T, w; = (—0.7106,—0.0196,0.7034)T, w},T = (—0.2459,0.8459,—0.4733)T. Results from the modified ERA are not comparable as it used only mode and could not generate mode shapes since only one sensor output with “strong” signal strength [99] was utilized. 118 6.3.1.1 Noise effects Next, the same system was sub jected to an 8—bit quantization noise which is a typical value considered in reference [91]. Proceeding exactly the same way as explained in section 2.4, random noise was added to the system. The only difference here is that the differentiation step size was chosen to be "d = 16 and r = 16 to ensure compatibility for ITD. Also, the step and sample sizes were the same as shown in the example above. Modal parameter estimation is presented in Table 6.2. SVMD gets good estimates in both frequency and damping values and has about 1% of maximum frequency error and about 5% maximum damping estimate error. The ITD results in almost identical frequency values with the system frequency values and also shows good damping estimates except the first mode for which the damping estimate is off by z 35%. The ERA has the best estimates both in terms of frequency and damping with almost no errors and up to 4th decimal place accuracy in estimation but did not generate any mode shapes. The SOD has maximum error of about 9% in the frequency estimates. Table 6.2: Modal parameters estimation from various decomposition methods com- pared against the theoretical eigenvalue problem for a 3 DOF discrete system with 8-bit quantization noise. Method System ITD SVMD ERA SOD Qt Freq. Damp. Freq. Damp. Freq. Damp. Freq. Damp. Freq. Damp. col 0.4209 0.0119 0.4210 0.0161 0.4208 0.0125 0.4208 0.0118 0.4625 - cog 1.0000 0.0050 1.0000 0.0050 0.9958 0.0050 1.0000 0.0050 0.9877 - tog 1.6801 0.0030 1.6802 0.0031 1.6601 0.0029 1.6801 0.0030 1.6441 - The eigenvectors obtained by SVMD and ITD match the system modes to The modes estimated by SOD are also reasonably close to the LNMS as can be seen in w? = the 4th decimal place. (0.7003,0.6180,0.3573)T, wg‘ = (0.6987,0.0019,-0.7154)T, wg‘ .—_ (0.2148,—0.8358,0.5053)T. The mass weighted POD shows similar trends without considerable deviation from the noise-free case 119 x2 x3 k x 16 k 2me C at]??? s s Figure 6.1: The mass-spring-damper model. The dashpots are figurative to represent the presence of damping, and do not accurately correspond to the example damping matrix. and resulted in W? = (0.3428,0.6128,0.7121)T, wg‘ = (—0.7121,—0.0188,0.7018)T, w; = (—0.2490, 0.8396, —0.4828)T. In summary, all the decomposition methods work very well for both noise-free and noise contaminated cases for frequency estimations. In the noisy case, performance in the ITD can be improved; that is, to get a better damping estimate for the first mode, a larger differentiation step size (for instance "d = 32) can be employed, but that results in somewhat decreased performance for all of the other identification schemes. 6.3.2 forced response: 21 sixteen 909' system exampfe For randomly forced response, a sixteen DOF mass-spring—dashpot system shown in Fig. 6.1 is explored. In this example, the same mass and stiffness matrices as presented in [137] and also shown in section (3.3) are utilized. These are given as - . ’ 2 —1 ' 2 0 0 0 _1 2 _(1) 3 8 01 0 0 M: ; -. _ _. . ,K= 0 —1 2 —1 30 03 0 1 ' ' ° ' " ' . .16 16 _ >< _ 0 0 1 um)<16 (6.20) and the damping matrix is chosen to be C 2 CM, where c = 0.01. In this example we considered the system already contaminated with measurement noise as explained in the following section. 120 6.3.2.1 Noise effects The current example is similar to that of reference [102] except that the current system is a sixteen DOF system and is excited by two independent Gaussian white noise excitations. These forcing functions are applied at the first and sixteenth masses. In addition, the displacement output ensemble is corrupted with additional sensor noise having the signal to noise ratio (SNR) of 40 dB. The system was simulated for 1000 seconds (later we found out it was about 15 modal periods of first frequency) with SIMULINK toolbox in Matlab, with a fixed-step Dormand Prince (a member of family of Runge-Kutta methods) differential equation solver [109] was used to evaluate the response of system. The programs default error tolerances were used (103 for relative error and 106 for absolute error). White noise forcing was generated using the Gaussian white noise generator function that produces discrete-time normally distributed random numbers with sampling time step matching the solver step size chosen as 0.1, resulting in generation of 10,000 data points. The forcing was observed to have a mean approaching zero. Both displacement and forcing matrices were saved to the Matlab workspace for further processing. In the decomposition, the V ensemble was formed with centered finite differences with a total step of two samples, such that difference matrix D was NU x N, and V = XDT was n x NU, where N1, = N — 2. From the data decomposition eigenvalues, estimates of the natural frequencies are compared to the true modal frequencies in Table 6.3. As mentioned earlier in section 6.2.2.1, for this large data set the SVD of the Hankel matrix was too large and was problematic to obtain. Cast this way, ERA is an expensive method and estimates were not computable. Reduction in the data set resulted in significant estimation errors and thus is not reported. The system frequency estimates for all schemes are good. The error estimates for com (which contains the largest error) reduced to about 2% for all methods when N = 100,000 points were used. Nonetheless, all schemes show similar trends and work 121 Table 6.3: System frequencies estimated from decomposition methods with white noise forcing compared against structural eigen frequencies Method System ITD SVMD SOD Qt Freq. Error Freq. Error Freq. Error Freq. Error “’1 0.0951 - 0.0926 2.60% 0.0926 2.60% 0.0934 1.76% wg 0.2832 - 0.2847 0.55% 0.2847 0.54% 0.2840 0.29% w3 0.4644 - 0.4708 1.37% 0.4705 1.31% 0.4704 1.27% W4 0.6345 - 0.6358 0.21% 0.6354 0.14% 0.6354 0.14% w5 0.7933 - 0.7883 0.62% 0.7874 0.73% 0.7874 0.74% “’6 0.9482 - 0.9456 0.27% 0.9441 0.42% 0.9440 0.44% (117 1.1035 - 1.1045 0.09% 1.1022 0.11% 1.1023 0.10% tug 1.2557 - 1.2557 0.00% 1.2524 0.26% 1.2511 0.36% tug 1.4000 - 1.4039 0.27% 1.3993 0.05% 1.3991 0.06% “110 1.5330 - 1.5337 0.04% 1.5278 0.34% 1.5279 0.33% “’11 1.6522 - 1.6453 0.41% 1.6391 0.79% 1.6392 0.79% can 1.7558 - 1.7559 0.00% 1.7481 0.44% 1.7482 0.42% w13 1.8423 - 1.8470 0.25% 1.8378 0.24% 1.8377 0.24% (.214 1.9107 - 1.9147 0.20% 1.9044 0.32% 1.9039 0.35% w15 1.9601 - 1.9698 0.49% 1.9635 0.17% 1.9622 0.10% w16 1.9900 - 2.0898 5.01% 2.1181 6.40% 2.1072 5.89% well for all the modes (even those with closely spaced frequencies, in this case the last three). The ability to work well in closely spaced frequencies is a basic advantage of time-domain methods over the frequency domain based identification schemes. The modal assurance criterion (MAC) [110, 111] is now applied to evaluate consis— tency of the estimated modes with the system modes. The normalized inner products (squared) between estimated and true modes are computed using the method. Values of near unit magnitude indicate modal vectors that nearly line up. For visualization, the modal vectors from the state-variable modal decomposition and the structural eigenvalue problem using the modal assurance criterion are compared in Fig. 6.2. A1- 122 though all modes estimates are quite good, two of the modes do not show the MAC value of identity, these are the same modes which had higher errors in frequency estimates, too. MAC value Figure 6.2: The MAC values for all SVMD modes. MAC value of] shows vectors match up. Modes are not in any particular sequence Modes from other schemes (SOD, ITD and mass weighted POD) are also shown in figures 6.3, 6.4 and 6.5. Barring POD, all the modes are identified well in all schemes. POD has good values except may be 3-4 modes that may be due to noise in the system (this requires further investigation). 6.4 Clamped-free cantilever beam experiment Though numerical simulations provide a good picture of the theoretical framework, theoretical assumptions may not always hold and the system may have features such as non—stationary signals, nonlinearity, effects of filtering, spurious modes, contamina- tion due to noise, unknown model order, and reduced observability, which can only 123 MAC value Figure 6.3: The MAC values for all sixteen modes for SOD. MAC value of 1 shows vectors match up. Modes are not in any particular sequence 0.8 cu 2 g 0.6 0 § 0.4 0.2 Mode number 10 0 Figure 6.4: The MAC values for all ITD modes. MAC value of 1 shows vectors match up. Modes are not in any particular sequence 124 -0.8 I 0.6 0.5 MAC value OO 13 5 . 15 11 10 ' ' Mode number ' 1 3 Mode 15 Figure 6.5: The MAC values for all mass-weighted POD modes. MAC value of 1 shows vectors match up. Modes are not in any particular sequence be properly understood by thorough testing. Nonetheless. the experimental beam studied in section 4.3 is investigated using the stated modal identification schemes. 6.4.1 Identification results In order to accurately compare the experimental results for SVMD, ITD, ERA. SOD and POD, the number of samples were reduced to the first N = 1000 samples. The data were then high—pass filtered at 0.5 Hz. These values were selected based on discussions in section 4.3.1. For SVMD, the correlation matrices P = YYT/Nd and Q = YWT/Nd were then formed. and the SVOMS and SVOVS were extracted. The two delayed ensemble matrices were obtained from the Y matrix ensemble for ITD. Extraction of SOMs and SOVs was accomplished by forming the two correlation ma- trices WT and XXT. For ERA, only one of the displacement time histories was em- ployed (the time history of the 8th sensor was used here, the choice being completely arbitrary). The obtained frequency identification results for the first five modes are 125 shown in the Table 6.4. Damping estimates for the first three modes were computed using the log decrement method applied to carefully excited, dominantly single-mode responses, and are compared against the identification schemes in Table 6.5. Table 6.4: Experimental system frequencies (Hz.) estimated from various decom- position methods for a linear beam free response compared against the undamped structural eigen frequencies Parameter Theoretical FFT ITD SVMD ERA SOD “’1 2.74 2.75 2.93 2.78 3.04 2.23 avg 17.16 17.20 17.18 17.18 17.17 17.49 w3 48.05 47.21 47.52 47.91 47.43 49.53 4114 94.16 94.28 92.17 93.55 96.83 56.97 L425 15668 155.9 153.57 153.96 156.05 144.47 Frequency identification for all schemes is relatively good with the exception of some estimates by SOD, in particular the 4th modal frequency. Note that the data set used in this estimation does not cover the entire first modal period, hence errors are expected as discussed in section 5.5. ERA was also able to estimate the sixth frequency as 228 Hz (where the theoretical undamped frequency and the FF T frequency peaks are 233.7 Hz and 230.7 Hz respectively) but was somewhat erroneous in estimating the first frequency. ITD and SVMD were relatively consistent for all modes and their estimates were matching the most among all. Table 6.5: Experimental modal damping estimated from various decomposition meth- ods for the linear beam free response. Only first three damping estimates are com- puted by using dominantly single mode log decrement method Parameter Theoretical ITD SVMD ERA (1 0.0048 -0.042 -0.057 -0. 1069 (2 0.0040 0.0039 0.0039 0.0035 C3 0.0036 0.0029 0.0028 0.0038 (4 - 0.0506 0.0097 0.024 (5 - 0.0055 0.0016 0.0015 The damping estimate for the first mode is poor for all of the schemes (again, this is because the time record only covers about half of the first modal period). (While 126 using a larger set of data points for the same test run with 12,000 sample points, the estimates were recorded as 0.0051 and 0.0049 for ITD and SVMD respectively. How- ever, in that case only the first two modal frequencies were identifiable, a discussion of this can be seen in section (4.3.2). The second and third modes are very good for all schemes. ERA also computed the damping estimate for the sixth mode at 0.0022. 0.6 I I I I ‘1' l l 0.3 - Mode1 0.2- l l 2 4 6 8 10 12 14 16 SensorPosMon Figure 6.6: Identification methods are compared for the first beam. mode. POD is shown with dotted line — -, SOD is shown with crosses x, SVMD with circles o, and [TD with the solid line —. Mode shapes are now compared primarily for the SVMD, SOD, and ITD methods along with POD shown in some plots. The eigenvectors obtained from these methods are normalized and are plotted separately for each mode in Figures 6.6, 6.7, 6.8, 6.9, and 6.10. These plots, in general, accord with the frequency estimates, with ITD and SVMD showing almost identical plots. SOD shows good results for lower modes since the lower modes were dominant, but errs at higher mode shape estimates. POD has the same trend and is only shown for the lower modes. For higher modes, it has huge error and thus is not. reported. The log plot of POVs is shown in Fig. 6.11 that shows 127 0.5 r r l l I I I 0.4 — 0.31 0.21 0.1- Mode2 ‘? _0'5 I l I I I 0 2 4 6 8 10 12 14 16 SensorPosMon Figure 6.7: Identification methods are compared for the second beam mode. POD is shown with dotted line - -, SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —. about four significant modes. . This however, does not mean that POD and SOD in general cannot estimate higher modes. For instance, when the same beam was excited with a different response that showed a higher mode (in that case a dominant third mode), both POD and SOD were indeed able to get a good mode shape estimate for the third mode as shown in Fig. 6.12. In essence, SVMD and ITD are able to estimate modal parameters up to the first five frequencies and five mode shapes very well in this experimental setup for the given set of data samples. ERA for the same data set was able to estimate the first five frequencies as well as a sixth frequency but at the cost of almost missing the lowest frequency, and was unable to estimate mode shapes. SOD was good for lower 128 "’“T‘T—‘q 4 II 1 0.4 I I l I l I l 0.3— 0.2 - ‘06 I I l I I l O 2 4 6 8 10 12 14 16 Sensor Position Figure 6.8: Identification methods are compared for the third beam mode. SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —. modal frequencies and associated mode shapes but did poorly for the higher ones. It was however, verified that the SOD produced modes were orthogonal ‘I’TRW = I. In general, for all identification schemes, it was observed that when larger data were used, lower modal estimates improved but high mode estimates degraded and vice-versa. (A complete discission on effects of time record lengths is presented in section 5.5). The particular data set was selected to get reasonably high modes without degrading performance too much for the lower modes for most schemes, and to be able to compare all schemes (hence the 1000 sample points). Tests were also done on other beam samples (not shown). Results with those cases contribute to our general assessment of the methods. 129 0.6 I I I I I I I 0.5 - _ 0.4 - - 0.3 - 0.2 r Mode 4 O _ .4 I I I I 0 0 2 4 6 8 10 12 14 16 Sensor Position Figure 6.9: Identification methods are compared for the fourth beam mode. SOD is shown with crosses x, SVMD with circles o, and ITD with the solid line —. 6.5 Summary We have made comparisons for various time-domain and decomposition based modal identification methods that extract modal information from responses of generally damped linear multi—(legree—of—freedom systems. In each case, input to the system was not measured and only the response was measured. For each method, an eigenvalue problem was constructed, and the resulting eigenvalues and eigenvectors contained frequency plus damping and mode shape information, respectively. Three kinds of systems were evaluated. The two numerical simulations were com— pared for free and random response cases that also involved quantization noise studies. Lastly, we performed an experiment on a real life structure and observed modal pa- 130 0.6 | | I I l l l 0.4 — - 0.3 r Mode 5 o 6 8 10 12 14 16 Sensor Position Figure 6.10: Identification methods are compared for the fifth beam mode. SOD is shown with crosses X, SVMD with circles o, and ITD with the solid line —. 2 I I I I I I I 0 0- _ _2_ _ -4.— _ log POV _6_ O ‘100 10 12 14 16 Figure 6.11: The log plot of POVs shown. The plot shows about 4 significant modes. 131 0.4 I I I I I I I 0.1— Dominant mode 3 response :5 7‘ l l 2 4 6 8 10 12 14 16 Sensor Position "0.6 I I I I 0 Figure 6.12: SOD and POD mode shape identification for a higher mode with the system excited in a way to get a dominant 3rd order beam mode response. SOD is shown with the solid line —, and POD is shown with a dotted line — -. rameter estimates obtained through identical circumstances (same number of samples employed, filter conditions, etc). Among all the methods, SVMD and ITD show closest resemblance in parameter estimation. Their frequency and mode shape estimates are reasonably good and the methods work very well in low noise conditions. SOD generally requires a larger data set, in particular for estimations of higher modes, and works well for lightly damped cases. ERA is a reliable but computationally expensive method and it would be interesting to be able to use ERA with more sensor histories for mode shape estimates (as original ERA is conducted). There is a trade-off between number of modal parameters to be estimated and their accuracy: and in general, accuracy of 132 lower modes increases with increasing number of sampling points, at the cost of the higher modes. These observations were also based on tests with other beams, which were not presented. 133 7 .1 Synopsis rthogonal decomposition methods for experimental modal analysis have been presented in this work. A novel approach called the state- variable modal decomposition (SVMD) was analyzed for free and random response cases for linear multi—degree—of—freedom structural problems. Also addressed was the extension of smooth orthogonal decomposition (SOD) for random response. These methods fall under the output—only time-domain category of modal analysis that does not require measuring the input forces. Both methods are exten— sions of the well known proper orthogonal decomposition method (POD) and over— come some of the inherent limitations of the POD. These methods tie up the gap between current time—domain schemes and the POD. We will briefly summarize work presented in each chapter and conclude with a discussion on potential future work. 7.1.1 free response A modal decomposition strategy based on state-variable ensembles was formulated. A nonsymmetric, generalized eigenvalue problem was constructed. The data-based eigenvalue problem was related to the generalized eigenvalue problem associated with 134 free-vibration solutions of the state-variable formulation of linear multi-degree-of- freedom systems. For linear free-response data, the inverse-transpose of the eigen- vector matrix represents the state—variable modal eigenvectors, and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state-variable model. As such, the eigenvalues lead to estimates of frequencies and modal damping. The in- terpretation holds for linear systems with multi-modal free responses without the need of input data, whether damping is large or small, or modal or nonmodal. Simulations for noise contaminated cases also showed satisfactory performance for underdamped system parameter estimations. 7 .1.2 Random response The smooth orthogonal decomposition (SOD) and the state-variable modal decompo- sition SVMD methods were extended for modal analysis of randomly excited systems. Modal parameter estimation in terms of natural frequencies and mode shapes was studied using the SOD method. We showed that under certain conditions, the SOD eigenvalue problem formulated from white noise excited response data could be tied to the unforced structural eigenvalue problem, and thus could be used for modal parameter estimation. Using output response ensembles only, the generalized eigen- value problem was formed to estimate modal frequencies and modal vectors for an eight-degree-of-freedom lightly damped vibratory system. The estimated frequencies were compared against system frequencies obtained from the structural eigenvalue problem and estimated modal vectors were checked using the modal assurance crite- rion. Simulations showed that for light damping, satisfactory results were obtained for estimating both system frequencies and modal vectors even in the presence of sensor noise. While the work was primarily geared towards the SOD analysis, the SVMD was briefly touched upon to show a similar connotation as the SOD analysis. 135 7.1.3 Experiments Since in general, real life systems can deviate considerably from ideal theoretical assumptions, applicability of SVMD method was evaluated for free responses by mul- tiple experimental setups. We evaluated the SVMD method for structures modeled as both single and multi-degree-of—freedom systems. Many beam structures were tested and we were successfully able to estimate modal parameters for multiple modes. This presents SVMD as a viable modal parameter estimation scheme. We observed that the time record of the data plays critical role in the parameter estimation. The trend found was that shorter length data was good for estimating higher modes, while longer length data were good for estimating lower modes. Modal coordinates together with mode shapes and modal frequencies were found useful to determine the quality of the decomposition and also to distinguish actual modes from fictitious ones. Another important factor was appropriate filter selection. Filter cut— off frequency selected as half (or less) of the fundamental frequency (computed by F FT) removed low noise drifts that may otherwise appear in the response, and in integrations of measured accelerations. The experimental investigation also revealed limitations of the SVMD method. It was observed that at best, the SVMD method is only as good as the number of sensors utilized for measurements. If the number of sensors is less than the active system modes, SVMD can only identify a few modes with accuracy. 7.1.4 @antitative evaluation In previous sections, we used information in modal coordinates qualitatively to exam- ine the effectiveness of the decompositiOn results and to help separate spurious modes. In order to establish a more efficient way of addressing the decomposition quality, we developed two new tools. These tools quantify the decomposition effectiveness. The spectrum assurance and spectrum difference tools use the FRFs obtained from both 136 the modal coordinates and the identification results (from decomposition methods or FF T) to generate a number that ranges between 0 and 1. Both could be used together to eliminate a “close but spurious” candidate. These criteria were first explored us- ing some numerical case studies. System parameters such as frequency and damping values were varied to establish any specific trends. It was seen that both criteria are less sensitive to changes in damping errors than to frequency errors. Then these-tools were applied to experimental data to quantify the decomposition results. Finally, we evaluated the data length effects on the identification. This was ex- plained via a numerical study. It was observed that with addition of noise to the system, results suffered if the data length was either too short or too long. At least one modal period should be used for frequency identification. While the damping es- timates showed similar trends, we found that useful information was contained within the oscillations peaks as well. Modal settling time was found to be the other useful measure. When time record was greater than modal settling time, the noise effects started to become significant and identification results were found to err. 7.1.5 Comparison to other methods A number of common time-domain methods are available for modal parameter es- timation. We compared the decomposition based methods SVMD, SOD and POD against the two methods of significant impact, the Ibrahim time domain method and the modified eigensystem realization algorithm. We simulated two numerical exam- ples for free and random response cases. The free response case used a three DOF system whereas the random response case employed a sixteen DOF structural system. Both cases were contaminated with noise at the output. All methods were compared against the theoretical frequencies and damping ratios. In the free response case, mode shapes were directly matched while in the random response case the modal as- surance criterion was used. Lastly, we made comparisons on the experimental beam 137 studied in the previous chapter. In numerical simulations with light damping, all of the methods generally pro- duced good estimates of the system frequencies even in the presence of noise. Also, mode shape estimates accorded well with the frequency estimates. The SVMD and ITD methods produced the most closely matching results, especially in the experi- ments. The estimation by using a modified ERA in the experimental test was slightly better for frequencies and damping estimates, but it was unable to produce any mode shapes. The SOD worked well for the lower modes but ran into issues at the higher mode estimations. 7.2 Recommendations for use of the decomposition methods In this section, we will briefly describe the strategy for any interested user to imple- ment the decomposition methods for physical experiments. The decomposition methods are output-only methods so collection of “clean” data cannot be overemphasized. From numerous studies, we have established that the decomposition methods work very well for noise-free systems. Given that the physical systems would have some form of noise, every possible effort should be made to minimize the effects of noise on response data. Thus, use of noise filters, low sensitivity voltage amplifiers, proper selection and proper installation of acclerometers is highly recommended. Decomposition methods are evaluated for free and random response cases. A near-ideal impulse or a broadband random excitation is required for methods to work the best. Care should be taken to implement the near-ideal excitation on the test structure. To construct decomposition method ensembles, integrations should be preferred over differentiations where possible. 138 An F FT of the response data can establish a range of system frequencies of interest since F FT is regarded as more accurate for estimating frequencies, and useful in determining candidates for extracted true modes. If decomposition methods have frequencies that may be “in the neighborhood” of the probable mode, SAC and SD tools should be applied to distinguish the real and spurious modes. SAC in general, produces more clear candidates but SD narrows down the choice. It is perhaps best to use these together. If the interest is in identification of lower modes, use of “longer” time record length (that is at least a modal period) is recommended. For low noise cases, the 2% settling time criterion of the lowest mode can be used as a guide for maximum time record length. In general, these criteria may not be a known priori but FFT can be used to estimate the lowest modal frequency. If the interest is in higher modes, since these modes generally damp out quickly, a “shorter” time record can identify the higher modes. At least one modal period is required though for adequate estimates. The decomposition methods estimate low to moderate damping in lower modes with relatively long data. High damping case or damping in higher modes were not verified experimentally in this work. For very light damping, all decomposition methods work equally well. SVMD, ITD and ERA can estimate low to moderate damping estimates. ERA is the most expensive scheme but perhaps can identify higher modes. ITD and SVMD have the most common features and most matching results. 7.3 Future work The SVMD method has been evaluated as a viable output-only time-domain modal identification scheme. The method is an extension of the POD method applied in 139 frfirnfirw ‘ linear structural system analysis. The method has been analyzed in free response and random response cases. Below, we suggest some of the potential directions for broadening the capabilities of the method. 7 .3 .1 Damping estimation for random excitation In section 3.4.1, we presented the analysis of the SVMD for MDOF systems subjected to random response. The analysis suggests that the SVMD should produce modal pa- rameter estimates for randomly excited systems including frequencies, damping ratios and mode shapes. However, in section 3.4.2, we observed that this was not entirely true. Indeed, the SVMD method produced good estimates of modal frequencies and the mode shapes, but failed to pickup any damping ratio estimates. In the same section it was noted that this may be due to differentiation errors associated with the Euler derivative applied to generate ensemble matrices V and A. We also ob- served that by using the ensemble matrices that are generated by SIMULINK (that uses high order numerical integration schemes), accurate damping estimates were obtainable by the SVMD method. While it may be tempting to use a higher order (Newton-Cotes [138]) derivative scheme, the stated observations suggest that a higher derivative scheme may or may not work. A better way is to integrate the acceleration ensemble to obtain the velocity and displacement ensembles. This is especially easier in experiments. We also suggest experimental verification of the ideas presented in chapters 4 and 5 to verify applicability of both SOD and SVMD methods to responses to random exciatation. 7.3.2 Continuous systems In deriving the SVMD method for modal analysis of linear MDOF systems, we used discrete models. The ensembles, numerical derivatives, etc., were all discretized. How- ever, in actual experiments we used continuous systems (beams). Even though the 140 decomposition worked well for such systems, it was never shown analytically that SVMD works well for a continuous system. Hence it would be useful to conduct an analytical study that addresses SVMD analysis for continuous systems. This can be done for both free responses as well as randomly excited systems. 7.3 .3 Nonlinear systems Throughout in the modal analysis by SVMD, we assumed the systems under study to be linear, time invariant, and causal. As mentioned in reference [3], It is appropriate to include a consideration of the possibility that not all the systems or structures encountered in practice will be linear. The SVMD can be used to identify the linearized frequency content present in a nonlinear system. To see this we consider a 2 DOF system that has an undamped harmonic oscillator attached to a Van der Pol oscillator [139]. The governing equations of motion are .131 + h‘(2:131 —- £132) = 0 (7.1) i2 + 1.13:2 — x2) — 6.152(1— m3) = 0 (7.2) where the mass and stiffness parameters are all normalized to unity. Varying the parameter 6, the fixed point dynamics can be varied. However, we used small 6 values, such that 0 < e < 0.5, for the SVMD analysis. The system was simulated using SIMULIN K toolbox for 10 seconds. The step size was chosen as At = 0.004977, the sample size was N = 2000, and the initial conditions were x(0) = (0.000001, 0)T, and v(0) = (0,0)T. The e value for this particular case was chosen as 0.3. The time record approximately matches one period of the lowest- frequency mode of the linearized system. The undamped natural frequencies of the linearized system are col = 0.6205, and tag = 1.6116. 141 2E—3 I I I I N o 2 0- - > a) 'ZE'30 500 1000 1500 2000 2500 Samples N 5E—3 T I r I T ‘2’ ;> 0 _ U) X E 65 435-3 -1E-3 -0.5E-3 0 0.5E-3 1E-3 1.5E-3 SVMC 2 Figure 7.1: The modal coordinates plot for Log and their phase space are plotted for 10 second simulation time. The decomposition was performed in the usual manner on the collected data. We found that even for this small time period, the SVMD was able to perfectly estimate the system frequencies and the damping ratios. Note that SVMD was able to identify negative damping. The SVMD identified mode shapes (normalized) were willw = v'vg: (0.6354 — 0.1043i,1.0554 — 0.058i)T, w; = w}: (—0.0671 - 0.67862”, —0.0487 + 0.4118i)T. These perfectly matched with those of the structural eigenvalue problem. By construction, Q = aQ. Using this relationship, we took the real part of time history of modal coordinate corresponding to (.02, and plotted it against the real part of a2 x Q(w2) (where a2 corresponds to the eigenvalue associated with the second mode). This plot is shown in Fig. 7.1. When the simulation time was increased to 400 seconds, the system response closely reaches the limit cycle, the SVMD identified the system frequencies as col = 0.6073, and tag = 1.8244. The SVMD was able to identify the modal vector associated with the col as wit" = w; = (0.6579 — 0.0016i, 1.0731 - 0.0013i)T, the real part being 142 _§I.'cf L N 0.2- 0 II 2 0 I I II I ‘ > 03-02— 0 20000 40000 60000 80000 Samples N 0.5 I | I I 1 r U E o_ _ (I) x , d _ .8 l l l I l L - .4 -0.3 —0.2 —01 0 0.1 0.2 0.3 sVMC2 Figure 7.2: The modal coordinates plot for (oz and their phase space are plotted for simulation time of 400 seconds. close to structural modal vector. However, the modal vector associated with 0.22 was erroneous. This vector was identified as w; 2 WI: (—0.0014 — 0.3941i, —0.0001 + 0.5236i)T. That the modal vector of the large motion is not a good match with the linearized modal vector is not surprising, since with large motion the nonlinearities participate significantly. Again, we took the real part of time history of modal coordinate corresponding to tag, and plotted it against the Q2 x Q(w2) (where ag corresponds to the eigenvalue associated with the first mode). This plot is shown in Fig. 7.2. We see that response has reached a limit cycle. These are early findings and we are unclear at this point whether the modal coordinates can contribute to dynamical phenomenon such as period doubling. In essence, SVMD can identify the linearized frequencies hidden in a nonlinear system, but needs further exploration in identification of nonlinear systems. 143 BIBLIOGRAPHY [1] P. 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