EXPERIMENTAL APPLICATIONS OF MODAL DECOMPOSITION METHODS TO A NONUNIFORM BEAM By Rickey A Caldwell Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Mechanical Engineering 2011 ABSTRACT EXPERIMENTAL APPLICATIONS OF MODAL DECOMPOSITION METHODS TO A NONUNIFORM BEAM By Rickey A Caldwell Jr. The goal of this research is to compute the mode shapes and in some cases the natural frequencies of a lightly damped freely vibrating nonuniform beam using sensed outputs, via accelerometers. The methods applied are reduced-ordered mass weighted proper decom- position (RMPOD), state variable modal decomposition (SVMD) and smooth orthogonal decomposition (SOD). A permutation of input impulse magnitudes, input locations, signal length, and acceleration, velocity, displacement ensembles were used in the RMPOD decom- position to gain some experience regarding the effects of input parameters and signal types on modal estimations. An analytical approximation to the modal solution of the Euler- Bernoulli beam equation is developed for nonuniform beams. In the case of RMPOD the theory is pushed into the experimental realm. For SVMD and SOD the science is also ex- tended into the experimental realm and is additionally applied to nonuniform beams. The results of this thesis are as follows: the analytical approximation accurately predicted the mode shapes of the nonuniform beam and can accurately predict frequencies if the correct material properties are used in the computations. RMPOD extracted accurate approxima- tions to the first three linear normal modes (LNMs) of the thin lightly damped nonuniform beam. SVMD and SOD extracted both the natural frequencies and mode shapes for the first four modes of the thin lightly damped nonuniform beam. Copyright by RICKEY A CALDWELL JR. 2011 I would like to dedicate this achievement to my mother, Glenda Caldwell, and my sister, Ke- nesha Caldwell. Additionally, there are countless others too numerous to name who believed in me and gave me a chance. To name a few Ms. Flecher, Ms. Horton, Mr. Brusick, Mr. Richard Welch, Mr. David Reed, Dr. A. Wiggins, Theodore Caldwell, M.Ed., Dr. S. Shaw, Hans Larsen, Dan and Tammy Timlin, Sloan Rigas Program, AGEP and other supporters. Finally, to all those who fought, were bitten by dogs, beaten, threaten, murdered, ridiculed, ostracized, and paid the ultimate sacrifice so that I might have the chance to pursue higher education, a million thanks; there is no way to repay my debt to you, so I honor you and the sacrifices you made for me. I truly stand on the shoulders of giants. To Carl. iv ACKNOWLEDGMENT Thank you Dr. Brian Feeny for your guidance and support. You astutely and masterfully led me on a journey of professional and personal development, with great temperance and patience like a benevolent Zen master. That's why I call you Yoda. Additionally, I would like to thank Dr. C. Radcliffe and Dr. B. O' Kelly, without whose help I would not be writing this now. To the land grant philosophy- a worthwhile endeavor! This work was supported by the National Science Foundation grant number CMMI- 0943219. Any opinions, findings, and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation. Additional support was received from the Diversity Programs Office and the College of Engineering at Michigan State University. v TABLE OF CONTENTS List of Tables .............................. viii List of Figures .............................. x 1 Introduction 1 1.1 Background . 2 1.1.1 Analytical Modal Analysis . 5 1.1.2 Experimental Test Modal Analysis . 8 1.2 Thesis Preview and Contribution . 9 2 Beam Experiment 11 2.1 Overview . 11 2.2 Experimental Setup and Procedure . 12 2.3 Additional Data Processing . 14 3 Analytical Approximation 16 3.1 Motivation . 16 3.2 Development . 16 3.2.1 Example . 20 4 Modal Decomposition Methods 26 4.1 Introduction . 26 4.2 Reduced-order Mass Weighted Proper Decomposition . 28 4.2.1 Motivation . 28 4.2.2 Reduced Mass Matrix of a Beam . 29 4.2.3 Experimental Results . 30 4.3 State Variable Modal Decomposition . 32 4.3.1 Background . 32 4.3.2 Mathematical Development . 39 4.3.3 Experimental Results . 42 4.3.4 Contribution . 48 4.4 Smooth Orthogonal Decomposition . 48 4.4.1 Background . 48 4.4.2 Mathematical Development . 49 4.4.3 Experimental Results . 51 4.4.4 Contribution . 54 4.5 Method Comparison . 55 vi 5 Conclusions 57 Bibliography .............................. 61 vii LIST OF TABLES 2.1 Beam width at sensor locations. 12 2.2 Equipment list. 14 2.3 Accelerometer calibration data. 15 3.1 Assumed material properties for the beam. 20 3.2 βL's for assumed modes. 21 3.3 Comparison of natural frequencies computed from the analytical approxima- tion compared to experimental data. 24 3.4 Torsional frequencies computed from the FFTs of the accelerometer signals. 24 3.5 Comparison of natural frequencies for discretization values n = 5, 10, 15, and 20.......................................... 24 3.6 MAC values for two-pair combinations of n values at n = 5, 10, 15 and 20 for the first five modes. 25 4.1 MAC values for RMPOD when compared to the approximate analytical modes. 32 4.2 POD and SVMD. The first row contains the ensemble matrices. The second row contains the expanded ensemble matrices. The third row contains the correlation matrices. Finally, the last row contains the eigensystem problems. 38 4.3 SOD vs POD case study. 49 4.4 MAC values for decomposition methods when compared to the discretized analytical analysis mode shapes. 55 4.5 Cross comparison of decomposition methods using MAC values. 55 4.6 SVMD and SOD extracted frequencies. 56 viii 4.7 Pros and cons of each decomposition method. 56 ix LIST OF FIGURES 1.1 Effects of damping on free vibrations. 3 1.2 Mass-spring-dashpot (MSD) system. 5 2.1 Experimental beam. 13 3.1 Analytical approximations of discretized mode shapes for n = 20, top: first mode, bottom: second mode. 23 3.2 Analytical approximations of discretized mode shapes for n = 20, top: third mode, bottom: fourth mode. 25 4.1 RMPOVs: mode (2) 1.2591, mode (3) 0.0562, mode (4) 0.0346. 33 4.2 Top: second mode shape extracted by RMPOD (o) plotted with the analytical approximation's discretized mode shape (line). Middle: 2nd modal coordinate acceleration from RMPOD. Bottom: fast Fourier transform of modal coordi- nate acceleration. 34 4.3 Top: third mode shape extracted by RMPOD (o) plotted with the analytical approximation's discretized mode shape (line). Middle: 3rd modal coordi- nate acceleration from RMPOD. Bottom: fast Fourier transform of modal coordinate acceleration. 35 4.4 Top: fourth mode shape extracted by RMPOD (o) plotted with the analytical approximation's discretized mode shape (line). Middle: 4th modal coordinate acceleration from RMPOD. Bottom: fast Fourier transform of modal coordi- nate acceleration. 36 4.5 Top: seventh mode shape extracted by RMPOD (o) plotted with the analyt- ical approximation's discretized mode shape (line). Middle: 7th modal coor- dinate acceleration from RMPOD. Bottom: fast Fourier transform of modal coordinate acceleration. 37 4.6 The second, third and fourth modes extracted by SVMD. 44 x 4.7 Top: second mode shape extracted by SVMD (o) plotted with the analyt- ical approximation's discretized mode shape (line). Middle: second modal coordinate of SVMD. Bottom: fast Fourier transform of the second modal coordinate. 45 4.8 Top: third mode shape extracted by SVMD (o) plotted with the analytical ap- proximation's discretized mode shape (line). Middle: third modal coordinate of SVMD. Bottom: fast Fourier transform of the third modal coordinate. 46 4.9 Top: fourth mode shape extracted by SVMD (o) plotted with the analytical approximation's discretized mode shape (line). Middle: fourth modal coordi- nate of SVMD. Bottom: fast Fourier transform of the fourth modal coordinate. 47 4.10 SOD extracted second mode (o) compared to the analytical approximation (solid line). 52 4.11 SOD extracted third mode (o) compared to the analytical approximation (solid line). 53 4.12 SOD extracted fourth mode- (o) compared to the analytical approximation (solid line). 54 xi Chapter 1 Introduction For beams freely vibrating in their linear elastic range with small amplitudes and known initial and boundary conditions, it is effective to describe the beam's dynamics using its mode shapes, natural frequencies, and modal damping. Generally the calculus of this information is derived from the Euler-Bernoulli beam equations or more generally the Timoshenko beam equation. The most significant difference between the two beam theories is that Timoshenko beam theory allows for warping of the cross sections and shear stress in the cross sections, and Euler-Bernoulli beam theory assumes that deformations occur in bending only and that cross-sections remain plane. In order to derive the dynamics from these beam theories one needs the material properties such as mass per unit length, Young's modulus, Possion's ratio, and geometry information such as the area moment of inertia of the cross section. If one considers discrete mass systems such as mass-spring-dashpot systems, then the mass, spring, and damping matrices must be known. In both of these cases, continuous beam and discrete mass systems, one needs to know the material properties and the geometry to compute the mode shapes and natural frequencies which can then be used to compute the dynamics of 1 the beam, such as displacement, velocity, and acceleration. The focus of the thesis is on decomposition methods where an engineer could capture dis- placement time histories or its derivatives and use that information to find the mode shapes, and in certain cases, the natural frequencies and modal damping coefficients.