This is to certify that the thesis entitled THE IMPACT OF MODAL ANALYSIS ON THE ENGINEERING CURRICULUM presented by H. METIN NUS RIZAI has been accepted towards fulfillment of the requirements for Master of Science Date May 5' 1980 07639 degree in Mechanical Engineering ¢\ ‘ quor professor mm: 25¢ per do per 1t.- RETUMUG LIBRARY MTERIAL§= Place in book return to move chem free circulation records THE IMPACT OF MODAL ANALYSIS ON THE ENGINEERING CURRICULUM By H. Metin Nus Rizai A THESIS Submitted to Michigan State University in partial fulfiliment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1980 ABSTRACT THE IMPACT OF MODAL ANALYSIS ON THE ENGINEERING CURRICULUM By H. Metin Nus Rizai Modal analysis is a procedure for describing the motion of a structure by identifying its modes of vibration. A brief des- cription of modal analysis techniques is presented as a prelude to a discussion of modal testing. This leads to a detailed discussion of modal testing technology and the potential impact of this technology on the Mechanical Engineering Curriculum at Michigan State University. DEDICATION To my major professor and my good friend Jim Bernard, my parents, Mr. and Mrs. Andrew TenEyck and my mother, who gave purpose to this work. ii ACKNOWLEDGEMENTS I particularly wish to express my gratitude to Dr. James E. Bernard, my major professor, for all his assistance and guidance as a teacher and a friend, on this thesis and throughout my M.S. program. Many thanks to Bill Grafton from Gen-Rad Corporation, who helped me a great deal to do the experiments and also to Dr. R.C. Rosenberg and Dr. N.V. Nack for their valuable comments throughout this work. Thanks also to all the friends who helped make my graduate studies an enjoyable experience. Finally many, many thanks to my parents Mr. and Mrs. Andrew TenEyck who brought me to where I am today. Without their help I could not have made it. And thanks to my brothers and sisters who gave me a great support during my work. TABLE OF CONTENTS LIST OF TABLES ............................................ vi LIST OF FIGURES ........................................... vii Chapter l. INTRODUCTION ......................................... l 2. ANALYTICAL APPROACH .................................. 2 2.1 The Finite Element Method ....................... 2 2.2 Diagonalization ................................. 3 2.3 Modal Analysis .................................. 4 2.4 Damping ......................................... 5 2.5 Dynamic Substructuring .......................... 6 2.6 Some Final Remarks .............................. 7 3. MODAL TESTING ........................................ 8 3.1 Transfer Function and Frequency Response Function ........................................ 8 3.2 Digital Signal Processing ....................... l2 3.3 Excitation Methods .............................. l2 3.4 Modal Data Identification ....................... l6 3.5 Noise and Distortion ............................ 24 3.6 Measurement Resolution .......................... 28 3.7 Some Other Considerations in Modal Testing ......................................... 29 3.7.1 Aliasing ................................. 29 3.7.2 Leakage .................................. 31 3.7.3 Windowing ................................ 33 3.8 Summary ......................................... 35 4. AN EXAMPLE OF MODAL TESTING .......................... 36 4.l Geometry Definition ............................. 36 4.2 Excitation Technique ............................ 38 4.3 Frequency Response Data Analysis ................ 4l 4.4 Modal Parameter Estimation ...................... 45 4.5 Mode Shape Display and Interpretation of Results ...................................... 49 iv 4.6 The Validity of the Modal Data .................. 53 4.7 Summary ......................................... 54 5. MODAL ANALYSIS AT MICHIGAN STATE UNIVERSITY .......... 55 5.1 The Undergraduate Curriculum .................... 55 5.1.1 ME 346: Instrumentation Laboratory (3 credits) .............................. 56 5.1.2 ME 464: Computer Assisted Design II ...... 56 5.1.3 Other Undergraduate Courses .............. 57 5.2 The Graduate Curriculum ......................... 60 5.2.1 A Graduate Course in Modal Analysis ...... 60 5.2.2 Other Graduate Courses ................... 6O 6. SUMMARY AND RECOMMENDATIONS .......................... 64 6.1 Summary ......................................... 64 6.2 Recommendations ................................. 64 APPENDIX .................................................. 66 LIST OF REFERENCES ........................................ 67 GENERAL REFERENCES ........................................ 69 3.1 4.l 4.2 5.1 5.2 5.3 5.4 LIST OF TABLES Different Forms of Transfer Function for Mechanical Structures ................................ 11 Modal Frequency Bands ................................ 44 Mode Parameters ...................................... 49 Outline for ME 464 ................................... 58 Undergraduate Courses Related to Modal Analysis ............................................. 59 Outline for ME 824 ................................... 62 Graduate Courses Related to Modal Analysis ............................................. 63 vi 3-l 3-2 3-ll 3-12 3-l3 LIST OF FIGURES Modes of Vibration ................................... 5 Impact Testing Method ................................ 14 Time History of Impact Force Using Three Different Hammer Tips .......................... 15 Frequency Domain of Impact Force Using Three Different Hammer Tips .......................... 15 Frequency Response of Multi Degree of Freedom System ....................................... 18 The Effects of Lower and Higher Modes ................ 18 The Difference Between Light Modal Overlap and Heavy Modal Overlap .............................. 19 Alternative Forms of Frequency Response and Single Degree of Freedom Curve Fitting ............... 20 Alternative Forms of Frequency Response and Single Degree of Freedom Curve Fitting ............... 21 Multi Degree of Freedom Curve Fit .................... 23 Measurement of Signal plus Noise ..................... 25 The Use of the Coherence Function and the Effects of Averaging ............................. 26 The Use of the Coherence Function and the Effect of Averaging .............................. 27 Base Band vs. Zoom Transform ......................... 3O Illustration of Aliasing. Cosine waves at 2 Hz and 99 Hz. Sampling Frequency less than 2x99 ....................................... 32 Illustration of leakage .............................. 33 vii 3-14 4-1 4-2 4-3 4-4 4-5 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-11 4-12 4-l3 Hanning Window and Esponential Nindow ................ 34 Z-Beam With the Points that Define its Geometry ...... 37 Typical Harmer Kit for Modal Analysis ................ 38 Z-Beam with the Instrumentation on it ................ 39 Time History and Frequency Domain of of Impact Force Using a Nylon Tip .................... 40 Difference Between Good and Bad Data ................. 42 Difference Between Good and Bad Data ................. 43 Example of Node Frequency Selection .................. 44 Circle Fit 360-500 Hz ................................ 46 Circle Fit 600-700 Hz ................................ 46 Determination of Modes Between 550-720 Hz via Circle Fit ....................................... 47 Multi Degree of Freedom Curve Fit .................... 48 Modes of Z-Beam; 3 Frames ............................ 50 Modes of Z-Beam; 3 Frames .............. ' .............. 51 The Illustration of Modal Response at a Mode ............................................ 52 The Modal Fit on an Arbitrary Transfer Function Data ........................................ 53 viii CHAPTER I INTRODUCTION Modal analysis is a procedure for describing the motion of a structure by identifying its modes of vibration. The motion of the structure is assumed to be linear and a mode of vibration may be thought of as a property of a structure. Each mode has a specific resonant frequency, damping factor and mode shape which identifies the mode spatially over the entire structure. Once these properties are known, the response of the structure to any input force can be predicted and, if necessary, modified. This thesis will discuss several different approaches to the determination of the motion of a structure. Chapter 2 gives a brief overview of analytical techniques, including dynamic sub- structuring and complications due to damping. Chapter 3 discusses the details of modal testing and Chapter 4 presents an example of testing techniques. Chapter 5 is the study of the potential impact of modal analysis on the Mechanical Engineering curriculum at Michigan State University. Chapter 6 presents a summary and recommendations. CHAPTER 2 ANALYTICAL APPROACH In recent years, computer based analytical techniques to aid in the understanding of dynamics of mechanical structures have become more sophisticated. These techniques may be purely ana- lytical, starting with a mathematical model and proceeding through desired calculations. Or they can start from measured results, and perform calculations which cast these results into more useful form. Initially, I will discuss a common purely ana- lytical approach, the finite element method. 2.1 The Finite Element Method In order to get accurate results, one has to depend on an accurate mathematical model. Since analytical techniques work with mathematical models, modeling has become an important part of analysis. Because of the requirement for a generalized method for modeling the dynamics of large, complex structures with nonhomo- geneous physical properties, an analytical technique called the finite element method [1] has been developed and used as a model- ing tool. The object of the finite element method is to sub- divide a structure into many smaller elements such as plates, beams, etc. Then the equations describing structure are con- structed from equations describing each of the individual ele- ments plus all the boundary and loading conditions on the model. When the finite element method is used for vibration prob- lems, the model leads to a set of simultaneous second order linear differential equations which describe the elastic motion of a complex mechanical structure. These equations are often written as: [M]{x(t)} + mmm = mm (24) where: Acceleration vector [M] = Mass matrix {x(t)} [K] = Stiffness matrix {x(t)} Displacement vector {F(t)} = Applied force vector If the system has n degrees of freedom, then the matrices are n by n, and vectors are n-dimensional. The matrices are real and symmetric. 2.2 Diagonalization A common method to solve these equations is to diagonalize them [2]. This is done by transforming the equations of motion to a new coordinate system called generalized coordinates in diagonal or uncoupled form as shown in equation (2-2). 2 T 21G.) x1 0. . go.) nn ”1' - f(t) I 21:“) + o x: q2(t) = 11215: tilt) (2-2) The transformation relating the generalized coordinates to the original coordinate system is a matrix, the columns of which are the eigenvectors of the system. x1(t) [ ”11 012.... q](t) x2”) = ”21 U22 “2“) (2‘3) Therefore diagonalization involves finding the eigenvalues, A1, and eigenvectors, uij' Once the equations of motion are in diagonal form it is much easier to understand them and to solve for the motion resulting from applied forces [2,3]. 2.3 Modal Analysis Modal analysis may be defined as the process of charac- terizing the dynamics of a structure in terms of its modes of vibration. These are the eigenvalues and eigenvectors of the mathe- matical model. That is, the eigenvalues of the equations of motion correspond to frequencies at which the structure tends to vibrate with a predominant well¢defined deformation. The rela- tive deformation is specified by the corresponding eigenvector. Therefore each mode of vibration is defined by an eigenvalue (resonant frequency) and corresponding eigenvector (mode shape). A schematic diagram of the first two modes of a cantiliver beam is shown in figure 2.1. Each of these modes corresponds to motion at a particular natural frequency. The natural frequencies and modes of vibration of a struc- ture are very useful information, for they tell the frequencies at which the structure can be excited easily and relate the excitation to the applied forces. This information in many cases is sufficient to indicate how to modify the structural design in order to deal with its noise and vibration effectively. / - \ 2nd MODE “~ \ 1StMODE “\ \ I ‘1= z (s, + .) (3-5) k=] pk S-pk where _ th pk - k root of Det[B(s)] * pk = Complex conjugate of pk [Ak] = Residue matrix for the kth root [Ak*] = Complex conjugate of [Ak]. The roots of Det [8(5)] can be written as: ll pk 3 "Ok + IWk pk = "Ok ' iwk (3‘6) where °k modal damping wk = damped natural frequency Equation (3-5) yields two of the three modal parameters, the resonant frequency and the damping. The modal vectors (eigen- vectors) are also needed. They are the solution to the homo- geneous equation: [8(pk)]{Uk} = 0 (3-7) The eigenvectors are proportional to the residue matrix in equation (3-5) [6], and the modal vectors represent a deformation pattern of the structure for a particular frequency. The de- flected deformation of a structure which describes a natural mode of vibration is defined by known ratios of the amplitude of the motion at the various points on the structure. TABLE 3.] DIFFERENT FORMS OF TRANSFER FUNCTION FOR MECHANICAL STRUCTURES Dynamic Compliance = Disp/Force Stiffness = Force/Disp Mobility = Vel/Force Mech. Impedance = Force/Vel Acceleration = Acc/Force Dynamic Mass = Force/Acc l2 3.2 Digital Signal Processing The main function of modal testing is to analyze the fre- quency response functions of mechanical structures. The general scheme for measuring frequency response functions consists of measuring simultaneously an input excitation and response signal in the time domain, Fourier transforming the signals and then forming the transfer functions by dividing the transformed res- ponse by the transformed input. This procedure is based upon the use of digital signal processing. The development, within the last decade, of both digital hardware and computer algorithms for the various transform techniques has made digital signal processing practical for the solution of structural dynamics problems. The area of digital signal processing is very broad. Here the focus is limited to those topics which are useful for es- timating the frequency response and modal properties. These topics are (a) inter-relationship between the time, frequency and s-domain [7], (b) Fourier transform, discrete Fourier transform [8,9], (c) signal sampling [8,10], (d) Correlation and Power spectrum [8,10], (e) Transfer Function and Coherence Function. 3.3 Excitation Methods There are various types of input excitation methods in- cluding random (pure, pseudo, periodic), sinusoidal, transient (impact, step relaxation). They each have their advantages and disadvantages. Reference [11] discusses each method in some detail. In this thesis, the emphasis will be put on impact testing. l2 3.2 Digital Signal Processing The main function of modal testing is to analyze the fre- quency response functions of mechanical structures. The general scheme for measuring frequency response functions consists of measuring simultaneously an input excitation and response signal in the time domain, Fourier transforming the signals and then forming the transfer functions by dividing the transformed res- ponse by the transformed input. This procedure is based upon the use of digital signal processing. The development, within the last decade, of both digital hardware and computer algorithms for the various transform techniques has made digital signal processing practical for the solution of structural dynamics problems. The area of digital signal processing is very broad. Here the focus is limited to those topics which are useful for es- timating the frequency response and modal properties. These topics are (a) inter-relationship between the time, frequency and s-domain [7], (b) Fourier transform, discrete Fourier transform [8,9], (c) signal sampling [8,10], (d) Correlation and Power spectrum [8,10], (e) Transfer Function and Coherence Function. 3.3 Excitation Methods There are various types of input excitation methods in— cluding random (pure, pseudo, periodic), sinusoidal, transient (impact, step relaxation). They each have their advantages and disadvantages. Reference [11] discusses each method in some detail. In this thesis, the emphasis will be put on impact testing. 13 Impact testing is fast, easy to perform and requires less time for the setup than the shakers which are used in other methods. The steps used in impact testing are shown in Figure 3.l. The figure illustrates a hand-held hammer with a load cell mounted to it to impact the structure. The load cell measures the input force and an accelerometer mounted on the structure measures the response. The frequency content and duration of the input force can be altered by using a softer or harder hammer tip. In general the longer the duration of the force impulse the lower the frequency range of the excitation. Therefore a hammer with a hard tip can be used to emphasize higher frequency excita- tion whereas a softer tip can be used to emphasize lower fre- quency excitation. Figures 3-2 and 3-3 present the force impulse of different hammer tips in the time and frequency domain. . Figure 3.1 also illustrates the accelerometer which measured the instantaneous acceleration of a vibrating structure. Reference 12 gives a good explanation of the use of accelero- meters in these measurements. The process of measuring a set of responses (i.e. transfer functions) may be either mounting a stationary accelerometer on the structure and moving the input force from point to point, or exciting the structure at one location and moving the accel- erometer from point to point. In the former case, a row of the transfer matrix is being measured, whereas in the latter case, column of the transfer matrix is being measured. Either a row or a column contains enough information to construct the rest of the transfer matrix [5]. l4 /+ + 4- r IMPULSE RESPONSE IMPULSE o) 'rqu; 'TNMEZ FREQUENCY FREQUENCY Aerx ' f p '7th V , |cunve FIT J MODAL PARAM. OFREOUENCY . MODE SHAPE "DANWNNGi Figure 3-1 Impact Testing Method. 15 2.DBE 88 M 8 b N u ' g x *1: x m, x t- , -2.BBE BB O.BDE-Bl TIME (SEC) 4.88E-D3 AleAMMER FORCE CDNT=HARD 0=NEDIUH X=SOFT A3 HAMMER FORCE 848430-008800 HISTORY -REAL PART A4 HAMMER FORCE 848498-388888 212+ 1R+ Figure 3-2 Time History of Impact Force Using Three Different Hammer Tips. 2 ODE-O2 O a C N f. U-m D‘ 2 E l‘. ”"-—-«~L_\ 0.0BE-Bl ETT‘H—Isfu~ -e HT O.DOE-Ol FREQUENCY _£HE? s.uu Do AleAMMER FORCE CONT=HARD O=MEDIUN DOTo=30FT azannmsa FORCE asaaae-egaaee FREDRESP-MODuLUS A4 HAMMER FORCE u404BB—BUUBUB 212+ 1R+ Figure 3-3 Frequency Domain of Impact Force Using Three Different Hammer Tips. 16 The impact testing has some advantages. It is easy to use and fast. It gives good accuracy and it can have very good frequency resolution. But it also has some drawbacks. The amplitude of input force is not easily controlled. It's energy density may not be high enough to excite the entire structure. More energy can be provided by hitting the structure harder but damage may result. Despite these disadvantages, impact testing provides fast solution for trouble shooting vibration problems. For large variety of mechanical structures this method gives satisfactory results. 3.4 Modal Data Identification When a structure is excited by a broadband input force, many of its modes are excited simultaneously. Since the structure is assumed to behave linearly, its transfer functions are the sum of the resonance curves for each of its modes as shown in Figure 3.4. Therefore at any given frequency the transfer func- tion represents the sum of motion of all the medes which have been excited. The response in a certain frequency range can be approximately described in terms of the "Inertia Restraint“ of the lower modes of vibration, the modes of vibration which are resonant in that frequency range, and the "Residual Flexibility" of the higher frequency modes (see Figure 3.5). The effects of lower and higher modes can be represented in additional stiffness and mass matrices [l3]. 17 The amount of overlap from one mode to another depends on (a) frequency separation, (b) damping of the structure and (c) nonlinear effects. Figure 3.6 shows transfer functions for the difference between light and heavy damping. In cases where modal overlap is light the transfer function data can be treated in the vicinity of each peak (resonance) as if it were a single degree of freedom system. In other words it is assumed that the contribution of the tails of adjacent modes near each modal resonance is negligibly small. In these cases, single degree of freedom curve fitting algorithms may be used to identify the characteristics of each resonance. Figure 3.7 presents alternative forms of single degree of freedom curve fitting. The first method is the so-called co- quad method where the modal frequency can be obtained by simply taking the frequency of the peak of the imaginary part of the transfer function or the frequency where the real part of the transfer function is zero. And the residue can be estimated by using the peak value of the imaginary part of the transfer func- tion. The second method is magnitude phase technique, where the modal frequency is the frequency of the peak of the transfer function magnitude or it is the frequency where the phase angle is 90 degrees. The third method, so called "circle fitting", gives the most accurate results. It is a way of estimating the modal parameters by least squared fitting of the parametric form of a circle to MAGNITUDE Figure 3-4 18 ‘A=.-_> F REQU EN Y Frequency Response of a Multi-Degree-of- Freedom System. ‘ ‘i‘flh‘u mean/1 nesrmli or» LOWER M0955 \ Figure 3-5 I I l FREUIHE) (LI FREOUNC‘I RANGE OF INTEREST I I 'I 2:: .d -"I H ezie The effects of Lower and Higher Modes. MAGNITUDE FREQUENCY a) Light Modal Overlap. MAGNITUDE FREQUENCY b) Heavy Modal Overlap. Figure 3-6 The Difference Between Light Modal Overlap and Heavy Modal Overlap. 20 h.) .BOE 82 (I a . ODE-01 JAR 1i __,n—o/."I‘_ Apr/(i Eh. S.BDE BB I.BDE 03 E-BERM 3 948388-888838 FREDR SP-MODULUS O4OEBB-OBBUOB ?2- 22+ a) Real and Imaginary Plots of Frequency Response Function (Co-Quad method). f— Wk‘lf— fi‘m J"_DLL 1 BFE 83 In" 11 , f I 51 .---ft ‘1' I f I -- .95.. 5’ 1.00E-BI /’ ‘4 5.88E BB 1.08E 83 Z-BEAH 5 848389-608888 FREORESP-BODE OdfiSBB-BOBBBO FZ- ”7 ad” b) Bode Plot and Phase Plot of Frequency Response Function 21 ?2- 112+ MODE SHAPE O: SCALE 31.82 MODE COEFFICIENT REAL -I.13 IMAG -1.3l ANPL 1.31' LIMITS 425.56 ':I:I.'L.'P\}QJCIEIISIEJ13* f 1. 1‘ ff 5 _,.,.~?‘ \‘H .fa’jd - ‘“_4 c) Nyquist Plot (Circle Fit) Figure 3-7 Alternative Forms of Frequency Response and Single-Degree-of—Freedom Curve Fitting. 22 the measurement data in Nyquist form. These methods are ex- plained detail in references [15] and [16]. If the modes are closely spaced, then multi-degree of free- dom techniques give much more accurate results [17]. These techniques involve curve fitting a multiple mode form of the transfer function to a frequency interval of measurement data containing several modal resonance peaks. In the process, all the modal parameters for each mode in a given frequency range are simultaneously identified. Figure 3.8 shows the polynomial form of the transfer function which can be used for curve fitting and an illustration of it on a transfer function data. The coef- ficients of the polynomials in the numerator and denominator are identified by curve fitting, and roots of the polynomials which contain modal parameters are found by a root finding routine. This and other multi degree of freedom methods are explained in detail in references [l6], [17], [18].. Many times, modal coupling or noise on the measurement may make it difficult to identify the number of modes and and their parameters from any single measurement. In these cases a curve fitting procedure that identifies modal parameter from the mul- tiple set of measurement should be used. In other words, mul- tiple row or column of transfer matrix should be measured by mounting more than one accelerometer to get more accurate re- sults. This technique is discussed in more detail in Reference [19]. 23 MULTIPLE MODE METHODS Polynomial Form: A +A s+A 52+...+A sm 0 l 2 m H(s) - 2 n + . Bo+81s+st ...+an s 3 3" 2.335 32 n O 5 {Pt I _.v' '\I I5 rlfl ’/ Kim I; ,1 \K / RM. f}... .i;.:" x .. 1/ a I..= r -‘lf Ii I _ 1 .2 - 2 “'38E 3 3.902‘92 (HZ) 3.9es as AlzZ-BEAN ~e.7- . - asefso-aaeeea FREDRESP-HODULUS ”“'“ 35“" I 346439-933339 72- 112+ Figure 3-8 Multi-Degree-of-Freedom Curve Fit. Dotted Line. The Curve Fit. 24 3.5 Noise and Distortion Another important matter in modal testing is the extra- neous noise which is included in the measurement along with the desired signal. Since we are interested in identifying modal parameters from measured input and output, the reliability of the parameter estimates is reduced in proportion to the amount of noise in the measurements. In general, we measure input and output signals and obtain an estimate of transfer function. However, since there is always noise to be considered, the trans- fer function is obtained in more accurate fashion as shown in Figure 3.9 [16]. The effect of noise is reduced as the number of averages grows (the noise term in the Figure 3.9 gets smaller) and the ratio of output to input more accurately estimates the time transfer function. This effect can be quantified in the coherence function. The coherence function is the ratio of response power caused by applied input to measured response power. As the number of averaging goes up, the coherence function becomes much smoother (see Figure 3.10). Whenever transfer functions are measured on a digital Fourier analyzer, the coherence function can also be calculated in terms of averaged input and output autopower and crosspower spectrums [8,9]. The coherence function indicates whether the response is being caused by the input. Values of coherence function less than 1 indicate that an amount of extraneous noise is being measured with the signal. Coherence is used to determine how much averaging is necessary to effectively remove the effects of noise from the measurement. 25 F(w) X(w) INPUT H(w) I _ + y(w) SIGNAL TRANSFER FUNCTION#] OUTPUT - - SIGNAL N(w) NOISE G— = Averaged Auto Spectrum G— G—- ff H = :Yi — Jif- where: fo 6?? Gyf = Averaged Cross Spectrum 2 2 16- l COHERENCE FUNCTION (7 ) 1?”: f yy Figure 3-9 Measurement of Signal Plus Noise. 2E5 se.ee* ULTSfULTS i I I 112 Ii J ‘eow‘H. "e MA: I "U2? I: I‘ 11¢f I -‘ FIEH‘ —I "I"; l g l , r 1 l 7 I c r 1 F 7 1 l 1 g a. raca «HZ? iLlN' «at. a) Frequency Response Data After 2 Averages. are (55 4 n at out sun Tait-auto RAN 953 FREQ e.ggeeea «s s 2 st . a: 2 CHR+282F OFF 499 UQLU 9.99e: . I. o 1 1 1 i I I I i I L I . ' -' - r—r ' “Mfr—v V‘YIQ 1 ~ . “I. . ’- a W .H Y . NH it ' (NH I1 .IT l’llli I (III I; II II?! ~g ten 2 3; III I :2 II : CLTSIULTS :1 ”l i --..—— b) Coherence Function of the Same Data After 2 Averages. 27 48.UE mus-wt; Ji. :1 ,11 :x2 ; f A : 1 vi, ‘ r I ,4" ' 2 ' TI] 1 : i! 1 is as.# if f .‘.:I I :I l -.B if“ T I T T T l g l 1 1 1 I I V I a FREQ (HZ) (L2H) I844 c) Frequency Response Data After 8 Averages. 53H CH9 4 U RC RUG SUN TRIG-RUTD HRH RES FREQ 5.385859 CHE/R CHA 2 U RC 9/ 3 CHR+292P OFF 499 URLU 9.9556 1 139174le T: "_ I I l Ifl l L-I_$e I i_§I 1 'I 1 I l I 5‘93 -II I if I 1: 1 HI TC .I” TC . i5: .-' he») ; d) Coherence Function of the Same Data After 8 Averages. Figure 3-10 The Use of the Coherence Function and the Effects of Averaging. 28 Distortion or nonlinear motion is another important subject to consider in vibration measurements. Since modal analysis techniques are based on assumed linearity of the dynamic model, the measurements should not reflect any nonlinear motion. Power spectrum averaging does help to reduce this kind of an effect [16]. In addition, different types of excitation techniques to use for testing in order to reduce nonlinear effects [11]. 3.6 Measurement Resolution Since the accuracy of modal parameters depends on the ac- curacy of the transfer function measurements, frequency resolu- tion is extremely important. In addition, curve fitting a1- gorithms are heavily dependent on adequate resolution. In the past, many Fourier analyzers have been limited to Base Band Fourier Analysis (BBFA), i.e., the Fourier transform is computed in a frequency range from zero to some maximum frequency Fmax' This digital Fourier transform is spread over a fixed number of frequency lines which limits the frequency resolution between lines. Therefore BBFA provides uniform frequency reso- lution from O to Fmax and the frequency resolution can be ex- pressed as Af = F /(N/2), where N is the number of sampling max points. From a practical point of view, in many structures modal coupling is so strong that increased frequency resolution is a necessity for achieving reliable results. In BBFA, the only way to obtain better resolution over this bandwith is to use a larger memory, collect more data and compute the spectral functions using more points which would increase the processing time. 29 More recently the implementation of Band Selectable Fourier Analysis (BSFA), the so-called "zoom“ transform, has made it possible to perform Fourier analysis over a frequency band whose upper and lower frequency limits are independently selectable. The resolution obtained in the frequency band of interest is approximately of = BW/(N/Z) where bandwith is the frequency region of interest. Therefore a narrow region of interest would increase the frequency resolution without increasing the number of spectral lines in the computer. However, the processing time gets longer as the bandwith gets narrower. Figure 3.11 shows the comparison between BBFA and BSFA. Reference 20 gives a good discussion about both BBFA and BSFA. 3.7 Some Other Considerations in Modal Testing There are several factors that contribute to the quality of actual measured transfer and coherence function estimates. I have already discussed several important ones such as the excitation method, noise, distortion and frequency resolution. There are a few more considerations to be mentioned in modal testing. They are: 3.7.1 Aliasing Sampling a signal at discrete times introduces a form of amplitude distortion called aliasing that converts high frequency energy to lower frequencies. If the sampling rate for an in- coming signal is not greater than twice the highest frequency of any component in the signal, then some of the high frequency a) b) Figure 3-11 ‘ P i 1 L 1 l i I I 1 I l ' ._ l . 1 I I A I l A; :8 , U54 I i I : E I i 's i. f". « l I i I ‘ g . '- I H I 5 f I . s . ' V‘ ’ . I ’ ’ . i . A. -. I I I - I . : ’, P: . 1 L A A 1 v V : . If _ __ I Y. . . , ’ : I I ' I. ' 3 1 . x! 5 ’ . , l l . ‘ 1 I 1 I I I I I I I I I 9 ‘ . 3 i l I . i l ‘: ‘ . I 'I it 4 i I x | l :1 ’ I i I I I. I I I g i i 7‘ I' I I .~ g ' j 1 I I, i I ; ,' g Q 1 i ‘1 -. _ x L ._ l , IV 7 I r r 1 r Y I l ' 't 1 I' j 5‘ l .I I :- 5 ’ I . I . . ‘ . I 1 I I f I r 1 3 I 3 IR I ‘ a II 1 l I . i, - . I .- I A A L A_A' Is. I I I. If i ‘v‘ *. . s r . 5' ; 2 ._i ‘f\/‘ I,_’, j -39 ;g I - fee ‘I 5 If i u E l I l :- 1 1 I x I . n , 8 FEE? HZ 'LINi SVBB Base Band Transfer Function. 3 Z:F' ' I A I A t *é’lxt? I i ,. l - I ‘ .. .u. I - -' 'd‘qyu‘f 'q ' ‘- 2"‘*.r“h . «(V U'" f" : u' I I I .h '2 P1 ‘_ . .. . ,-.,' '~.,»..-' {LA was-oi” w “m” "“429- "a . W '5’ 1 -. JP. '51: I 7 l I l 4-5- ’-~ ~~~f' - 2r;' 7.;; can : : EU; fit: 4 teen. .~ .ioo BSFA Transfer Function (Zoom). Base Band vs. Zoom Transform. 31 components of the signal will be effectively translated down to be less than one half of the sampling rate [8]. This translation may cause serious problems with interference between high and low frequency components. In Figure 3-12, cosine waves with two different frequencies are shown in the frequency domain. Since sampling frequency was not greater than twice the highest frequency, it caused that frequency to appear in the lower frequency region. To avoid these interference effects, the signals can be sampled at a sufficiently high rate and/or a low pass filter can be put to reduce the amplitude of the higher frequency component so they are not longer large enough to be troublesome. References [8], [9] and [l0] give more detail about aliasing and filtering. 3.7.2 Leakage When a signal of finite length is sampled and Fourier trans- formed, the resulting transform is representative of a periodic signal for which the sampled signal is one period. If the origi- nal signal before sampling was not periodic, it will cause a smearing of data in the vicinity of peaks in the spectrum, which introduces another type of distortion called leakage. A simple example of what can happen is shown in Figure 3.l3, where a sinewave has been sampled. The discontinuity at the ends leads to set of components in the analysis that may interfere with the components of interest. Several techniques have been developed to reduce the effects of "leakage", one of which is Hanning window. Reference [l0] gives a good discussion of leakage. MAGNI TU DE 32 80 l 68— 40 d 20 .. a I IT TITI III III ITT 0 10 20 38 4e FREQUENCY Figure 3-12 I11ustration of AIiasing. Cosine Waves at 2 Hz and 99 Hz. SampIing frequency 1ess than 2 x 99. 33 flfl mg, I \J V \J \J ITvmoow common I Figure 3-l3 Illustration of Leakage. 3.7.3 Windowing The purpose of windowing is to remove unwanted character- istics of the signals. The most commonly used windowing tech- niques are Hanning and exponential [9,l0]. When Hanning is used, the data at the ends of the window are ignored since they are multiplied by a value near zero (Figure 3.14). It is important that the data window be wide enough to ensure that the important behavior is centered within the window. Another commonly used technique is exponential weighting, which multiplies both the input and output signals by an ex- ponentially decaying envelope (see Figure 3.l4). In the case of impulsive excitation, in which the signal/noise ratio is greater at smaller values of time, the weighting rejects most of the noise. This procedure leads to more consistent determination of resonant peak amplitudes. But it also makes the determination of closely spaced modes more difficult. Therefore "zoom" trans- 34 [\ /\ [\ / . V \/ \/ SINE WAVE A . HANNING WINDOW SINE WAVE MULTIPLIED BY THE HANNING WINDOW a) Hanning Window. b) Exponential Window. Figure 3-l4 Hanning Window and Exponential Window. 35 form analysis may be required in some cases to allow sufficient resolution of closely spaced modes. 3.8 Summary Modal testing is based on frequency response information of the structure. The general scheme for measuring frequency res- ponse functions consists of measuring simultaneously an input excitation with a load cell mounted to a hammer in impact testing or a shaker in other methods of excitation, and the response signal with a transducer, preferably accelerometer, mounted to the structure. Digital signal processing techniques are applied to these signals. Then the modal parameters such as resonant frequency, damping and mode shapes can be obtained with using single degree-of-freedom or multi degree-of-freedom.curve fitting algorithms. There are several factors that should be considered to obtain accurate test results, including (a) the selection of input excitation method, (b) the selection of curve fitting algorithm, (c) noise and distortion, (d) Measurement resolution, and (e) aliasing, leakage and windowing. CHAPTER 4 AN EXAMPLE OF MODAL TESTING Modal testing was performed on a Z shaped aluminum beam (Z- Beam) using Gen-Rad 2508 Structural Analysis System [Zl] uti- lizing the SDRC MODAL PLUS software. The Case Center for CAD (Computer-Aided Design) at Michigan State University has a Gen- Rad 2507 [Zl] which is very similar to Gen-Rad 2508. SDRC MODAL PLUS is a joint software product of SDRC (Structural Dynamics and Research Corporation), Cincinnati, Ohio and Gen-Rad, Inc. AVA Div., Santa Clara, California. The Modal Plus software is also included in Gen-Rad 2507, Structural Analysis System. This chapter uses the Z-Beam as an example to demonstrate the procedure of modal testing. The implementation of the methods in MODAL PLUS requires the following steps: definition of the geometry of a structure, excitation of the structure and data acquisition, computation of the frequency response function, estimation of modal parameters, and generation and display of mode shapes. Each of these subjects is covered in this chapter. 4.l Geometry Definition The implementation of modal analysis depends on the geometric definition of the structure. That is, a coordinate system must be selected and several points on the structure must be defined. 36 37 Additional points yield better estimates of mode shapes, but more points require that more data must be collected. The structure in this case was a "Z" shaped aluminum beam (Z-Beam) which was modeled with a 28 points. The base of the Z-Beam was clamped to the ground (see Figure (4-l). Figure 4-1 Z-Beam with the points that define its geometry. 38 4.2 Excitation Technique In order to estimate frequency response function from mea- sured data, one must supply an excitation function which is rich in energy at all frequencies of interest. In this case, the impact method was used with the hammer kit shown in Figure 4.2. The kit includes the hammer, a load cell attached to the hammer and an accelerometer. In this experiment, the accelerometer remained at one loca- tion and the hammer was used to impact the 28 points (see Figure 4.3). A nylon tip was used for the hammer (medium tip). A time history and the corresponding frequency function of a hammer impact are shown in Figure 4.4. Figure 4-2 Typical Hammer Kit for Modal Analysis. 39 (SerURad Figure 4-3 Z-Beam with the Instrumentation on it. 40 l' u an *‘I m G! MOE—{HICIIEX ISI -2.DEE DD D.BDE-BI TIME (SEC) 1.68E-D2 94:2-BEAH a-BEAM HAHN ER RESPONSE E43423 Egaaao H113TDRY -REAL PART u4u43u- uuuBuB 212+ 1x+ a) Time History 1.86E-32 H E b H I T U c. E yfibfifi 3%; ‘-._.._ ,_ . - — —"_"—!.— a N R“ B.DDE-81 , E EE E- U1 FEEDUEMIIV I1HZ.: 1.38E 83 H4zE-BEHN Z-BEHH HHMMER RESPONSE 9484:33- 883683 FREDPESP- MDDULUS 44E4 au- BBuuuB 212+ b) Frequency Domain Figure 4-4 Time History and Frequency Domain of Impact Force Using a Nylon Tip. 41 4.3 Frequency Response Data Analysis After preliminary calculations it was decided that O-lOOO Hz frequency was adequate. Data was for accelerations in the ver- tical direction. Data was taken at each point with 5 averages and the coherence function was analyzed before accepting transfer function data. Figure 4-5 illustrates the difference between good and bad transfer function data with the aid of coherence function. Frequency response functions such as Figure 4-6 were in- spected for resonant peaks to determine at what frequencies modal estimates should be obtained. The numbers on the peaks indicate the use of a digital cursor to obtain the frequency and magnitude values listed on the left side of the plot. Inspection of this plot resulted in the selection of the frequency bands listed in Table 4.l as the probable location of significant modes of vibra- tion. b) 1.86E 32 M H E N I T U D E a... ‘71 I}; IS: 0“ a) 42 A .R—U— “QUE BB FEEDUEMCY (HZ) 1.83E 33 FREQPESP- -%%DULUS I; 'b- Good Transfer Function Data. 1". IS: —-1H 2 '3‘: I» 3 '3‘ p..- i3.“ ~— o.— K1“~an Mmpm (V l "‘14“ “‘— - m c:- ._...... y J~m 3. EE- 31 44.2-35am 3.88E 88 '5 FREQUENC? Good Coherence Function. (H2) @4121 Dan 1.38E 33 :PEETPUN MDDULUS 112+ 3 53: as; a 4: —h) 1.886 82 III—033 B.BBE-Bl RizZ-BERN C) p—o 0;. ISI "’1 IS: moc: ~iH 2:7: 1:3 (‘1' D.BBE-Dl n4zz-aenn d) Figure 4-5 43 I U l I I % cAWWWrmL ‘W ‘1 AMW‘ V“ saws DD FREQUENCY (H2! 1.886 83 a4a4sa-aoaoee Fpeoeg;sp nooyLus 846438-388833 .-- 11¢+ Bad Transfer Function Data “-3 in 5.83E BB ‘3 Bad Coherence Function FREQUENCY ( HZ} 1.38E B3 spsxztggui HDDULUS lb 112+ Difference Between Good and Bad Data. 5.366 21 n 3 b 1F: 3.4315 9:" 1n: 6.961E 99% 2F: 7.3545 913 an: 1.2955 815 4 SF: 9.444: 81 3n: 1.5088 01 R 4F: 4.3335 82 421: 2.7352 61 3 A 5F: 5.5395 32 11 A an: 3.4352 68 f i \ fi5 9.305-91 -—-——e=,_e_uzf‘ 3””“dd_i: n: , Benn 5.335‘99 FREQUENCY (HZ) 3.388 a. :go ‘ 4 . 9-933329 FREQRE”P-HODULUS 343233-393999 7i1 172+ Figure 4-6 Example of Mode Frequency Se1ection. TABLE 4.] MODAL FREQUENCY BANDS Mode Number Frequency Band (Hz) 1 29 - 39.8 2 72 - 86 3 89 - 100 4 428 - 450 5 648 — 670 45 4.4 Modal Parameter Estimation Two methods were employed in MODAL PLUS in extracting modal parameter; (1) Circle fit (2) Multiple degree of freedom curve fit. Figure 4-7 and 4-8 are two examples of circles fitted to data. Figure 4.7 presents data in the frequency range from 360 to 500 Hz. The points are quite dense, indicating a well re- solved spectral analysis. However, in Figure 4-8, which is in the frequency range of 600 to 700 Hz, there appears to be a second mode of smaller magnitude at the right hand side. Con- sidering this point, circle fit was used in the frequency range of 550 to 650 Hz and 655 to 720 Hz, and two seperate modes were found between 550 and 700 Hz. (See Figure 4-9.) Multi degree of freedom curve fitting method was employed in the frequency range of 500 to lOOO Hz and l0 to l000 Hz (see Figure 4-l0), the mode which was noticed in Figure 4-10 was also noticable in the 500 to lOOO Hz frequency range. Table 4.2 shows the modal parameters for the modes of Z-Beam from Figure 4-6. 46 F2- 112+ MODE SHRPE B: SCRLE 31.32 NUDE COEFFICIENT RERL B.BBBBBE—Bl IMQS -1.383665 31 RHPL 1.33366E BI LIMITS 363.833 586.938 (n.‘L.’R}QIC}z}SJEJI)# If a Figure 4-7 Circle Fit 360—500 Hz. FZ- 112+ HUGE SHQPE B: SCALE 99.99 NUDE COEFFICIENT RERL 0.68398E-31 IMRE -4.12529E BI RMPL 4.125285 Bl LIMITS 688.883 ?BB.BBB {H.‘LJR!QJCJEJSJEJI3% # Figure 4-8 Circle Fit 600-700 Hz. 47 ?2- 112+ MODE SHRPE O: SCRLE 99.99 MODE COEFFICIENT RERL 8 88888E-81 IMHO -3.9?28?E 81 QMPL 3. F287E 81 LIMITS 655.888 728.888 lig.‘LJRIQ.ICIZJS!EII)* a) Circle Fit 655-720 Hz. SCALE 99.99 8 REQL 8.88888E-81 IMHO 3.35458E 81 QMPL 3.35458E 81 LIMITS 558.888 658.888 {RJLJRIQJCIEJSIEJI :‘# b) Circle Fit 550-650 Hz. Figure 4-9 Determination of Modes Between 550-720 Hz via Circle Fit. 48 —-7‘.— 1m: 1.335 32 n B L? N 1 U 23.x” o E 1 3‘ 1" )1 H T 1 335—32 3533 5.335335 FREQUENCY (H2) 1. 335 33 .1:._ 32:3-5533 s 343333-333333 55535555 3355 343433-333333 32- 113+ a) Multi Degree of Freedom Curve Fit SDO-lDDO Hz. sen . -0 . 1 335 32 n 3 5 N 1 5 F J 3 E 53 I 1 4"!) W l M3 L1 I '3 0 _ ., ‘5 1.3335531 5125311511511 1:321 1.935 9;, Siii-BEH' 34333 333333 55535 7533 3335 343433 333333 - 115+ b) Multi Degree of Freedom Curve Fit lO-lOOO Hz. Figure 4-l0 Multi Degree of Freedom Curve Fit. Solid Line-Data Circles Fit. 49 TABLE 4.2 MODE PARAMETERS MODE FREQUENCY DAMPING AMPLITUDE PHASE 1 33.831 0.007358 10.33 1.7396 2 82.246 0.043136 179.7 -1.1411 3 98.664 0.74359 134.3 -2.6723 4 439.221 0.013872 507.0 -1.4423 5 658.719 0.015534 2542.0 -1.8938 4.5 Mode Shape Display and Interpretation of Results Once the modal parameters are estimated for all points on the geometry of a structure, then these parameters may be asso- ciated with the structural geometry. This facilitates the dis— play of animated mode shapes and global visualization of the structural vibration. For the Z-Beam, the mode shape display task software in MODAL PLUS was employed. Figure 4-11 shows four mode shapes of the Z-Beam. Figure 4-lla presents three frames of a mode at 32.3 Hz. This mode has a modal node very close to the position where the accelerometer was attached which causes it to almost disappear from the "driving point" frequency response data. This is illustrated in Figure 4-12. a) b) 50 l, i I ,9” I .--".-f .x’ .' .l .' ~-~-- ------- ..................... ..---......_,,, _, ..... -"' “fin-"v” . , ‘,_ -.. ”...; mum-Mf::_:,’“ ""' / / .mflwmnmsmswy~~“‘ '"“~¢L/ ,/ I" ’1' 1_ 7'2- CI..MP.-F= 32.338 HZ ( 8.9. 2.8: $3.13, 1:38.BIJ=UIEH Mode at 32.3 Hz. “...-......” _, ...... - —-—._.. a" """ \. -... _ :5" ~" «...—...... ,- "" ~s .. .../f .;,!l" .. ..... ..--.. .--'°:f.~_r 'va - ...“ ......— - J o . u l i 0. n 1" a __..- f -..-...... . . I0. I. .' . UUUUUUUU ‘I... 4'. l" ... f 3' J i. ., _. o n' J r" J. I: I f 2' ." .' 0" I. I, n. 1‘ t' .v ,r .' .r' f. .. .. '. . . ,P’ __c I‘ .“ aJ .' r" I" u“ z' '0. .- - cat-.-.“um l"’-" I... I...) 1.- {-13'1" £93". .’.-'a’ .4-1.’ ’1 F’Z- IZIJMP1F= 31.13813 H2 '5. 13.8 2.13; 8.0.: 1:38.BIJ=UIEH Mode at 81.00 Hz. 51 “' .- I. . " _ 'I. ’1’ —" .. nnnnnnnnnnnn {/I/ o.— ”-I" -'I-. I . ~ “...-m w‘-‘ "humayv .... .- 5“ r .. ”‘t _. '- ”...-..."."T‘ .6- .. -‘ ’5? 4: 72- COMPlF= 434.888 HZ ( 8.8; 2.8; 8.8; 138.83=UIEH c) Mode at 434 Hz. . . .. ... '- .0 .0 . ... .._ .... .... ...- ... . .. . .."- ... A. ... . .....""/ .. 0‘ 0" of O i 1' .’ ."z .-_.» I .........................l .- .5 z _. 7. , ,. _a _- .4 2‘ .".-" .I‘ I .- - ...-”.1 . . . _} 39;." n2»? ._.‘,‘_. ’1' 5: FE- CDMPJF= 654.88 (SI I IN! A IS! IS?! 2.8. 8.8; 188.8)=UIEN d) Mode at 654 Hz Figure 4-ll Modes of Z-Beam; 3 Frames. 52 --~... _ . Jan 2.88E 82 " 11 g 3&1 1 1 :5 :1 // .. .4 P‘ E ”Ff—A" Uh. 1 b) point 7. Figure 4-12 2.935—92 . 1.655sea FREQUENCY 1H2) 1.335 83 g‘ ‘7BERN 848388-888898 Feeoegsp-qu‘ 848438-888888 rc— re- a) Transfer Function at the Driving Point. ”‘f-* I AIIFA -:7 ~ —=‘I - 2.395 92 ~ n 8 L: N 1 .Ir 1 1 l. ‘1. 1 I . 1“ U I ‘4 1'. f ; ”1,4 0 1 5 »/”‘x. E ,/ k \ { x” ‘3, 11 14 WLHH. l g ‘5‘“\ ! Audit | 1 -a\N~J/2‘Js/ ' 1 2.885-82 55 m 1.335539 FREQUEHC? (H2) 1.395 93 “1.:- n - H‘ ' 545335-295593 55595535-3055 - c4a4aa—aoaaae 52- 115+ Transfer Function Input at Point ll response from The Illustration of Modal Response at a Node. 53 4.6 The Validity of the Modal Data The validity of the modal data can be assessed by syn- thesizing various response functions, and then comparing them with measured data. If the modal data is accurate, the fit from it should be fairly close to any frequency response function of the structure. In this case, the modal data was obtained by analyzing 5 modes in frequency band of 0-1000 Hz. i.e., the Z-Beam frequency response representation was the sum of the 5 modes. Figure 4-l3 shows the frequency data and the fit from the modal data, between points 7 and l9 where the accelerometer is at point 7. The fit indicates that the modal data matches fairly well with the modes for this frequency response data. "TOG—4H2 ’11 3 '90.. J“;- ,l'“ 4‘." ,r e if H’“ / "'2 ; l “R: Tail 7] 13 X1 2.595—52 9 5.105000 1.5u5 a1 FREQUENCY (HZ) 1.355 53 §_:2-555n s u 5» I I' 945339-955329 55595555-qu5 545433-359539 .5- 135+ Figure 4-l3 The Modal Fit on an Arbitrary Transfer Function Data. Circles-Fit. Solid Line- Data. 54 4.7 Summary A typical modal test was performed on a Z shaped beam using GenRad 2508 Structural Analysis System with the SDRC MODAL PLUS software. The procedure was to define the geometry, choose an excitation techniques, and identify the characteristic of the dynamics of a system by analyzing its frequency response data and using curve fitting algorithms. The last step was to display the mode shapes, interpret the results and verify the modal data. CHAPTER 5 MODAL ANALYSIS AT MICHIGAN STATE UNIVERSITY The use of analytical and experimental modal analysis has become popular and widely used during the l970's primarily be- cause of the advances in the computer technology. Today, modal analysis has an important role in the design and development of mechanical systems. New developments and better techniques are being researched in industry [4,22] and in the university en- vironment [4, 23]. Various subjects, including vibration theory, matrix al- gebra, Laplace and Fourier transforms, transfer functions, and electronic instrumentation are involved in applications of modal analysis. Most of these topics are covered, at least in part, as part of the required undergraduate curriculum in mechanical engineering at Michigan State University [24]. But in order to gain a satisfactory knowledge of modal analysis, the student should be able to combine these topics. This chapter presents a scenario for the coordinated introduction of modal analysis into mechanical engineering undergraduate and graduate cur- ricula. 5.l The Undergraduate Curriculum The mechanical engineering undergraduate curriculum requires 180 credits, of which l28 credits are related to mathematics, science and engineering. I want to consider only those courses which relate modal analysis theory and its applications. 55 56 These courses will give students the opportunity to blend various topics on this subject. 5.l.l ME 346: Instrumentation Laboratory (3 credits) The prerequisite for this course is ME 351 which presents an overview of several mathematical methods which are essential to the understanding of the topics covered in modal analysis theory. The object of the course ME 346 is to present to students the principles and applications of instrumentation in Mechanical Engineering. Two weeks of the course are devoted to modal test- ing including equipment calibration, set up of an experiment, modal testing, and the display of animated mode shapes. Since the time is limited, a very simple modal experiment should be presented which emphasizes the impact of the instrumentation on modal testing. Thus, ME 346 will introduce modal testing tech- niques. 5.1.2 ME 464: Computer Assisted Design II The objectives of this course are to introduce the modal analysis theory along with the digital signal processing and to help students to understand modal testing techniques by present- ing several experiments. Table 5.1 presents the outline of the course. In addition, if time permits, a comparison between a modal testing of a sample structure and a computer simulation of the same structure should be demonstrated. 57 The present prerequisites are ME 455 (Mechanical Vibrations), which presents material covering up to two degrees of freedom in time domain, and ME 463 (Computer Assisted Design I) which ensures that the students understand interactive graphics. I feel that, in addition, the students also need ME 458 (Control Theory) as a prerequisite. ME 458 gives an overview of Laplace domain, time domain and frequency domain relationships along with the appli- cations of transfer function analysis. Another course which is strongly recommended prior to ME 464 is MTH 334 (Theory and Applications of Matrices). The structure to be tested should be small in size because of space limitations and have simple geometry since any elastic structure would be sufficient for the purpose of learning testing techniques. These considerations will simplify the set up of the experiments and make it easier to feed test point coordinates to the computer for the animation of mode shapes. It would also make it easier to construct a mathematical modal or a simple structure for computer simulation purposes. 5.1.3 Other Undergraduate Courses Table 5.2 summarizes the courses which are related to Modal Analysis in the mechanical engineering undergraduate curriculum at Michigan State University. In my view, it would be appropriate for any of these courses to use the modal equipment on a demon- stration basis, and perhaps, for ME 499, for hands on experiments. 58 TABLE 5.1 OUTLINE FOR ME 464 Analytical Approach to Modal Analysis i) Analysis of the mathematical model ii) Eigenvalues, eigenvectors and their applications iii) Normal mode shapes, complex mode shapes. Fourier Analysis and Digital Signal Processing i) Time domain, Laplace domain and frequency domain. ii) Fourier transform iii) Signal sampling, power spectrum analysis iv) Aliasing, leakage, windowing. Modal Testing i) Transfer function, frequency response function ii) Modes of vibration iii) Excitation methods iv) Modal data identification (i.e., curve fitting algorithms) v) Reduction of measurement noise and coherence function vi) Measurement resolution Presentation of Several Laboratory Experiments and Class Participation in Interpreting the Results. ME ME ME ME ME ME ME ME 346: 422: 455: 458: 463: 464: 499: 59 TABLE 5.2 UNDERGRADUATE COURSES RELATED TO MODAL ANALYSIS COURSE Instrumentation Laboratory Mechanical Engineering Analysis Mechanical Design Projects Mechanical Vibrations Control Theory Computer Assisted Design I Computer Assisted Design 11 Independent Study TOPICS Instrumentation for mechanical engineering applications in- cluding demonstration of modal analysis. Essential mathematical methods to the solution of engineering problems. Application of design concept. Team design project. Basic theory of mechanical vibra- tion and it's applications. One degree and multiple degree of systems, time varying systems. Control systems, application of transfer function analysis, time, Laplace, frequency, domains and stability. Basics of interactive graphics, line fitting and surface develop- ment, finite element analysis. Modal Analysis techniques. Individual Projects. 60 5.2 The Graduate Curriculum The purpose of the graduate program in modal analysis is to take a number of qualified students and lead them to a deeper level of understanding of modal analysis techniques. The grad- uate program will include more mathematical sophistication and an enhanced ability to exploit the laboratory facilities in complex applications. The present Mechanical Engineering graduate curri- culum is given in the Appendix. 5.2.1 A Graduate Course in Modal Analysis (e.g., ME 824: Modal Analysis - Theory and Measurement Techniques) The objectives of this course will be to present the analy- tical and experimental approach to Modal Analysis by building a mathematical foundation from the analytical and experimental point of view and to participate in the "hands on" experiments on various structures using the Modal Analysis testing facilities and Prime 750. The students should understand the material presented in ME 823 (Theory of Vibration I) and ME 464 (Computer Assisted Design II) prior to this course. It is also recommended to take MTH 831 (Matrix Theory) to gain an in depth facility with linear algebra. Table 5.3 presents a course outline. 5.2.2 Other Graduate Courses Table 5.4 shows the graduate courses that are related to Modal Analysis theory in the Engineering College of Michigan State University. 61 By participating in such courses in the graduate curriculum, an engineering student will gain in depth knowledge of a type of structural testing that is becoming increasingly useful to in- dustry. Therefore experience gained in studying modal analysis is often viewed as an asset by employers. This subject also carries exceptional academic utility, since it promotes under- standing of multi-degree of freedom vibrations, and demonstrates the related applicability of several topics such as Laplace and Fourier Transforms and linear algebra. 62 TABLE 5.3 OUTLINE FOR ME 824 l. Modal Testing - Its Techniques and Applications i) Fourier analysis ii) Signal sampling and its considerations iii) Frequency response analysis iv) Modes of vibration v) Excitation methods vi) Curve fitting algorithms vii) Reduction of measurement noise, coherence function 2. Analytical Approach to Modal Analysis 1 Eigenvalues, eigenvectors and their applications ) 11) Effects of various types of damping ) ) 111 Dynamic Substructuring 1v Modal Perturbation Analysis, i.e., modifications on the mathematical model with "what if“ type of questions, e.g., what if we make changes in M, C or K matrices?’ How would the modified structure behave dynamically? 3. Group Projects Involving modal testing and computer simulation of a structure, and correlating the results. 63 TABLE 5.4 GRADUATE COURSES RELATED TO MODAL ANALYSIS COURSE ME 823: ME 824: ME 890: ME 899: MMM 805: MMM 809: SYS 826: Theory of Vibrations I Modal Analysis and its Applications Special Topics Research Strain and Motion Measurements Finite Element Method Linear Concepts in System Science TOPICS Discrete and continuous systems with linear and nonlinear characteristics. Analytical and experimental ap- proach to Modal Analysis, modal analysis of complex structures. Special problems, projects in modal analysis. Research projects in Modal Analysis. Laboratory training in strain and motion measurement. Study of strain gages and accelero- meters. Theory and application of the finite element method. State space and frequency domain models of interconnected systems, solution of continuous and dis- crete time linear systems. CHAPTER 6 SUMMARY AND RECOMMENDATIONS 6.1 Summary This thesis is a study of the theory and an approach to experimental modal analysis of structural dynamics. Initially, a brief overview of analytical techniques was given to show the relationship between purely analytical and purely experimental approaches. Then the experimental approach known as modal ana- lysis was studied in some detail. Various excitation methods were presented and the impact testing method was discussed in detail. Curve fitting methods to identify modal data were presented. Several factors that contribute to the quality of testing were discussed and an example was given to illustrate the technique. Since modal analysis has wide application in industry and since it is topic worthy of academic study from several points of view, the impact of modal analysis on the Mechanical En- gineering curriculum at Michigan State University was dis- cussed, and several courses at different academic levels (i.e., junior, senior, graduate) were proposed. 6.2 Recommendations Modal analysis offers the opportunity to establish an academic focal point for the department of Mechanical Engineering at Michigan State University. I recommend the establishment of several courses devoted to this area. This would bring to the department the 64 65 the following benefits 1) The students would gain an overview of several important tools which, though they are inherently worthwhile, are not now brought together in a cohesive unit. 2) A sophisticated overview of modal analysis is very much in demand throughout the industrial community. Currently, only the University of Cincinnati has a strong reputation in this area. MSU's department is now ideally suited through faculty, students, and equipment to become a major contributor in the field. APPENDIX APPENDIX MASTER'S DEGREE PROGRAM IN MECHANICAL ENGINEERING* (Beginning with Fall Term 1979) (Revised - Effective Fall Term l980) 1. Credit requirements Total credits: 45 (minimum) Plan A (thesis): 30 (minimum) at 800 level or above 8-12 thesis (These credits may count as part of the 30 credit minimum above. The booklet "The Graduate School Guide to the Preparation of Master's Theses and Doctoral Disserta- tions" is available from the Graduate School.) Plan 8 (report): 27 (minimum) at 800 level or above. Six credits of ME 890 are to be taken and applied toward a research/design project and a report. (In each plan, a minimum of 9 credits must be taken outside the Mechanical Engineering Department. Thesis and Report Form de- tails age available at the Mechanical Engineering Headquarters 200 EB. 2. Area requirements Each student must take at least one regularly scheduled 800 level course in each of four areas, chosen from the list below: a. Design: M.E. 826, 827, 828 b. Fluid Mechanics: M.E. 840, 841, 842, 843 c. Heat Transfer: M.E. 810, 813, 814, 817, 832 d. Systems and Control: M.E. 851, 860, 862 e. Thermodynamics: M.E. 815 f. Vibrations: M.E. 823 *The requirements listed here are in addition to those indicated in the college requirements document "MASTER'S DEGREE PROGRAM REGULATIONS COLLEGE OF ENGINEERING, MICHIGAN STATE UNIVERSITY." 66 LIST OF REFERENCES LIST OF REFERENCES l) Oesai, C.S., and Abel, J.F., "Introduction to the Finite Element Method", Van Nostrand Reinhold, New York, 1972. 2) Meirovitch, L., "Analytical Methods in Vibrations", Mac- millon-London Co. 3) Thompson, N.T., "Theory of Vibration with Applications", Prentice Hall Inc. 4) Nelson, F.C., "A review of Substructure Analysis of Vibrating Systems", The Shock and Vibration Digest, Volume 11, No. 11, Nov. 1979. 5) Brown, D.L., I'Modal Analysis - Theory and Measurement Techniques“, A Short Course at the U. of C. Cincinnati, Ohio, Sept. 1977. 6) Richardson, M., and Potter, R., "Identification of the Modal Properties of an Elastic Structure from Measured Transfer Function Data", 20th 115, Albuquerque, New Mexico, May 1974, ISA Reprint. 7) Melsa, J.L., and Schultz, D.G., "Linear Control Systems", McGraw-Hill Book Company. 8) Beranek, L.L., "Noise and Vibration Control", McGraw- Hill Book Company. 9) Otnes, R.K., and Erochson, L., "Applied Time Series Anal- yses Vol. 1", Wiley-Interscience Publication. 10) Peterson, A.P.G., and Gross, Jr., E.E., "Handbook of Noise Measurements", Gen-Rad Corp., 8th Edition Santa Clara, California. ll) Ramsey, K.A., "Effective Measurements for Structural Dynamics Testing, Part 1 and 2", Hewlett Packard Company, Santa Clara, California. 12) Dove, R.C., and Adams, P.H., "Experimental Stress Analysis and Motion Measurement", Charles E. Merrill Publishing Co., Columbus, Ohio. 67 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 68 Klosterman, A.L., and McClelland, N.A., "Combining Experi- mental and Analytical Techniques for Dynamic System Analysis". Structural Dynamics Research Corporation (SDRC), Cincinnati, Ohio. Presented at the 1973 Tokyo Seminar on Finite Element Analysis, Nov. 1973. Pancu, C.D.P., and Kennedy, C.C., "Use of Vectors in Vibration Measurement and Analysis", J. Aeronautical Sciences, Vol. 14, No. 11, Nov. 1947. Rades, M., "Methods for the Analysis of Structural Frequency- Response Measurement Data", Shock and Vibrations Digest, Vol. 8, Pt. 1, 1976. Richardson, M., "Modal Analysis Using Digital Test Systems", Hewlett-Packard Co., Santa Clara, California. Reprinted from Seminar on Under-standing Digital Control and Analysis in Vibration Test Systems. Levy, E.C., "Complex-Curve Fitting", IRE Trans. Ac. 4, 1959. Klosterman, A.L., "A Combined Experimental and Analytical Procedure for Improving Automotive System Dynamics", Struc- tural Dynamics Research Corporation (SDRC), Cincinnati, Ohio. Richardson, M., and Kniskern, J., "Identifying Modes of Large Structures from Multiple Input and Response Measure- ments", Hewlett-Packard Co., Santa Clara, California. McKinney, H.N., "Band-Selectable Fourier Analysis", Hewlett- Packard Journal, Vol. 26, No. 8, April 1975. GenRad Co., Acoustics, Vibration and Analysis Division. Structural Analysis Systems. Structural Measurement Systems, Inc. "An Introduction to the Structural Dynamics Modifcation System", Santa Clara, California. Jennings, A., "Eigenvalue Methods for Vibration Analysis", The Shock and Vibration Digest, Volume 11, No. 11, Nov. 1979. Michigan State University 1979 Description of Courses, Michigan State University Publication. GENERAL REFERENCES 25) Bendat, J.S. and Piersol, A.G., "Random Data: Analysis and Measurement Proceedings", Wiley-Interscience 1971. 26) Bergland, G.D., "A Guided Tour of the Fast Fourier Transform", IEEE, Vol. 6, pp. 41-52, Ju1y 1969. 27) Bergland, G.D., "A Fast Fourier Transform Algorithm for Real-Valued Series", Comm. of ACM. Vol. 11, No. 10, Oct. 1968. 28) Davis, J.C., "Modal Modeling Techniques for Vehicle Shake Analysis", SAE No. 720045. 29) Dudnikov, E.E., "Determination of Transfer Function Coef- ficients of a Linear System from the Initial Portion of an Experimentally Obtained Amplitude-Phase Characteristics", Automation and Remote Control 29, 1, 1959. 30) Enochson, L. and Grafton, M., "An Example of Digital Modal Analysis on a GenRad Signal Analysis System", JACC, June 1979. 31) Halvorsen, N.G. and Brown, D.L., "Impulse Techniques for Structural Frequency Response Testing". 32) Hasselman, T.K., "Damping Synthesis from Substructure Tests“, AIAA Journal, Vol. 14, No. 10, Oct. 1976. 33) Heron, K.H., Skingle, c.w., and Gaukroger, D.R., "Numerical Analysis of Vector Response Loci", J. Sound and Vibs., Vol. 29, No. 3, 1973. 34) Heron, K.H. Skingle, c.w. and Gaukroger, D.R., "The Proces- sing of Response Data to Obtain Modal Frequencies and Damping Ratios", J. Sound and Vib., Vol. 35, No. 4, 1974. 35) Klosterman, A.L., "On the Experimental Determination and Use of Modal Representations of Dynamic Characteristics", Ph.0.)1hesis, Dept. of Mech. Engr., Univ. of Cincinnati 1971 . 69 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 7O Klosterman, A.L., and McClelland, N.A., "NASTRAN for Dynamic Analysis of Vehicle Systems", Structural Dynamics Research Corp. (SDRC), Cincinnati, Ohio. Klosterman, A.L., I'Modal Surveys of Weakly Coupled Systems", Structural Dynamics Research Corp. (SDRC), Cincinnati, Ohio. SAE No. 760876. Kuhar, E.J., and Stahle, C.V., "Dynamic Transformation Method for Modal Synthesis", AIAA Journal, Vol. 21, 1974. Long, G.F., "Understanding Vibration Measurements", Nicolet Scientific Corp. (1975). Norin, R.S., "Pseudo-Random and Random Testing", GenRad Time/Data Division. Peterson, E.L., "Integrating Mechanical Testing Into the Design and Development Process", Aerospace Meeting, Los Angeles, Dec. 1979, SAE Technical Paper Series. Potter, R. and Richardson, M., "Mass, Stiffness and Damping Matrices from Measured Modal Parameters", IIA-Conference and Exhibit, New York City, Oct. 1974, ISA Reprint. Sisson, T., Zimmerman, R., Marte, J., "Determination of Modal Properties of Automotive Bodies and Fourier Using Transient Testing Techniques“, Structural Dynamics Research Corporation (SDRC), Cincinnati, Ohio. Smith, C.C., Thornhill, R.J. and Miller, c.w., "Modal and Vibration Analysis - Theory and Measurement Technique", ASME Winter Annual Meeting, San Francisoc, Ca., December 1978. Strang, G., "Linear Algebra and Its Applications", Academic Press. Tolani, S.K. and Roche, R.D., "Modal Truncation of Sub- structures Used in Free Vibration Analysis", Journal of Engineering for Industry, August 1976. Nalgrave S.C. and Ehlbeck, J.M., "Understanding Modal Analysis", West Coast Meeting, Town & Country, San Diego, August 7-10, 1978. SAE No. 780695. Winfrey, R.C., "The Finite Element Method as Applied to Mechanisms", ASME. finite Element Applications in Vibration Problems. Presented at the Design Engineering Technical Conference, Sept. 1977, pp. 19-40. 31293 03196 7544 .11“ 1|“ “ '11“ I“ H E“ “1| “ ...—WI 1111111 “I "ll 1“ H|