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COUPLED WAVE PROPAGATION IN A ROD WITH A DYNAMIC ABSORBER LAYER by Jiulong Meng BSEE. University of Science and Technology of China, 1988 SUBMIT'ED TO THE DEPARTMENT OF OCEAN ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1991 Copyright @ Massachusetts Institute of Technology, 1991. All rights reserved. Signature of Author __ .1*- __ " _I leartment of Ocean E(gjeering Felo y, 1991 Certified by Professor Ira Dyer Thesis Supervisor Accepted by ProfesWor A. Douglas Carmichael Chairman, Department Committee COUPLED WAVE PROPAGATION IN A ROD WITH A DYNAMIC ABSORBER LAYER by Jiulong Meng Submitted to the Department of Ocean Engineering on February , 1991 in partial fulfillment of the requirements for the degree of Master of Science. Abstract This thesis experimentally tests the effect of a continuous longitudinal dynamic absorber layer on longitudinal wave propagation in a circular rod. Wavenumber-frequency solutions are derived analytically. The associated attenuation and phase velocity results are presented to show how their behavior depends on the loading treatment. Experimental results for the phase velocity are compared to a model developed by Dr. Dyer and Olivieri. A relaxation mechanism is developed to fully model the viscoelastic material. It is also shown that the interaction between longitudinal/flexural waves may lead to significant rates of transformation of the compressional wave energy into bending. Thesis Supervisor: Title: Professor Ira Dyer Weber-Shaughness Professor of Ocean Engineering, MIT Dedication I would like to express my gratitude to many people whose invaluable contribution made this thesis possible. Dr. Ira Dyer, for his wealth of insight, guidance and support for this work. Dr. Richard Lyon and Dr. John Lienhard for their meaningful discussion and advice. Ms. Marilyn Staruch and Mary Toscano. My colleagues at MIT: Joe Bondaryk, Djamil Boulahbal, John Briggs, Chifang Chen, Matthew Conti, Chick Corrado Jr, Joe Deck, Kay Herbert, Tarun Kapoor, Kelvin LePage, Lan Liu, Charles Oppenheimer, Dave Ricks, Ken Rolt, Hee Chun Song, Da Jun Tang; and Alan, Brain, Larry of EECS. A final note of thanks must go to my family. From them I derived the strength, determination and confidence to make it through and to continue moving forward. I dedicate this thesis to the memory of Dad, Guozhong Meng and Hao Tang. Table of Contents Abstract Dedication Table of Contents List of Figures List of Tables 1. Introduction 2. Analytical Model 2.1 Coupled Wave Equations 2.2 Resonance Characteristic and Relaxation Mechanism 2.3 Wavenumber Analysis 3. The 3.1 3.2 3.3 Experiment Experiment Design Signal Conditioning Data Aquisition and Spectrum Analysis 4. Results 4.1 Resonance Frequency and Loss Factor 4.2 Phase Speed From Cross Spectrum Function 4.3 Flexural-Longitudinal Wave Energy Ratio 5. Conclusions References Appendix A: Computer Program for Wavenumber Analysis Appendix B: Drive Point Impedance Diagram Appendix C: Cross Spectrum Data Appendix D: Longitudinal-Flexural Coupling Data(Symmetric Loading) Appendix E: Longitudinal-Flexural Coupling Data(Asymmetric Loading) 54 56 60 62 69 89 108 List of Figures Figure 1-1: Elements of a vibratory system Figure 1-2: Schematic diagrams of dynamic equivalent vibratory systems, Ref[3] Figure 1-3: Mass-spring-damper model of the dynamic absorber Figure 2-1: Semi-infinite rod with dynamic absorber Figure 2-2: Three-element spring and dashpot combination Figure 2-3: Attenuation vs. normalized frequency ratio for 3=3, r=--0.1,0.3,1.0 Figure 2-4: Attenuation vs. normalized frequency ratio for q=0.2, 3= 1,2,3 Figure 2-5: Phase Speed vs. normalized frequency ratio for (=3, r--0.l,0.3,1.0 Figure 2-6: Phase Speed vs. normalized frequency ratio for =--0.2, P3=1,2,3 Figure 2-7: Attenuation vs. (o, for 3=3, N=1, -1=0.1,0.34,1.0 Figure 2-8: Attenuation vs. 3, for =--0.2, N=1, P=1,2,3 Figure 2-9: Attenuation vs. o, for rl--0.2, 3=3, changing N factor Figure 2-10: Phase speed vs. 0o, for 3=3, N=1, Tr=0.1,0.34,1.0 Figure 2-11: Phase speed vs. o, for 1r=0.2, N=1, P=1,2,3 Figure 2-12: Phase speed vs. o,, for 3=3, =--0.2, changing N factor Figure 3-1: Experiment apparatus Figure 3-2: Resistance bridge with cancellation of flexural vibrations Figure 3-3: Resistance bridge with cancellation of longitudinal vibrations Figure 3-4: Decide the range of Rt under the most unfavorable combinations of resistors Figure 3-5: Six-channel circuit lay-out Figure 3-6: AD624 functional block diagram Figure 3-7: AD624 pin configuration Figure 3-8: Noise interference problem, initial testing of the conditioning circuit Figure 3-9: 60 Hz and its harmonic interfering noises,conditioning circuit with proper balance and ground Figure 3-10: Response of pure tone excitation with battery suplied bridge circuits Figure 3-11: Response of pure tone excitation with SNR larger than 30 dB Figure 3-12: Noise interference problem in data aquisition using the Concurrent Computer Figure 3-13: Data aquisition diagram Figure 3-14: Clock connections on the CK10 and SH16FA modules Figure 3-15: Sampling of a periodic timing signal Figure 3-16" LWB modules in the data flow diagram Figure 3-17: Synchronization virtual instrument Figure 4-1: Phase speed vs. normalized frequency,ao = 2nf o = 2cn*134.75 Figure 4-2: Longitudinal to flexural coupling wave energy ratio Figure 5-1: With wave propagation in the dynamic absorber layer 7 8 9 10 15 18 19 20 21 22 23 24 25 26 27 29 30 32 33 34 36 36 37 38 39 40 41 42 43 44 45 46 50 53 55 -6- List of Tables Table 3-1: AD624A specifications (@ V,= ±15v, Gain= 100, RI= 2kfl and TA = 25 35 OC Table 4-1: Resonance frequency and loss factor Table 4-II: Flexural/Longitudinal coupling wave energy ratio 49 52 Chapter 1 Introduction This study investigates the effect of a continuous longitudinal dynamic absorber layer on longitudinal wave propagation in a circular rod. Previous studies [21,22,36,42] relating vibration control to a continuous dynamic absorber layer focused on the attenuation of flexural or longitudinal wave propagation. In this study, an apparatus for measuring phase velocity and flexural/longitudinal wave coupling energy ratio is designed. In addition, a relaxation mechanism is employed to simulate the behavior of the isolator/dynamic absorber. One of the basic principles in engineering is to start analysis with simple cases. For that reason, modeling of the dynamic absorber in several simple combinations of vibratory elements is studied here. I L 9 Hooke Newton Maxwell Kelvin Figure 1-1: Elements of a vibratory system Zener -8- The mechanical response of viscoelastic bodies are poorly represented by either a spring or a dashpot, which obey Hooke's law and Newton's law, respectively. J.C. Maxwell suggested a series combination of the spring and dashpot elements, which is merely a linear combination of perfectly elastic behavior and perfectly viscous behavior. Another simple element which has been used frequently in connection with viscoelastic behavior is the socalled Kelvin or Voigt model, with a spring and a dashpot in parallel. Creep and stress relaxation studies[2,3,13,14,33,38,39,40,43] reveal that the response of either Maxwell model or Kelvin model to several kinds of deformation does not fully represent some real damping systems. Different combinations of vibratory elements continue to appear in their applications, as cited by S.H.Crandall in the foreword of [33]: vibration theory was essentially complete - except for a realistic treatment of damping. iLT (a) (a) I-. (c (C) (b) Figure 1-2: Schematic diagrams of dynamic equivalent vibratory systems, Ref[3] -9When a spring is used as a vibration isolator and damped with a dashpot in parallel (right, the Kelvin or Voigt model) the conventional analysis accurately predicts force transmitted, deflection and damping loss. But when the elastic element is adhesive vinyl foam tape (also known as weatherstrip) with internal damping the conventional analysis may be in substantial error. For such a visco-elastic material, representation with a relaxation spring added in series with the dashpot (left, known as Zener model) more precisely simulates the behavior of the isolator. It is also regarded as possessing "one and one-half " degrees of freedom[23]. F-F F.F 0 e Relaxat unit I I Y1 All xi Figure 1-3: Mass-spring-damper model of the dynamic absorber -10- Chapter 2 Analytical Model 2.1 Coupled Wave Equations We first consider a infinite slender elastic rod with a continuously distributed layer of similar masses, springs and dashpots, transporting longitudinal waves (see Figure 2-1). kI r ~ ' I1 Figure 2-1: Semi-infinite rod with dynamic absorber The equations of motion of this freely suspended rod with Kelvin coupling between u(x,t) and v(x,t) are: m 2-+ K,(u-v) + C-a(u-v)=O 2 aat at -11- m,-+K,(v-u) + C-(v-u)=E s, TX2 at at2 where E I, mi, p,, sl are modulus of elasticity, unit length mass, density and cross sectional area, of the rod, m ,, C, p 2, K, are the unit length mass, resistance, density and stiffness, of the dynamic absorber layer. Finally, u and v are longitudinal displacements of the rod and absorber, and wo which follows is the natural resonance frequency of the absorber: 02 -- K, m2 The damped resonance frequency is usually approximated as mo[7,43]: (2= 002(1- 2)= 020 with the viscoelastic damping factor typically small, where C2 - 4Kmn Since u and v are both space and time dependent, we assume the solution is harmonic and substitute -iofor the time derivatives 'x v = V.ei-(k -w -t) t) -w ' u = U-ei'(k'x in the equation: E l a2v pI at2 a'2 t 2 K, S(v-u) - - Inl which corresponds to, C nlat (v-u) = 0 -12- K2V - KI2V + --(V-U) - iC•-O(V-U)=O mE, m,E, where E= 1 We can normalize the above equation, with the following non-dimensional parameters: K y K, ratioof wavenumber =2 K, =-o - mass ratio 2 m, C stiffnless loss factor oom2 Wo normalizedfrequency Therefore the normalization yields the following equation: V- v +-P(V-U) - iP;l(V-U)=o The coupled equations can now be rewritten as: V-v+ (V-U)- ifp(V-U)=o tO. - U +-- U-V) - il(U-v)=o on On2 -13- This is a set of two coupled homogeneous linear equations in U and V. For a nontrivial solution to exist, the determinant of the coefficients must vanish. This leads to the dispersion relation: y-1+j3Y -j3Y -Y -1+Y =0 with Y = 1 (o) 2 i (on (y- 1+1Y).(-1 + Y)- P- = 72(-1 + Y) + 1- Y - Y = 0 1 1-ilco (on The roots of this equation in the wavenumber y represent a right going wave and a left going wave. Therefore, there are two different natural modes that can propagate in this semi-infinite rod with an absorber layer. Each mode, of course, can be left going and right going. -142.2 Resonance Characteristic and Relaxation Mechanism For realistic treatment of damping influence in the vibration isolation, when we look at the indirectly coupled viscous dampling Zener model[35], the complex ratio of stress or strain or, equally, the complex stiffness Kc of the three-element mounting may be written as Kc=K+ 1 (I/NK) + (l/i71() It is readily shown that the stiffness approximately equals K at low frequency, K + iro. near resonance, and K + NK at high frequency. Therefore this is consistent with the concept of the mass-control, damping-control and stiffness-control regions of a dynamic absorber[24]. 2- Kc - K+imoC - K(1 + isoC ) m K m mm K isoC +-(1"+')c 2K In our light damping situation, OC K is small near the resonance frequency. We shall therefore be able to approximate 2Oo = Following the analysis in the previous section, we can easily write down the wave equation for the three-element combination with Zener coupling as: m, m2 1 + K ( x t - x, ) + NK( x - x 3 ) = Es,x + K(x NK( x, -x 3 2 -x I ) + C ( X-x ) + C ( x,1 - x3 ) =0 3) =0 -15- where xI, x2, x3 are the displacements at points shown in Figure 2-2, N is the stiffness ratio of the relaxation spring over the main spring. X1 Figure 2-2: Three-element spring and dashpot combination Again we assume a harmonic solution, substitute -iO for the time derivatives, and normalize with the same non-dimensional parameters. X1 = XIei-(k-x-w-t) X2 = X.ei-(k-x-w-t) X3 = X 3 .ei-(k (Y - -X 2 x- w-t) 1)XI + I(X 1-X 2 )/on2 + N0(XI-X3)/0,n + (X:-Xl)/On2 - i0(X2-X3)/1, N(XI-X3)/,n 2 n 2 =0 =0 - ir(X2-X3)/m,, = 0 The wavenumber-normalized frequency solution is thus obtained: -16- 11+ irlon-N 2.3 Wavenumber Analysis We introduce a complex wavenumber K, describing propagation of a lightly exponentially decaying wave, Kc = Kr + iK. The imaginary part of the wavenumber representing the right traveling wave is separated to yield an exponentially decreasing amplitude envelope: ei'(kc-x-wt) = e-kix.e4i-(krx-w-t) The attenuation per wavelength in dB(dB/A) is stated in terms of the wavenumber components as Attn = 20-loge-K,-k where X is the wavelength at each frequency without coupling. We normalize this term to the corresponding wavenumber ratio = Yi Ki.-- and obtain: Attn = 54.6yi(dB/X) In such a dispersive wave propagation pattern, phase and group velocities are defined as: C• K -17- and do) gdK respectively. The velocities c and c9 can be determined from the dispersion relation. To avoid some complex algebraic manipulation, we write a computer program (enclosed in Appendix A) solving for both the imaginary component and the real component of the wavenumber from the above dispersion relation, for both the relaxation (Zener) and non-relaxation (Kelvin) cases. The following figures, which depict a lightly damped system with different mass ratios, show that both the peak and bandwidth of attenuation increase dramatically with increasing 1, for both relaxation and non-relaxation models. These figures also reveal that with increasing loss factor, the attenuation and the phase speed peak drops considerably, for the non-relaxation model, but the attenuation bandwidth widens. Furthermore, by increasing loss factor, the relaxation model predicts the attenuation and phase speed drop in the low damping region, increase in the high damping region, and possesses a transition frequency, which is referred to as optimum (attenuation is a maximum at the optimum damping point). Analytical results are also obtained when holding mass ratio and loss factor unchanged, increasing the stiffness ratio N factor, which causes the attenuation and phase speed to drop. The analytical model is therefore consistent with the concept of a dynamic absorber. -18- Attenuation vs. Normalized Frequency ,inn - eta=0. l,beta=3 180 '" '~'' i"'"" " " " " '"' '" ~~" " '" "~"~"~ 160 E 140 120 ~-- ............. · .··:· 100 ',eta=0.3,beta=3 I· . .. ........ r · 1 I 8 60 40 r r. Seta=l.Obeta=3. - 20 0on-relaxation I vI'I 10- .... . . . .. . . . .'... ..... - •.••,. . :. . ----------- " • - -•.. 10o Normalized Frequency,w/wO Figure 2-3: Attenuation vs. normalized frequency ratio for 03=3, =--0.1,0.3,1.0 . . . ._ -19- Attenuation vs. Normalized Frequency I' 180 160 140 120 100 o 80 0 60 40 20 0 10-1 100 Normalized Frequency,w/wO Figure 2-4: Attenuation vs. normalized frequency ratio for 11=0.2, 13=1,2,3 -20- Phase Speed vs. Normalized Frequency - 450n' .. . ....... ................ 4000 3500 3000 - I 2500 2000 ............. ... 1,beta=3 ~ • ... 135 0 - I .......... 1000 , . ·... ,··. ....,.. ·· . , ............!,,,,,! ,, " ......... ...... eta=1.0, beta=3 500 ......... . .............. Non-relaxation with loss factor 10-' 100o Normalized Frequency,w/wO Figure 2-5: Phase Speed vs. normalized frequency ratio for 3=3, rI=0.1,0.3,1.0 10 Phase Speed vs. Normalized Frequency .3LAJ 2500 2000 I 1500 10001 eta=0.2, beta=l1 500 .ta=0.2, beta=3' 0n -I 10" •ta--=.2, bea= 1 Non-relaxation with mass-factor . 100 Normalized Frequency,w/wO Figure 2-6: Phase Speed vs. normalized frequency ratio for 11=0.2, O3=1,2,3 10' -22- O~nn Attenuation vs. Normalized Frequency 8• "4 U 0 10o- I 100 Normalized Frequency,w/wO Figure 2-7: Attenuation vs. o,, for j=3, N=1, 1--0.1,0.34,1.0 10' -23- Attenuation vs. Normalized Frequency 0 U 10-' 100 Normalized Frequency,w/wO Figure 2-8: Attenuation vs. (. n for r~=0.2, N=1, 3=1,2,3 10' -24- 1mV 10-' Attenuation vs. Normalized Frequency 100 Normalized Frequency,w/wO Figure 2-9: Attenuation vs. o,, for 1=0.2, 5=3, changing N factor -25- Phase Speed vs. Normalized Frequency cfl,', a 10-' 10o Normalized Frequency,w/wO Figure 2-10: Phase speed vs. (o. for 3=3, N=l, 1=--0.1,0.34,1.0 10' -26- "Mnn 10-' Phase Speed vs. Normalized Frequency 100 Normalized Frequency,w/wO Figure 2-11: Phase speed vs. co, for 1--0.2, N=l, 0= 1,2,3 10' -27- ,2fwvl Phase Speed vs. Normalized Frequency 2500 2000 1500 1000 500 0 10- 100 Normalized Frequency,w/wO Figure 2-12: Phase speed vs. o, for P3=3,,=0.2, changing N factor 10' -28- Chapter 3 The Experiment 3.1 Experiment Design An experiment should be planned to bring into prominence those factors to be studied and to enable their effects to be assessed in relation to the unavoidable errors of experimentation. The first object to undertake an experimental investigation of wave propagation is to build an apparatus that closely resembles the analytical model: a thin elastic rod with a continuously distributed layer of masses, springs and dashpots, transporting longitudinal waves. A Delrin rod is a solid material used in Olivieri's experiment, which has a very low modulus of elasticity, an average density for a crystalline plastic and one-third of the compressional wave speed that is in steel or aluminum. The use of the Delrin rods was considered fixed, due to the desirability shown in Olivieri's experiment. To simulate one dimensional propagation of longitudinal waves in an infinite medium, we dampen the propagating waves at the end of the test rod with sand. Enough sand is placed around the end of the rod to reduce wave reflection. The other end of the rod (not immersed in sand) is drilled and fitted with a bolt, then connected tightly perpendicular to a Wilcoxon Research Fl shaker with a matching Z-602 impedance head. Input to the shaker was provided by a signal generator with gain provided by a McIntosh power amplifier. The frequency range of the signals, limited by the linearity of the shaker, power amplifier and signal generator, is considered to be low-bounded by 40 Hz. -29- U, 0 c0 a, E C Uo 0 E 0 S ca a, 0 E 0 0, c *0 0o Figure 3-1: Experiment apparatus Accelerometers, which were implemented in previous experiments[21,22,36,42] to measure attenuation spectra, are considered poor choices in measuring phase velocity and -30wave coupling energy ratios, because of their phase lag and sensitivity axis difference, and high mass. Instead, strain gauges are used. 3.2 Signal Conditioning In order to measure longitudinal and flexural waves separately, we need to investigate the strain gage balancing and amplifying circuits. All commercial strain indicators employ some form of the following Wheatstone bridge circuit to detect the change of resistance in the gage with strain. Vs Figure 3-2: Resistance bridge with cancellation of flexural vibrations In the above well-known Wheatstone circuit, vs and vo are the source and output voltage of the balancing circuit, respectively; Rgl and Rg2 are two matched gages connected as nonadjacent arms of the bridge circuit (with the same length leadwires, they maintained identical temperature-compensation); and R1, R2 are reference resistors on the two other arms. We find: -31- Rg 2 R2 R,+Rg2 RgI+R, Vo_ v, where Rg2 = Ro + A R t - A Rf Rgl =Ro+ AR,+ ARf R1 = R2 = Ro where A R, and A Rf donate to the resistance changes in the two strain gages mounted on the opposite side of the rod at the same distance from the drive point, which reflects the changes in resistance according to the longitudinal and flexural wave propagation. Then: vo AR, - ARy+Ro Ro vS 2Ro+AR - ARf 2Ro+AR, + ARf Ro 1 I 2 +(ARt-IAR.) 1+ 1 -0 [= 2Ro AR - AR 2 Ro )(,- 1+ (ARI + ARf) 2Ro ARI - SAR1 - AR,AR- ARf 2 Ro 2Ro R 2Ro ARI+AR G(1 (1 )] 2Ro AR +ARf AR,+AR Ro 2Ro With the obvious assumption Ao << 1, we obtain vo ARi vS 2Ro The change of resistance in the strain gage due to longitudinal strain is proportional to the gage factor Fg and actual strain E as ARt = Fg-Ro-• so the bridge circuit output voltage is vo=Vs -T-. -32Recall in this case that the measurement cancels out the flexural modes and only contains the longitudinal modes. In order to investigate the amount of flexural wave energy coupled from longitudinal excitation, the following circuit configuration is used: + Vs Figure 3-3: Resistance bridge with cancellation of longitudinal vibrations where "~ with the << 1 assumption. In practice, a Wheatstone bridge is never precisely balanced as a result of the finite tolerances of the bridge resistors. Consequently, some method must be introduced to slightly change the resistance ratios of one side of the bridge. Thus, the pot and trim resistors are introduced as shown: -33- + Vs Figure 3-4: Decide the range of Rt under the most unfavorable combinations of resistors Given that the resistors RI, R2, R3, R4 are 350.0( ±+0.3% (for Micro-Measurement CEA series gages and carefully chosen reference resistors), the maximum value of Rt, for which the bridge can be balanced under the most unfavorable combination of resistors, is decided to be 8.7 kQ. The 50 kQ Rp resistor draws little current and acts simply to control the voltage on one side of Rt. We now construct this bridge circuit on a proto-board. With a Tektronix ocilloscope and a regulated power supply, we are able to balance the bridge to v o less than or equal to 25 gv. Considering that the contact resistance at mechanical connections within the bridge circuit can lead to errors in the measurement of strain, a "wiggle" test is made on wires leading to the mechanical connections. The actual change in balance does occur, so we decided to wire-wrap and solder the bridge circuit on a Vector-board to insure that good connections have been made. The following circuit layout diagram shows six channel balancing bridge circuits and their differential amplifers: -34- !..: ............ '.............. :.... :: i"- *: ::::....** 9 : * : "'......... . . -. ...- -....*...--* ... .. :._ . ......... .,.... . -.,,,-... • ... ..... ...,.. . ,, ...® . .. ,~~~~~~ 1 ii4r'1I . . . .... ..,+. . . . ..... .. Tc:. .... . '.-.. .. . ,I"... ...!.........! , . . 0, ... . .. . •.. . ... ... : : :: : : :: : : : :: : -.. . : :::: : u, ) .::.':.. .... O• 0 .. . . . ..... . . . . . . . . c. ......... . CD .. o - ..... .... , ......... . ., • . . . ., . ,... ... ...... . •.. ... .'j .. 99 .. ,L.. . .:.::::: :.. ...... .. . ... ... .:::..: .. :: .:::: . ., • iL S. •. . • • • •. . ... . .. .. . ..... . . .\ .• . . . . .. . ,, \ ,. . . . . (. D• . . ,. '.......... .. ............. . *. *. ..\ . ..•........ .. . .. .... ....... .. ... ....... o........, .7 . 7 . - , . ., ,, • _i .-- ~- I * ,. o, ... . . . . . .. .. '.i.' .... .. .. , .. /. . ., . . *, . ... ,.."- CO, . ....... ...+.. .+....g. ............. .... .. ,. . . . - I. •.. 4 ... • •••+. S...... .. .. •.... .. .4 .• ' ! .. ......... t . T . CCl . . 1 o• :::.... ::: ........ a i..it . . . . . 1.... .... CL,. •O. .,.. .. . . . .•. . . . .o. . .,V /• - . . . . . ' •::: 1: . - . . . . •••,:: .. . .,• I 0. . ..... ... •. *. . ... . ..,4-,* . .. .. . .. . - - . . . " ... ,. .. . .. ,. . ...... . . . .. .. .. . . . . . . . , . . . , . . o•......... ,................... +......... •......... +......... ,........., | #feetl @Avg l~t Figure 3-5: Six-channel circuit lay-out -35Table 3-I: AD624A specifications (@ V, = 15v, Gain = 100, R, = 2k• and TA = 25 OC Speci- Value Un- fication it Gain Max Error ±1.0 % Gain Nonlinearity ±0.005 % v ±10 Input voltage range (Max Differ. Input Linear) Output rating Dynamic response (small signal -3dB) Slew Rate ±10 25 v kHz 75 ps Power supply range Min ±5 v Max ±18 Typ ±15 The operational amplifier we chose is a AD624 precision instrumentation amplifier. The AD624 amplifier is designed primarily for use with low level transducers (including strain gages), with low noise, high gain accuracy, and low temperature coefficient. For the adjustable pretrimmed gain of 1000, the linearity range of the dynamic response is DC to 25 KHz. The 5V/gs slew rate and 15 gs settling time permit the use in our multiple channel, high sampling rate data acquisition applications. -36- -INPUT G 100 G = 200 SENSE G = S00 RG, OUTPUT RG, + INPUT REFERENCE Figure 3-6: AD624 functional block diagram - INPUT 1" RG, I 15" OUTPUT NI ILL + INPUT r2 RG, - AD624 1" OUTPUT NUILL AD624 G = 100 INPUT NULL E4 7 INPUT NULL E- E G - 200 II G - 500 REFERENCE -Vs 7 * Vs SHORT TO RG, FOR 1D SIRED GAIN 101 SENSE "IJOUTPUT FOR GAIN OF 1000 SHORT RG, TO PIN 12 AND PINS 11 AND 13 TO RG, Figure 3-7: AD624 pin configuration -37Too often in experiment designs, noise is considered to be one step down from the weather: hardly anyone even talks about it. Yet it is the noise level in a measurement circuit that ultimately limits the ability of that circuit to transmit faithfully the information carried by the signals being processed. To avoid a "noise-limited" statement that would likely appear in the "conclusion and discussion" chapter, we shall now incorporate into discussion the possible noise sources and their effects. W12 AUTO SPEC CH.A Yo -6. 5[d /1. OOV 2 Xa 54.00Hz - 100Hz SETUP S1* #As 10C MAIN PWR 80dB LIN X& Ys -U. 5dB 61. 50OHz N *W I) 10-4'--. .---- ..... 60 SETUP 70 ........ r...... ...-........... 60 90 100 110 120 130 140 S1 MEASi.:REMENT: TRIGGER: DELAY: AVERAGINGs DUAL SPECTRUM AVERAGING FREE RUN CH. A-*B: 0. Cms LIN 100 OVERLAPs MAX FRED SPAN: CENTER FREQs WEIGHTINGs 100Hz AF: 125mHz ZOOM 104Hz RECTANGULAR CH. A: 8V 3Hz DIR CH. 8: 800mV 3Hz DIR GENERATOR, VARIABLE SINE SINE GENERATOR FREQ.a 203. 218Hz Ts 8s FILT: 25. 6kHz FILT:25. 6Hz ATL 7. 81ms 1V/V V.,V Figure 3-8: Noise interference problem, initial testing of the conditioning circuit 150 -38The interfering signals initially were dominated by the interference at 60 Hz and harmonics of 60 Hz, introduced at the Wheatstone bridge circuit. A test signal of a pure tone sine wave at 203 Hz is buried under the 60 Hz, and its harmonics (see Figure 3-9): CE j12 AUTO SPEC CH.A2 PWR Y: -49.1dB /I.0OV LIN X: GHz - 800Hz #As 100 'OD'-~t~~--~l-~..--~-JL -- L ' · - INPUT MAIN Ys XI 203Hz 8OdB -71. 7dB 1 hi jo-2 ' 10-2 TU 10-4 10-a 0 SETUP 100 200 400 300 500 700 600 'A'i MEASUREMENTs TRIGGER: DELA Y: AVERAG INGs DUAL SPECTRUM AVERAGING FREE RUN O. COms CH. A-,8: MAX OVERLAP: 100 LIN FREC SPAN, CENTER FRECQ ElEGHTING%, 800Hz AF BASEBAND RECTANGULAR CH. As 5CmV + 30mV CH. Bs VARIABLE GENERATOR: SINE GENERATOR FREQ. s 1Hz 3Hz 3Hz SINE DIR DIR T, Is FILTs 25. 6kHz FILT, 25. 6 Hz ATs 488js I V/V IV/V 203. 218Hz Figure 3-9: 60 Hz and its harmonic interfering noises,conditioning circuit with proper bala 800 -39- By carefully studying the coupling between the power lines and the experiment apparatus, along with the use of power transformers and a regulated DC power supply, the interference is reduced by about 10 dB from Fig3-9 (as shown in Figure 3-9, Figure 3-10). With the battery supplied bridge circuits and DC transformer supplied operational amplifier, the signal to noise ratio is satisfactorily increased to larger than 30 dB. W12 [AUTO SPEC CH. A C Ya -68. 7d- -/1. C351.0 PWR Xi OHz + 400Hz L.IN #As 100 3 INPUT -105.4dB _ L____ -A 100 MAIN Yv X8 203Hz 80dB N E 10-2 10 I" In"-4 0 50 SETUP 100 150 200 250 350 Wi DELAY: AVERAGINGs DUAL SPECTRUM AVERAGING FREE RUN CH. A-B: 0. OOms LIN :00 CVERLAPs MAX FREQ SPAN: CENTER FREQa WEIGHTINGs 800Hz AF: 1Hz BASEBAND RECTANGULAR MEASU'REMENT: TRIGGER: CH. As CH. Bs 300 30mV + 30mV GENERATOR: VARIABLE SINE GENERATOR FREC.: 3Hz DIR 3Hz DIR SINE 203. 0OOHz Til s FILTsBOTH FILT: BOTH AT: 488Js 1V/V IV/V Figure 3-10: Response of pure tone excitation with battery suplied bridge circuits 400 -40- W12 I0,'NCISE RATIO INPUT MAIN Y: 34. 9dB Xi 203Hz Y, 27. 7dB 80dB Xs OHz + 400Hz LIN SETUP 01 #As 100 20 0 -20 -40 - .... ,. .. .,4 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 1.0 0. 8 0. 6 0. 4 0. 2 C 412 CCHERENCE 1. C Y: X2 OHz - 400Hz SETUP 01 #A, MAIN Y: X: 800Hz 6. 92m LIN 100 Figure 3-11: Response of pure tone excitation with SNR larger than 30 dB -41- The drifts in power supplies and amplifier offsets are controlled by the balancing adjustment in the circuit. The stray capacity fluctuations and electronic device noises are problems in data acquisition with the Concurrent Computer, as shown in Figure 3-12. With careful shielding, chase and signal grounding, wax-sealing the trimpot, and using the bandpass and low-pass filters, we finally achieve an excellent degree of noise isolation in the measurement apparatus. Power-lin ground ng ac noise source (motor, computer, fan,relay, etc.) Sour Netwo V gnIy C iSA YIuuIIU Load ground -42- 3.3 Data Aquisition and Spectrum Analysis The strain gage output is collected and digitized at the Acoustics and Vibration Lab using the Concurrent Computer. %OW I I IF%-16-8 Figure 3-13: Data aquisition diagram -43The analog input from the resistance bridge is amplified by the AD624 operational amplifier. After passing through the Frequency Devices 9016 programmable low pass filter, it is sampled, digitized and displayed with proper triggering, anti-aliasing, synchronizing and clipping. The AD12FA analog/digital converter, along with a SHI6FA sample and hold module, are used to digitize the data. Two analog/digital channels (Channel 0 and Channel 5) contain amplified signals for flexural and longitudinal waves, respectively. syne pulse to external de sync pulse From external or external pulse For exter tr ggered sweep! Inputs pulses to 0/A converter pulses to A/D converter Figure 3-14: Clock connections on the CKIO and SHI6FA modules -44- sweep rate or exeternal sync pulse clock 0 sweep length clock I frame rate clock 2 framne length burst rate resuleteng timing sequence Figure 3-15: Sampling of a periodic timing signal clock 3 clock 4 -45- Figure 3-16: LWB modules in the data flow diagram -46- Figure 3-17: Synchronization virtual instrument -47- The data acquisition on the Concurrent Computer takes place inside the Lab Workbench (LWB) environment. The analog signal is demodulated using a demultiplexer module to separate channel 0 from channel 5 ( and multi-channel demodulating, when applicable). The channel 0 signal provides the input for a trigger module that controls the data flow in both channels. The synchronization enables us to measure phase speed in the time domain. The trigger threshold and intervals are adjustable. This is of importance for future experimental investigations of multi-mode wave propogation problems. The power spectra, defined as the Fourier transform of the input time series, are calculated and displayed with time series for both channels. We can now measure the ratio of energy transformation as a result of flexural wave coupling with longitudinal excitation. -48- Chapter 4 Results 4.1 Resonance Frequency and Loss Factor The quarter wavelength resonance frequency for a free rod of 3.10 m length, with a longitudinal nondispersive wave speed of 1161 m/s, is determined to be 93 Hz. Three tests are conducted to decide the resonance frequency for the Delrin rod with the attached dynamic absorber layer. An impedance head is installed between the shaker and the contacting surface of the rod. The acceleration and force gage output from the impedance head are taken to the B & K spectrum analyzer. The drive-point impedance (defined as force over velocity, which comes from integration of the acceleration) is obtained to decide the actual resonance frequency and loss factor. The first actual resonance peak occurs at a much lower frequency than predicted and is considered to be caused by the resonance frequency of the shaker and sand termination problem, as addressed in Larry Olivieri's report[21]. In order to experimentally decide the loss factor, we conduct a test of the resonance frequency fo and the half power bandwidth (-3dB down from both sides of fo). The loss factor 11 is defined as fo for a system consisting of a short rod with a single isolator located very closely to the drive point(mass ratio is 3.2). -49Three measurements are conducted and reveal the following results (Plots are enclosed in Appendix B): Measurement freq. span fto (Hz) f 3dB low(HZ) 800 135 111 400 134.5 109.5 200 134.75 107.75 Tf Hz =134.75 (Hz) 3dB high 157 .341 156.5 .349 155.25 .353 1F =.348 Table 4-I: Resonance frequency and loss factor 4.2 Phase Speed From Cross Spectrum Function The Fourier transform of the cross-correlation function, which is the expected value of the product of two time series, is defined as the cross-spectral density function (Cross Spectrum). R, (r) = E [y(t) x(t+t) ] S, (f) R,('t)-.e-i-2 'ftd r, The phase speed of the wave propagation can be determined from the frequency and the phase lag, through phase function 0,,(f) of the cross spectrum 9,(f)= i,(f)l e- i x(f) Using the B&K 2032 dual channel signal analyzer to measure the frequency response -50- of a series of pure tone longitudinal excitations to the rod, the phase speed is obtained and compared to the analytical prediction, where the crosses represent the experimental value. Phase Speed vs. Normalized Frequency I U 10-' 100 Normalized Frequency,w/wO Figure 4-1: Phase speed vs. normalized frequency,o, = 27f,o = 2n* 134.75 10' -51- The experimental phase speed is dispersive, with shape as predicted, provided that N = 1.4. The good agreement in shape confirms the analytical model used and also that the attached mass system acts as a continuous longitudinal dynamic absorber. Although an independent measurement of N was not carried out, the Zener model seems to be a significant improvement over the Kelvin model. 4.3 Flexural-Longitudinal Wave Energy Ratio Power densities referring to longitudinal and flexural wave energy are obtained from the output of two independent sets of stain gages, measuring simultaneously at the same distance away from the drive point. The ratio is presented in dB vs. normalized frequency, and the results reveal that the coupling from longitudinal excitation to a bending wave is much stronger at low frequency than at high frequency. The estimated spectrum is calculated by Fourier transforming the auto-correlation function of the time series from a sample function, in conjunction with a "window" which is a weighting function applied to data to reduce the spectral leakage associated with the finite observation intervals. By applying a Hamming window, the power spectra we calculated achieves -30dB down sidelobe level and good frequency resolution of .01 Hz, as shown in Appendix D and Appendix E. In the symmetric loading case, the flexural wave is considered to be induced by the slight misalignment at the drive point, supporting fishing line, sand termination, and any other imperfections for longitudinal wave propagation. The experiment results show that the longitudinal to flexural wave coupling is induced significantly by the asymmetric(disc mass adding through half circle weatherstrip to the rod) loading of the resiliently mounted masses. It is also shown in Figure 4-2 that both power spectra decrease linearly in logarithmic frequency scale. -52Above resonance frequency, the flexural wave diminishes(-30dB per decade)in the symmetric loading case, while the longitudinal to flexural wave coupling grows (20dB per decade) throughout the investigated frequency span for the asymmetric loading case. Asymmetric Loading Symmetric Loading Vertical Flex./Long. Energy Ratio frequency (Hz) Symmetric loading (dB) Asymmetric loading (dB) 19.531 9 18 39.063 2 13 58.594 -13 8 78.125 -8 5 117.19 -24 -3 156.25 -40 -8 195.31 -35 -25 410.16 -44 -20 800.78 -53 -15 Table 4-II: Flexural/Longitudinal coupling wave energy ratio -53- Fkex./ong. wave negy ra•io vs. Nrmalized Frequency A0U 10 - - -. 0 ,.. -10 -20 -30 -40 -50 60I 10-I ......... .. 100 10' Normalized Frequency,w/wO Power spectrum measurements for a series of pure tone longitudinal excitations to the Delrin rod with a dynamic absorber layer attached, where the crosses represent asymmetric loading and circles for symmetric. Figure 4-2: Longitudinal to flexural coupling wave energy ratio -54- Chapter 5 Conclusions For zero damping, a stop band exists for the range 1 < /coo < (1 + (3)n, in which the wavenumber y is pure imaginary. Realistic treatment of damping is applied and the effects of damping parameters to the longitudinal wave propagation through the dispersion relation is verfied from the experiment. The results confirm the analytical model used and that the attached mass system acts as a dynamic absorber. The three-element combination(Zener model) does stiffen by a small amount as the frequency increases and, by association, is said to possess a transition frequency. This model gives better prediction than the Kelvin model when the loss factor is not too small(1r> 0.2 in our case). It is also shown that the interaction between longitudinal and flexural waves may lead to significant rates of transformation of the compressional wave energy into bending, as the coupling is much stronger in the asymmetric loading case than symmetric. Future work may be suggested to have a continuous isolation layer, with consideration of wave propagation in the isolator layer. Also an investigation of multi-mode wave propagation within one layer would appear worthwhile. -55- - Velocity, displacement > Velocity, displacement - Figure 5-1: With wave propagation in the dynamic absorber layer In complex structure testing, e.g. fluid loaded cylindrical shell, use of rubber as a mounting material is generally expected. For this, it is important to fully model the viscoelatic behaviour. Experimental investingation of the stiffness ratio N-factor may thus be important. -56- References [1] Abdulhadi, M.I. Stiffness and Damping Coefficients of Rubber. Ingenieur-Archiv, 55, pp 421-427, 1985. [2] Aklonis, J. J. and Macknight, W. J. Introduction to Polymer Viscoelasticity. John Wiley & Sons, New York, 1983. [3] Alfrey, T. and Doty P. The Methods of Specifying the Properties of Viscoelastic Materials. J. Applied Physics, 16(11):700-713,, 1945. [4] Allen, P. W., Lindley, P. B. and Payne, A. R. Use of Rubber in Engineering. Maclaren and Sons LTD, London,, 1967. [5] Baer, E. EngineeringDesignfor Plastics. Reinhold Publishing Corp., New York, 1964. [6] Bendat, J. S. and Piersol, A. G. EngineeringApplication of Correlationand Spectra Analysis. Wiley, New York, 1980. [7] Bendat, J. S. and Piersol, A. G. Random Data:Analysis & Measurement Procedures. John Wiley & Sons, New York, 1986. [8] Brown, R. P. Physical Testing of Rubbers. Applied Science PublishersLTD, London ,, 1979. [9] Cremer, L.. Heckl, M. and Ungar, E. E. Structure Borne Sound. Springer-Verlag, New York, New York, 1988. [10] Feinberg, M. Vibration-IsolationSystems. Mach. des. 37(18):142-149, 1965. [11] Freakley, P. K. and Payne, A. R. Theoryn, and Practiceof Engineeringwith Rubber. Applied Science Publishers LTD, London, 1978. [12] Fung, Y.C. Foundationof Solid Mechanics. Prentice-Hall, Englewood Cliffs, N.J., 1965. -57[13] Gent, A. N. and Rusch, K. C. Viscoelastic Behavior of Open-Cell Forms. Polymer Conf. Series, Wayne State Univ. , May, 1966. [14] Gurnee, E. F. Dynamics of Viscoelastic Bahavior. Polymer Conf. Series, Wayne State Univ. , May, 1966. [15] Harris, Fredric. On the use of Windows for Harmonic Analysis with the discrete Fourier Transform. Proc.IEEE, V. 66, No. 1,, 1978. [16] Hopkins, I. L. Resonance as Observed by Fitzgerald in Relation to characteristics of the SpecimenAparatus System. Polymer Conf. Series, Wayne State Univ. , May, 1966. [17] Junger, M.C. and Feit, D. Sound, Structures and Their Interactions. The MIT Press, Cambridge, MA, 1986. [18] Kaul, R.K. and McCoy, J.J. Propagationof Axisymmetric Waves in a CircularSemi-infinite Elastic Rod. JASA, 36, pp.653-660, 1964. [19] Kerwin, E. M. Jr. and Ungar E. E. Discussion of paper entitled 'Damping StructuralResonances Using Viscoelastic Shear-DampingMechanisms' by J.E. Ruzicka. Trans. ASME, J. Eng. Ind. 83(4):424, 1961. [20] Klyukin, I. I. Influence of the Elastic DissipationParameters,Load characteristics,and Degree of LateralConstraintof Vibration-IsolationElements on Their Damping Properties. Soviet Physics Acoustics, 28(1):46-49, 1982. [21] Olivieri, L.A. The Effect of Dynamic Absorbers on Longitudinal Wave Propagation in A Circular Rod. S.M. Thesis, MJ.T. , September, 1989. [22] LePage, K.D. The Attenuation of Flexural Waves in Asymmetrically Masses Loaded Beams. S.M. Thesis, M.I.T. , September, 1986. [23] Levitan, E. S. Forced Oscillation of a Spring-Mass System having Combined Coulomb and Viscous Damping. JASA, 32(10):1265, 1960. -58[24] Lyon, R. H. MachinaryNoise and Diagnostics. Butterworths, Stoneham, MA, 1987. [25] McNiven, H.D. Extensional Waves in a Semi-infinite Elastic Rod. JASA, 33, pp.23-27, 1961. [26] Meirovitch, L. Elements of VibrationAnalysis. McGraw-Hill, Inc., 1975. [27] Morse, P.M. and Ingard, K.U. TheoreticalAcoustics. Princeton University Press, Princeton, 1986. [28] Nishinari, K. Longitudinal Vibration of High-Elastic Gels as a Method for Determining Viscoelastic Constants. Jap. J. A. Phy. 15(7):1263-1270, 1976. [29] Nolle, A. W. Methodsfor Measuring Dynamic Mechanical Propertiesof Rubber-Like Materials. J. App. Physics, 19(8):753-774, 1948. [30] Plunkett, R. MechanicalImpedance Methods. The American Society of Mechanical Engineers, 1958. [31] Ruzicka, J. E. Forced Vibration in Systems with Elastically Supported Dampers. S.M. Thesis, M.I.T. , June, 1957. [32] Ruzicka, J. E. Resonance characteristics of Unidirectional Viscous and Coulomb-Damped Vibration Isolation Systems. Trans. ASME, J. Eng. Ind. 89(4):729-740 ,, 1967. [33] Ruzicka, J. E. Influence of Damping in Vibration Isolation. Technical InformationDivision, Naval Research Lab, Washington, D.C. ,, 1971. [34] Senturia, S.D. and Wedlock, B.D. Electronic circuits and Applications. John Wiley & Sons,. [35] Seybert, A. F. Estimation of Dampingfrom Response Spectra. J. Sound Vib. 75(2):199-206, 1981. [36] Smith, Timothy, L. The Attenuation of Flexural Waves in Mass Loaded Beams. S.M. Thesis, M.I.T. , May, 1985. -59[37] Snowdon, J. C. Vibration Isolation:Use and Characterization. Rubber Chem., 53(5):1041-1087, 1980. [38] Snowdown, J. C. Vibration and Shock in Damped MechanicalSystems. John Wiley & Sons, Inc., 1968. [39] Soroka, W. W. Vibration Isolators. Prod.Eng. 141-145 , July, 1957. [40] Sperry, W.C. Rheological-ModelConcept. JASA, 36(2):376-385, 1964. [41] Steven, Kay and Stanley, Marple. Spectrum Analysis - A Modem Perspective. Proc. IEEE, V.69, No. 11 ,, 1981. [42] Taylor, P.D. Wave Propagation in a Beam Having Double Resonant Loading. S.M. Thesis, M.I.T. , August, 1987. [43] Tse, F. S., Morse, I. E., and Hinkle, R. T. Mechanical Vibrations : Theory and Applications. Allyn and Bacon, Boston, 1978. [44] Tse, F. S. and Morse, I. E. Measurement and Instrumentation in Engineering. Marcek Dekker, New York, 1989. [45] Ungar, E. E. and Kerwin, E. M. Jr. Loss Factorsof Viscoelastic Systems in Terms of Energy Concepts. JASA, 34(7):954, 1962. [46] Welch, P. D. The Use of FFT for the Estimation of Power Spectra. IEEE Trans.Audio & Electroacoust,V.AU-15, No.2, June, 1967. [47] William, H. Press, etc. Numerical Recipes, & Example Book (Fortran). Cambridge Univ. Press, 1989. -60- Appendix A: Computer Program for Wavenumber Analysis for k-1:250: tl(k)-(2*k+1)/50.0; rl(k, ) -l+beta/(l-(tl(k)*tl(k))/(1+i*ata*tl(k)*N/(i*ata*tl(k)-N))); r(k,1) -sqrt(rl (k,1)); att (k, 1) 54.6*imaqg (r (k, 1) ) ; c (k, 1) -1161/(real (r (k, 1))); end; for k-1:250; tl(k)-(2*k+1)/50.0; rl(k,2) -1+beta/(1- (tl(k)*tl(k)) / (+i*ata*t(k)*N/(i*ata*tl(k)-N))); r(k,2)-sqrt(rl(k,2)); att(k,2)-54.6*imag(r(k,2)); c(k, 2)-1161/(real(r(k,2))); end; for k-1:250; tl(k)-(2*k+1)/50.0; rl(k, 3)-1+beta/(1-(tl(k)*tl(k)) / (l+i*ata*tl(k)*N/(i*ata*tl(k)-N))); r(k,3)-sqrt(rl(k,3)); att(k,3)-54.6*imag(r(k,3)); c(k,3) -1161/(real (r (k,3))); end; text(2.2,2900,'N-0.1,eta-0.204,beta-3.23') >> axis([-1,1,0,30001) >> semilogx (t1, c) >> grid >> axis([-1,1,2,4]) >> loglog(o,c) >> title('Phase Speed vs. Normalized Frequency') >> xlabel('Normalized Frequency,w/w0') text(2,1500,'Non-relaxation') >> text(2.2,2500,'Relaxation') >> ylabel('Phase speed (m/s)') text(1,40,'eta-0.204, beta-3.23') text(.3,0,'Kl=.6 K2,eta-0.204,beta-3. 2 3') text (2.2,2900, 'N-0.1') 3 23 text(.2,50, 'Relaxation with N-factor,eta-0.204,beta- . ') >> print('oe -h') text(.7,1800,'Non-relaxation') text(.3,2500,'Relaxation with N-1) text(1,40,'Loss factor 0.284, Mass ratio 3.23') text(.7,2800,'Relaxation with N-.3') text(.7,2200,'Relaxation with N-1.5') semilogx (tl, c) semilogx (tl, att) axis ([-1,1,2,41) axis ([-1,1,0,30001]) axis([-1,1,0,2001) -61- text(.12,20, 'Relaxation with N-factor,eta-0.2 N-1') >> clg >> semilogx(ti,att) >> title('Attenuation vs. Normalized Frequency') >> xlabel('Normalized Frequency,w/wO') >> ylabel('Attenuation (dB per wave length)') >> text(.12,20, 'Relaxation with Mass-factor,etaO0. 2 N=1') >> text(1.2,150,'Beta-3') >> >> >> >> >> >> >> >> title('Attenuation vs. Normalized Frequency') xlabel('Normalized Frequency,w/wO') ylabel('Attenuation (dB per wave length)') grid text(l.2,190,'eta=0.1,beta=3') text(1.2,100,'eta-0.3,beta=3') text(1.2,40,'eta=1.0,beta=3') text(.1,20,'Non-relaxation') >> axis((-I,1,0,200]) axis ([-i,1, 0,3000]) for k=1:250; tl(k)=(2*k+1)/50.0; rl(k,l) =l+beta/(1-tl (k)*tl(k) / (l-i*ata*tl(k))); r (k,l)=sqrt(rl (k,1)); c(k,1)=1161/ (real (r (k, 1))); att(k,i)=54.6*imaq(r(k,1)); end; for k=1:250; tl(k)=(2*k+1)/50.0; rl(k,2)=l+beta/ (1-tl(k)*tl(k) r(k,2)=sqrt(rl(k,2)); c(k,2)=1161/(real(r(k,2))); / (1-i*ata*tl(k))) att(k,2)=54.6*imag (r(k,2)); end; for k=1:250; tl (k)=(2*k+l) /50.0; rl(k,3) =l+beta/(l-tl(k)*tl(k) / (1-i*ata*tl(k))); r(k,3) sqrt(rl(k,3)); c(k,3) 1161/(real (r(k,3))); att(k,3)=54.6*imag(r(k,3)); end; ; -62- Appendix B: Drive Point Impedance Diagram Measurement freq. span f a (Hz) f 3dB Iow(HZ) f (Hz) 3dB high 800 135 111 157 .341 400 134.5 109.5 156.5 .349 200 134.75 107.75 155.25 .353 Hz fT =34.75 "=.348 -63- 0 " N NtI O a 0 -4J 0 O 0O O NO * moa °H .N L-n ao0C z o ) Ix 44 4.0)0 .C z 4-> o Iz uc 4)Q 4 2 oo 0 OqN x1 w w Z w w O O o0 0 w WO 4,4) m w -. ma N •" * X9 So 0 a a o e o 0 d d ' 1 d N Iel 4 w d 00 r-4 (Y o 00 lg 0 N 6 Z Q. >L 0 0C 0mV 4, CO) C 10 xO E 0 U o 0 4-o0 -64- N m 0 I-O0 O CONO N lo o 0 0 0 0 C) z (r/0• <a O4 $4 0 NV-6 44 r.40 O0OI 3 (z SJ- 0 O0 -I 0I w U Z • 0< 4N=NJ ]= g,'oa. 0 0 4 + O -0m m NNN 10013 a O ) Uo rY CO •'• o ••- •L <t a w N CO o * .- d d f N d a 0 ro 4•0 o L m S'm .0 0 00) o 0"-0 10 -65- a C N 0 'i' 'i I 'I o ' C S M 0- O 0 O N eN 0Z * 01. 0 9,.No >c; N 0 O*N N o0 C) 0 z wo 0 • N0 0 0 z '.4 c *00 O OO I a S0 IO C I NO< I in + I a * a ¶u O Om 4J 0 N$ IN L OO 4 -o *N X003 . o. IL OL .. r .C NO O C, W 3k 0 I 4 N ( 0 V Ko ~ ?i -. 1 o . 0 0d - co . .**. . *.........I # 0 0 a * 4 •) fU) 0 0 '-9 1 IO *._ C n > a C) P w00 NO 44izN S g O m 0 ry0) a0 X As. a Ncy 0 41' * roE - e0 E U 0 CO 0 it 4-)CO r o oi U)c -66- em ........ aa m "0 cJ N m '-4 a a N3 I 7 < o JO o 'U XI0 V 1 N <S< , 0 x +N a i N+I -4 L- :o : ) L - a . a 0 d . . a . a. C1. . O . L0 0 N * m Oz I . -. 0 IO 1 J I t -67- 8 m '3 "I (I ,,N ,N I In ;V1 z m <,a 7X xx N IL o. Z 8 N 3 a ly 0 a 2 0 N 0< 0< Z + SN Nt SWOI 0 . . U'QL N I- wq a-- am(j :3:J.-x 3>-XU) 0 a g 9 0 a . S. m HX N d d 3 ca ; C | i ;I a a30Aw1 1 rj -A. d a d E E 11 L I I I m .0 o N N 0 m MN E -I N)N 0 0 In >- C D Ix N a0 0 z 0 U 0? 0 aC 0 0 NO cN 0 · N0 0 O N< C N 0 U N I X WJ N , IN WOO- 03IN 0. -O. N I- I 0 ' 0 N 0 0 0 U - c B :, z 6 r d O 13 aL p n CLN 0 Ex m U I d d' t d N d' -69- Appendix C: Cross Spectrum Data Cross spectrum data = 3.23, 11= 0.34 8, f o0 134.75 Hz f (Hz) phase delay (degree/120.0cm) (m) phMe speed (m/s) 0.20 26.625 31.0 13.94 371.0 0.31 41.687 34.4 12.56 523.5 0.56 74.00 91.2 4.737 350.5 0.63 84.00 79.0 5.468 459.3 0.70 93.00 79.4 5.441 506.0 0.73 97.00 88.4 4.887 474.0 0.77 103.00 100.5 4.299 442.7 0.84 112.00 104.5 4.134 463.0 0.99 132.00 107.8 4.007 529.0 1.27 168.25 77.1 5.603 942.7 1.41 187.25 67.9 6.362 1191 1.73 229.5 53.4 8.090 1857 1.80 239.0 50.9 8.487 2028 1.92 256.0 59.7 7.236 1852 2.68 356.0 101.2 4.269 1520 3.43 456.0 145.0 2.979 1359 4.24 564.0 180.0 2.400 1354 4.80 638.0 255.9 1.688 1077 5.66 753.0 279.3 1.547 1165 f f -70- 0 0N 0 a ,7.N >.. .D1 iU Z N N Xx 0 SzN I= 4- WZ I 0. NO ID 0L) ; .+ + o U Z N WI ON MN . 'l ON WOON - N NX IrW 0.. NO Oa o sro •O•1 0 N 0 O O I zZ 0 m a.00 ) ) a 0 O I .0 0U IO0 O d 0 v· -71- (3 -- LN · I II -u - 9' .0 5 rd .e Z I . r .. 1 ZX - (n a. IU) L 1 " Z - '1-4 . --. 4I. -* 0. · w °° .-. *t U (flI p- f ,- z w O:cr ** I* c| (LI :3 C. 0. u .tjL 32 0 N N..... S-c. p ... •i. C; P 4) el L *1L .j 's. :- o 00 0. U u (I 10 L Li -'J -72- m 8 .·h-La o w 0 N 0 0 II0 O 0 o 0 Z z -4 0 IX IX U o s0 g a- I, a z t Z Lf <J N a I au 2 -I J -4 4 NO NO O0 O S N 3>X 1. . 0 o o N x fm a N a* S 0 0° a m 0 I I - -- m i N mW a0 " d N m O N 6 I- 00) aCr M . m - ( ON WCON 0 . . 0 a 0 L ( .··+ 0 L C w +k NU Z N wZ u o 07 @'o to a d q d N a -73- T AA U w 0 0 -j a;; NN II 0 I 0 0 .0 )xe0 co Z S <a ber 'a x- z S w t fl <-I I 3: 0~ N I NO < WnOI iVa f OLX TO 0 j1 I I 4 Z N i! *I . d 8 I x o z SS 0 a.N1 cc m L 0 x 10 d d oa d o wz w ONV WOONY IooR II s-~ m NZ - .ma IA w .F 7 ^ -74- 0 0 a; II is C 0 U W 0 13 Q 0 0 N - 0 0 0 3 0 U '.. m o Z tZ 9a 4 C 0 I.. Ex .0 * U 0: t to <a I- V3 U 0 9, Cu Z 1NO "1" NO FioI i 00 0-40 t0 0O i' SN W U Z 10 c (ONL au N ON w OON 0 .a U-OL N N a .1d. m I.. a t N x ý-x o St a - L L m a m 0C o CL 0. ce m 0W 0 4m00 d N d c d 0 -75- o w a 9 aW IN I 0 0 a 0 ScN a, z xx a AC a S 8q U m a0x I- 8 3 0 w0 I .0 0. ft No 0. 8 II 9 UN a-t < U a Iv w N 1003t N Noi rW Nam a L mN IW ----a) U " d ---------* d d Iwr I ft d N m a cl 6 ) (L q 4J c m U) 41a 4U x' 00 E E 0 a,^f 76- a f""' U a ' Pd i O e i 0o 'I S N o II ( S a O 0 a S 0 0 o z ,-r0 i. 4. *a Ix (L z a a aY Pd Pd au a a a <n- < I 2 a j O S a -J ab V O NO U C. w U Z I a N< W U m a o N a a a O .4 O L aiL ) I N Qa .- 3t )- L a S * O d L o0 + WOON ] 0 0 ry a N Q. N< .0 0"3 10 IQ I 4. P -77- - - a a 0 b. a a .0 .A 4.. a -. a .0 .& 'NY w 0 .0 0 U) weaa U 'SI <~iN II C; .0 0 0 0 .0 04 >- 04N o 0.. 9 U IX 4 .4 IX .0 J 4, 0. zCL p0 mi4 be 4 4 0o a9N a O 0S 0r I S I U *0 0 0 S. i QI 9 uA w N xU * Nr I ' I) D a w 3 :a U) Nu ! LV1 Ca-; L 6'0 CL d N N - a Li; 0 a 0~ a a a o . o* N N B "4 '0 N S L 0 z Oa 0) .0 C 0) 10 s a a a I Z> >:U) 9 N c O -78- ,.··· - U·. 'lv .6~··r,· 0 .3 S 0 *N ¶ IX V4 ,] -a 'U 0 4 -a 4 -a 0 XX 'Iv z Z 4 -a w z I L S.J 0 NO N<0 * .3 -aN 7 U NO + N I W N 0. UOS N N u0- , & a- a Al aJ e O a Ga N.I a ~to a*-0a :3 A. O on(U x~V L o S I.. M 6 z0 - 0 - cL in w xS 0) s"o -79- .. . .. ... . wu N .. . . . .. . .... .... .. - . . . N0 0 II a8 U m Li N, Y xx O 1zn 'I a' U, <1 I tO 0 -0 .5 NO Li Bku Un wu I 44ý( zy, * Q a. C',~) N - Oxi f 2oul N '081-00 LI i:I N NtO ; 4 x Li mW 4 N * d11I o o4J nco em u, C 81111 6i E - ~i C .jJ -80- n CON II " - N 0 ., . I, . "N t : rX 0,* 1.1 -0 in0 -.0 O 4:, flx wSIt< " Ir4 .* EnDx t70 -- U " . z * c ." = . -. 5 N L " j. QiU f4 C3 • I" .... 0 •. a ! C0 ) 0 3> ,j f , L0 4a (n *T .! I -81- -- , *l. *1 . 4 ~J Q 0 w C -,' in IN I a) Ln z * , 1 N N (M < xx IX -0 w IiJ 'I' U Nr z *0 Uc a F1 *6O .0 4 * ICL a. 'O- - .r13( -u) x 1f 0 UU ZN U, W MUiO T N* 'ai0 ,1 * 0.. N N 0 m 1... . U . Al S m S m 0 m I0(D Q IM O .3 a *l ,. 13 I, I .t2.1 a .0 IO : t j E C 0 U ii a- -82- *7 'C' CC? o - ) In d cc In * U,cl IN IO -- 0 • $1 ccinC Cm z IX I- " .n a. nn 60) .o.0 *, cc, r' -n u' ala O rL I n n.r. S0 LU Z U NW1 N.. m *I 0 r: n, cl *1 r e fi L am 0 11) ) a, Cu 0 1 0.0m r. ci |'" I, 3t'' • -83- -- --- ---- ---- ---- a2a w . . 100 UU nn IN ri C, Q z<a U*) S Ix C3 U)8 - 0 uU, CL' z0~ w - U) < m3 CL N0 3c% 80 L ttC* rl c+ 3C N l. U "1 Zn,,t w IM: cl IP N... Na. 0 a N . . a 2 aC .. aCI zC 0 . C. 0 0 ... 0 I e. ul1 E -84- ................ • ,L r· C · · · ~·~···.····~~a a .· wV O 11* O N o* O N I rS. I a 0 In z u 0 .0*C m *0a .0C) In aw a. z >10 w .0) 0) < I 4-l w 4.. I . . . 0 0. - U. 0< C Nl Imp Uu * 1 .0 z w zoI I N, . . .. f* .. . . . . . . M -.-" O 0 0 - t O' w a . N m UN I- N.. XOP O0 z * 0.0 0 S.0 -85- - a a U a u w * * a a t. a O 0 0 N -I N a a >.1 Ln I.Z xx.1 Z U X I-4 a 0 0 0U) 1. .: N im jN. 10 N UO + +N ZO iN I- <.3 gCi-OG. W +N N.. m N a*x 3t 3b- N a>@ 0 a a oo 0a - a 0 - d . d 4J CD L ED 0M 10l 0u am . .0IO I i- .1*3 m %.Iii d d I -86- o d N C- ,. 1I I0 Z IC < V C' a C, lx C' a. z UI I Li ..00 W _ * C, _ O& L Q .•- . OX w.0-c-- w '.. N N... 31 $ In S cl n Cl 1 a C .. C r C' !L * m 0 - CL. C c.r 0) 0- U) xO e 0S -? c C d c6 -87- .1 W 0 O N m z x4 N 0 I, 0O <, I:3 a-a JO 0 N. WO IN u at 0U-IOL .1 ON n N 3).XIDU xU ,x W - NXI 0 '38 S d i' N d d L 6 z Lo 41 o -- @0 i i -88- m m a 4 I% w 0 Si N N N m i i z n z lx I" < z N) Ij <a -*.Jo "o 00 o a + +O N 0..I z X - 1 1I~ 010 O 11 ; L p0 mer ÷N W IjLL I 0 0XW I )O~NL N SON<N.Q 0O , Wr O· I K U) N.. zu Plr UOOL m P '1004 ir Na ai 3t >X I 0 N N L ow. >, 0 0 - 0 L Ni zr~rr N z . 00 -.. 001 0 , ,, - . M 0 "c . 0 0 d a 0 0 0 E o i -89- Appendix D: Longitudinal-Flexural Coupling Data(Symmetric Loading) Power Spectrum Data -90- V I I --- -~N to -ýEE N I 0 ,- o Ln o3 00 0i N I V n ..0 r E 4J IT -,-4 (d 4) 3 0 2 0 III C C-4 0 o € III U i c-h X ,-4 4a) 0 0 0 r_ 3 ac a 0 4) u -4 c i • r. o 0 0 l4 4-4 o 4) 0 0 I w >1 C W 4) 4-4 O 0 I II H ) 4) w 0 0 0 *1- O 4) 4) (J2 -H 0 a 0 u -91- 0 0 f exurl I 0.001) 0. S P e 1 c AA% AA, 10 100 le- Hz Longitudinal wave power spectrum for pure tone excitation @ 19.531 Hz 0 , II ongltUdinal 5o1 le-05 I s 1e-06 p e 1 e-07 C 1e-08 e-09 10 100 Hz Flexural wave power spectrum for pure tone excitation @ 19.531 Hz e+ -92- N en LOl -,C< 0 c 111111 1 11111111 I 1111111 U-I4 0 0. ) O 0 0 4) 4) 'l 0 0 C) 4H C 5) >1 4) E4 4) $4- IIIII ....... huuI IIuItI II IIII = i ....... . ... lli " o 0) 4) " .1- (D LO I 0) (D 4) *f. U aa) u -93- Longitudinal wave power spectrum for pure tone excitation @ 39.531 Hz =7 flexurol 1 0.0001~ 1 e-05 j e-06 s p I Ie-07 !c0 H i e-08 ir ji I ei0 nhh -0 0 i, I) Hz Flexural wave power spectrum for pure tone excitation @ 39.531 Hz h~ I I -94- E U E M7=17 Z II r I ! ' ' • m F N L'4 ox 4, 11111111 hut111 iI lulII I1 -,-i .,- U) wftý ... mmmm.-.& 0 4) o C4 0 ,-- u i 0 4 .,- hu iii, iii111 11 I1111 i uui -4 -- V) .. J 4) -95- Longitudinal wave power spectrum for pure tone excitation @ 58.594 Hz f exuroI ie-06 E 1e-07 E 1e-08 1e-09 A I I.0 I !00 HZ Hz Flexural wave power spectrum for pure tone excitation @ 58.594 Hz I II 1 le+ 3 HCR le+ II -96- FT-- ~4J1JZZIZI9~T 1 11 110111 i I IIIItIII I jlititi I 0 i I II :>mll a 0 u 3 =PEF I -- -L 0 S S -C 0 0 0 0 iT C SIl 1O Ililli11 111111II I1111 Lt 0 I E1 511 Itl 1 t I 0 0 I 0 I uc a I CO0 0 I N -97- longitudinal S00 001 r le-05 te-06 s P e C le-07 1e-08 -09 I It\I ' ' ' ' , 1 I I I I 1 Longitudinal wave power spectrum for pure tone excitation @ 78.125 Hz f exurol Ie-05 :e-06 5 s P ie-07 - e C - Sj e+i 'e-08 1e-09 7 L 10 - - ----- , 100 ----------- Flexural wave power spectrum for pure tone excitation @ 78.125 Hz 1 -98- N 01 a' .9-4 GD r-4 0 -4 E4) *H 0 *0 '0 0/ aou• .) O (U *c 0 4, O O 5.4 4 '-4 30 co 0 C4) O 4-4 *H 4) a 0, 0o c c *H 5.4 4) 111111 iI LL···-· 11111 ·1······ II I· ········ - · IIIII I I 0 0 4) 4) V) L 4) 0 -99lIongitudinal 0.000 1r Kf I1- le-059 s 1e-06, 6 e c 1e-07 iei-08 ~e-09 10 100 HZ Longitudinal wave power spectrum for pure tone excitation @ 117.19 Hz 1e-06 r 1e-0 7 Se-08 I e-09 i- 100 Hz Flexural wave power spectrum for pure tone excitation @ 117.19 Hz j le+ -100- IEIg JZlZZ itI o0 0 0 N IL SI Le ,,. .., -, € i . 0 .,4 I 4) III -U Ii' Eq 41 0 *.4 ! i I i r 41 0c o ý4 4E 04 4) ,-4 -, 4) *1* -101- longi tudinal 0.0001_ 1e-05 1e-06 s P e 1 e-07 C 1a-08 IA 1e-09 I e-O9 !0 100 1e+ Hz I Longitudinal wave power spectrum for pure tone excitation @ 156.25 Hz l f exuro I - - I ! 1 e-07 :1 f e-08 S I P e I ee- II cif 10 100 Hz Flexural wave power spectrum for pure tone excitation @ 156.25 Hz I I I I I 1e+ -102- MIMI 1 114m i" I• I • /· I J II OD r 00 0a) 0a) 0a) ) 1 I 0a u I! f -- 1 I i- O I or 0 00 0 a) V) Y O a0 ) 0- a( u -103- Longitudinal wave power spectrum for pure tone excitation @ 195.31 Hz Flexural wave power spectrum for pure tone excitation @ 195.31 Hz -104- N GD r-I o ,-I .H 1111111 I .......................... 111111111 (11111111( ..1 i 0 0 c4 c I4 4) 4-I U) 4) IU), 4) 111111 I111111 I 1111111 I -00 0 O 0 S0 0) Q. O) -105- LJ long i tud i no 1 re-05 1e-06 F P e 1e-07 = c 1e-OB 91 l e-09 Fi '0100 1e+O. HZ Longitudinal wave power spectrum for pure tone excitation @ 410.16 Hz ff!exuralI II If ie-07 'i le-08 s I le-09 1e-1O Ii l-11 1 0 100 •0Hz Flexural wave power spectrum for pure tone excitation @ 410.16 Hz e+ 30 -106ro N O 3 1 1 1 1 1111111 11111111 )1111111 o) (0 r,- I I 0 0 0 0 0 x 41 (d GO I 0 c 41 I *u c·0C-i C 0 w O 04· O 44 Q4 K 0 O C 0 Go 11 44 E4 *rl 4) 111111 1i11111 i 1 1i1i1 (n 0. ) I O U) r4) 4) -107- ongi tudinal le-05 1e-06 F s P le-07 c . 1e-08 le-09 10 100 1e+ 3 Hz Longitudinal wave power spectrum for pure tone excitation @ 800.78 Hz flE flexuroi 1e-08r 1e-09 s P 1e-10" e c 1e-1 1 le-12 -, 10 , 100 I 1le) Hz - Flexural wave power spectrum for pure tone excitation @ 800.78 Hz - -108- Appendix E: Longitudinal-Flexural Coupling Data(Asymmetric Loading) Power Spectrum Data Asymmetric loading frequency (Hz) longitudinal ( x10 5 ) flexural ( xlo0 Energy ratio F/L ) (dB) 19.531 2.0705 133.37 18' 39.063 1.9188 40.502 13 58.594 1.1283 7.7021 8 78.125 1.6532 4.7221 5 117.19 1.6897 0.92276 -3 156.25 1.1827 0.20135 -8 195.31 0.83341 2.8741 x10- 3 -25 410.16 0.4790 4.7251 x10 3 -20 800.78 0.86933 2.8937 x10 2 -15 -109- N o< LO r-4 0 C) *4 x 4) .4 C0 '-4 C O U, 44 0 C O E--4 U) i,0) 4) -e-ý .4- .- 4 0 CL W u -110ongi tud i naoI -I 0.0001 I Ie-05 S e C l e-06 ½ l e-07 -i i 10 I I i I I 1 1 1 100 1 1 Hz t. Longitudinal wave power spectrum for pure tone excitation @ 19.531 Hz Flexural wave power spectrum for pure tone excitation @ 19.531 Hz 1 1 11 • leý+ -111- N 0 .-)O q< 1111111 II 0·l· 0 0 0 4C,-I 0 C a) O r4 C 0 0 44 N .4 U) <~ ~ C) .74 N: 4) U) 4) .7 IIIIII II' """ ) Q. C -112- F 3 long tudinal=L 0.0001ý IIL 1e-05t~ S p 1- 0 1 e-06 i 1r I le-07 '0 H 100 le+0 Hz Longitudinal wave power spectrum for pure tone excitation @ 39.531 Hz ieexurol u•! 0.001 0.0001 re-05 le-06 10 ,· 100 Hz Flexural wave power spectrum for pure tone excitation @ 39.531 Hz 111 le+ -113- N uD c .U) C U-) .4 1 1 ll II I 111111 11111111 11111111 6111 O hLlLLU C C .r4 CL) a o u 0 0 0 0 E,U) 4, lI 1111111 1 1111111 m rn ( CL lI I I1 *-4 i 4, E 0 C t -114- longitudinal 0.0001 le-05 S p 1e-06 1e-07 I 10 I I I I I I 100 HZ Longitudinal wave power spectrum for pure tone excitation @ 58.594 Flexural wave power spectrum for pure tone excitation @ 58.594 Hz 1 1 " 1e+ -115- -116- Longitudinal wave power spectrum for pure tone excitation @ 78.125 Hz flexurol 0.001 ' 5 -- 0.0001 c 1e-05 e-06 ii 10 100 Hz Flexural wave power spectrum for pure tone excitation @ 78.125 Hz le+d I -117- I _1_ N 0 T- 0 1-I C 0 C4) 41 I11111 I Ihutll IIiii ttu 4) H i C 0 *4 (n) C.) O 0 .4-) 4) 0 44 Co 0 Co Q4 >1 a) E- Co U) 4) w a i 11I ti Ii ]III1 I tutu11 .H v 1 C (D o -118longi tudinolI 0.o0001 ii r Ie-05 F- s le-06 I !I le-07 i 10 100 le+, HZ Longitudinal wave power spectrum for pure tone excitation @ 117.19 So0.0001ooo -- I e-05 Ie-06 1e-07 r - I I I I I I 100 Flexural wave power spectrum for pure tone excitation @ 117.19 Hz 111,111il -119- I I "t" N LO CN 0 U) II N -I I 0 All O 0 9- t1il il llll 1n lil ln I ! 0 o 0 0 V V .4. 0 O G) a) I-I o 'U 0 c *1- II I ) Q 0 V C) ,, I 4) I O 00o C 44 0 C, 0as 0 I Co W 4) 54 w >1 w O T- 11111111 (1111111 V V ) V )11111111 ) 44 w U) - 4) .94) U) (A CL o - E .9- -120longitudinal I I =I1M 0.0001 1e-05 p C 1e-06 l e-07 10 100 1e Hz Longitudinal wave power spectrum for pure tone excitation @ 156.25 Hz flexurol Ie-05 I s I- t e-06 p SII 1e-07 E i1 iI Ie-08 10 100 II le+ 1 HZ ii Flexural wave power spectrum for pure tone excitation @ 156.25 Hz IU -121- N Cr) a-) C O .,H 4 (U X 0 44 0 4) *H VC) 0 ) O C -,04 C, C, U) Qa 0 O -122- r I= I, loni tudinal In - e 1e-06 C I e-07 le-08 1111 10 100 Hz Longitudinal wave power spectrum for pure tone excitation @ 195.31 Hz Flexural wave power spectrum for pure tone excitation @ 195.31 Hz le+ -123-- p. DIDEEEI 11 1zIE~~ U I U 3 L I' C S It' ri I, I Llr I -124- ' o - ,longitudinal -l Fe-05 EI L P 1e-06 0F .e-07 le-08 O i 1 10 1 l I . 100 Hz Iil. le+ Longitudinal wave power spectrum for pure tone excitation @ 410.16 Hz I I -~ 1,fex7ro- Ie-07 1 e-08 s e c le-09 le-10 10 100 Hz Flexural wave power spectrum for pure tone excitation @ 410.16 Hz le+ I -125- r r r EUE r ri | il N co U 0 o0 0 0 0 o 0 0 N e,0 I IIIII III K !(1111 *4- x I*UlI I 2 4, aa .r4 - 0 11 4, a -7 II 0U (0 Lfl 0 0 II (D 0) oCL 3 0( 04) U o I • • t0) O o 0, 44 04, V) 0 O unu-JlfnAL-iLLZ O I V U) 4, U) I" O ý4 4, 3 ~II I S 6 0 00 E-4 4, 4, -F- 0) )0 cn 0 0 0) a U) ) C) -126~3 - long itud ina I le-05 1e-06 p c le-07 l e-08 "1 I 1I 10 100 A I Hz Longitudinal wave power spectrum for pure tone excitation @ 800.78 Hz Sflexural 1e-06 sC I e-07" ir 10-08~ ie-ogh~ I 00-090LI 10 100 HZ Flexural wave power spectrum for pure tone excitation @ 800.78 Hz i1 l 1e+3