COUPLED WAVE PROPAGATION IN A ROD WITH DYNAMIC ABSORBER …

COUPLED WAVE PROPAGATION IN A ROD WITH ADYNAMIC ABSORBER LAYER

by

Jiulong Meng

BSEE. University of Science and Technology of China, 1988

SUBMIT'ED TO THE DEPARTMENT OF OCEANENGINEERING 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*- " leartment of Ocean E(gjeering

Felo y, 1991

Certified byProfessor Ira DyerThesis Supervisor

Accepted byProfesWor A. Douglas Carmichael

Chairman, Department Committee

__ __ _I

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 partialfulfillment of the requirements for the degree of Master of Science.

AbstractThis thesis experimentally tests the effect of a continuous longitudinal dynamic absorberlayer on longitudinal wave propagation in a circular rod. Wavenumber-frequency solutionsare derived analytically. The associated attenuation and phase velocity results are presentedto show how their behavior depends on the loading treatment. Experimental results for thephase velocity are compared to a model developed by Dr. Dyer and Olivieri. A relaxationmechanism is developed to fully model the viscoelastic material. It is also shown that theinteraction between longitudinal/flexural waves may lead to significant rates oftransformation of the compressional wave energy into bending.

Thesis Supervisor:Title:

Professor Ira DyerWeber-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

AbstractDedicationTable of ContentsList of FiguresList of Tables

1. Introduction

2. Analytical Model2.1 Coupled Wave Equations2.2 Resonance Characteristic and Relaxation Mechanism2.3 Wavenumber Analysis

3. The Experiment3.1 Experiment Design3.2 Signal Conditioning3.3 Data Aquisition and Spectrum Analysis

4. Results4.1 Resonance Frequency and Loss Factor4.2 Phase Speed From Cross Spectrum Function4.3 Flexural-Longitudinal Wave Energy Ratio

5. ConclusionsReferencesAppendix A: Computer Program for Wavenumber AnalysisAppendix B: Drive Point Impedance DiagramAppendix C: Cross Spectrum DataAppendix D: Longitudinal-Flexural Coupling Data(Symmetric Loading)Appendix E: Longitudinal-Flexural Coupling Data(Asymmetric Loading)

545660626989

108

List of Figures

Figure 1-1: Elements of a vibratory systemFigure 1-2: Schematic diagrams of dynamic equivalent vibratory systems,

Ref[3]Figure 1-3: Mass-spring-damper model of the dynamic absorberFigure 2-1: Semi-infinite rod with dynamic absorberFigure 2-2: Three-element spring and dashpot combinationFigure 2-3: Attenuation vs. normalized frequency ratio for 3=3, r=--0.1,0.3,1.0Figure 2-4: Attenuation vs. normalized frequency ratio for q=0.2, 3= 1,2,3Figure 2-5: Phase Speed vs. normalized frequency ratio for (=3, r--0.l,0.3,1.0Figure 2-6: Phase Speed vs. normalized frequency ratio for =--0.2, P3=1,2,3Figure 2-7: Attenuation vs. (o, for 3=3, N=1, -1=0.1,0.34,1.0Figure 2-8: Attenuation vs. 3, for =--0.2, N=1, P=1,2,3Figure 2-9: Attenuation vs. o, for rl--0.2, 3=3, changing N factorFigure 2-10: Phase speed vs. 0o, for 3=3, N=1, Tr=0.1,0.34,1.0Figure 2-11: Phase speed vs. o, for 1r=0.2, N=1, P=1,2,3Figure 2-12: Phase speed vs. o,, for 3=3, =--0.2, changing N factorFigure 3-1: Experiment apparatusFigure 3-2: Resistance bridge with cancellation of flexural vibrationsFigure 3-3: Resistance bridge with cancellation of longitudinal vibrationsFigure 3-4: Decide the range of Rt under the most unfavorable combinations of

resistorsFigure 3-5: Six-channel circuit lay-outFigure 3-6: AD624 functional block diagramFigure 3-7: AD624 pin configurationFigure 3-8: Noise interference problem, initial testing of the conditioning circuitFigure 3-9: 60 Hz and its harmonic interfering noises,conditioning circuit with

proper balance and groundFigure 3-10: Response of pure tone excitation with battery suplied bridge

circuitsFigure 3-11: Response of pure tone excitation with SNR larger than 30 dBFigure 3-12: Noise interference problem in data aquisition using the Concurrent

ComputerFigure 3-13: Data aquisition diagramFigure 3-14: Clock connections on the CK10 and SH16FA modulesFigure 3-15: Sampling of a periodic timing signalFigure 3-16" LWB modules in the data flow diagramFigure 3-17: Synchronization virtual instrumentFigure 4-1: Phase speed vs. normalized frequency,ao = 2nfo = 2cn*134.75Figure 4-2: Longitudinal to flexural coupling wave energy ratioFigure 5-1: With wave propagation in the dynamic absorber layer

78

910151819202122232425262729303233

3436363738

39

4041

4243444546505355

-6-

List of Tables

Table 3-1: AD624A specifications (@ V,= ±15v, Gain= 100, RI= 2kfl and TA 35= 25 OC

Table 4-1: Resonance frequency and loss factor 49Table 4-II: Flexural/Longitudinal coupling wave energy ratio 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.

Hooke Newton Maxwell Kelvin Zener

Figure 1-1: Elements of a vibratory system

9

I L

-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 so-

called 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.

(a) iLT

I-.

(C)

(b)

Figure 1-2: Schematic diagrams of dynamic equivalent vibratory systems, Ref[3]

(a)

(c

-9-

When 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 0 e F-F

Relaxatunit

Y1

xi

Figure 1-3: Mass-spring-damper model of the dynamic absorber

I I

All

-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)=Oat

2aat

-11-

m,-+K,(v-u) + C-(v-u)=E s,at2 at TX2

where EI, 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:

- K,02 -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 -io for the time derivatives

v = V.ei-(k 'x - w -t)

u = U-ei'(k'x-w't)

in the equation:

E l a2v a'- K, CS(v-u) - - (v-u) = 0

pI at 2 t2 2 Inl nlat

which corresponds to,

-12-

K2V -KI2V + ---(V-U) - iC•-O(V-U)=OmE, m,E,

where

1 E=

We can normalize the above equation, with the following non-dimensional

parameters:

Ky - ratio of wavenumber

K,

=2 mass ratio

K, = -o 2m, stiffnless

C- loss factoroom2

Wonormalized frequency

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)-

- U + -- U-V)On 2

ifp(V-U)=otO.

- il(U-v)=oon

-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

=0with Y = 1 i

(o)2 (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.

-14-

2.2 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

1Kc=K+

(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].

Kc K+imoC K isoC2 - - - (1 + )

m mm K

K isoC+-(1"+')cm 2K

In our light damping situation,

OCK

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, 1 + K ( x t - x, ) + NK( x - x3 ) = Es,x

m2 + K(x 2 -x I ) + C ( X-x 3 ) =0

NK( x, -x 3 ) + C ( x, - x 31 ) =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-(kx - w-t)

(Y - 1)XI + I(X 1-X2 )/on2 + N0(XI-X3)/0,n 2 = 0

-X 2 + (X:-Xl)/On2 - i0(X2-X3)/1, n = 0

N(XI-X3)/,n 2 - ir(X2-X3)/m,, = 0

The wavenumber-normalized frequency solution is thus obtained:

-16-

1-

1+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

eta=0. l,beta=3

i"'" '" '~'' " " " " " '"' '" " '" ~~" "~"~"~

',eta=0.3,beta=3

Seta=l.Obeta=3.

- -----------.... . .'... ..... . . . .:. . . .. . . •.••,. " • - -•.. . . . _

0 on-relaxation

10o

Normalized Frequency,w/wO

Figure 2-3: Attenuation vs. normalized frequency ratio for 03=3,=--0.1,0.3,1.0

180

160

140

120

100

8 60

40

20

vI-10- 'I

,inn -

E

............. ·~--

. .... .. .. ..

.··:·

I·

r ·

1

I

rr .

-

I

-19-


100


Figure 2-4: Attenuation vs. normalized frequency ratio for 11=0.2,13=1,2,3

I'

180

160

140

120

100

o 80

0 60

40

20

010-1

-20-

Phase Speed vs. Normalized Frequency

4000

3500 -

3000 -

I 2500 -

2000 ... .............

135 0 - I .......... ,,

1000 ......... " ......

eta=1.0, beta=35 0 0 ......... . ..............

Non-relaxation with loss factor10-' 100o 10


Figure 2-5: Phase Speed vs. normalized frequency ratio for 3=3,rI=0.1,0.3,1.0

450n' - .. . ............. ............. ...

1, beta=3• ~ · ·· · ·.. , ... ... , . . . .,.. . ,

.. .. .. .. .. ..!, ,, ,, !


100


Figure 2-6: Phase Speed vs. normalized frequency ratio for 11=0.2,O3=1,2,3

.3LAJ

2500

2000

1500I10001

500

0n

eta=0.2, beta=l1

.ta=0.2, beta=3'

Non-relaxation with mass-factor•ta--=.2, bea= 1

10"-I 10'.

-22-


10o- I 100


Figure 2-7: Attenuation vs. o,, for j=3, N=1,1--0.1,0.34,1.0

O~nn

8•"4U

0

10'

-23-


10-' 100


Figure 2-8: Attenuation vs. (.n for r~=0.2, N=1,3=1,2,3

0

U

10'

-24-


100


Figure 2-9: Attenuation vs. o,, for 1=0.2, 5=3,changing N factor

1mV

10-'

-25-


10o


Figure 2-10: Phase speed vs. (o. for 3=3, N=l,1=--0.1,0.34,1.0

cfl,',

a

10-' 10'

-26-


10-' 100


Figure 2-11: Phase speed vs. co, for 1--0.2, N=l,0= 1,2,3

"Mnn

10'

-27-


100


Figure 2-12: Phase speed vs. o, for P3=3, ,=0.2,changing N factor

2500

2000

1500

1000

500

010- 10'

,2fwvl

-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,

0c0a,EC

Uo0

E0

S

0caa,

E0

0,c

*00o

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

-30-

wave 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-

Vo_ Rg2 R2

v, R,+Rg2 RgI+R,

where

Rg2 = Ro + A Rt - 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 RovS 2Ro+AR - ARf 2Ro+AR, + ARf

1 Ro I2 +(ARt- IAR.) (ARI + ARf)

1+ 1+2Ro 2Ro

1 AR - AR ARI - R ARI+AR[= -0 G(1 )(,- (1 )]2 Ro 2Ro 2Ro

SAR1 - AR, AR- ARf AR +ARf AR,+AR2 Ro 2Ro 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-.

-32-

Recall 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 combinationsof 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 vo 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"-

ii4r'1I

• _i

.-- ~-

Figure 3-5: Six-channel circuit lay-out

'.............. ............

:.... : : ::::....** * : : * " '......... .

9 ....: . ... .. :._ .. .- - *...--* .-.

CD o- ..... .......... . ..:: :::: ....::: : .. .. .. . :.. ...... .:.:::::

....... .. ....... .. ... * *. .............

.........,~~~~~~ ...® .. . .. . ,, ...,- .,.... . -. ,,,-... • ........

.... . .- ' .. .. . ,I"... .. Tc: . . ..... ..,+. . . . . .....

.. . . • . .. .. . , . . , ...0 . .. ... !.........!

u, ) : .:: .':.. .... O• : : :: : -.. . : :::: : : :: : : :: :

1 .. . . , ..... .... 0 . .. c. .. . . . .... . . . . . . . . .

.. ..9 9 S . . . . . , . • . , . . . . . • • • • • ., • i L ., .. .. ..... .. . ... .,L ... . .. •.. . ... .'j ...... . .( ... ..D• ,, . . .\ .• . . . ... . .. . . ..\ . ..•........ . . ..'.......... . ,. \ ,. . . . .

o........, . . . , .4 . - ..7 7 - CO, ... ,.."- . *, . . . .

! .. ... .. .... .• ' I . ....... .... '. ' / i. . .. t . T . ' . . . .. ., ,, .. •.... . . . . .. .. .. .. , . CCl ., . . .. . -.. .. .. +.. .. . .+.. .. g. ............. O. • .,.. CL,. 1.... ....

S...... •.. 4 ...• •••+. . . . .•. . . . .o. . .,V /• - . . . . . .

.. . . I * ,. . . ........ a i..it o• 1 :::.... ::: 1: •::: •••,:: .. .,• .

,. . . o, ... ..... . .. . . - - . .0. . "I

. . .. . . . . . .

•. . ... ,4-,* *. . .. ... . .. .. ... ,. .. . .. ,. . ...... . .. .. . , . . . , . .

o•......... ,................... +......... •......... +......... ,.........,

| @Avg

#feetl l~t

-35-

Table 3-I: AD624A specifications (@ V, = 15v, Gain = 100, R, = 2k• and TA = 25 OC

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.

Speci- Value Un-fication it

Gain Max Error ±1.0 %

Gain Nonlinearity ±0.005 %

Input voltage range ±10 v(Max Differ. Input Linear)

Output rating ±10 v

Dynamic response 25 kHz(small signal -3dB)Slew Rate 75 ps

Power supply range Min ±5 vMax ±18Typ ±15

-36-

Figure 3-6: AD624 functional block diagram

- INPUT I

+ INPUT r2

RG, -

INPUT NULL E4

INPUT NULL E-

REFERENCE

-Vs 7

* Vs

AD624

1" RG,

15" OUTPUT NI

1" OUTPUT NU

7 G = 100

E G - 200

II G - 500

101 SENSE

"IJ OUTPUT

ILL

ILL

SHORT TORG, FOR

1D SIREDGAIN

FOR GAIN OF 1000 SHORT RG, TO PIN 12AND PINS 11 AND 13 TO RG,

Figure 3-7: AD624 pin configuration

-INPUT

G 100

G = 200

G = S00

RG,

RG,

+ INPUT

SENSE

OUTPUT

REFERENCE

AD624

-37-

Too 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 MAIN Ys -U. 5dBYo -6. 5[d /1. OOV 2

PWR 80dB X& 61. 50OHzXa 54.00Hz - 100Hz LINSETUP S1* #As 10C

10-4'--. ..... .---- ........ ...-........... r......

60 70 60 90 100 110 120 130 140 150SETUP S1

MEASi.:REMENT: DUAL SPECTRUM AVERAGINGTRIGGER: FREE RUNDELAY: CH. A-*B: 0. CmsAVERAGINGs LIN 100 OVERLAPs MAX

FRED SPAN: 100Hz AF: 125mHz Ts 8s ATL 7. 81msCENTER FREQs ZOOM 104HzWEIGHTINGs RECTANGULAR

CH. A: 8V 3Hz DIR FILT: 25. 6kHz 1V/VCH. 8: 800mV - 3Hz DIR FILT:25. 6Hz V.,VGENERATOR, VARIABLE SINESINE GENERATOR FREQ.a 203. 218Hz

Figure 3-8: Noise interference problem, initial testing of the conditioning circuit

N

W*

I)

-38-

The 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):

j12 AUTO SPEC CH.A CEY: -49.1dB /I.0OV 2 PWR 8OdBX: GHz - 800Hz LIN

#As 100

0 100 200

INPUT MAIN YsXI 203Hz

300 400 500 600 700 800

SETUP 'A'i

MEASUREMENTsTRIGGER:DELA Y:AVERAG INGs

FREC SPAN,CENTER FRECQ

ElEGHTING%,

CH. AsCH. BsGENERATOR:

DUAL SPECTRUM AVERAGINGFREE RUNCH. A-,8: O. COmsLIN 100 OVERLAP: MAX

800Hz AF 1HzBASEBANDRECTANGULAR

5CmV + 3Hz DIR30mV 3Hz DIRVARIABLE SINE

SINE GENERATOR FREQ. s

T, Is

FILTs 25. 6kHzFILT, 25. 6 Hz

203. 218Hz

Figure 3-9: 60 Hz and its harmonic interfering noises,conditioning circuit with proper bala

-71. 7dB

jo-2' 10-2

T 10-4U

10-a

ATs 488js

I V/VIV/V

-- L ' · - 1'OD'-~t~~--~l-~..--~-JL

hi

-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 3 INPUT MAIN Yv -105.4dBYa -68. 7d- -/1. C3 51.0 PWR 80dB X8 203HzXi OHz + 400Hz L. IN

#As 100

100 -A L____ _

N

E 10-2

10

I n"-4

0 50 100 150 200 250 300 350 400

SETUP Wi

MEASU'REMENT: DUAL SPECTRUM AVERAGINGTRIGGER: FREE RUNDELAY: CH. A-B: 0. OOmsAVERAGINGs LIN :00 CVERLAPs MAX

FREQ SPAN: 800Hz AF: 1Hz Til s AT: 488JsCENTER FREQa BASEBANDWEIGHTINGs RECTANGULAR

CH. As 30mV + 3Hz DIR FILTsBOTH 1V/VCH. Bs 30mV 3Hz DIR FILT: BOTH IV/VGENERATOR: VARIABLE SINESINE GENERATOR FREC.: 203. 0OOHz

Figure 3-10: Response of pure tone excitation with battery suplied bridge circuits

I"

-40-

W12 I0,'NCISE RATIOY, 27. 7dB 80dBXs OHz + 400Hz LINSETUP 01 #As 100

INPUT MAIN Y: 34. 9dBXi 203Hz

20

0

-20

-40

- .... ,. .. .,40 50 100 150 200

Figure 3-11: Response of pure tone excitation with SNR larger than 30 dB

250 300

1.0

0. 8

0. 6

0. 4

0. 2

C

350 400

0 50 100 150 200 250 300 350

412 CCHERENCE MAIN Y: 6. 92mY : 1. C X: 800HzX2 OHz - 400Hz LINSETUP 01 #A, 100

400

-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 band-

pass and low-pass filters, we finally achieve an excellent degree of noise isolation in the

measurement apparatus.

Power-linground

ac noisesource

(motor,computer,fan,relay,

etc.)

SourNetwo

V

ng

Load groundiSAgnIy C YIuuIIU

-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.

Figure 3-13: Data aquisition diagram

%OW I I IF%- 16-8

-43-

The 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!

pulses to 0/A converter

pulses to A/D converter

Inputs

Figure 3-14: Clock connections on the CKIO and SHI6FA modules

-44-

sweep rate clock 0or exeternalsync pulse

sweep length clock I

frame rate clock 2

framne length clock 3

burst rate clock 4

resuletengtiming sequence

Figure 3-15: Sampling of a periodic timing signal

-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).

-49-

Three measurements are conducted and reveal the following results (Plots are

enclosed in Appendix B):

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

Measurement (Hz)freq. span fto (Hz) f 3dB low(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 Tf = 134.75 1F =.348

-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.


IU

10-' 100 10'


Figure 4-1: Phase speed vs. normalized frequency,o, = 27f,o = 2n* 134.75

-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.

-52-

Above 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

Flex./Long.

Symmetric Loading

Energy

frequency Symmetric loading Asymmetric loading(Hz) (dB) (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

Vertical

Ratio

-53-

Fkex./ong. wave negy ra•io vs. Nrmalized Frequency

100 10'


Power spectrum measurements for a series of pure tone longitudinalexcitations 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

A0U

10

0

-10

-20

-30

-40

-50

60I

10-I......... ..

- - -.

,..

-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.Engineering Design for Plastics.Reinhold Publishing Corp., New York, 1964.

[6] Bendat, J. S. and Piersol, A. G.Engineering Application of Correlation and 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 Publishers LTD, 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-Isolation Systems.Mach. des. 37(18):142-149, 1965.

[11] Freakley, P. K. and Payne, A. R.Theoryn, and Practice of Engineering with Rubber.Applied Science Publishers LTD, London, 1978.

[12] Fung, Y.C.Foundation of 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 Specimen-

Aparatus 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.Propagation of Axisymmetric Waves in a Circular Semi-infinite Elastic Rod.JASA, 36, pp.653-660, 1964.

[19] Kerwin, E. M. Jr. and Ungar E. E.Discussion of paper entitled 'Damping Structural Resonances Using Viscoelastic

Shear-Damping Mechanisms' by J. E. Ruzicka.Trans. ASME, J. Eng. Ind. 83(4):424, 1961.

[20] Klyukin, I. I.Influence of the Elastic Dissipation Parameters, Load characteristics, and Degree

of Lateral Constraint of Vibration-Isolation Elements on Their DampingProperties.

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.Machinary Noise 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 Vibration Analysis.McGraw-Hill, Inc., 1975.

[27] Morse, P.M. and Ingard, K.U.Theoretical Acoustics.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.Methods for Measuring Dynamic Mechanical Properties of Rubber-Like Materials.J. App. Physics, 19(8):753-774, 1948.

[30] Plunkett, R.Mechanical Impedance 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 Information Division, 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 Damping from 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 Mechanical Systems.John Wiley & Sons, Inc., 1968.

[39] Soroka, W. W.Vibration Isolators.Prod. Eng. 141-145 , July, 1957.

[40] Sperry, W. C.Rheological-Model Concept.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 Factors of 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')text(.2,50, 'Relaxation with N-factor,eta-0.204,beta-

3.2 3')>> 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)));att(k,2)=54.6*imag (r(k,2));end;

/ (1-i*ata*tl(k))) ;

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 f (Hz)freq. span f a (Hz) f 3dB Iow(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-

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Appendix C: Cross Spectrum Data

Cross spectrum data= 3.23, 11= 0.348, f o0 134.75 Hz

f f f phase delay phMe speed(Hz) (degree/120.0cm) (m) (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

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Appendix D: Longitudinal-Flexural Coupling Data(Symmetric Loading)

Power Spectrum Data

-90-

I

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-108-

Appendix E: Longitudinal-Flexural Coupling Data(Asymmetric Loading)

Power Spectrum DataAsymmetric loading

frequency longitudinal flexural Energy ratio F/L(Hz) ( x10 5 ) ( xlo0 ) (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

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