COUPLED DYNAMIC

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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.
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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.
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[23]
Levitan, E. S.
Forced Oscillation of a Spring-Mass System having Combined Coulomb and
Viscous Damping.
JASA, 32(10):1265, 1960.
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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.
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[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.
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Nolle, A. W.
Methodsfor Measuring Dynamic Mechanical Propertiesof Rubber-Like Materials.
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Plunkett, R.
MechanicalImpedance Methods.
The American Society of Mechanical Engineers, 1958.
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Ruzicka, J. E.
Forced Vibration in Systems with Elastically Supported Dampers.
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Ruzicka, J. E.
Resonance characteristics of Unidirectional Viscous and Coulomb-Damped
Vibration Isolation Systems.
Trans. ASME, J. Eng. Ind. 89(4):729-740 ,, 1967.
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Ruzicka, J. E.
Influence of Damping in Vibration Isolation.
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Senturia, S.D. and Wedlock, B.D.
Electronic circuits and Applications.
John Wiley & Sons,.
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Seybert, A. F.
Estimation of Dampingfrom Response Spectra.
J. Sound Vib. 75(2):199-206, 1981.
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Smith, Timothy, L.
The Attenuation of Flexural Waves in Mass Loaded Beams.
S.M. Thesis, M.I.T. , May, 1985.
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Snowdon, J. C.
Vibration Isolation:Use and Characterization.
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Snowdown, J. C.
Vibration and Shock in Damped MechanicalSystems.
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Soroka, W. W.
Vibration Isolators.
Prod.Eng. 141-145 , July, 1957.
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Rheological-ModelConcept.
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Steven, Kay and Stanley, Marple.
Spectrum Analysis - A Modem Perspective.
Proc. IEEE, V.69, No. 11 ,, 1981.
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Taylor, P.D.
Wave Propagation in a Beam Having Double Resonant Loading.
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Tse, F. S., Morse, I. E., and Hinkle, R. T.
Mechanical Vibrations : Theory and Applications.
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Tse, F. S. and Morse, I. E.
Measurement and Instrumentation in Engineering.
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Ungar, E. E. and Kerwin, E. M. Jr.
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Welch, P. D.
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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
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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
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74.00
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4.887
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104.5
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463.0
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529.0
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168.25
77.1
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187.25
67.9
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Appendix D: Longitudinal-Flexural Coupling Data(Symmetric Loading)
Power Spectrum Data
-90-
V
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to
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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
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10
100
Hz
Flexural wave power spectrum for
pure tone excitation @ 19.531 Hz
e+
-92-
N
en
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-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
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ir
ji
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Hz
Flexural wave power spectrum for
pure tone excitation @ 39.531 Hz
h~ I I
-94-
E
U
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iii, iii111
11 I1111
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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
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-97-
longitudinal
S00 001 r
le-05
te-06
s
P
e
C
le-07
1e-08
-09
I It\I
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,
1
I
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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
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Sj
e+i
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1e-09 7
L
10
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-
-----
,
100
-----------
Flexural wave power spectrum for
pure tone excitation @ 78.125 Hz
1
-98-
N
01
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1e-06,
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c
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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+
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41
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0.0001_
1e-05
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s
P
e
1 e-07
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1a-08
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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
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I
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1 e-07
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I P
e
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cif
10
100
Hz
Flexural wave power spectrum for
pure tone excitation @ 156.25 Hz
I
I
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I
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1e+
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1
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-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
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I
..........................
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long i
tud i no 1
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1e-06
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P
e
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c
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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
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41
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-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<
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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+
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N
0
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q<
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p
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le-07
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H
100
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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)
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rn
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4,
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0
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t
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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_
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0
T-
0
1-I
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0
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41
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0.o0001
ii
r
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s
le-06
I
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i
10
100
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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
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N
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0
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I
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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
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Ie-08
10
100
II
le+ 1
HZ
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Flexural wave power spectrum for
pure tone excitation @ 156.25 Hz
IU
-121-
N
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C
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r
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loni tudinal
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1e-06
C
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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+
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p.
DIDEEEI
11 1zIE~~
U
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3
L
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1e-06
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.e-07
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i
1
10
1
l
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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
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r
r
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r
ri
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co
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0
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1e-06
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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
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10
100
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Flexural wave power spectrum for
pure tone excitation @ 800.78 Hz
i1 l
1e+3
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