Viscoelastic Polymer Analysis: Experimental,
Data Analysis, and Modeling Techniques Applied
to Cellular Silicone Foam
by
Richard Matthew Hanna
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degrees of
Bachelor of Science
..............
SJ- TITUTE
OFTECHNOLOGY
and
Master of Science
at the
OCT 25 2002
I LIBRARIES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2002
© Massachusetts Institute of Technology 2002. All rights reserved.
A uthor ...................
Department of Mechanical Engineering
A
Certified by ..............
7/
May 24, 2002
Kamal Youcef-Toumi
Professor of Mechanical Engineering
_...
Thesis Supervisor
A ccepted by .....................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
2
Viscoelastic Polymer Analysis: Experimental, Data Analysis,
and Modeling Techniques Applied to Cellular Silicone Foam
by
Richard Matthew Hanna
Submitted to the Department of Mechanical Engineering
on May 24, 2002, in partial fulfillment of the
requirements for the degrees of
Bachelor of Science
and
Master of Science
Abstract
Through the study of a viscoelastic foam, cellular silicone, experimental and data
analysis techniques designed for the characterization of dynamic viscoelastic response
are developed and evaluated. With basic viscoelastic theory as a starting point, two
experimental techniques and associated data analysis methods are developed from a
conceptual viewpoint. These methods are then implemented and evaluated, leading
to the establishment of a consistent method for dynamic response testing at frequencies below 300Hz via a servo-hydraulic test bed. A higher frequency test technique,
utilizing an induction shaker head, is evaluated and suggestions for its improvement
are provided. The development and evaluation of these methods, based on the testing
of cellular silicone, has yielded a phenomenological functionality between the material properties Young's modulus and loss factor and the test variable strain frequency.
Basic trends are also established for the test variables initial compression and strain
amplitude.
Thesis Supervisor: Kamal Youcef-Toumi
Title: Professor of Mechanical Engineering
3
4
Acknowledgments
This research was performed under the auspices of Lawrence Livermore National
Laboratories. The following contributed substantially to its genesis:
Charles Chow
Tom Woehrle
Bob Sanchez
Al Shields
Steve DeTeresa
Pat Hargrove
5
6
Contents
1
13
Introduction
15
2 Methods and Concepts
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Viscoelastic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3
General Testing Methodology
. . . . . . . . . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . . . . . . . . . .
21
2.4
2.3.1
Temperature Effects
2.3.2
Servo-Hydraulic Test Methodology
. . . . . . . . . . . . . . .
22
2.3.3
Canister Test Methodology . . . . . . . . . . . . . . . . . . . .
25
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Sum m ary
3 Implementation: Experiments and Data Analysis
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2
Servo-Hydraulic Experimentation . . . . . . . . . . . . . . . . . . . .
32
3.2.1
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.2
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . .
37
3.3.1
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.3.2
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.4
Experimental Differences . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.5
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3
4
31
Canister Experimentation
43
Results and Discussion
7
5
4.1
Introduction . . . . . . . . . . . .
. . . . . . . . . . . . .
43
4.2
Quasi-Static Results
. . . . . . .
. . . . . . . . . . . . . . . . . .
43
4.3
Dynamic Servo-Hydraulic Results
. . . . . . . . . . . . . . . . . .
45
4.3.1
Modulus Functionality . .
. . . . . . . . . . . . . . . . . .
49
4.3.2
Loss Factor Functionality
. . . . . . . . . . . . . . . . . .
54
4.3.3
Strain Rate vs Frequency.
. . . . . . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . . . . .
58
4.4
Canister Test Discussion.....
4.5
Summary
. . . . . . . . . . . . .
61
Conclusions
A Servo-Hydraulic Test Results
63
B Canister Test Results
69
C Canister Fixture Diagrams
71
D Matlab Source Code
77
8
List of Figures
2-1
Stress and Strain Sine Curves offset by phase angle 6 . . . . . . . . .
17
2-2
(a)Stress vs Strain, in phase (b) Stress vs Strain, 6 = 0.5 out of phase
20
2-3
Servo-Hydraulic Test Concept . . . . . . . . . . . . . . . . . . . . . .
24
2-4
Schematic Diagram of Canister Test System . . . . . . . . . . . . . .
26
2-5
Basic System Model for Canister Test . . . . . . . . . . . . . . . . . .
28
3-1
Servo-Hydraulic Test Schematic . . . . . . . . . . . . . . . . . . . . .
32
3-2
(a) Quasi-Static Load Curve (b) Dynamic Data Inset . . . . . . . . .
35
3-3
Force and Displacement Curves for test at 36% compression, 5 mil
displacement, and 10Hz
3-4
. . . . . . . . . . . . . . . . . . . . . . . . .
36
Force vs Displacement for test at 36% compression, 5 mil displacement,
and 10Hz ........
.................................
37
3-5
Schematic Diagram of Canister Test System . . . . . . . . . . . . . .
38
4-1
Experimental Quasi-Static Loading Results . . . . . . . . . . . . . . .
44
4-2
Complex Magnitude of Young's Modulus for all servo hydraulic tests
46
4-3
Loss angles for all servo hydraulic tests . . . . . . . . . . . . . . . . .
47
4-4
Comparison of test data with classic viscoelasticism (a) Classic Viscoelastic Curves [3, p. 72] (b)36% Compression Servo-Hydraulic Test
R esults
4-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model . . . . . . . . . . . . . . . . . . . . . .
4-6
48
51
36% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model . . . . . . . . . . . . . . . . . . . . . .
9
52
4-7
48% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model . . . . . . . .
. . . . . . . . . . . .
53
4-8
Classic Viscoelastic Models . . . . . . . . . .
. . . . . . . . . . . .
55
4-9
Loss Factor Models (a) 24% (b) 36% . . . .
. . . . . . . . . . . .
57
4-10 48% Compression Loss Factor Data . . . . .
. . . . . . . . . . . .
58
4-11 Complex Modulus Magnitude vs Strain Rate
. . . . . . . . . . . .
59
C-1 Canister Fixture Hardware . . . . . . . . . .
72
C-2 Canister Fixture Hardware . . . . . . . . . .
72
C-3 Canister Fixture Hardware . . . . . . . . . .
73
C-4 Canister Fixture Hardware . . . . . . . . . .
73
Canister Fixture Assembly . . . . . . . . . .
74
C-6 Canister Fixture, Shear Arrangement . . . .
75
C-5
10
List of Tables
Initial Test Matrix Proposal . . . . . . . . . . . . . . . . . . . . . . .
22
A.1 Servo-Hydraulic Test Table. . . . . . . . . . . . . . . . . . . . . . . .
63
. . . . . . . . . . . . . . . . . . . .
69
2.1
B.1
Canister Test Result Calculations
11
12
Chapter 1
Introduction
Viscoelastic foams have been used in Lawrence Livermore National Laboratories designs for years. Their value lies in their native damping characteristics for packaging,
defect stack-up, and thermal expansion applications. Specifically, LLNL has made
significant use of a series, designated M97xx, of viscoelastic cellular silicone - a foamed
silicone polymer. Under stress, cellular silicone exhibits stiffnesses and damping factors that are highly dependent upon compression, strain amplitude, and strain rate
and frequency. These dependencies are highly non-linear, and thus require experimentation over a broad range to pin down the dependency curves. Strain rate and
frequency dependencies, are, in particular, quite difficult to characterize.
The lack of good correlations between conditions and material behavior has led
designers to work with data garnered through quasi-static and resonance testing; data
which, because of the non-linear nature of cellular silicone's response, cannot be considered complete. In order to optimize designs for specific conditions, it is necessary
to have more complete characterizations of the material. Because the cellular silicone
in question was designed by and for the Lawrence Livermore National Laboratories,
there have been no outside efforts to characterize its behavior. There has, of course,
been much work done , theoretically and practically, to characterize and describe
the behavior of many other viscoelastic materials. In part, this work attempts to fit
cellular silicone's response characteristics with those of similar materials, upon which
more extensive work has been done[1] [3] [2].
13
Another essential goal of this work is to evaluate methods for the characterization
of similar foams. The cellular silicone used for this work is manufactured in 1mm
thick slabs of 60 percent open-celled porosity. Foams of the same base rubber but
different porosities and thicknesses (1.5mm, 2mm) are also used in Livermore designs.
The base rubber itself was not available for experimentation. The establishment of an
efficacious set of testing and analysis methods serves to shorten the analysis time for
the material considerably. The technical difficulties of working with the foam pads
need not be revisited.
The pads are just 1mm thick, compressed to 64% of original thickness (a standard
testing parameter used in this work); the strain amplitudes tested are approximately
.000025 m, a displacement which is quite difficult to measure under high frequency
conditions. The experimental work focuses on eliminating possible sources of experimental error and creating conditions stable enough to acquire accurate data measurements. The challenge of the data analysis was to create a body of code which
could handle the massive amounts of data collected, with minimal effort from the
analyst. The results from this semi-automated analysis are then free for comparison
and correlation with expected results from existing viscoelastic theory.
This thesis details first the ideas and concepts behind the approach to this study,
explaining the methods used and the reasons for their usage. This is followed by
a more detailed explanation of the exact experimental apparatus, hardware, and
procedures. The results from these experiments are then presented, along with the
comparisons to existing viscoelastic theory. Finally, conclusions are presented.
14
Chapter 2
Methods and Concepts
2.1
Introduction
The goal of this work is two-fold. First, it establishes the dependencies of cellular
silicone on the design variables initial compression, strain frequency, and strain amplitude. The inter-relationships of these dependencies are explored via a matrix of
different testing conditions. Additionally, it creates and critiques testing and data
analysis methods for future work on similar foams.
The genesis of the method behind this work lies in a review of previously recorded
data and the testing techniques behind it. This previous work
1 reveals
strong de-
pendencies of the Young's Modulus and damping factors (collectively, the complex
stiffness modulus) on strain frequency and amplitude, as well as initial compression
of the material. Strain frequency, rather than strain rate, is used in these tests because, operationally, the material is subject to steady state vibrations and not impact
loads. Some tests were also performed in order to validate this assumption. These
dependencies are explored through the experimental work. What follows is the experimental and data analysis theories used in this work. Chapter 3 will explain the
actual application of these techniques, where they succeeded and where they failed.
'This unpublished work was performed at Lawrence Livermore National Laboratories by Charles
Chow, Steve Deteresa, Thomas Woehrle, and Pat Hargrove.
15
2.2
Viscoelastic Theory
In simple elastic mechanics, the relationship between stress and strain is a simple
linear function based on Young's Modulus (in uniaxial strain). Strain x E = Stress
or, cE =- a. One can see that this simple relationship conserves energy. The ratio
between the stress and the strain is just a single number, any energy created by strain
will be stored as potential energy and released when the material decompresses. In
practice, no material is 100 percent elastic, and a perfect relationship must have
room to represent the dissipated energy. In the case of viscoelastic materials, the rate
of energy dissipation becomes significant in an engineering sense, and the equations
must change to obtain accurate models. Here, we adopt a complex Young's Modulus
of the form shown in Equation (2.1) to characterize both the stiffness and damping
characteristics of the material. E' is known as the storage modulus, and represents
the modulus term that is entirely elastic. 17is referred to as the loss factor, and creates
an imaginary term (denoted by (i)) that mathematically represents the dissipative
modulus term [3]. Because of its usage in vibration damping applications, this work
examines cellular silicone from a dynamic point of view. Any vibrational excitation
can be described as a series of sine waves of varying amplitude and frequency. These
excitations translate into strains. The goal of the experimentation will be to offer a
sound functional relationship relating the dependence of E*, and q to strain frequency,
amplitude of strain, and initial compression.
E* A E'(1 + 7 i)
(2.1)
Viscoelastic materials, cellular silicone included, when strained, respond with an out
of phase stress. In the following equations, c and - represent the actual strain and
stress, while E' and a- represent the respective amplitudes of these sine waves. W is
the frequency in radians, t is time, and 6 is known as the loss angle.
E = E sin(wt)
16
(2.2)
1,5
6
L
1
-0.5
- -----
0
. ...
1
-------
2
.......
4
3
...
5
6
time
Figure 2-1: Stress and Strain Sine Curves offset by phase angle 6
u = &sin(wt + 6)
(2.3)
That is, the stress leads the strain, as in Figure 2.2, and the two reach their
maximum and minimum levels at different points in time. One can think of the
material as responding somewhat slowly to forces placed on it, the deformation that
result from a certain force does not happen at the same instant in time as the force
itself. To establish the dependency between a, c, and E*, multiply Equation (2.1)
by Equation (2.2) to yield Equation (2.3). This may not be readily obvious, and a
conversion to exponential format may be helpful in the elucidation.
*
=
oeWte
o-A ao e
t-u
Im(6*)
-
Im(u*)
E* x E* =E-- eett + Elqieit
(2.4)
(2.5)
(2.6)
00
E = - cos 6
sin6
cos 6
(2.7)
(2.8)
These relations, all considered together, yield Equation (2.9). When Equation (2.9)
17
is expanded, it is composed of four terms, two of which are real, and two of which are
imaginary. The only part of interest is the imaginary part, as that is how we specified
E. The imaginary part of that expression is shown in Equation (2.10) and a simple
trigonometric identity reveals in Equation (2.11) that a-and E are related by E*.
E* x E* = o-cos 6e
t
+ - sin 6ie
(o- cos 6 sin wt +-' sin 6 coswt)
osin(wt + 6)
=
(2.9)
=
a-
(2.10)
(2.11)
This method is somewhat backwards, as we started with knowledge of the relation
between the stress and the strain. Without that knowledge, E* can be arrived at
through simply dividing the stress and the strain.
CO
0-*
*
E* =
-e
o
=
E*
o-
+ -%sin
-COS
(2.12)
6
(2.13)
El
C cos 6
(2.14)
E2
(
sin 6
(2.15)
E60
E*= El + iE 2
(2.16)
E2
El
E2
= tan 6
(2.17)
E tan6
(2.18)
,= tan 6
(2.19)
E* = El (1 + ir7 )
(2.20)
Equations (2.14) through (2.16) illustrate the introduction of the terms El and E2 ,
known as the storage and loss moduli, respectively. The storage modulus represents
18
the relationship of the in phase stress and strain; that is, the stress that produces
potential energy, which is stored and returned when the material returns to its initial
length. The loss modulus represents the relationship of the out of phase stress to the
strain. This stress produces energy that is not stored, but lost to dissipative heat
generation. Equations (2.17) through (2.19) illustrate the notational simplification
that is carried out. It is important to note that E* = E*(w, length, c). That is, E*
shows strong dependency on strain frequency, initial length (compression percentage),
and even strain amplitude. Establishing these dependencies is central to the final data
analysis. The expression for a, a-= (O cos 6sinwt + a' sin 6 cos wt) explains dynamic
viscoelasticity. When the phase difference, 6, between the stress and the strain is
small, most of the stress is in phase with the strain (i.e. cos 6 -+ 1 as 6 -+ 0), at an
amplitude that varies by E = 1. This is the simple elastic definition. As 6 becomes
non-negligible, however, the in-phase portion decreases and the out of phase portion,
which represents the energy lost, increases. For this reason, r7 is also known as the
loss factor, and sometimes the loss tangent. It represents the portion of the input
work that is not returned as usable energy, but dissipated within the material. The
mechanisms for this dissipation are not completely understood, but stem from the
curling and uncurling of the polymers that make up the material [3]. This relationship
can also be observed by the hysteresis loop: the graph of stress vs strain. In essence,
this is a plot of force vs distance, or energy. Figure 2-2(a) shows in phase stress and
strain, essentially a line. The integral is 0, because it is not one line but two, right
on top of one another. Figure 2-2(b) shows a stress and strain with a 6 = 0.5. An
ellipse is plotted; this time the integral is not zero. This integral represents the energy
not returned by the system [1]. In the first case, all of the energy added during the
loading phase is returned during the unloading phase, and in the second, some energy
is irretrievably lost. In the second case, it should be noted that the material follows a
different curve whether it is loading or unloading. This hysteresis is evident in cellular
silicone when one views the quasi-static test data, as the material demonstrates that
its loading and unloading curves are different.
19
1.5
1.5
r
T
T
r -r--
0
5
1.5
1.5
-1.5
-1-5
-1.5
--....
..
1
.5
1.5
-
.5 0.5
0
__
__
1
Strain
Strain
(b)
(a)
Figure 2-2: (a)Stress vs Strain, in phase (b) Stress vs Strain, 6
2.3
. -
0.5 out of phase
General Testing Methodology
The major operative dependency explored through this experimentation is the frequency dependence, with additional testing to explore basic trends caused by initial
compression and strain amplitude. The general trend is one of increasing stiffness with
increasing frequency, although most viscoelastic materials display a section where the
rate of stiffness increase is much higher. Understanding the bounds of that section,
as well as the slopes of the stiffness-frequency and loss factor-frequency curves before
and after it are essential to creating an accurate predictive model. Two different tests
are devised to cover the entire dynamic frequency range to be explored: 5 decades
long, from 0.1
-
1000 Hz. Additionally .01 Hz quasistatic level testing is included in
the testing matrices to provide basic information on material loading and unloading
characteristics. The quasistatic testing consists of slow cycling from 0 compression
to complete lock-up 2 on a servo-hydraulic test bed. Each cycle spans 100 seconds.
While this creates a frequency of .01 Hz, these tests are not considered within the
2
Lock-up is defined as enough compression that the porosity of the material has been compressed
to nothing and any further displacement is essentially compression of the base rubber. The approach
to lock-up shows quickly increasing material stiffness and the lock-up point can usually be reached
at a compression percentage equal to the porosity percentage.
20
1. 5
dynamic range because loading cycles from 0 compression to full compression are inherently different than sine wave displacement inputs with just .000025 m amplitude.
The canister testing method, a method of testing that utilizes an induction shaker
head, is designed to explore the entire frequency range, while the low end (<300 Hz)
is explored through a series of tests on a servo-hydraulic test machine. The 300 Hz
cap on this series of tests is necessitated by the basic frequency response of the servohydraulic test bed. It was neither possible to run the machine at a higher rate than
300 Hz, nor to gain access to another machine that might be capable of this sort of
testing. The goal in overlapping these testing methods is to use each as a comparison
and accuracy check of the other. Table 2.1 is a table of the entire planned testing
matrix, detailing the full array of tests to be performed. In the table, f represents the
strain frequency in Hertz (QS indicates quasistatic). 24%, 36%, and 48% represent
the material compressions at which the tests were to be run. The numbers in mils
represent the total displacement range, twice the strain amplitude. The quasistatic
tests are listed at 1 mil because that was the data range examined to establish quasistatic moduli at the different compression levels. Exact testing procedures will be
detailed in Chapter 3.
2.3.1
Temperature Effects
Before continuing, a brief mention of temperature effects should be made. Frequency
dependence is a very important factor in the behavior of the material, but equally important, in an analogous way, is temperature. This introduces two potential variables
into the mix. The first is environmental temperature during testing. The second is
potential heat build up in the material, as the primary mechanism for energy dissipation is heat generation. The first of these is not very significant, as cellular silicone's
temperature dependent properties are very constant near room temperature. For all
intents and purposes, a 70' room is the same as a 750 room. Even so, efforts were
made to maintain constant room temperature throughout testing. The problem of
heat generation within the foam is also not very significant, as the actual damped
energy is exceptionally small during these tests and all material specimens had a large
21
f
24%
36%
I mil
QS
.1
2 mil
48%_
I mil
2 mil
3 mil
V/
5 mil
7 mil
V
V
1 mil
2 mil
V
.5
5
V
10
30
Vx
V/x
_
V
_
V
V/x
50
V
V
V
V
V
V
V
V
_/
V
75
100
150
200
275
V
Vx
Vx
Vx
V
_
V/
V
300
VVx
V
x
x
1000
Key: V: Servo-Hydraulic Test Point x : Canister Test Point
Table 2.1: Initial Test Matrix Proposal
surface area of contact with highly conductive heat sink materials.
2.3.2
V/
Servo-Hydraulic Test Methodology
The basic servo-hydraulic test consists of compressing a material specimen between
two very hard, very flat, platens, driven by a hydraulic piston. The exact apparatus
and instrumentation will be described in chapter 3. The material's compression levels
are recorded through displacement gauges and the force required is recorded through
load cells. The advantages of this testing method lie in the high degree of user control
afforded by this technique. Both displacement and frequency are easy to control and
maintain, these are the input variables. Force, easily translated to stress, is the
output. The tests, and the results, are straightforward and easy to understand: a
certain strain requires a certain stress. Conducted with accurate instrumentation,
these tests are quite reliable and very repeatable. Over the servo-hydraulic frequency
range, excepting the quasi-static level tests, 3 compression levels were tested as well as
multiple strain amplitudes (and therefore rates) at a given frequency and compression.
22
V
Additionally, three different samples were run for each set of test conditions. The
quasi-static testing consisted of a few very slow cycles from zero displacement to
lock-up .
Servo-Hydraulic Data Analysis Methodology
In general, the data acquired from the servo-hydraulic machine is of the forms found
in Equations (2.21) and (2.22). F and d represent the actual force and displacement
curves, while F' and d' represent the amplitudes of their respective sine waves. a
and -y represent the phase angle of each of these data sets.
F= F sin(wt + a)
(2.21)
d= d'sin(wt + y)
(2.22)
These can be reduced to the e and a- forms from Equations (2.2) and (2.3) by
Equations (2.23) and (2.24).
F = o
area
d
length
(2.23)
(2.24)
Some conventions are adopted here and used throughout this work. The length,
1, is defined as the compressed thickness of the specimen tested. This is the thickness
of the specimen before undergoing any dynamic strains. This convention has been
adopted because, in engineering practice, cellular silicone is installed at a specific
compression value, and expected to damp very low strain vibrations. The tests performed are intended to mimic those conditions as closely as possible. The area is
the total area of the specimen in the dimension to which the force is being applied.
Figure 2-3 illustrates the concept of these tests.
When comparing two sine waves, their offsets (a and 7) with respect to zero are
of no interest because any t=0 is arbitrary anyway. Thus, to simplify calculations, y
can be subtracted from each sine wave's offset which results in:
23
Specimen
Length
Fi
Area
Figure 2-3: Servo-Hydraulic Test Concept
24
E = e sin(wt)
(2.25)
a- =osirn(wt+ 6)
(2.26)
6= a - Y
(2.27)
With E and -, all that is required is the simple division demonstrated in Equations
(2.12) through (2.20) to establish a value for E* and r7. These values are different
for each test, and the collection can then be used to establish the dependencies that
are sought. The quasi-static modulus results are computed by taking the tangent of
the slope of the stress/strain curve at any point. The platens are moving too slowly
during this test, and over too great a distance to get anything that might usefully be
called a loss factor, so a real modulus is used here.
2.3.3
Canister Test Methodology
For the canister tests, the approach must be somewhat more complicated. The canister test method has strengths, as well as drawbacks. The 'canister' itself, Figure
2-4 is a cylindrical aluminum fixture, easily mountable to an induction shaker head.
Inside the canister, there is room for a mass, which is mounted atop a slab of material
specimen and then closed in by a bolted down lid. Between the lid and the mass is
placed a well characterized wave disk spring. Through a combination of accelerometers, load cells, and displacement sensors, placed at the indicated locations (ax and ay
), this system can be completely instrumented. The load on the load cells is spread
over all three load cells by means of the spanning plate, and the three displacement
sensors are placed so as to read the mass where it overhangs the spanning plate. It
is only truly necessary for two of four instruments to be used, but redundancy was
built in to test the efficacy of various measurement techniques.
With full knowledge of the system at any two of these four locations, the properties of the specimen can be deduced by a simple model of the entire system. It is
quite important to look at the entire system in the modelling effort, because, unlike
25
Figure 2-4: Schematic Diagram of Canister Test System
26
the servo-hydraulic setup, the composition of the system itself will contribute to the
response. The simplest spring-mass-dashpot system shows strong frequency dependency, eventually reaching a resonance point before the response decays. By looking
at this system in its entirety, this system response can be separated from the material
response.
The advantage of the canister test method lies in the fact that the canister provides
a better approximation of true operating conditions. Cellular silicone is typically used
in applications that require it to damp free and forced vibrations. The canister test
is a far better approximation of this usage than is the carefully induced strain of the
servo-hydraulic tests. The canister test also has a greater frequency range than the
servo-hydraulic. The disadvantages of this method lie in the practice: it is actually
quite difficult to implement, as we shall see in chapter 3. While the canister test
method allows for a greater range of frequencies, it is comparatively limited in its
allowance for varying initial compressions, and the strain amplitude is not an input
value, but a consequence of the input acceleration.
Canister Test Data Analysis Methodology
Because of the redundancy of the data provided by the canister tests, there exists
more than one way to examine this data to produce the intended results. All of
the examinations begin with a simple system model of the canister test setup, they
differ in the final solution when it is decided which data variables to use. Figure 2-5
displays the basic system model from which the following equations are derived
3.
Using Newton's first law, we arrive at Equation (2.28). Just as we adopted a complex
Young's modulus earlier, here we adopt a complex spring constant 4 k* of the same
form. The truly useful factor in this adoption is that this complex constant can
replace k in the basic Hooke's law equations, simplifying the computations. Equation
(2.32) demonstrates the basic system model arrived at. From there, the derivation
3
This system model is clearly a simplification of the actual experimental hardware. The experimental hardware is detailed in chapter 3, and the experimental justifications for this simplification
are also provided
4I use the term 'constant' here because that is the generally accepted name for this term, although
the term itself is not at all constant
27
k
m
k*y
Figure 2-5: Basic System Model for Canister Test
can go in different directions, depending on which data is used. The variables in these
equations are best understood through a look at Figure 2-5. x and y represent the
displacements of the mass and the fixture, respectively. m is the mass of the mass.
k* and k are the stiffnesses of the material specimen and wave disk spring, and
fk
fk*
are the forces within those springs. Also, k' is analogous to E 1 .
mx = A
-
fA
k* =k'(1I +'i)
fk*
fk =
(2.28)
(2.29)
k*(y - x)
(2.30)
k(x - y)
(2.31)
m, = k*(y - x) + k(y - x)
(2.32)
The terms x and y are not explicitly defined by the instrumentation. The accelerometers give ,%and
i,
the load cells give fk*, and the displacement sensors give
(y - x). The simplest method here can be seen in Equation (2.30). The displacement
sensors and load cells give two of the variables here, it only remains to divide to get
k*. Using the accelerometer data is somewhat less straightforward; the relations are
shown in Equations (2.33) through (2.40). Because of the system perspective, it is
convenient to take a LaPlace transform here and work in the frequency domain. The
system transfer function, Equation (2.34), can be computed from the integrals of the
28
accelerometer data, Equations (2.35), (2.36) 5 and solved for k* . In these last two
equations, the a and 3 represent the amplitudes of the sine waves being integrated;
in this case, the amplitude of the the acceleration sine waves.
ms 2 X(s)
=
Y(s)(k* + k) - X(s)(k* + k)
k*+k
H X(s)
=nS
Y(s)
H
2
+
k*
-
-
=M
y
k
f= a sin(wt)dt
= ft
3 sin(wt + #)dt
=-
(2.33)
(2.34)
sinwt2
w(2.35)
sin(wt +
(2.36)
After the integration step, the magnitude and phase of the system transfer function
are well defined. These can be used to produce the transfer function H and, subsequently, k* can be solved for. The 6 in Equation (2.38) is the same 6 from earlier
equations where it represented the loss angle.
M - mag(H) = a
(2.37)
6 = angle(H) = -0
(2.38)
H = Mei6
(2.39)
k* = Hns2 +Hk-k
(2.40)
1 - H
There is a third possibility for the derivation of k*, one that involves using data from
the accelerometer at x and the displacement sensors. If the (y - x) term in Equation
(2.32) is replaced with the variable z, representing the displacement sensor data, k*
'Integrating data from an accelerometer is not often a thoroughly sound practice to get position,
because integration introduces a constant, and accelerometers cannot always provide boundary conditions. Integrating the acceleration can only return the velocity and position changes associated
with that acceleration, it gives no information about the state of the velocity or position before the
acceleration occurred. In this case, the boundary conditions are known; the shaker head induces
a sine wave displacement, and thus sine wave velocity and sine wave acceleration. The constants
introduced through integration cannot take the form of a sine wave, and thus must be 0.
29
can be derived fairly easily.
k* = m. - kz
z
(2.41)
The reasoning behind the redundancy of the instrumentation and data analysis techniques is that the hardware for these experiments needed to be designed and built.
Because of the design and build time frame, it seemed prudent to not rely on any
single set of instrumentation, as it was unclear as to how reliable and accurate each
data set might be.
2.4
Summary
This chapter has set forth the the design concepts for the dynamic cellular silicone
analysis. With a starting point in linear viscoelastic theory, a series of tests and accompanying techniques to understand the data acquired have been developed. These
tests and techniques are proven here to be theoretically sound. The next chapter will
explore the practical implementation.
30
Chapter 3
Implementation: Experiments and
Data Analysis
3.1
Introduction
The methodology set forth in chapter 2 provides a blueprint for the viscoelastic experimentation. With this fully established approach, implementation is the next step.
As with any plan, its ultimate proof is in its execution, which is often far more difficult
than creating the plan itself. The methods and techniques outlined in the previous
chapter must be put into actual practice in order to obtain real results. This chapter
is a discussion of this implementation, its difficulties and obstacles, and the solutions offered to counter these. The organization is analogous to the previous chapter,
with sections and subsections dealing with the tasks of experimentation and analysis
techniques for each of the two testing methods.
31
-
-
717§
-
Li.
Extensometer
t
t
t
t
Figure 3-1: Servo-Hydraulic Test Schematic
3.2
Servo-Hydraulic Experimentation
3.2.1
Experiments
Hardware
Figure 3-1 schematically diagrams the servo-hydraulic system utilized for the low
frequency tests. Its basic make up is a simple 2000 lb-force servo-hydraulic piston
operated by an MTS Test Star Classic II system. The Test Star system is controlled
through a PC running Testware SX and acquires data at 5 kHz. The force is measured
in two places: an Interphase Strain Gauge Load Cell is located in a global position,
and a Kistler Quartz Load Cell is located locally. The displacement is measured by
an MTS Clip Extensometer, attached just wide of the platens themselves. For each
test, the test specimen is placed between the two platens, which are 4340 steel with
Rockwell Hardness C between 40 and 45. The pistons are then used to provide a specific amount of compression at a specified frequency. Before any data is recorded, the
system is allowed to reach steady state. Appendix A contains the specific information
about each actual test, in accordance with the testing matrix set forth in Chapter 2.
32
Experimental Difficulties
One of the major advantages of using the MTS Test Star controlled servo-hydraulic
for these tests is the relative stability and ease of experimentation. This test bed is a
well established method of evaluating many materials, and much work has gone into
improving it. Although the test system is limited in its frequency range, its control
system allows the user a high level of control over the experimental conditions. One
potential problem arises from the possibility of compliance within the machinery itself.
Because the measured displacements are so small, even a minute amount of global
machine compliance would ruin the data. This possibility is countered by the use
of the local instrumentation, the only compliance that could potentially affect the
outcome is that of the steel platens. Those platens are relatively much harder and
stiffer than the material being tested between them. The other problem is encountered
when the machine is run at higher frequencies. The response of the piston at higher
frequencies becomes very unreliable at frequencies approaching 300Hz, and the system
often becomes unstable. Careful adjustment of the control software's parameters can
push this frequency higher, but that adjustment was used to push the maximum
useful frequency from 100 to 300Hz. Higher than this, the machine is not useful.
3.2.2
Data Analysis
The data acquired from this system lends itself quite well to the data analysis approach described in chapter 2, the actual implementation is carried out through the
Matlab scripts included in Appendix D. Not all of the scripts are included, as there
were thousands of lines of code used to perform the data analysis. Much of this code
however, is very similar, as identical codes were adjusted slightly for variations in the
test data. The included code implements all of the major algorithms used in the data
analysis for the general cases of test data and so represents the basis from which all
of the scripts are derived.
The magnitude of each sine wave is computed by finding local maxima and minima
of the signal and then taking the mean. For each signal, every crossing of the zero axis
33
is recorded, and the absolute maximum between one zero and the next was recorded.
The mean of this data is the signal amplitude.
The phase difference is a somewhat trickier calculation, for many reasons. One
possible solution consists of recording the phase that each signal is in at recognizable
amplitude points (maxima, minima, and zero crossings), averaging these phases and
comparing with similar data from the other signal. The non-linearity of the response
makes this method hard to justify. A more reliable method approaches the problem by
looking at the energy lost, and then converting that into a phase angle for notational
convenience.
It is the energy lost that is of importance anyway. The lost energy
is calculated through a numerical Matlab integration between the two signals. This
gives the work, which is the area integral between the force and displacement curves
when they are plotted against one another. The work done in a single cycle can be
computed by Equation (3.1) and the angle can be computed by Equation (3.2) where
F represents the force and x represents the displacement.
work = rxFsin6
(3.1)
aresin work]
(3.2)
6
Translating from a force/displacement to a stress/strain representation is trivial, and
with the magnitudes and phase angles in hand, computation of E* is also trivial.
A bit of discussion about the form of the data returned is necessary here. Calling
the data a sine wave is really an approximation for small strains. Examining the
quasistatic loading curve in Figure 3-2(a), it can be readily seen that cellular silicone's
loading curve is non-linear from zero strain all the way through lock-up. A close
examination of Figure 3-3 and Figure 3-4 shows that the sine waves from these tests
are not true sine waves. These are the normalized force and displacement curves for
a typical servo-hydraulic test, in this case the 10Hz test at 36% compression and 5
mils displacement. Given true sine curves Figure 3-3 would show a pair of curves
that have a constant phase difference and Figure 3-4 would show a perfect ellipse.
Because of this non-linearity, which becomes more marked with increasing strain
34
Stress (Pascs
INSET
Quasi-Static Loading Curve
210
..........
1.510, .............. ..............
..............
..............
.................
.............. ----------
............... ................. -------- -------- ........
............. ................. ................ ............... .
. ............... ................. ................. ................. ----------------
I 10'
......... ........ ...... ..................
...........
..................
.............. . ......
..............
------------- .............. ..............
51o,
0
0
.....................
01
0.2
0.3
0.4
0,5
.................
..........
.................
................. .........
...............
................... ...... ;
--- ------- .. ..........
..........
..............-
0.6
Strain
(a)
(b)
Figure 3-2: (a) Quasi-Static Load Curve (b) Dynamic Data Inset
amplitude, it becomes difficult to compare the phase difference; instead, the energy
loss is used, and a loss factor r is computed which corresponds to this loss. Note that
this loss factor does not represent the actual tangent of the phase difference, as phase
difference is a somewhat meaningless term when the shape of the data is considered.
Energy loss, and not the actual phase difference, is of primary interest in any case.
Using the phase angle is just a notational convenience, which, while not describing
the actual curves perfectly, still describes the energy lost perfectly. Attempting to
utilize non-linear theory instead of the small strain approximation made for linear
theory is quite impractical, as non-linear viscoelasticity is complicated and unwieldy
in practice. Figure 3-2(b) is a close-up of the outlined box in Figure 3-2(a), with the
hysteresis loop from Figure 3-4 overlaid. When viewed together like this, the reason
for the shape of the hysteresis loop becomes quite clear.
35
I-----
I
displacement
Load and Displacement Ys Time
0.003
40
30
20
..........
....
..........
.. .......
........................
..................... .- ----- I---
..... ..............
10
.......................... ..
. ............... .......... .
--------- -------- -- -------------- ........
.. .........
-----------I ---- --
--- ---------
.......
0.002
A ..
------ - --L
..........
0.001
..... .
----------------------
0
----
------ ---- ----
. ....... ..
------........
M
3
-10
--------------------- -------.
-------------------- ---------- --
.............
...................... ........
.
............ .........
-20
.
.
.
.
-0.001
.
.
.
........ . ............ .......... ........ ...............
.
.
.
.
.
...... ....... --- --------- -- ---- -------- ----
-30
............................
-40
11 8.2
.................
-0.002
. ............
-------------- -----.
.
.
...
......... ......
............ . . ...
.
118.2
118.3
118.3
118.4
-U.UU-j
118.5
Time
Figure 3-3: Force and Displacement Curves for test at 36% compression, 5 mil displacement, and 1OHz
36
Hysteresis Loop
....1..
............... .............
. ....
displacement
Figure 3-4: Force vs Displacement for test at 36% compression, 5 mil displacement,
and 10Hz
3.3
3.3.1
Canister Experimentation
Experiments
Hardware
The schematic concept for the canister experimental hardware has been displayed in
Figure 2-4, and is reproduced here again in Figure 3-5; its realization in hardware is
detailed further in Appendix C, a collection of the actual models used for the eventual
part drawings. The fixture contains three Kaman eddy current displacement sensors,
three Dytran load cells, and a pair of Endevco accelerometers. The data acquisition
system was an HP-4000. The canister fixture is almost exclusively 6061-T6 aluminum,
excepting the mass, which is 1020 steel. In addition to the parts displayed in the
schematic, appendix C includes the drawings for the parts required to adapt the
canister set-up for shear and vacuum testing. Time constraints prevented these tests
from being conducted.
37
Figure 3-5: Schematic Diagram of Canister Test System
38
Experimental Difficulties
The canister test system model presented in Chapter 2 is a simplification of the
actual experimental apparatus. Any part of the fixture that is not of a piece with
the main fixture must be considered to have a certain compliance in its displacement
with regard to the main fixture. This means that the lid of the canister and the
load cell spanning plate should both be considered as separate masses in the system
model. Clearly, this would significantly complicate the system. Instead of a single
mass between springs, the system would consist of three masses, all separated by
springs. These parts may be assumed to be part of the fixture if it can be shown
that they don't move independently of the fixture during testing. Preliminary testing
of the system with accelerometers mounted to both the lid and the spanning plate
validate this assumption, and the system model presented in Chapter 2 is an accurate
model of the actual system.
The canister test has a major drawback in that it has not seen significant use
for testing of this nature; there exists a lot of uncertainty as to whether the testing
method is sound, as it hasn't seen extensive use. If successful though, the method
has many advantages over the servo-hydraulic testing apparatus. These advantages
extend from the canister's better approximation of real-life situations and the speed
with which these tests can be run. It requires approximately 30 minutes for each
specimen/compression combination for set up and testing, regardless of how many
frequencies are tested. In contrast, each servo-hydraulic test requires between 5 and
10 minutes to conduct.
There are a few obstacles involved with the canister test which make the tests
very difficult to run as designed. The first of these can be found in the eddy current
displacement sensors. These sensors are quite sensitive and have a high enough resolution to detect the exceptionally small movements of the mass at high frequencies.
The problem is in the calibration. Because these sensors induce a magnetic field
and detect changes in it to measure distance, it is essential that they be calibrated
inside their fixture and using the actual target. It is also essential that the target be
39
positioned precise and consistent distances from the sensor heads during the long calibration process. Finally, the sensor heads must not move in the fixture at all during
or after calibration, something which becomes particularly difficult when the fixture
head is shaking at 1000 Hz. These factors, and perhaps others, contributed to make
the data from these sensors unusable. A quick look back at the analysis techniques
shows that only the accelerometer only derivation works without the displacement
data, so that is the approach taken in the Matlab codes.
The other difficulty in these tests is understanding the exact compression of the
specimen.
Before each test is run, the specimen, mass, and wave-disk spring are
enclosed in the canister, with the lid set with shims at a precise distance from the
floor of the chamber. Because of the way the material's stiffness varies with frequency,
it should be expected to push back against the wave-disk spring with more or less
force according to the frequency, and thus it should be found that its compression
is a function of its current frequency dependent stiffness.
This is just one more
complication introduced into the data analysis. Clearly, the data returned from such
an experiment won't fit in exactly with the rigidly defined experimental parameters
set out for the servo-hydraulic testing.
3.3.2
Data Analysis
The data analysis code (found in Appendix D) implements the methodology set out
in Chapter 2. Many of the same problems with the signal analysis as in the servo
hydraulic tests existed here as well, and there is a lot of code reuse for the signal
analysis sections. The algorithms for determining the amplitudes and phases of the
data curves are identical to those used for the servo-hydraulic tests, slightly adjusted
for smaller overall signal amplitudes. Additionally, there is code to use the material
stiffness to compute the specimen thickness. This is essentially the data that the
displacement sensors would have provided, but now must be calculated after the fact
to arrive at the complex modulus.
The data analysis of the accelerometer data reveals that the canister experiments
failed; there is virtually no useful data. The actual numbers, included in Appendix
40
B, clearly don't quite work in a real world sense. The previously presented analysis
of the canister test is flawed, incomplete. The key problem is related to the working
compression of the material. At higher frequencies, the specimen becomes stiffer, and
therefore pushes back harder against the opposing spring. At lower frequencies, the
specimen is softer and will compress more under the force from the opposing spring.
The spring was chosen in such a way as to match the moduli that showed up in the
servo-hydraulic tests. These tests, however, took place under a much higher strain
amplitude than the shaker was able to produce. A higher strain amplitude implies
a higher strain rate, which in turn implies a higher elastic modulus. Because of the
low amplitude of the strain produced by the shaker head, the elastic modulus of the
foam was much lower than expected, and therefore was unable to provide an adequate
balance for the spring. The foam thus underwent compressions
(
> 60%) that were
too high to provide useful results. The only test that returned a result that might
be valid was the highest frequency (1000 Hz) test, but without surrounding data to
correlate it with, it isn't a particularly valuable data point.
3.4
Experimental Differences
The differences in experimental design were originally incorporated to allow for an
overlap in the tests performed, as a cross check. If a certain test yields the same
results on two entirely different set-ups, it certainly helps to corroborate the validity
of both sets of experiments. Unfortunately, this turned out to be impossible because
of the canister test failure, and might have proved exceptionally difficult even had the
canister test produced reliable data. The difference in the strain amplitude parameter
as well as the inability to exactly specify initial compression would serve to make a
comparison difficult in any case.
41
3.5
Summary
This chapter presents the detailed implementation of the test and data analysis
methodology designed in chapter 2. The testing methods are fully realized in hardware and procedures, and the analysis methods are fully realized in Matlab code.
The servo-hydraulic testing method presents a few minor difficulties which can be
overcome through clever data analysis. It also reveals that the linear viscoelastic
analysis discussed in chapter 2 is an approximation, and that the non-linearity of the
results increases with as the strain grows. This approximation is acceptable for two
reasons. First, because the results it gives are still quite useful; it still manages to
describe the material stiffness and energy loss characteristics, despite the fact that it
doesn't quite describe the actual test curves. Second, non-linear viscoelastic theory
is exceptionally complicated and unwieldy and has not been developed to the point
of practical usefulness. The canister test design is shown to be flawed.
42
Chapter 4
Results and Discussion
4.1
Introduction
Presented here are the results acquired from testing and data analysis presented in
Chapter 3. Additionally, those results are analyzed, and the functionalities of the
complex modulus and loss factor with respect to strain frequency, strain amplitude
and initial compression are discussed and explored. This exploration comes in the
form of data models based on the work of Ahid Nashif [3].
4.2
Quasi-Static Results
The quasi-static results displayed in Figure 4-1 represent the first step to understanding the viscoelastic behavior of cellular silicone. There are three distinct lines on this
graph, labelled a,b, and c. The line labelled a is the first cycle loading curve, where
the material shows somewhat more resistance to strain. The lines labelled b and c
are the cyclical loading and unloading curves, respectively. It is quite clear that the
material displays a stiffness that has a highly non-linear functional dependency on
compression. It is also clear that the material exhibits different stiffnesses in the processes of loading and unloading. These two ideas are very important to understanding
the dynamic behavior discussed in the later sections of this chapter. The first cycle
loading is of little importance to the dynamic response this work is chiefly concerned
43
with.
Stress (Pascals)
I
2106
------ ----
------
----
E.
-----
----
-------
------- ------ -------I------
- ----- --- ------- ----
----
-
a_
-------------
-----
4-
I
CO
5 10 6
0
---I-a
............------------- ------
-----
E
E
-5105
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Strain (infin)
Figure 4-1: Experimental Quasi-Static Loading Results
The behavior of the stress-strain relationship of the cellular silicone under quasistatic loading has a potential explanation in the cellular nature of the foam. As the
foam is compressed, the initial stages of compression (region I in Figure 4-1) show a
basically linear stress/strain relationship. The slope of this curve becomes shallower
when the material has been compressed to approximately 90% of its original thickness,
this is the beginning of Region II. The transition from Region I to Region II represents
the buckling of the walls of the cellular matrix, this causes the material to become
slightly softer. The silicone then exhibits another approximately linear stress-strain
44
dependency, requiring very little additional stress to create more strain throughout
Region II. At approximately 45% compression the slope takes a dramatic upturn
and continues to rise until lock-up. Region III represents the densification caused by
compression levels at which the pores in the cellular matrix are flattening and the
mechanism providing the stiffness is the material itself and no longer the material
structure. At lock-up, nearly all of the cells have been compressed, and any further
strain is operating on the base silicone.
The quasi-static results are easily handled; a designer simply needs to know the
required static load bearing capacity before being able to calculate how much material
will be necessary to carry that load and how much strain it will experience '. This,
of course, only works under static loading conditions; under dynamic conditions a
wholly different approach is required.
4.3
Dynamic Servo-Hydraulic Results
The data garnered from the dynamic servo-hydraulic tests can be broken up and
viewed in many ways, depending upon the functionality to be explored. The first
two, Figures 4-2 and 4-3 display log-log plots of the complex modulus magnitude and
the loss factor against the frequency. The full magnitude of the complex modulus is
used here because it is the best way to incorporate the imaginary part in the view.
Displayed are the average results from the three specimens tested. The error bars here
represent the span of the three specimen's results. Figure 4-4 is a comparison plot
between the complex modulus magnitude and loss factor results from this testing,
and from Nashif's text on vibrational damping [3]. The similarity is striking.
The dynamic performance can be broken into three regions for explanation. These
regions, determined by the strain frequency, are analogous to the polymeric temperature regions rubber, transition, and glassy. In the rubber region, Region I, both the
storage modulus and loss factor are low. The material is not very stiff and most of the
'Actually, this approach is further complicated by the viscoelastic material's tendency to creep.
That is, under a constant stress, the strain will increase as a function of time. This functionality,
however, requires a wholly different set of tests and models.
45
Modulus Magnitude (Pascals)
['Q
1.OOE+07
0-
AL
A
14k
*
+
* 36% Compression
0-
II III
I
24% Compression
* 48% Compression
*
0
0
0
*
r:j~
CD
,
'-1
0
+
*
T
t
*
1.OOE+06
0.01
10
1
frequency (Hz)
100
1000
Loss Factors, delta (rads)
1.000
-
1
---
0
A--
24% Compression
36% Compression
48% Compression
11I11
I
c-c
-1
-i
-
---
0.100
0.01
0.1
1
10
Frequency (Hz)
100
1000
BEHAVIOR AND TYPICAL PROPERTIES OF DAMPING MATERIALS
E
0
E
0
U
0
U,
U,
0
-j
C1
Frequency (log scale)
(a)
36% Loss Factor
36% Real Part of Modulus
Experimental Results
0.3
:3
0
0.2
310 6
0_n
'
L
'
'
'
'
'I
0
0
aU
....-- ........--. ....................
....
..... ....
[L
0.1
0.09
0.08
2106 0
0. 1
10
I
100
.- L ON.07
10
Frequency (log scale)
(b)
Figure 4-4: Comparison of test data with classic viscoelasticism (a) Classic Viscoelastic Curves [3, p. 72] (b)36% Compression Servo-Hydraulic Test Results
48
response is in phase with the input. The frequency of the strain is low enough that
the molecular curling and uncurling of the polymers can easily follow the material
displacement, causing the material to behave in a rubbery manner. In the transition
phase, Region II, as the frequency increases, the molecular curling and uncurling begins to occur out of phase with the input strain, providing a mechanism for energy
dissipation. Additionally, the storage modulus increases rather dramatically during
this stage. Finally, the material reaches the glassy phase, Region III. The loss factor
drops away as the molecular curling and uncurling simply does not have time to occur
under the very high frequency strain. The added polymeric stiffness serves to create
very high moduli. The rate of growth of the elastic moduli levels out here as the
material polymers have little additional stiffness to provide [3].
4.3.1
Modulus Functionality
The striking similarity between the results obtained here and the curves that Nashif
presents has led to the use of Nashif's equations [3] as a starting point in constructing
a relationship between the frequency and the stiffness and loss moduli. The goal is
to construct an equation which will display the same frequency dependence as the
actual material, as demonstrated in Figures 4-2 and 4-3.
E* = E(w)[l + irj(w)]
(4.1)
E1 (w) = E + E[1 - #};
(4.2)
1
#
1+ (0,)"n
=(4.3)
These equations create the elastic modulus dependence on frequency.
the lowest observed value of the dynamic storage modulus. t +
maximum value of the storage modulus. As
# varies
E
E
represents
represents the
between 1 (when w = 0) and 0,
E(w) varies between the maximum and minimum values of the storage modulus [3].
This particular data model manages to remain quite true to the actual data. The
drawback of this equation is that it models the modulus as asymptotically approaching
49
the maximum value as w grows towards oc. It doesn't do a very good job of modelling
the slight slope of the data before and after the drastic change in the middle. A
new parameter, r will be introduced here to make the appropriate adjustment, as
seen in Equation (4.4). For the computation of this new data model, referred to
as the slope-adjusted data model,
E
andE were considered variable parameters. In
multiple performances of a data regression with different initial values, these terms
were consistently equal to
E
and approximately one half of Nashif's value for E,
respectively. This is reflected in Equation (4.4). Figures 4-5, 4-6, and 4-7 demonstrate
the data model values, and the resultant curves for the 24%, 36%, and 48% data sets,
respectively [3].
E (+
1
El (w) =E+ 2
1
)*o
w
(4.4)
Both models fit the data exceptionally well, with the second producing slightly better
results, as can be seen from the R values displayed on the Figures. The larger potential
benefit of the slope adjusted data model is that it stands a better chance of predicting
higher frequency moduli, as it doesn't assume that the moduli will depend very little
on frequency in Region III. There is no strong evidence against this model holding true
past the 300 Hz maximum achieved in these tests, and both basic linear viscoelastic
theory, and the work done by Nashif and Jones [1] on other materials at higher
frequencies, suggest that a functionality that holds true into the beginning of the
region III glassy phase will continue to hold to higher frequencies. Still, prudence
demands skepticism in cases such as these. In Figures 4-5, 4-6, and 4-7, the data
model is represented by a solid line and the data by circles at each data point and a
straight line between them.
It is not valuable to attempt to relate these curves to the initial compression values.
It is most likely the case that 3 and n, as well as
E
and k are indeed functions of
the initial compression, but with just three data points, there are too many potential
functionalities to begin suggesting them here.
An exhaustive search of literature pertaining to linear viscoelastic theory reveals
little evidence that any one model is better than another.
50
Jones, a co-author of
24% Slope Adjusted Data Model
101
a)
C:
C,
x
a)
CO
E
106
0
1
10
100
'
1000
10
E
(a (logscale)
n
K
R
1.287 x 106
4.126 x 10 5
0.00382
2.2523
0.00676
0.96943
(a)
24% Nashif Data Model
10
co
C,
CD
M
.2
CO
106
0.1
1
100
10
.c
1000
10'
(logscale)
$
2.13 x 106
6.88 x 10'
0.0089447
0.927
n
xxx
K
0.96147
(b)
Figure 4-5: 24% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model
51
36% Slope Adjusted Data Model
10 1
CU
'0
CM
ca
I
I
I
10
100
-
____
En
CO
E
10
0. 1
1
1000
10'
E
2.13 x 106
6.88 x 105
'3_ 0.0027499
n
2.2523
II
0.00676
R 1 0.98471
CO (bg scale)
(a)
36% Nashif Data Model
101
r'E
10
0.1
10
100
10'
1000
O(logscale)
2.13 x 106
E 6.88 x 105
/3 0.00421
1.1241
n
xxx
.97292
R:
(b)
Figure 4-6: 36% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model
52
48% Slope Adjusted Data Model
10'
C ,
a)
_0
2
C:
zM
CU
0
a)
cm
E
106
0. 1
10
1
100
'
1000
10
E
6 x 106
1.5 x 106
/3 0.054773
1.1915
n
K .028797
0.9947
R
CO (log scale)
(a)
48% Nashif Data Model
101
CO)
E
6
10f
0.1
1
10
100
1000
6
1.5 x 106
10'
/3
to (log scale)
6 x 10
n
K
R
.031928
0.85271
(b)
xxx
.99325
Figure 4-7: 48% Compression Storage Modulus Equation (a) Slope Adjusted Data
Model (b) Nashif Data Model
53
Nashif's, presents the fractional derivative model in his latest text, The Handbook of
Viscoelastic Vibration Damping and uses it for the entirety of his text. Some authors
have preferred the classical models of viscoelastic behavior, Figure 4-8 demonstrates
some of these. All of these models, however, come down in the end to choosing
parameters to fit experimental data. There has been little offered to validate any of
these theories other than their coincidence with the data. This, of course, doesn't
make any of these models less useful; it but points out that the "best" model is the
one that fits the data most accurately, predicts new data points most consistently, and
is easier to use than the others. The more parameters a model requires (the multiple
standard element model requires parameters for every element used), and the more
complicated (in the case of the fractional derivative model, complex parameters) they
are, the more difficult the model is to use
4.3.2
[1].
Loss Factor Functionality
It does little good to predict the material stiffness without being able to predict the
loss factor as well. The loss factor represents the most important functionality of
the material: that is, energy dissipation. Again using an adjusted version of Nashif's
suggested functionality [3], the equation for qj's dependence on frequency is shown in
Equation (4.5). Two new variables are introduced here. C is a necessary parameter
to account for the offset from zero of the loss-factor curve, and a takes the place of a
2 in Nashif's equation.
r()=C +
nrrE(/3w)"
n~k&n(4.5)
4E(w)[1 + (Ow)n]o(
Figures 4-9(a) and 4-9(b) show the regressions for the 24% and 36% data sets. As
with the modulus models, the model is shown by a smooth solid line and the data
by an interconnected set of circles. Figure 4.3.2 displays the loss factor data for the
48% compression tests, without a regression because the data shows a significantly
different pattern than the other two data sets. This deviation from the other data
sets is probably due to the increased material densification at such a high compression
54
k1
00]
c2
(a) Maxwell Model
k1
C1
p.-
(b) Kelvin Model
k1
c1
7
k2
.......
111,1
A 1A1
1\
(c) Standard Element
6r C2
k2
k3
C
(d) Multiple Standard Elements
Figure 4-8: Classic Viscoelastic Models
55
level. As close as that is to lock-up, it is not unexpected for the material to show a
deviant behavior. The loss moduli in this range are not small, they are still larger
than those of the smaller compressions, but as a percentage of the storage modulus
they are much smaller, which, of course, means that the loss factors will be smaller.
4.3.3
Strain Rate vs Frequency
There is some question as to whether the dependencies established here are truly
frequency dependencies, or simply strain rate dependencies.
The two notions are
similar, but clearly not identical. An attempt to establish this was conducted during
the servo-hydraulic tests. Multiple strain amplitudes were tested at identical frequencies, creating a series of different strain rates at the same frequency. The data from
these tests is presented in Figure 4-11.
The graph shows that tests conducted at higher strain rates, but identical frequencies display higher moduli. However, at similar strain rates, the higher frequency
always displays a higher modulus. This information alone isn't enough to decide what
is actually going on, but a close look at the response curves for each series of tests
shows increasing non-linearity with increasing strain amplitude. Looking back at the
quasi-static data curve in Figure 4-1, it can be seen that the modulus 2 is greater
when a larger strain amplitude is considered. This implies that the modulus is a
function of strain amplitude and frequency, not strain rate directly. For this reason,
frequency has been used as the independent variable in these regressions.
4.4
Canister Test Discussion
As discussed earlier, there is no data to report from this set of tests. This does not,
however, invalidate the canister testing method as a concept. Despite the factors that
were overlooked in the original design, the method itself still has much to recommend
it. The two most important items here are the larger frequency range that is available
for testing, and the more true-to-life approximation this test method provides.
2
At any given point, tangent to the slope
56
24% Loss Factor Model
1
CO
W
0
0.1
1
0.
10
100
10 0'
1000
Ca (logscde)
C
a
R
0.12679
1.1794
0.99336
(a)
C
a
R
0.12182
1.7577
0.95857
(b)
36% Loss Factor Model
I
.
-J
C.
0.1
0.1
1
100
10
10'
1000
ca (log scale)
Figure 4-9: Loss Factor Models (a) 24% (b) 36%
57
48% Loss Factor
r
0.1
0.1
1
10
100
1000
10'
o (log scale)
Figure 4-10: 48% Compression Loss Factor Data
The failed tests could easily be conducted again, with different opposing springs to
be used for different frequency tests. Using a less stiff spring would serve to alleviate
the over-compression problem that was experienced at low frequencies during the
earlier tests. This simple adjustment would correct the previously described errors.
It might also be feasible to test the system without the canister lid, allowing the mass
to move freely up and down on the specimen. This last would require very low input
accelerations, because the cellular silicone material is not designed to work in tension.
4.5
Summary
The data resulting from the servo-hydraulic tests agrees closely with the work of
others in the field, specifically Ahid Nashif [3] and David Jones [1]. This agreement
leads to a starting point (the dependencies proposed by Nashif) for the development
of an empirical model to describe the storage modulus and loss factor dependencies on
strain frequency. From there, these models are adjusted to more accurately describe
58
Modulus vs Strain Rate
4.OOE+06 150Hz
200Hz 300Hz
* + +
+2 mil displacement
350E+06
-
- 3 mil displacement
A6 mil
100Hz
displacement
*
*
x 7 mil displacement
75Hz ,
S300E146
5OHz+
30Hz#
101l
2 50E+06
O.5Hz
O.1Hz
*
0 2.00E06
00.1H-z
5Hz
1Hz
1OHz
+~
+
0.5H-z
5Hz
Hz
30HZ
@
x10Hz
30Hz
$3OHz
1OHz
1Hz
1 50E+06
1 flfEF ftG
1.OOE-02
1.lOE-01
1.OOE+00
1.00E+01
1.00E+02
Strain Rate (strain/second)
Figure 4-11: Complex Modulus Magnitude vs Strain Rate
the observed response of cellular silicone. The end results are models that show
accuracy within the frequency range tested and, show promise to be useful above
that range.
59
60
Chapter 5
Conclusions
Viscoelastic materials are exceptionally difficult to characterize. Not only are their
basic properties more complicated than that of simple elastic materials, but those
basic properties are functions of multiple independent variables. At any given time,
these materials are sensitive to strain frequency, strain amplitude, temperature, compression, and perhaps other factors that have not been considered. The work accomplished here establishes a part of that functionality for the viscoelastic material
cellular silicone.
The basic frequency dependencies for both modulus and loss factor have been
established at multiple compression levels, giving the designer a good deal of freedom
when incorporating cellular silicone into a design. The data regressions included here
are quite accurate over the frequency range tested and show promise to hold true
outside of that range. The dependencies established herein are a solid option for
design up through and perhaps even beyond 300 Hz. They may not be as accurate
at higher frequencies, but they are certainly better than quasi-static values.
Additionally, the methods and techniques established here can easily be duplicated
without any further experimental design, and can be applied to any viscoelastic material. In the case of the canister test, only a slight redesign of the system is required
to allow for it to become a feasible testing method. The data analysis codes included
in Appendix D serve the purpose of expediting and standardizing the data reduction
phase of any further research, giving the researcher more time to spend on serious in61
terpretation of the actual results. This work serves not only as an analysis of cellular
silicone, but as a primer in basic experimental and data analysis techniques for work
with viscoelastic materials in general.
The characterization of cellular silicone requires a few things to be considered
complete. Both the servo-hydraulic and the canister tests can be altered to accommodate testing in shear, making the analysis more complete and the design possibilities
greater. Additionally, the canister test can be altered for low vacuum testing, this
could provide a better idea of the actual damping mechanisms within the foam and
whether or not the air filled cellular matrix plays a significant role in the material's
dynamic properties. While more data is required for a complete analysis and thorough design parameter definition, the work presented here represents a significant
step towards that goal.
62
Appendix A
Servo-Hydraulic Test Results
This appendix contains the experimental test tables and reduced results for the servohydraulic tests. Included are the specimen number, test number, frequency (in Hertz),
compression %, input test displacement (in meters), computed complex modulus
magnitude (in Pascals), and loss factor.
Table A.1: Servo-Hydraulic Test Table
Specimen # I Test #
f
Compression
Displacement
E*
i
77
1
2
0.01
24
2.540E-05
2.76E+05
5
2
0.01
24
2.540E-05
2.91E+05
11
2
0.01
24
2.540E-05
2.53E+05
1
2
0.01
36
2.540E-05
6.36E+05
5
2
0.01
36
2.540E-05
7.15E+05
11
2
0.01
36
2.540E-05
5.87E+05
1
2
0.01
48
2.540E-05
2.62E+06
5
2
0.01
48
2.540E-05
3.10E+06
11
2
0.01
48
2.540E-05
2.83E+06
1
3.1
0.10
24
5.080E-05
1.27E+06
1.21E-01
5
3.1
0.10
24
5.080E-05
1.19E+06
1.23E-01
11
3.1
0.10
24
5.080E-05
1.22E+06
1.23E-01
63
fI
Compression
Displacement
E*
3.2
0.10
36
5.080E-05
2.19E+06
1.30E-01
3.2
0.10
36
5.080E-05
2.03E+06
1.42E-01
3.2
0.10
36
5.080E-05
2.11E+06
1.08E-01
3.3
0.10
48
5.080E-05
6.57E+06
1.OE-01
3.3
0.10
48
5.080E-05
5.53E+06
1.25E-01
3.3
0.10
48
5.080E-05
5.91E+06
9.93E-02
3.4
0.10
36
7.620E-05
1.92E+06
1.08E-01
3.4
0.10
36
7.620E-05
1.77E+06
9.94E-02
3.4
0.10
36
7.620E-05
1.85E+06
1.08E-01
4.1
0.50
24
5.080E-05
1.33E+06
1.19E-01
4.1
0.50
24
5.080E-05
1.26E+06
1.19E-01
4.1
0.50
24
5.080E-05
1.30E+06
1.18E-01
4.2
0.50
36
5.080E-05
2.26E+06
1.03E-01
4.2
0.50
36
5.080E-05
2.16E+06
1.06E-01
4.2
0.50
36
5.080E-05
2.23E+06
1.05E-01
4.3
0.50
48
5.080E-05
6.78E+06
1.17E-01
4.3
0.50
48
5.080E-05
6.22E+06
1.26E-01
4.3
0.50
48
5.080E-05
6.68E+06
1.20E-01
4.4
0.50
36
7.620E-05
1.91E+06
1.08E-01
4.4
0.50
36
7.620E-05
1.89E+06
1.08E-01
4.4
0.50
36
7.620E-05
1.96E+06
1.06E-01
5.1
1.00
24
5.080E-05
1.30E+06
1.27E-01
5.1
1.00
24
5.080E-05
1.25E+06
1.30E-01
5.1
1.00
24
5.080E-05
1.29E+06
1.26E-01
5.2
1.00
36
5.080E-05
2.24E+06
1.11E-01
5.2
1.00
36
5.080E-05
2.16E+06
1.15E-01
5.2
1.00
36
5.080E-05
2.21E+06
1.1E-01
5.3
1.00
48
5.080E-05
7.O±E+06
1.24E-01
5.3
1.00
48
5.080E-05
6.60E+06
1.39E-01
Specimen # I Test #
64
Specimen
#
f
Compression
Displacement
E*
r7
5.3
1.00
48
5.080E-05
6.34E+06
1.29E-01
5.4
1.00
36
7.620E-05
1.97E+06
1.13E-01
5.4
1.00
36
7.620E-05
1.91E+06
1.16E-01
5.4
1.00
36
7.620E-05
1.97E+06
1.13E-01
6
5
24
5.080E-05
1.43E+06
1.39E-01
6
5
24
5.080E-05
1.38E+06
1.42E-01
6
5
24
5.080E-05
1.42E+06
1.40E-01
7
5
36
5.080E-05
2.42E+06
1.35E-01
7
5
36
5.080E-05
2.38E+06
1.43E-01
7
5
36
5.080E-05
2.43E+06
1.35E-01
8
5
48
5.080E-05
7.91E+06
1.27E-01
8
5
48
5.080E-05
7.57E+06
1.24E-01
8
5
48
5.080E-05
7.57E+06
1.28E-01
9
5
36
7.620E-05
2.24E+06
1.42E-01
9
5
36
7.620E-05
2.20E+06
1.49E-01
9
5
36
7.620E-05
2.27E+06
1.42E-01
10
5
36
1.270E-04
2.20E+06
1.46E-01
10
5
36
1.270E-04
2.15E+06
1.54E-01
10
5
36
1.270E-04
2.21E+06
1.45E-01
11
5
36
1.778E-04
2.31E+06
1.44E-01
11
5
36
1.778E-04
2.27E+06
1.51E-01
11
5
36
1.778E-04
2.30E+06
1.46E-01
12
10
24
5.080E-05
1.45E+06
1.58E-01
12
10
24
5.080E-05
1.39E+06
1.61E-01
12
10
24
5.080E-05
1.43E+06
1.57E-01
13
10
36
5.080E-05
2.41E+06
1.66E-01
13
10
36
5.080E-05
2.38E+06
1.78E-01
13
10
36
5.080E-05
2.43E+06
1.66E-01
14
10
48
5.080E-05
8.10E+06
1.25E-01
Test
#
65
Specimen
#
Test #
f
Compression
Displacement
E*
48
5.080E-05
9.46E+06
1.19E-01
48
5.080E-05
7.59E+06
1.24E-01
36
7.620E-05
2.32E+06
1.72E-01
36
7.620E-05
2.29E+06
1.84E-01
36
7.620E-05
2.32E+06
1.73E-01
36
1.270E-04
2.27E+06
1.75E-01
36
1.270E-04
2.26E+06
1.87E-01
36
1.270E-04
2.26E+06
1.75E-01
36
1.778E-04
2.39E+06
1.67E-01
36
1.778E-04
2.39E+06
1.73E-01
36
1.778E-04
2.38E+06
1.67E-01
24
5.080E-05
1.52E+06
2.36E-01
24
5.080E-05
1.47E+06
2.56E-01
24
5.080E-05
1.51E+06
2.44E-01
36
5.080E-05
2.63E+06
2.69E-01
36
5.080E-05
2.65E+06
2.90E-01
36
5.080E-05
2.64E+06
2.75E-01
48
5.080E-05
8.49E+06
1.26E-01
48
5.080E-05
8.21E+06
1.23E-01
48
5.080E-05
8.88E+06
1.24E-01
36
7.620E-05
2.53E+06
2.67E-01
36
7.620E-05
2.57E+06
2.80E-01
36
7.620E-05
2.56E+06
2.69E-01
36
1.270E-04
2.55E+06
2.43E-01
36
1.270E-04
2.56E+06
2.44E-01
36
1.270E-04
2.60E+06
2.42E-01
36
1.778E-04
2.71E+06
2.11E-01
36
1.778E-04
2.74E+06
2.06E-01
36
1.778E-04
2.77E+06
2.07E-01
66
Specimen
#
Test #
f
Compression
Displacement
50
5.080E-05
2.97E+06
2.75E-01
50
5.080E-05
3.01E+06
2.78E-01
50
5.080E-05
3.02E+06
2.77E-01
75
5.080E-05
2.94E+06
2.75E-01
75
5.080E-05
3.14E+06
2.25E-01
75
5.080E-05
3.17E+06
2.38E-01
100
5.080E-05
2.91E+06
2.85E-01
100
5.080E-05
3.01E+06
2.55E-01
100
5.080E-05
3.48E+06
2.67E-01
150
5.080E-05
3.54E+06
2.13E-01
150
5.080E-05
3.57E+06
1.92E-01
150
5.080E-05
3.65E+06
2.02E-01
200
5.080E-05
3.32E+06
2.45E-01
200
5.080E-05
3.46E+06
2.25E-01
200
5.080E-05
3.53E+06
2.31E-01
300
5.080E-05
3.33E+06
2.14E-01
300
5.080E-05
3.28E+06
1.90E-01
300
5.080E-05
3.54E+06
1.99E-01
100
5.080E-05
9.18E+06
1.34E-01
100
5.080E-05
9.21E+06
1.29E-01
100
5.080E-05
9.29E+06
1.28E-01
100
5.080E-05
1.83E+06
3.71E-01
100
5.080E-05
1.89E+06
3.72E-01
100
5.080E-05
1.90E+06
3.60E-01
300
5.080E-05
1.88E+06
3.13E-01
300
5.080E-05
1.74E+06
2.59E-01
300
5.080E-05
1.86E+06
2.84E-01
275
5.080E-05
9.34E+06
1.04E-01
275
5.080E-05
9.48E+06
1.25E-01
67
Specimen #
Test #
f
Compression
Displacement
E*
E
11
33
275
48
5.080E-05
9.08E+06
68
I
rI
1.23E-01
Appendix B
Canister Test Results
Table B.1: Canister Test Result Calculations
Frequency
Mils Disp
Eta
K* Mag
Gap Mils
E* Mag
Compression
10
4.67E+00
0.027068307
1.179E+06
126
1.582E+05
76.777%
30
1.61E+00
0.296815868
7.993E+05
126
4.314E+04
90.659%
100
2.98E-02
0.108369463
5.336E+05
126
-1.171E+04
103.798%
300
3.98E-02
0.198529808
1.512E+06
126
2.822E+05
67.689%
1000
5.86E-04
0.00822983
7.729E+06
126
3.525E+06
21.078%
10
2.13E+00
0.00219007
1.265E+06
126
1.885E+05
74.206%
30
4.59E-01
0.15583151
9.250E+05
126
7.729E+04
85.538%
100
9.03E-02
0.095624469
4.729E+05
126
-2.008E+04
107.349%
300
2.09E-02
0.249193379
1.476E+06
126
2.680E+05
68.569%
1000
2.17E-04
0.032398143
7.239E+06
126
3.251E+06
22.288%
10
4.54E+00
0.015542071
1.236E+06
131
2.229E+05
68.785%
30
4.25E-01
0.159240747
8.509E+05
131
9.289E+04
81.106%
100
8.47E-02
0.062279152
5.671E+05
131
2.151E+04
93.434%
300
2.79E-02
0.246976704
1.203E+06
131
2.105E+05
69.705%
1000
1.86E-04
0.032693697
7.201E+06
131
3.307E+06
20.522%
10
2.70E+00
0.016935875
1.238E+06
131
2.238E+05
68.725%
30
3.40E-01
0.098442114
9.868E+05
131
1.352E+05
76.284%
69
Frequency
Mils Disp
Eta
k* Mag
Gap Mils
E* Mag
Compression
100
6.68E-02
0.106441915
4.683E+05
131
3.647E+03
98.652%
300
2.78E-02
0.277448545
1.170E+06
131
1.985E+05
70.634%
1000
1.99E-04
0.030887109
6.620E+06
131
2.982E+06
22.026%
These are the Matlab calculated results from the canister tests. k* is in newtons/meter,
and E* is in Pascals. Mils Disp is the displacement of the mass in mils, and gap mils
is the total 'gap' in mils. The 'gap' is the combined thickness of the opposing spring
and the material specimen. The apparent stiffnesses, computed from the acceleration
of the mass, were used to calculate the theoretical compression percentage, given that
stiffness. For reasons discussed earlier, these results are basically useless. The compression percentages for all but the highest frequency test are well above the lock-up
range, pushing the data outside the range of usefulness.
70
Appendix C
Canister Fixture Diagrams
The following are a series of views of the canister fixture design. This is the hardware
designed to implement the canister schematic displayed in Figure 2-4. The first four
figures display the canister itself in various views. The fixture consists mainly of an
aluminum cylinder with a base designed for mounting to the induction shaker head.
Inside the cylinder rests the mass, placed atop a specimen of material which is, in
turn, atop the spanning plate. On the underside of the canister, there are a series
of round holes in triangular arrangements, these are designed to contain the load
cell and displacement sensor instrumentation. The slot in the side of the canister is
designed to allow the mass accelerometer room to move with the mass. Figure C-5 is
an exploded view of the entire assembly before mounting to the shaker head. Figure
C-6 shows the canister fixture converted to operate in the shear mode.
71
Figure C-1: Canister Fixture Hardware
Figure C-2: Canister Fixture Hardware
72
Load Cell Mounts
A
Displacem
Sensor Mounts
rometer
Figure C-3: Canister Fixture Hardware
Load Cell Mounts
Displac
nt
Sensor Mounts
0
Figure C-4: Canister Fixture Hardware
73
Lid
mass
Spanning Plate
Load Cells
Can-Ister
Mounting
Hardware
Figure C-5: Canister Fixture Assembly
74
0
00
F
0
00
Figure C-6: Canister Fixture, Shear Arrangement
75
76
Appendix D
Matlab Source Code
This appendix contains the Matlab source code implementations of the major data
analysis algorithms utilized in this work.
Datatrim.m allows the user to examine
the data visually to ascertain that nothing went drastically wrong during testing.
It also allows the user to trim off entrance and exit lengths of useless data points.
Freqfind.m examines a single sine wave, returning its frequency, amplitude, and phase
angle. Compare.m uses freqfind.m to run a comparison of two sine waves, returning
the modulus between the two. Massprocessv2.m is an automated processor of large
amounts of servo-hydraulic data, stored in text files of a certain format as specified in
the comments. Essentially, it uses compare to calculate the moduli and loss factors of
all the data sets. Throughout, it displays the data visually so the user can check its
progress. Lossfact.m corrects the deficiency massprocessv2.m has in its calculation of
loss angles. Lossfact.m calculates loss angles based on the energy method. Finally,
kstar.m analyzes the canister test data from a systems view, calculating the stiffness
modulus K* for each set.
77
XDatatrim
is written to trim off entrance and exit lengths of data.
%It shows the user a basic plot of the data, and prompts the user t
%to select beginning and ending points
%Input t:time scale, x: first data set, y: second data set
%output a: first usable data point, b: last usable data point
function
[a,
b]=datatrim(t,x,y);
figure;
subplot(2,1,1)
grid
plot(x)
subplot(2,1,2)
grid
plot(y)
'Select the x-axis endpoints:\n'
[tt,yy]=ginput; %after this, we're going to redo the comparison,
%this ginput command allows the user to cut off either end of the data
if length(tt)>1
a=round(tt(1));
b=round(tt(2));
end
if length(tt)==1
a=round(tt(1));
b=length(t);
end
if length(tt)==O
a=1;
b=length(t);
end
close
78
%freqfind is designed to look at a sine wave data set and pull out all
%of its information - if the sin wave is a*sin(wt+phi),
a, w, and phi are returned
%as well as the number of data points and the total time
%this only works for integer valued frequencies
%works much better with more data points, more than 5 or 6 times the frequency...
%the fewer data points, the more likely it is to miscalculate the frequency...
%phi is calculated in two ways - phi3 looks at the phase angle of the peaks
Xout3
is phi calculated by the phase angle of the zeros
time data,
%Input: t:
%(if known,
x: amplitude data,
f: suggested frequency
enter actual frequency,
%if unkown, enter 0
houtput:outl:frequency, out2:amplitude, out3: phase angle (zeros),
% phi3: phase angle (peaks)
%out4= total number of datapoints,
function
[outi,
out5:
total time span
out2,out3,phi3,out4,out5]=freqfind(t,x,f);
figure;
plot(t,x);
hold
step=t(length(t))/(length(t)-1); %this calculates the time step interval
s=sign(x);
test=0;
i=1;
%find the first zero crossing
while test == 0
test=s(i+1)-s(i);
i=i+1;
end
cycles=0;
beginner=t(i);
79
b=i;
plot(t(i) ,0'g+');
c=1;
indices(c)=i-1; %indices is a vector of the zero crossing indices,
% c is the index of the indices vector
Xeacn
index is the one the first point before the actual crossing
%populate indices:
while i<length(t)
test=s(i+1)-s(i);
if test ~= 0
c=c+1;
indices(c)=i;
ender=t(i);
e=i;
%plot(t(i)
,0,'g+');
cycles=cycles+1;
end
i=i+1;
end
if f==0
frequency=round((cycles/2)/(ender-beginner));
else
frequency=f;
end
%the algorithm for determining the amplitude works like this:
%b and e represent the indices of the beginning and end of the data set
%starting at b, and stepping an amount based on the index of each zero crossing
80
%based on the period, and then taking the absolute maximum in each of those sets
%phi3 is calculated in the same loop, by the same basic method used to calculate
%out3, which happens a litle later in the code and is explained there
sum=0;
T=1/frequency;
rads=(2*pi*t)/T;
anglesum=0;
figure;
plot(rads,x,'b.');
hold
for c=1:(length(indices)-1)
[Y,I]=max(abs(x(indices(c):indices(c+1))));
plot(t((I+indices(c)-1)) ,Y,'ro')
sum=sum+Y;
ang=rads(I+indices(c)-1);
plot(ang,Y,'ro');
anglesum=anglesum+mod(ang,pi);
end
phi3=anglesum/(length(indices)-1);
Xphi3
now equals the average position of each peak, on a 0-pi basis.
%the phase angle will be whether that point leads or lags pi/2.
%unfortunately, doing it this way could result in an 180-out-of-phase.
phi3=phi3-(pi/2);
%to correct for that 180 out possibility... when the angle is between:
%0-pi/2 - use the normal, when the angle is between pi/2 and pi, use this
%with the angle between pi and 3pi/2, use this,
%between 3pi/2 and 2pi again, use regular,
if and(phi3<0,phi>0)
81
with it
phi3=pi+phi3;
end
if and(phi3>0,phi<O)
phi3=phi3-pi;
end
amplitude=sum/(length(indices)-1);
out1=frequency;
out2=amplitude;
%now, the program will use the frequency data, along with the time step data
%, to convert the x-axis into radians.
%how many seconds for 2pi radians? 2pi radians is one hertz, so at frequency f,
%it takes 1/f=T period for 2pi radians
T=1/frequency;
%so in the span T, 2pi radians are covered...
%the goal is to start numbering at 0,
and increment so that T and 2pi coincide...
rads=(2*pi*t)/T;
%this gives an array in radians that matches the array in time, index for index.
%so, using the indices array, containing the indices of where it crosses 0
%we can take the radian
%numbers on either side of point, and take an average, this should be phi
%but we can only use the negative to positive crossings...
%because they represent the begining of each cycle
total=0;
figure;
plot (rads,x);
hold;
grid;
for c=1:(length(indices))
82
if x(indices(c))-x(indices(c)+1) < 0 %ie,
a negative to positive crossing
cross = (rads(indices(c))+ rads(indices(c)+1))/2;
plot(rads(indices(c)),x(indices(c)),'ro');
plot(rads(indices(c)+1),x(indices(c)+1),'go');
angles(c) = mod(cross,2*pi);
total = total+angles(c);
end
end
phi = 2*total/(length(indices));
%now...
if phi is bigger than pi, then it can be a notational problem at
%some point down the road... so, for any phi bigger than pi, we'll make it
%a negative that's less than pi
if phi>pi
phi=(phi-2*pi);
end
%here's the part where we fix phi3
out3=phi;
out4=length(x);
out5=t(length(t))-t(1);
83
%compare receives two sets of data.
%the first set is is basically strain (displacement) data,
% the second is force (stress) data
%the program doesn't care about that, it simply uses freqfind
% to return the important data on each set
%and to give back a transfer sort of function.
%this is designed to work with stress and strain, but will compute
% the complex modulus between any two
%things...
%input:
%tl: time base of first data set
%t2: time base of second data set
first data set (strain)
%xl:
x2: second data set (stress)
%f:
expected frequency
%output
%eql,
:
e2: information about each dataset in the following format:
Xeql=[frequency,
amplitude, phase angle, phase angle] - these two phase angles
%are each calculated differently
%eq3 : comparison info [G1 G2 G* delta eta magnitude delta2 delta3]
function [eql,eq2,eq3l=compare(tl,xl,t2,x2,f)
%eql will return the vitals of the first set
Xeq2 will return the vitals of the second set
%eq3 will return their comparison...
[frequencyl,amplitudel,phil,pphil,dpl,ttl]=freqfind(tl,xl,f);
[frequency2,amplitude2,phi2,pphi2,dp2,tt2]=freqfind(t2,x2,f);
eql=[frequencyl,amplitudel, phil,pphil];
eq2=[frequency2,amplitude2, phi2,pphi2];
delta=phil-phi2;
delta3=pphil-pphi2;
84
eta=tan(delta);
%the complex modulus G*=stress/strain * cos delta + i sin delta stress/strain
G1=(amplitude2/amplitudel)*cos(delta);
G2=(amplitude2/amplitudel)*sin(delta);
Gstar=G1+i*G2;
mag=abs(Gstar);
%delta 2 is the phase angle data on the other side of the sin wave...
%the best way, to do that is to run freqfind for the other sides
%of the curves...
no problem there..
xreverse=-x1;
yreverse=-x2;
[f,a,anglel,pal,dx,tx]=freqfind(tl,xreverse,f);
[f,a,angle2,pa2,dx,tx]=freqfind(tl,yreverse,f);
delta2=anglel-angle2;
eq3=[G1 G2 Gstar delta eta mag delta2 delta3];
85
%massprocessv2 is the automated servo-hydraulic data reduction code
%it
performs
_all_ of the data reduction for all
% of the servo-hydrualic test data
%commented out at the beginning is the code necessary to
% select individual files
%otherwise, this script automatically sets up the arrays
%necessary to
%process all
the data (they are currently arranged
% for the 2001 servo-hydraulic
%testsrun by Richard Hanna and Al Shields
%after each run through, it returns a results file,
% the name of which needs
%to be changed before the next iteration
% so it
doesn't get overwritten
%there are no explicit inputs or outputs, but implicitly,
%massprocessv2s inputs are:
%files, in the matlab path, with the titles 'test#xQ.txt' #
% 0 - test number
%the output is a file
called output.txt containing
%the following data (in this order:
%specimenid, test number, frequency, compression %,
% thickness of undeformed specimen
%thickness of deformed specimen, diameter of specimen
X4a,b,d,e,c,f
%where a,b equal frequency,amplitude phase angle of
%strain and stress, strain
%based on the compressed thickness
%c,d equal f,a,phi of strain, stress, strain based
%on uncompressed thickness
Yc,f are the data comparisons as produced by compare
86
-
specimen id
%from a,b(compressed thickness strain)
%and d,e (uncompressed thickness strain)
%each contains:
[G1 G2 G* delta eta magnitude delta2 delta3]
function []=massprocessv2();
%r=input('Enter Instron File name:\n' , 's');
%1=input('Enter Instron File extension:\n', 's');
%thick=input('Enter specimen thickness:\n');
%compress=input('Enter mean test thickness:\n');
%diam=input('Enter specimen diameter:\n');
Xid=input('Enter specimen ID Number:\n');
%test=input('Enter test number:\n');
%fre=input('Enter test frequency:\n');
Xtime=input('Time
data in column?:\n');
hforce=input('Force data in column?\n');
%disp=input('Global Displacement in Column?:\n');
ldisp=input('Local Displacement in column?:\n');
%d='.';
r='test';
fill='x'
specl=[1;
.0396; 2.272;];
spec5=[5;
.0392; 2.27;];
specll=[11;
.0394; 2.262];
specs=[specl spec5 spec11]; %specs is now basically
%a record containing the data on each
%specimen...
first row id, row 2 thickness, row 3 diameter
%now, process with a loop that'll run through all the tests ...
tb=input('Test to begin with?\n');
te=input('Test to end with?\n');
specid=input('Specimen ID?\n');
87
if specid == 1
specscount=1;
end
if specid == 5
specscount=2;
end
if specid == 11
specscount=3;
end
for testnumber=tb:1:te %for all the inputted test
% numbers... this is the major loop
%filenams will all be in this format: testXxY.txt
% where X=specimen id and Y = test number
if testnumber<10
filename=strcat('test',num2str(specid),'xO',num2str(testnumber),'.txt');
else
filename=strcat('test',num2str(specid),'x',num2str(testnumber),'.txt');
end
dataset = load(filename);
Xperc=1-(compress/thick);
thick = specs(2,specscount);
diam=specs(3,specscount);
testnumber
perc=input('is the test number, enter the compression %:\n');
compress=(1-(perc*.01))*thick; %this is the compressed thickness
fre=input('Frequency?\n');
time = 1;
88
force = 3
if testnumber>25 %for the higher tests,
% a quartz force cell was used, and the tests
%force data has ended up in the 5th column...
force = 5;
end
disp =2;
ldisp = 4;
%basically, up to this point, this script is accepting the data from one of the
%instron files... next, it will plot the two displacements, so that the user
%can decide which one to use, then it will plot stress and strain at the same
%time, so the user can decide if everything is 'ok'
forcedata=dataset(:,force);
timedata=dataset(:,time);
timedata=timedata-timedata(1);
%i want the first one to start at time=O
local=dataset(:,ldisp);
%globe=dataset(:,gdisp);
%because its just the amplitudes of these curves that matter,
% not the actual displacements,
%im going to subtract off the DC contribution
forcedata=forcedata-meanfind(forcedata);
local=local-meanfind(local);
%globe=globe-meanfind(globe);
displacement=local;
%next, the program will plot a normalized displacement versus
% a normalized force, just to make it
89
%clear to the user whether that data set is ok, if it is,
% the program will continue, it will take _all_
Xof the data involved, convert displacement and force
% into stress and strain,
and then produce
%a processed file which would contain all of the
% basic data on the specimen and the test
%as well as the processed results.
%the first step is to convert the displacement
% data into strain data... strain=displacement/compress;
strain=displacement/compress;
strain2=displacement/thick; %the strain might look
% much higher simply b/c of the crushedness
%now stress (engineering) in pascals....
strain
%of course, is dimensionless
radius=(diam/2)*.0254; %radius in meters
area=(radius/2)^2*pi;
newtonforce=9.80665*(forcedata*.453592);
stress=newtonforce/area;
%before any comparisons, we'll run a datatrim just to make sure...
[firstpoint lastpoint]=datatrim(timedata,strain,stress);
timedata=timedata(firstpoint:lastpoint);
strain=strain(firstpoint:lastpoint);
strain2=strain2(firstpoint:lastpoint);
stress=stress(firstpoint:lastpoint);
timedata=timedata-timedata(1);
0
%%%/
0 <-------------------------->
XXXthis
is the section where the program will
%reverse stress and strain, looking
Xat them in a compressive=positive sense
strain=-strain;
90
strain2=-strain2;
stress=-stress;
[a b c]=compare(timedata,strain,timedata,stress,fre);
Ed e f]=compare(timedata,strain2,timedata,stress,fre);
%this set of data is using the real strain...
%because we do...
if we entereed 0 in place of fre,
%it would find the frequency - 'cept to do that
%doesn't work for fre<1
%so now,
a = freq, amplitude,
and phi of strain,
% b= f,a,phi of stress, and c=G1, G2, Gstar, delta, eta, and G(mag)
%the plan now is to graph these things,
%make sure all
is nice and happy,
then write to a file
%first, graph a normalized set of stress vs
% strain just to see how the data works...
nstress=stress/b(2);
%a(2) is the amplitude, if we divide the whole thing by that, we're golden
nstrain=strain/a(2);
figure;
subplot(2,1,1);
plot(nstrain(1:500),'r');
hold;
grid;
plot(nstress(1:500),'b');
subplot(2,1,2);
plot(nstrain(length(nstrain)-500:length(nstrain)),'r');
hold;
grid;
plot(nstress(length(nstress)-500:length(nstrain)),'b');
legend('Normalized Strain','Normalized
keyboard
91
Stress');
tempstring=input('Is this ok?\n','s');
%this is where the user cuts it off and does it manually if the answer
% is no
if tempstring == 'n'
testnumber
'is bad.
Make note of this and come back later.
Mass processing continuing:\n'
end
[tt,yy]=ginput; %after this, we're going to redo the comparison,
this ginput command allows the user to cut off either end of the dta
if length(tt)>1
a=round(tt(1));
b=round(tt(2));
end
if length(tt)==1
a=round(tt(1));
b=length(timedata);
end
if length(tt)==O
a=1;
b=length(timedata);
end
ntime=timedata(a:b);
ntime=ntime-ntime(1);
stress=stress(a:b);
strain=strain(a:b);
[a b c]=compare(ntime,strain,ntime,stress,fre);
nstress=stress/b(2);
92
%a(2) is the amplitude, if we divide the whole thing by that, we're golden
nstrain=strain/a(2);
figure;
plot (nstrain, 'r');
hold;
grid;
plot(nstress,'b');
legend('Normalized Strain','Normalized Stress');
%ok, now that we've seen normalized stress
%and strain, we're going to plot sinwaves generated
%by the info pulled out, just to make sure they match
t=timedata(length(timedata))-timedata(1);
figure;
wavel=singen(a(1),a(2),length(strain),t,a(3));
wave2=singen(b(1),b(2),length(strain),t,b(3));
figure;
plot(timedata(1:500),strain(1:500),'ro',wavel(1:500,1),wavel(1:500,2),'b')
legend('Data Strain','Ripped Strain');
figure;
plot(timedata(1:500),stress(1:500),'bo',wave2(1:500,1),wave2(1:500,2),'r');
legend('Data Stress',
'Ripped Stress');
save output.txt specid testnumber fre perc thick compress diam a b d e c f -ascii
strcat('results',num2str(specid),'_',num2str(testnumber),'v2.txt')
afterall=input('Rename output.txt file:\n', 's');
close all
end
93
-
XLossfact
corrects a deficiency in the phase angle calculations performed
%by massprocessv2.
there is an automated mass-process version
%of this as well
%lossfact calculates the phase angle between two approximately sin
%wave data sets, using the 'energy' method
%the phase angle calculated is basically an approximation, based on the
%concept that phase angle phi = atan eta, where eta is the energy loss factor
%Input:
t: time base
%xl: data set 1
x2: data set 2
%f:
expected frequency
%Output:
%phi: phase angle
function[phi]=lossfact(t,xl,x2,f)
%this clearly won't work if we don't reset the data to have a mean
%of zero...
xl=xl-mean(xl);
x2=x2-mean(x2);
[firstpoint lastpoint]=datatrim(t,xl,x2);
t=t(firstpoint:lastpoint);
xl=xl(firstpoint:lastpoint);
x2=x2(firstpoint:lastpoint);
t=t-t(1);
[a,b,famp]=shortfreq(t,x2, f);
[a,b,dispamp]=shortfreq(t,x1, f); %shortfreq is a faster running version
%of freqfind
xl=x1(a:b);
94
x2=x2(a:b);
t=t(a:b);
t=t-t(1);
tt=t(length(t));
cycles=tt*f;
figure
plot(xl,x2,'.');
hold;
plot(x1(1),x2(1),'ro');
plot(xl(length(xl)),x2(length(x2)),'go');
%for this to work properly, the first
%and last x values must be equal, it doesn't make a difference
%however, if the data is centered...
%so, rather than bothering with data-centering, I'll
%just use a mean calculation to make the amplitude calculation,
%then I'll use the zero crossings
% data from shortfreq to trim to the proper lengths
area=trapz (xl, x2);
A=area/cycles;
%need the amplitudes to finish this calculation...'
sinphi=A/(pi*dispamp*famp);
phi=asin(sinphi);
[linearfit otherstuff]=polyfit(x1,x2,1);
%this last section accounts for the
%problem introduced by phi>pi/2
if
linearfit(l)<O
if phi>O
phi=pi-phi;
end
if phi<O
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phi=-pi-phi;
end
end
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%kstar examines two data sets
%from the canister tests, and returns kstar,
%Input: t - time base
%y - acceleration of base
%x - acceleration of mass
%f - expected frequency
%Output:
Ukstar : K* complex
%system: system transfer function
%trfu:
system transfer function, computed for magnitude and angle
%at the given frequency
%amplitude
: amplitudes of both accelerations
function[kstar,system,trfu,amplitude]=kstar(t,y,x,f);
%x is the accelerometer on the mass
%y is the accelerometer on the base
mass=.7427;
k=6e4;
w=f*2*pi;
[fill,yamp,ay,by]=canfreqfind(t,y,f);
[fill,xamp,ax,bx]=canfreqfind(t,x,f);
amplitude (1) =yamp;
amplitude(2)=xamp;
%amplitudes are calc'd, reset curves so
%that curve y starts at 0
y=y(ax:bx);
t=t(ax:bx);
x=x(ax:bx);
t=t-t(1);
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%quick bit of code here to adjust t so that x really crosses 0
%at t=0
tadj=asin(x(1) /xamp) /w;
t=t+tadj;
xsin=[xamp w 0];
ysin=[yamp w 0];
%integration part:
xint(1)=-xsin(l)/(xsin(2)^2);
xint (2) =xsin(2);
xint(3)=xsin(3);
yint(1)=-ysin(1)/(ysin(2)^2);
yint(2)=ysin(2);
yint (3) =ysin(3);
phi=canloss(t,x,xamp,y,yamp,f);
amplitude (3) =xint (1)
mag=xsin(1)/ysin(1);
ang=-phi;
%this gives the magnitude and angle of the left
%side of the system transfer function
Xuse that to construct the complex value
H=mag*exp(i*ang);
trfu=H;
S=i*w;
kstar=(H*mass*S^2+H*k-k)/(1-H);
s=tf('s');
system=(kstar+k)/(mass*s^2+kstar+k);
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Bibliography
[1] David I.G. Jones.
Handbook of Viscoelastic Vibration Damping. Wiley, West
Sussex, England, 2001.
[2] Roderic S. Lakes. Viscoelastic Solids. CRC Press, Boca Raton, Florida, 1999.
[3] Ahid D. Nashif. Vibration Damping. Wiley, Reading, Massachusetts, 1985.
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