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MI CHI IGAN STATE

|III I III

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

300913 4812

This is to certify that the

thesis entitled

Characterization and Enhancement
of the Damping Within Composite Beams

presented by
Daniel John Trahan

has been accepted towards fulfillment
of the requirements for

Master of Science degree in Mechanical Engineering

 

 

 

Date February 21, 1991

 

0-7639 MSU is an Affirmative Action/Equal Opportunity Institution

 

 

I

LEQRARY I

Michigan ‘3’ate I
University

 

PLACE IN RETURN BOX to remove We checkout from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

   

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* ]I_\I I

MSU Is An Affirmative ActIorVEquel OpportunIIy Institution
chiI—nt

 

 

 

 

 

 

 

 

 

 

 

 

fl.—

CHARACTERIZATION AND ENHANCEMENT OF THE
DAMPING WII'HIN COMPOSITE BEAMS

By

Daniel John Trahan

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering
1991

ABSTRACT

CHARACTERIZATION AND ENHANCEMENT OF THE
DAMPING WII'HIN COMPOSITE BEAMS

By

Daniel John Trahan

The dynamics relating the transverse motions of a clamped-free graphite/epoxy beam are
investigated. A Bemoulli-Euler beam model incorporating viscoelastic damping is used to
predict the transfer function relating clamped end displacement to the displacement of the
free end. Three damping strategies are considered. One strategy relies upon the internal
damping abilities of the fiber and matrix. A second approach utilizes strips of viscoelastic
damping material within the laminate. A third design considers optimal placement of the
damping material to achieve the greatest damping enhancement with minimum strength and
Stiffness reductions. Each strategy is applied to two layup geometries, unidirectional and
crossply. The experimental apparatus is configured to measure specimen end displacements
resulting from random excitement of the clamped end. Collected data are analyzed to
evaluate the utith of the model structure for capturing the essential system dynamics and the

effectiveness of the damping strategies considered.

DEDICATION
To my parents for their enduring and continued love, understanding, and support during
my years of study. Their help was genuinely appreciated although it often appeared that I

was oblivious to the assistance.

ACKNOWLEDGEMENTS

During the past year my research has been assisted by numerous people whose help was
essential in completion of this project. I am indebted to the 3M company for the donation of
the carbon/epoxy prepreg and the viscoelastic material, and to the Composite Materials and
Structures Center at Michigan State University for assistance in specimen fabrication.
Naturally I owe a world of thanks to my advisor, Dr. Philip FitzSimons, who provided
countless hours of advice and support as well as an understanding of the “big” reseach
picture. Thanks are in order for Dr. Clarke Radcliffe who offered timely advice regarding
experimental matters. Lastly, special thanks to Scott Emery for assistance with the

Macintosh computers and enjoyable nerf games.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................. vii
LIST OF FIGURES ........................................................................... ix
NOMENCLATURE ........................................................................... x
1.0 INTRODUCTION ........................................................................ 1
2.0 DAMPING IN COMPOSITE LAMINATES ......................................... 4
2.1 Laminate Methods of Internal Energy Dissipation ........................... 4
2.2 Laminate Shear Analysis ........................................................ 5
2.3 Damping Strategies .............................................................. 6
2.3.1 Undamped Specimens ................................................. 7
2.3 .2 Generally Damped Specimens ........................................ 7
2.3.3 Optimally Damped Specimens ........................................ 8
3.0 DEVELOPMENT OF THE CANTILEVER BEAM MODEL ....................... 15
3.1 Bernoulli-Euler Model .......................................................... 15
3.2 Boundary Conditions ........................................................... 17
3.3 Eigenfunction Solution .......................................................... 17
3.4 Eigenfunction Selection and Normalization of Constants ................... 19
3.5 Solving for z(x,t) ................................................................ 22
3.6 Transfer Function Solution Process ........................................... 25
3 .7 Modal Parameter Values ........................................................ 26
3.8 Analysis of Results .............................................................. 26
4.0 EXPERIMENTAL MATERIALS ...................................................... 28
4.1 Experimental Materials .......................................................... 28
4.1.1 Material Aging .......................................................... 28
4.1.2 Aging Effects Upon Specimens ....................................... 30
4.2 Specimen Fabrication ........................................................... 30
4.2.1 Lay-up Orientations and Damping Material Locations ............. 30
4.2.2 Fabrication Equipment ................................................. 32
4.2.3 Fabrication Methods .................................................... 32
4.2.4 Problems Unique To Fabrication ..................................... 33
5.0 EXPERIMENTAL PROCESS ........................................................... 34
5.1 Experimental Set-up ............................................................. 34
5.2 Equipment descriptions ......................................................... 34

5.3 Excitation Method ............................................................... 37

5.4 Specimen Linearity .............................................................. 37

5.5 System Damping ................................................................. 38

5.5.1 Apparatus Damping .................................................... 39

5.5.2 Aerodynamic Damping ................................................. 39

5.6 Input Displacement Content .................................................... 40

5.7 Specimen Quality ................................................................ 41

6.0 EXPERIMENTAL RESULTS .......................................................... 42

6.1 Data Collection ................................................................... 42

6.1.1 Sampling Rate ........................................................... 42

6.1.2 Sampling Process ....................................................... 42

6.2 Data Manipulation ............................................................... 43

6.2.1 Laser Calibration ........................................................ 43

6.2.2 Kaman Calibration ...................................................... 43

6.3 Parameter Identification ......................................................... 44

6.4 Results ............................................................................ 47

6.4.1 Parameter Values ....................................................... 48

6.4.2 Parameter Magnitude ................................................... 52

6.4.3 Analysis of Time Responses .......................................... 53

6.4.4 Damping Parameter Convergence To Zero .......................... 56

7.0 CONCLUSIONS and RECOMMENDATIONS ..................................... 57

7.1 Conclusions ...................................................................... 57

7.2 Recommendations ............................................................. . . 57

REFERENCES ................................................................................. 59

Appendix A .................................................................................... 63

Appendix B .................................................................................... 64

Appendix C .................................................................................... 65
Appendix D .................................................................................... 66 '

Appendix E .................................................................................... 69

Appendix F .................................................................................... 97

LIST OF TABLES

Nummr Title Bag:
2-1 Position of optimal damping strips. 14
3-1 Eigenvalues and truncation error. 20
5-1 Specimen linearity. 38
6-1 Crossply-generally damped dimensionless damping parameter

and natural frequency values values. 48
6-2 Unidirectional- generally damped dimensionless damping parameter
and natural frequency values values. 48
6-3 Crossply-optimally damped dimensionless damping parameter
and natural frequency values values. 48
6-4 Unidirectional-optimally damped dimensionless damping parameter
and natural frequency values values. 48
6-5 Crossply-undamped dimensionless damping parameter
and natural frequency values values. 49
6-6 Unidirectional-undamped dimensionless damping parameter
and natural frequency values values. 49
6-7 Damping enhancement ratios. 50
6-8 Aluminum 6061 modal dimensionless damping parameter values
and natural frequency values, test A. 51
69 Aluminum 6061 modal dimensionless damping parameter values
and natural frequency values, test B. 51
6-10 Aluminum 6061 modal dimensionless damping parameter values
and natural frequency values, test C. 51
61] Aluminum 6061 modal dimensionless damping parameter values
and natural frequency values, total case. 52

6-12 Crossply- generally damped modal dimensionless damping parameter
values and natural frequency values, test case A. 52

6-13 Crossply-generally damped modal dimensionless damping parameter
values and natural frequency values, test case B. 52

6-14 Crossply- generally damped modal dimensionless damping parameter
values and natural frequency values, test case C. 52

6-15 Crossply- generally damped modal dimensionless damping parameter
values and natural frequency values, total case. 52

6-16 2-norm of measured displacement signal. 54
6-17 2-norm of the one mode approximation error signal. 54

6-18

6-19

6-20

6-21

2-norm of the two mode-one damping parameter approximation
error signal.

2-norm of the two mode-two damping parameter approximation
error signal.

2-norm of measured free end displacement and the error between
predicted free end displacements and measured free end
displacements, crossply-generally damped specimen.

2—norm of measured free end displacement and the error between

predicted free end displacements and measured free end
displacements, aluminum 6061 specimen.

54

55

56

56

LIST OF FIGURES

Hiram: 1119

2-1 Laminate side view for the generally damped strategy.

2-2 Beam configuration for energy analysis.

2-3 Beam element free body diagram.

2-4 Moment magnitude along clamped-free beam of
normalized length.

3-1 Differential beam element.

3-2 Four mode approximation of magnitude and phase responses
with varying dimensionless damping parameter values.

5-1 Experimental set-up.

I?
\O \l ('0

10

12

16

27

35

.Oe-w

am “1

f(x,t)
Hi(S)

NOMENCLATURE
specimen cross sectional area
calibration removal scale factor for Kaman measuring system
Kaman measuring system “guage” factor
viscoelastic damping parameter
specimen width
constant used in the eigenfuncu'on
Young’s modulus
square of 2-norm of error signal
square of 2-norm of measured signal
matrix representation for the eigensolution
load per unit length applied to specimen
beam transfer function
area moment of inertia
vibration mode
function minimized in optimization algorithm
complex variable
mode specific parameter for optimization of experimental data
arbitrary specimen length
laminate bending moment
beam element bending moment
beam mass
beam element mass

modal participation of qi(t)

parameters in the modal response equation
Laplace transform of the mode participation parameter
mode participation parameter

complex frequency variable

beam response function

laminae thickness

Laplace transform of the input variable
generalized description of beam input
beam element shear force

variable for separation of variable method
potential energy

dissipated energy

independent spatial variable

Laplace transform of the specimen flee end displacement
neutral axis vertical displacement

laser displacement measrnements

eddy current displacement measurements
measured flee end displacement

predicted flee end displacement

beam element vertical velocity

beam element vertical displacement

transformed displacement term

half-power bandwidth
differential beam element length

time partial derivative

8x

7».

l

p

(Dru)

é

Subscripts

t E
X E

Superscripts

I

()

0

spatial partial derivative
eigenvalue
specimen density

eigenfunction solution

dimensionless damping ratio

time derivative

spatial derivative

spatial derivative

time derivative

LLINIRQILUCIIQN

Performance requirements for many components in the aerospace and automotive
industries require the combination of high stiffness and damping characteristics in one
material. A need exists for a material capable of enduring large applied loads while
possessing the ability to dissipate vibrational energies. Historically, monolithic materials
have been utilized to meet these requirements, but of late their dominance has been
challenged with the introduction of composite materials. The term “composite materials” is
used here in its broad definition in that it describes the integration of two or more dissimilar
constituent materials into one material of superior mechanical properties and thus improved
performance.

Enhancement of a material’s ability to dissipate energy via damping can be advantageous
to the system in which it resides. This is true whether the system is a commercial aircraft or
the extremity of a robot. In an environment where acoustic energy is unwanted, dissipation
within structural materials results in less noise in areas where it is undesirable. This benefit
of energy dissipation within the material can eliminate secondary manufacturing operations
traditionally designed to minimize noise levels. Increased material damping can also reduce
the physical motion of a system. Reduction of internal vibration within structural members
can reduce unwanted oscillatory motions during starting and stopping. An example of this
phenomena is the replacement of an aluminum robot arm with a composite arm by Magolan
[1]. A reduction in the amount of internal vibrations can result in less physical motion of the
system and thus an increase in the fatigue life of the structure.

Numerous attempts have been made to improve the damping of structural composite
materials. In general, fiber/epoxy composite materials have better damping than metallic
monolithic materials. Rivin [2] constructed thin-layer rubber-metal composite materials
such that impact loads were better cushioned and a reduction in peak sound pressure levels

was attained.

The type of composite material focused upon in this study is composed of carbon
reinforcing fibers embedded in an epoxy matrix. Carbon/epoxy composite materials are
furnished with the means for internal dissipation of vibrational energy. In general, it is
assumed that energy is removed at three locations within a composite laminate. Vibration
losses occur in the reinforcing fibers, within the matrix, and at the interface common to the
fibers and matrix. The damping characteristic of single carbon fibers is quantified by
Adams [3].

Desired mechanical and aesthetic performance levels of a composite material are
determined by the selection of the constituent fibers and matrix, and their volumetric
percentages. The size, type, and orientation of reinforcing fibers can determine the damping
properties of the laminate. Comparisons between continuous and discontinuous reinforcing
fibers indicates that material damping is larger for discontinuous fiber composites than
continuous fiber laminates [4, 5]. Another methodology to improve laminate damping is via
optimization of fiber orientation and ply thickness [6, 7]. Liao et al developed an algorithm
that determines a maximum specific damping capacity when provided values for ply
thickness and fiber orientation [8].

Further attempts to improve material damping have explored the addition of a
constrained viscoelastic layer within the composite material. The exclusive purpose of this
layer is the dissipation of energy. Basic versions of this constrained layer ideology consider
three ply laminates in which the middle ply is responsible for damping and the outer two
layers are responsible for stiffness and constraining [9, 10]. Finite element models have
been designed for the three ply, constrained layer laminates that predict strategies of minimal
natural flequency and maximum modal damping [11]. Lall et al used the Rayleigh-Ritz
method to assess damping and modal loss factors of a simply supported plate with a partial
layer of constrained viscoelastic material. Results indicate that a decrease in the shear
modulus of the viscoelastic layer produces an increase in the laminate loss factor and lower

resonant frequencies [12]. ,Beam specimens equipped with a constrained layer of

3

viscoelastic material along the length axis have been the focus of studies [13-15]. Maximum
material damping has been acknowledged via both Optimal placement of damping strips
along the specimen and optimal ratio of damping strip length to beam length.

Many authors have performed the task of modeling the damping proficiency of
composite laminates [16-19]. A common approach used when analyzing damping
properties is to compute values for the material’s specific damping capacity and loss factor;
these quantities are typically determined for quasi-static conditions. Algorithms used to
compute dynamic characteristics typically originate as the classical laminate theory.
Investigators have modified these algorithm’s such that the predicted values for dynamic
properties include hygrotlrermal effects and allow for consideration of hybrid composite
laminates [20, 21]. An experimental observation worthy of note evaluated the dynamic
properties of 8-2 glass/epoxy unidirectional, cantilever specimens excited over a 30 Hz to
1000 Hz band. Loss factors spanning 5 x 10'3 to 10 x10'3 were measured using the half
power bandwidth method [22].

The current study will identify and assess the ability to modify the damping properties of
carbon/epoxy composite laminates. Specimens used to generate experimental data represent
three damping strategies including carbon/epoxy laminate, carbon/epoxy laminate with a
general damping enhancement, and a carbon/epoxy laminate with optimal enhancing of its
damping mechanisms. Each strategy is applied to a unidirectional and crossply layup
geometry. The transfer function of an unloaded Bernoulli-Euler clamped-flee beam
considering viscous damping is developed; this relation is used to construct a modal
response equation. The modal response relation is used in conjunction with experimental
data to acquire values for the dimensionless damping parameter and the natural flequency.

Specimen modal damping values are indicative of their material damping measured in a
dynamic environment. Specimen modal damping values are compared. Deviations and
regularities of damping parameter magnitudes are addressed as well as inconsistencies

between experimental data andpredicted results.

W
A composite laminate is produced by stacking a desired number of prepreg layers upon

one another. The number of prepreg layers and the orientation of their fiber angles dictate
the material properties of the laminate. A prepreg layer, also termed a lamina, is a thin sheet
of continuous, unidirectional fibers embedded in an epoxy matrix. The thickness of an
uncured prepreg layer is unchanged during the autoclave cure cycle. An autoclave cure
cycle uses elevated pressure and temperature to harden the prepreg layers to their end
consistency. The fibers typically provide a laminate with its strength and structural integrity
while the epoxy matrix protects the reinforcing fibers from damage due to the surroundings
and provides the means to transfer applied forces within the laminate. Fibers are generally
composed of carbon, glass, or an organic substance termed Kevlar. As mentioned above,
the angle of a ply’s reinforcing fibers, termed ply orientation, will dictate the material
properties and dynamic response of a laminate. Desired material properties and dynamic
response can be achieved by altering the number of lamina and/or the ply orientations. The
presence of an interface region where the fibers and matrix bind together provides a laminate
with an internal means to dissipate energy; vibrational energy is lost across the fiber/epoxy
interface. Such a mode of energy dissipation within the internal structural system offers
composite laminates a distinct advantage over monolithic materials in the realm of energy
dissipation.

2.1 Lanrinate Methods of Internal Energy Dissipation

The main source of internal material damping in a composite laminate is due to the
viscoelastic phenomenon of the relative slipping between the matrix and fibers. The amount
of energy dissipated internally, and thus the amount of damping, is dependent upon the

following factors:
1) fiber volume flaction
2) fiber diameter
3) matrix properties

4) thermoelastic damping

5) fiber orientation within the laminate

6) fiber surface treatment

7) presence of debonds, inclusions and voids

The initial four modes of internal energy dissipation listed above are specific to the
component fiber and matrix materials selected, while the latter three means of energy
dissipation arise as a product of the fabrication process and the design ideals when
developing the manufactured item [23]. Factors 1 and 2 indicate that the amount of energy
dissipated corresponds to the amount of fiber/matrix interface area present within the
laminate. Increasing the fiber volume fraction or fiber diameter allows more interfacial area
in which energy dissipation can occur. The properties of the matrix relate to the amount of
friction between a sliding fiber and the stationary matrix. Thermoelastic damping is due to
the different thermal coefficients of expansion between the fiber and matrix.

Fibers oriented in the direction of an applied load will have a higher degree of mobility,
and thus exhibit more dissipation of energy to friction, than fibers oriented perpendicular to
the direction of an applied load. Fiber surface coatings regulate interface friction and
bonding strength which effects the amount of material damping. Lastly, the presence of
fabrication defects, voids, debonds, or delamination offer regions for additional energy
dissipation which are specific to the quality of fabrication and not the abilities of the

material.

2.2 Laminate Shear Analysis

In the fabrication of a composite laminate, the process of stacking and curing prepreg
layers introduces a distribution of shear through the laminate’s thickness. Transverse shear
deformation within a laminate will influence certain static and dynamic properties including
lateral deformation, bucking loads, and naun'al flequencies.

Transverse shear is the means by which a laminate internally dissipates energy due to its

motions. The amount of energy removed flom a laminate can be further increased with the

addition of an energy absorbing material at laminae interfaces. Inclusion of a secondary
material provides an auxiliary means by which vibrational energies are internally dissipated
as a function of shear. Addition of an object foreign to the laminate at an interface will cause
a reduction in overall laminate strength due to the interruption of the interfacial surface area.
Interfacial surface area is the means by which a laminate distributes loads across its
thickness and is the plane of failure when a laminate’s maximum load is exceeded. Thus a
trade off is recognized, positioning damping material at a laminae interface increases
laminate damping with an associated strength reduction penalty. It is desirable to position
the energy absorbing material at height positions corresponding to high shear values to best
utilize its damping abilities.

2.3 Damping Strategies

Three damping strategies varying in the amount and placement of viscoelastic damping
material will be combined with two layup geometries, crossply and unidirectional. The
three methodologies include no viscoelastic damping material, a general arrangement of
viscoelastic damping material, and optimally positioned damping material to dissipate
energies of the first and second vibration modes.

Damping strips are located about the laminate midplane because this is the region where
the largest shear values exist. Therefore, a maximum damping benefit is achieved while the
associated laminate strength reduction penalties are minimized. In the spirit of maintaining a
midplane symmetric laminate of eight plys, damping material may be added at one, three,
five or seven ply interfaces. The interruption of five or seven interfaces with the damping
material would offer increased damping at the expense of major strength reductions because
nearly all the planes of load transition, ply interfaces, are made weaker by the presence of
the damping material. Placing the strips of the viscoelastic material only at the midplane
interface would retain the laminate strength, but not produce notable increases in laminate

damping. Thus it was decided to employ the damping material at the three interfaces about

/— Damping Strips

Y /
/

 

 

 

 

 

X Midplane

 

 

 

 

 

 

Figure 2-1 Laminate side view of a generally damped specimen.

the midplane to achieve a desirable amount of damping without major strength reduction
penalties as shown in Figure 2-1. Strips of viscoelastic damping material were one-half
inch in width were laid perpendicular to the beam’s longitudinal axis. The one-half inch
damping strip width was selected to combine ease of fabrication with strips that were not
excessively wide yielding large segments of non-contact at laminae interfaces thereby

reducing laminate strength.
2.3.1 Undamped Specimens

Specimens without viscoelastic material were constructed to determine the damping
characteristics of the carbon/epoxy material. Undamped specimens allow for comparison of
the effect of different ply orientations on internal damping. The crossply laminate should
experience greater amounts of internal energy dissipation than the unidirectional laminate.
Furthermore, because these specimens do not have the viscoelastic material interrupting their
ply interfaces, they should have higher laminate strength properties than both the generally

damped and optimally damped specimens.
2.3.2 Generally Damped Specimens

The generally damped strategy equips laminates with strips of damping material located
at the three ply interfaces symmetric about the laminate midplane. The viscoelastic material

was dispersed with one-half inch spacing between the strips in a checkerboard pattern
through the laminate height as displayed in Figure 2-1. Spacing between damping strips
allows for both damping enhancement and lamina interaction to occur simultaneously at an
interface; ideally this method will achieve the desired improvements in damping with only a
minimal reduction in laminate strength.

2.3.3 Optimally Damped Specimens

In the design of the optimally damped specimens, internal energies were removed
according to the mode of vibration with which they were associated. For the clamped-free
arrangement, the percentage of total system energy associated with a mode generally
decreases as the mode number increases. This energy per mode relation mandates that the
energy dissipation associated with the first two modes is of larger magnitude than the
dissipation of the higher modes. Optimally damped specimens were equipped with damping
material at selected positions along their length axes to facilitate the dissipation of energies
associated with the first two modes of vibration. The optimally positioned viscoelastic
material also serves to dissipate the energies of higher vibration modes.

These “best” locations of the viscoelastic strips were determined by observing the
methods in which energy is stored and dissipated within a Bernoulli-Euler beam model.
The amount of vertical displacement is essential to the beam model’s internal energy storage
mechanism due to the shear distribution along its length. Distribution of shear along the
beam specimen will determine the amount of work that can be performed as the system
transitions from an initial, deformed configuration to a final undeformed position as
displayed in Figure 2-2. The beam specimen is geometrically designated using the variables
x 5 independent spatial variable, y -=.neutral axis vertical displacement, and b 5 beam
width.

A flee-body diagram for a beam element, of mass me and length Ax, is shown in Figure

2-3 with the associated shear forces, V, and bending moments, M. It is assumed that the

shear forces acting within the beam have a much greater effect upon the system dynamics
than the bending moments. This allows the bending moments to be neglected in the
determination of the internal energies.

It is desirable to determine the energy dissipated within a beam specimen for a selected
mode of vibration. Energy lost within a non-conservative beam system is accounted for via
shear forces acting upon an arbitrary beam element in motion.

As the beam oscillates through its range of motion, the balance of energy is determined
by investigating the kinetic, potential and dissipative energy mechanisms within the beam.

The potential energy stored, W

pm, is computed as the product of a vertical displacement, y,,

and the beam element shear force summation. A vertical displacement accounts for the
difference in position of the beam element for the two beam positions in question. This

displacement is independent of the path the beam element travels to arrive at the new

position. The rate of energy dissipation, WW is a product of the shear forces on the beam
element and a vertical velocity, yt, of the beam element. Equations (2-1) and (2-2) are
expressions for the potential energy stored and the rate energy is dissipated in a beam
element respectively. Note the displacement and velocity terms in the energy equations are

negative, this is a result of the direction of deformation arbitrarily selected in Figure 2-2 and

 

 

 

 

 

 

 

 

 

Y
A initial, deformed position
i
l > x
final, undeflected position

 

Figure 2-2 Beam configurations for energy analysis.

10

V(x) V(x +Ax)

M(x) M(x + Ax)
I.— .. _.|

Figure 2-3: Beam Element Free Body Diagram

 

 

assigning the y-axis as positive in the upward direction.

[V(X) - V(X + AX)]' (-Y.) = AW,“ (2-1)

[V<x)-V(x+Ax)1-<—y.) = AW... (2-2>
Dividing both energy equations by the length of the beam element and taking the limit as

the beam element becomes infinitesimally small we obtain differential equations relating the
change in shear to the differential energy components.

 

8W
a—‘é—S‘l-(m =75“- (2-3)
33:“) (y.)=—a§:“‘ (2'4)

The subscripts v and t, indicative of the virtual displacement and virtual velocity
respectively, will be removed to simplify the notation of subsequent equations.

Recall that the shear force, in general, is composed of two components. One component
is related to the stiffness of the beam and is proportional to the third spatial derivative of the
displacement term. This component is indicative of the potential energy stored in the beam
element. The other component is related to the damping of the beam and is proportional to

the time rate of change of the third spatial derivative. This component is indicative of the

11

rate at which energy is being dissipated within the beam element. The expressions that

result are:

Elymm y dx = dWP, (2-5) Blym y, dx = dwfi, (2-6)
Two integrations by parts on the left hand sides of the fourth order equations above
produce the energy equations, integrated along the beam of arbitrary length, L, which

account for the energy stored via potential energy, (2-7), and the rate of energy dissipation,

(2-8).

W.» EIIIYm'YII:+ly.-Yulli+Irma? ‘1‘} (2'7)

W”, = BI{[ym‘oyJIE-l-[yn-ymllt-I- I:(ym)2 dx} (2-8)
The first two terms on the right hand sides of the above equations represent the
boundary conditions for the beam specimen. End conditions for the clamped-flee beam
arrangement are listed in (2-9). The clamped-flee boundary conditions cause the first two
terms on the right hand sides of the above equations to be zero producing the desired
integral relations, (2-10) and (2-11), which indicate the amount of energy stored internally
within a clamped-flee beam as potential energy and the rate of energy dissipation,
respectively.

The damping material was placed at locations where the bending moments of the first

y(0) = 0 x.(L) = 0 (2-9)
y,(0) = 0 ym(L) = 0
L 2
w” = El [0 (ya) dx (2-10)

W... = BI [:(ymrdx . (2-11)

12

 

 

 

 

 

0 0:2 0:4 0:6 0:8 1
normalized specimen length

absolute moment value (second mode is divided by six)

Figure 2-4 Optimal Damping Strategy.

two modes of vibration are at maximum levels. Such placement of the damping material at
positions along the specimen having large moments allows the strips to maximize their
internal damping capabilities with minimal strength reduction penalties. A plot of the
bending moment versus the normalized beam length was generated to assist the
determination of regions where large moments exist. Moment values were obtained by
evaluating the second spatial derivative of the equivalent eigenfunction solution, <I>i(x); the
equivalent eigenfunction solution is developed in the following chapter.

Figure 24 displays the absolute moment magnitude versus normalized beam length for
the first and second modes of vibration. Moment values for the first mode are significantly
lower in magnitude than those of the second mode. To aid the comparison of mode
participation, the imbalance in magnitude was offset by dividing the second mode
magnitudes by a factor of six. Scaling of the second mode is used to interpret the general

trends associated with the moment values and normalized beam length positions. Inspection

13

Normalized Mode
Position Danced
0325950350 1St
0450930475 2“(1
0525950550 2“d
0.650SxSO.675 1St

Table 2-1: Position of optimal damping strips.

of the traces indicated positions of large moments for the second mode correspond to
minimum moment locations for the first mode. This may be attributed to the orthogonality
condition discussed latter. This poses the problem that a strip of damping material may be
designated to dissipate the energies of either the first or second mode, but not both. Results
of Figure 2-4 indicate first mode damping is best attained by positioning viscoelastic
material near the clamped end and second mode damping is achieved by locating the
viscoelastic material at the specimen mid-length. 'Ihese predictions are verified in Sun et a1
[24] .

To preserve laminate strength properties, a minimal number of damping strips were used
in the optimally damped specimens. Two strips were alloted for both the first and second
modes. The four strips were positioned symmetric to the specimen mid-length. Damping
material was not positioned at the specimens ends because maximum laminate strength is
needed in this region to support the substantial moments a clamped-free specimen
experiences during static loading. These regions, termed “end regions”, were designated in
the normalized length units as: 0 S x S 0.25 and 0.75 S x S 1.0. Locations chosen for the
damping material and the mode flom which they dissipate energy are listed in Table 2-1.
Selection of the Optimal damping strip locations was determined using a trace that was not
an exact representation of clamped-flee beam dynamics.

The rate of energy dissipation per damping strip for a given mode is quantified via the
equation listed below. Equation (2- 12) is integrated across the damping strip’s normalized

width. The viscoelastic damping term is assigned a large value, corresponding to the

l4

dimensionless damping of 0.025, when equation (2-12) is used to quantify the energy
dissipation associated with the strips of viscoelastic material.

2

wag, = BIL"”(<I> )dx (2-12)

Lu:

15

W
A Bemoulli-Euler beam model was used to predict the dynamic response of a clamped-

flee composite beam. The generic Bernoulli-Euler beam equation was derived such that a
frequency domain transfer function relating the displacement of the clamped end to the flee
end could be developed. Damping parameter values were assigned to correspond to the
three methods of internal damping and the two layup geometries.

3.1 Bernoulli-Euler Model

Certain assumptions and constraints must be applied to the beam system to assure the
validity of the Bernoulli-Euler beam model. It is assumed that both the radius of curvature
of the deflected beam is large relative to its thickness and that beam cross sections, which
are oriented perpendicular to the neutral axis prior to bending, remain plane during flexure.
Since the Bemoulli-Euler model is most accurate at low excitation frequencies, the
experimental work was performed using a random excitation signal between 0 Hz and 800

Hz. This excitation bandwidth contains the first four natural flequencies for the eight ply,

 

 

 

clamped-flee laminate specimens.
f(x,t)
Y V(x) V(x +Ax)
I K I m.
M(x) * M(x + Ax)

IHHI

 

>X

 

Figure 3-1: Differential Beam Element

16

The differential beam element flee body diagram of Figure 3-1 is used to display the
bending moments, shear forces, and applied loadings. The bending moment and shear
force relations are applied to establish equations (3-1) and (3-2). These initial equations are

combined to produce the shear-moment relation of equation (3-3).

M(x,t) = EIyxx + BIymu (3- 1)

WM) = M (3-2)
8x

V(x,t) = —[EIym + Brym] (3-3)

Evaluation of the shear forces acting upon the differential beam element yields equation
(34) and a force summation leads to equation (3-5). Equation (3-5) contains the specimen
density, p, and the specimen cross sectional area, a.

V(x+Ax,t) = V(x,t)+ g}. Ax+0(Ax2) (3-4)
[Egg-Ax + V(x,t)]- V(x,t)+ f(x,t)- Ax = -—(pa)yqu (3-5)

Addition of the two shear force terms on the left hand side of equation (3-5) reduces it to
a three-term equation (when higher order terms are neglected) in which each term includes
the differential beam element length. Dividing by the element length and taking the limit as
Ax -) 0 produces the general Bernoulli-Euler equation of motion presented in equation (3-
6). According to the sign convention shown in Tim 3-1, beam displacement is positive in
the upward direction. The force summation assumes downward forces are positive. The
externally applied distributed load, f(x,t), is in the positive force direction and will result in a
negative vertical displacement of the differential beam element and thus a negative beam
element acceleration. This negative acceleration is accounted for on the right hand side of

equation (3-5).

17

[Elym + Blym] + f(x, t) = - (pa)y, (3-6)
The equation of motion for the unloaded Bernoulli-Euler beam model is therefore given
by equation (3-7).

[Elym + Brym]+(pa)yu = o (3-7)

3.2 Boundary Conditions

The experimental procedure introduced a random wide band displacement input into the
clamped-flee specimen via a vibration exciter. This system driving input located at the
clamped end was characterized by the general time function, u(t), and is represented by the

clamped-flee time varying boundary conditions.

Claimed: Free:
y(0.t) = u(t) yu(1,t) = o (3-8)
3'40") = 0 y...(1.t) = 0

Note that henceforth x is considered to be the dimensionless length variable and the
requisite scaling is applied.

The random input motion at the clamped end causes the boundary conditions to become
non-homogeneous. A variable transformation, (3-9), is used to obtain a partial differential

equation with homogeneous boundary conditions, equation (3-10).

y(x,t) = z(x, t) + u(t) (3-9)
$[Erzm+Brzm]+(pa).[zu(x,t)+rim] = 0 (3-10)

3.3 Eigenfunction Solution
The separation of variables technique is used to develop an eigenfunction solution

describing the displacement of the resulting beam model. The transformed displacement

18

term, z(x,t), is separated into its spatial and time domain components according to equation

(3—11).

z(x, t) = W(x)- a“ (3-11)

The eigenfunction solution is obtained by solving the homogeneous portion of the
system governing equation. The homogeneous solution is acquired by equating the force
input u(t) to zero. The spatial and time derivatives of the variable separation equation are

incorporated into the now homogeneous equation resulting in equation (3-12).

EI - WWI:St + BI - w,m scS‘ + (pa)L‘W(x)s’c:St = 0 (3-12)
Dividing out the time domain exponential terms and consolidating the terms associated
with the fourth spatial derivative of the separated displacement variable, Wm(x), one
obtains a simplified form of the above relation, the general displacement solution, in

equation (3-13).

 

Wm - A‘wor) = 0 (3-13)
4 2
Hm

The general displacement written below is the solution of a homogeneous, fourth order

differential equation. Expressions will be determined to obtain values for the constants Ci.

W(x) = Qe‘ " + go“ 1‘+ (2,? " +C4c-p' " (3-15)

The first three spatial derivatives of the general displacement solution yield three new
relations. The transformed boundary conditions are then applied to the spatial derivative
relations to obtain homogeneous equations in terms of the constants Ci. These
homogeneous equations have been consolidated into a concise matrix form and displayed in

equation (3-16).

19

1 1 1 1 I “C,

1. -1. f). -p. c,

rze‘ 121:" 4.2a" 4.2!“ ' c, = 0 (3'16)
C4

)3 e” -7\.’ e” 117312“ 1136’"

L- _ h

 

 

 

 

The above homogeneous matrix relation is rewritten in the abbreviated form to simplify

the ensuing discussion.

[FICJ = 0 (3-17)
When establishing values for the constants Ci, one wants to avoid the undesirable,
trivial case when the constants are all equal to zero. This is precluded by stipulating that the

determinant of matrix F be equated to zero. This requirement is given by

Det.[F] = 4 + e-(“j )1 + 2H” )1 + ea”. )1 + co” )1 = 0 (3-18)
Algebraic manipulation of (3-18) produces the general eigensolution from which an
infinite number of mode specific eigenvalues, 3... can be obtained. The mathematical

development to obtain the general eigensolution is shown in detail in Appendix A.

cos(}ti) - coshOti) = —1 (3-19)

3.4 Eigenfunction Selection and Normalization of Constants

Eigenvalues are mode specific numbers used to represent the system’s dynamics. The
eigenvalues for the first five modes of vibration were computed by using the general
eigensolution and the symbolic manipulation software package “Mathematica”.
Eigenvalues, truncated to the sixth decimal position, are listed in Table 31.

Determination of the eigenvalues became cumbersome as we moved away flom the
fundamental mode, ten or more decimal positions were not sufficient to obtain the
eigenvalues for modes five through ten. Initially the eigenvalues were computed by using

only four decimal positions for accuracy, this resulted in unacceptable errors of 7% for the

20

third mode and 121% for the fourth mode. The numerical precision was then increased to
six decimal positions, with a large increase in computational time, which produced minimal
truncation errors in the first four vibration modes as listed in Table 3-1. The resulting
truncation error percentage is computed via the equation below. The “actual” term in the
error equation is the value obtained when the truncated eigenvalue was substituted directly

into the general eigensolution.

= -1- (actual) x 100

% error (3-20)

An eigenfunction solution form that is equivalent to the general displacement solution is

established next. This solution form is used to generate values for the constants Ci. An
equivalent eigenfunction expansion is given below. It may be shown that the eigenfunctions

are orthogonal, normal, and complete.

d>i(x) = C,cos(1.ix)+ C2 sianx) + C3 cosh(3.ix) + C4 sinhOtix) (3-21)
The transformed, homogeneous clamped-flee boundary conditions written below are
substituted in equation (3-21) to solve for the unknown constants. Note that spatial

derivatives are signified with a prime.

W: W:
<p,(0)=0 <I>”1(1)=0 (3-22)
<b’.(0) =0 <b”’1(1)=0

W neural m
1.875000 -0.999983 0.0017
4.694096 -0.999739 0.0261
7.854760 -1.003305 0.3305
10.995541 -0.992099 0.7901
14.137169 -l.420000 41.9999

MAUJNr-w-h

Table 3-1: Eigenvalues, and truncation errors.

21

The combination of the homogeneous boundary conditions with the equivalent
eigenfunction solution are listed below. The equivalent eigenfunction solution and its first

derivative establish relationships which eliminate the constants C3 and C4.

<1>im=O=C1+O+C3+0 (3—23a)
cg...) = o = 140+ 02 + o + c4] (3 - 23b)
or.) = o = til-c, cos(r,) - c2 sin(1.,.) + C,cosh(r,) + c,sinh(r,.)] (3 — 23c)
or. = o = 1.3,..[Clsin(2t,.) - c,cos(>.,.) + c,sinh(>.,.) + C4cosh(l,.)] (3 - 23d)

Elimination of two constants simplifies equation (3-23) to a two-equation, two-
unknown situation. The eigenvalue outside the brackets on the right hand side of the two
remaining equations is divided out allowing the trigonometric terms to be equated to zero.
This division of the eigenvalue shows the assumption that the eigenvalue does not equal
zero and is greater than zero for all vibration modes. The remaining trigonometric
expressions are consolidated into the following matrix form. Equating the determinant of

the two by two matrix to zero confirms the previously stated eigensolution.

cos(2.i)+cosh(2ti) “(MI-m0.) . C1 = -
[Sinai-smug 008(ki)+cosh(3.i)] [(2,] 0 (3 24)

The equivalent eigenfunction solution is used to select arbitrary values for the constants.
Manipulation of equation (3-24) produces a relation from which mode specific, non-

normalized values for the two remaining constants are obtained.

 

0080..) + cosh(2ti)] (3-25)

C2 = —Cl[ smOti) + sinh(7ti)

Non-normalized values for the constants are acquired by arbitrarily fixing the value of

one constant and computing the value of the second constant. Non-normalized values for

22

C2 over the first five modes were computed by fixing C1 = 1.0 and substituting the
eigenvalues into equation (3-25).

Normalization of constants consists of integrating the square of the equivalent function
solution over the generalized beam length, 0 S x S l, as displayed in equation (3-26).
Normalized values for C1 and CI for each vibration mode are computed by using equation
(3-27) and are listed in Appendix B.

1
normal = j ¢i(x) - ¢,(x)dx (3-26)
0
c. =__C.L_ c. =_&_ (3-27)
1'“ Vnormal 2‘“ Vnormal

3.5 Solving for z(x,t)

Having specified the constants in the equivalent eigenfunction solution, one can now
obtain a mode specific solution for the transformed beam displacement. The equivalent
eigenfunction solution, d>i(x), can replace the general displacement solution, W(x), in the
variable separation. Since the equivalent eigenfunction solution is dependent upon the
vibration mode in question, the exponential time domain term in the variable separation will
be replaced by the mode participation term, qi(t). These substitutions are displayed below.

The presence of the summation sign represents the most general form of any solution.

z(x,t) = 2¢i(x)-qi(t) (3-28)

To make use of the beam transfer function model, information at both ends of the
clamped-flee beam must be specified. The clamped end displacement is provided via the
system input term, whereas the flee end data is obtained flom the transformed displacement
term that is measured at the flee end’s normalized position, x=1. The mode participation
term is the only parameter that has yet to be specified.

23

The general equation of motion with the beam displacement described in the z-coordinate
system is repeated below.

E12,... + 1312.... + (pa)L‘[z (x,t)+ u(t)]: 0 (3-29)
Separation of the spatial and time domains of equation (3-28) are employed in the
general equation of motion to transform it into a more usable form. It is evident in the
variable description that qi is a function of time and (bi is a spatial function. Thus their
respective arguments, (t) and (x) will be neglected in the following derivation. Also, the

subscript derivative notations t and x will be changed to the derivative descriptions

1 and i respective]
dt dx y‘

as e.
BI —[Zq.<t><1> on] 33x.[2q.(t)¢.(x)]+

32 ee
(pa)L‘ - ¥[Zqi(t)¢i(x)]+ (pa)L‘ii = 0 (3-30)

       

Enclosing the summation terms in the brackets and consolidating the partial derivatives

with respect to x and t produces equation (3-31) shown below.

.. 4 2
2:[(r~:1qi + 31-51-31) - (“1:21) + (pa)L‘ 34$] +(pa)L‘ii = 0 (3-31)

Recalling that the eigenfunctions are a sum of sine, cosine, hyperbolic sine, and

hyperbolic cosine, one will note the validity of equation (3-32) which is used to produce
equation (3-33).

4
933" -?t’1d>i(x)=0 13-32]

:{[A‘. ”(El ql+ BI :H—+paL‘—‘1:t‘§—ri]¢(x)}+parjfi= 0 [3-33]

i

24

Recall that the notations d/dt and ( . ) both signify time derivatives. The different
notations are used to facilitate the reader’s understanding of the origin of that derivative.
The d/dt derivative is for the mode participation parameter and the ( . ) is for the input
description.

Eigenfunctions are selected as normal with a 2—norm of one. Orthogonality and
completeness of the eigenfunction is stated in the condition of equation (3-34).
Multiplication of equation (3—33) by the equivalent eigenfunction <l>j(x) and integration over

the length of the beam leads to equation (3-35).

‘ Oforiatj
o. .o. - = - 4
I .(x) ,(x)dx {Mom}. (33)
2 l
141- (El - qi + BI - 9%) + paL‘Ethii + paL‘ii_[¢i(x)dx = 0 (3-35)
7 o

The ¢i(x) term inside the integral is equivalent to the <I>j(x) according to (3-34). A

notation simplifying relation is substituted into equation (3-35) to obtain equation (3-37).

I
41 = I¢.(x)dx (3-36)
0
.. l dq. dzq.
=— .x‘nar +-—* MBI + aL‘—‘ 3-37
u 931,411 ) d.( )p 1.2] < >

In the development of the transfer function in the following section, a relation for the
mode participation parameter in the flequency domain is needed. This requirement is
fulfilled by taking the Laplace transform of (3-37) and consolidating the resulting Q(s) terms
to produce equation (3-38).

25

s’~Pi-paL‘-U(s)
1.4.51) + s- (1.4.131) + s2- paL‘]

 

Q.(s)=[( <3-38)
3.6 Transfer Function Solution Process
The transfer function relating the displacement of the clamped end to the flee end is now
developed. A complete and accurate transfer function includes an infinite summation of the
individual transfer functions for each mode of vibration. The analysis at hand will
determine an approximate transfer function based upon the first four modes of vibration

since this is the range over which the eigenvalues were most accurately computed.

The relationship in the flequency domain between the mode participation factor, Qi(s),

and the system input, U(s), are necessary in developing the transfer function. This

relationship is termed the beam response function, Ti(s). This relation provides the beam’s

response at the flee end due to a given input for a selected mode.

T1(5) = "(ti—((5)2 (3-39)

The relation of the free end displacement (output) to the clamped end displacement
(input) in the Laplace domain is:

Y(1.s)=2[<I>.(1)-Q.<s)]+U(s) 13-40)

Substitution of Qi(s) and moving the U(s) term to the left hand side of the equation

produces a transfer function, H(s).

H(s)=-‘I’J—‘2§=2[¢.a)rrs)]+1 13-41)

26

The Ti(s) term is expanded using equations (3-38) and (3-39) to obtain the form of the

following transfer function equation, which is used to generate the theoretical magnitude and

phase angle plots and define the modal response relation.

 

. . 4 2
«>10 P. (paL)s a (3412)

111(8) = [(231 E1) + 8° (7531) + sz.pa1.‘]

3.7 Modal Parameter Values

Material properties and beam dimensions need to be established so magnitude and phase
angle plots for select damping values can be generated as well as a definition of the modal
damping relation. Material properties of the 3M graphite/epoxy “Scotchply” material will be
used. Specimen beam dimensions are assigned as 28cm x 2.54cm x 0.112cm. Pertinent
material properties and specimen geometry values are listed in Appendix B. Values for the
damping parameter were selected to correspond to the dimensionless damping ratio, é,
typical of carbon/epoxy composite laminates. Appendix C shows the development of the
dimensionless damping ratio.

The transfer function equation was combined with the material properties to produce the
analytical flequency response for clamped-flee laminates. Calculations were performed in
using the MATLAB software package. Transfer function responses were developed for five
damping values. Magnitude versus frequency and phase angle versus flequency plots were
also generated. Magnitude and phase angle responses were evaluated across the flequency
band spanning l x 10"3 Hz to 1 x108 Hz, dynamic activity was noted in the 1 Hz to 1000 Hz

frequency band. This was consistent with experimental observations.

3.8 Analysis of Results
Plots comparing the one mode, two mode, three mode, and four mode magnitude and

phase angle approximations of the transfer function for varying values of the dimensionless

27

 

 

 

 

 

 

 

 

 

 

 

 

" 5
a
8
"E.
§-5 . . 1 l. “.1 . . . ....'
100 101 102 103
flequency (Hz)
’8 C - ‘1 \;43 t 2 3 2 3—3 i i 4
:63 ‘— a
a “W ._..
.1: c _.> "K
Q‘sm l I #141!!! l I Lllllll l I LJllll d
100 101 102 103
flequency (Hz)
Figure 3-2: Magnitude and phase angle approximations for the transfer function and

dimensionless damping equal to 0.025. In the traces, ‘a’ indicates the
one mode approximation, ‘b’ indicates the two mode approximation, ‘c’
indicates the three mode approximation, and ‘d’ indicates the four mode
approximation.

damping parameter are located in Appendix D. Magnitude and phase angles for a

dimensionless damping parameter equal to 0.025 are displayed in Figure 3-2. Magnitude

plots for varying dimensionless damping parameters exhibit an overall decrease in

magnitude for increased damping.

28

W

4.1 Experimental Materials

The crossply and unidirectional beam specimens were fabricated using materials donated
by the 3M Corporation. Beam laminates were constructed from aerospace grade
carbon/epoxy prepreg. This continuous filament prepreg is termed “Scotchply” SP-319 by
its producers. When properly cured, this thermoset material has a fiber volume flaction of
55%, a tensile strength of 245 kpsi and a density of 1.55 g/cc. The internal damping of the
carbon/epoxy laminates was enhanced via the introduction of strips of viscoelastic damping
material laid at the interface region between the laminate’s plys as shown in Chapter 2. The
viscoelastic material was designed to suppress internal laminate vibrations by dissipation of
mechanical energies. This viscoelastic material, also donated by the 3M Corporation, is

assigned the reference number SJ 2015X type 1202 and referenced by the trade name
“Scotchdamp”.

4.1.1 Material Aging

In handling prepreg materials one must consider their individualized storage
temperatures and shelf lives. The shelf life of the Scotchde viscoelastic material is of
minimal concern since it is a stable thermoplastic. Thermoplastic materials are composites
that can be repeatedly heated, formed, and cooled without significant losses to their
mechanical properties. On the other hand, the carbon/epoxy prepreg is a thermoset material
whose shelf life and storage temperature is of paramount concern. A thermoset prepreg
becomes a laminate after being exposed to the elevated heat and pressrn'e of the cure cycle.
A laminate will meet its stated mechanical properties if the cure process is performd within
the designated shelf life provided the prepreg has been stored below its specified storage
temperature. A thermoset material will “age” quicker when stored at temperatures that

exceed their individualized storage temperature. An aged prepreg will suffer a lack of

29

tackiness during fabrication and produce a laminate possessing reduced mechanical
properties.

Estimation of a prepreg’s useful shelf life, or time during which it retains its tackiness,
is essential to both fabrication and research. A prepreg is a single layer of fibers embedded
in a solution of partially crned monomers. Polymers are dissolved into solvents forming
solutions. Storing polymers in the solution form is essential to the prepreg fabrication
process in that the polymer evaporation rate decreases, the polymer solubility is improved,
and the polymer is stored and processed in a less volatile state [25]. Below a temperature
unique to a monomer, termed the storage temperature, monomers are prevented flom
reacting with one another. This absence of monomer interaction disrupts the solvent
integrity and causes a reduction in prepreg tackiness. This stabilizing temperature is
provided by the prepreg supplier. The amount of time a prepreg is stored at temperatures
exceeding its stabilizing temperature, termed “out time”, will cause the prepreg to loose its
tackiness, and material aging will occur. Loss of tackiness creates difficulties in the layup
of laminae during fabrication in that prepreg plys do not readily adhere to one another.
Sanjana [26] cited the loss of prepreg tackiness as the critical parameter in determining the
aging of prepreg.

Akay [27] performed an assortment of mechanical property tests on unidirectional, 63%
fiber volume fraction carbon/epoxy specimens having out times ranging flom 100 hours to
930 hours. The battery of tests, which included interlaminar shear and flexural modulus, do
not show a degradation of mechanical properties over the range of out times studied. The
same analysis was performed using specimens immersed in 70'C water for twenty one days
to evaluate the combination of out time and hygrothermal effects. In addition to the expected
reduction in mechanical properties for the wetted specimens, the test results showed that out
time did not have an effect upon lowering the mechanical properties for a hygrothermally

treated specimen. As noted by Akay, the only effect out time had upon mechanical

30

properties was an increase in the glass transition temperature by approximately 5'C for out

times of forty days.

4.1.2 Aging Effects Upon Specimens

The aerospace grade SP-319 carbon/epoxy prepreg must be stored at temperatures
below 40'F to prevent aging and retain material tackiness. The thermoset prepreg used in
this study was packaged in solid carbon dioxide to assure its material temperature remained
below its storage temperatrue during shipping; transportation delays caused the dry ice to
sublimate, exposing the prepreg to approximately 75°F temperatures for a period of time not
exceeding three days. According to the results stated in the Akay study, a prepreg out time
of three days will not affect the mechanical properties of the carbon/epoxy laminate used to
produce the beam specimens.

4.2 Specimen Fabrication

Unidirectional and crossply beam laminates were fabricated with the assistance of the
Composite Materials and Structmes Center at Michigan State University. The carbon/epoxy
prepreg layup was autoclave cured according to the cure methods outlined by the material
supplier. Viscoelastic damping material used in the general and optimally damped
specimens was added at the prepreg interfaces dining the lay-up process and subjected to the
elevated pressures and temperatures of the curing cycle.

4.2.1 Lay-up Orientations and Damping Material Locations

Tests were performed on six laminates that varied in the amount of viscoelastic damping
material and orientation of reinforcing fibers as discussed in Chapter 2. Specimens 2.54 cm
in width and 28 cm in length were cut flom laminates approximately 5.25 inches by 12
inches in size using a diamond blade circular saw. They were cut from the middle portion
of the plates to reduce the efl‘ects of edge discontinrrities that arise in fabrication.

The decision to enhance material damping by placing discontinuous strips of viscoelastic
material at ply interfaces distinguishes this study flom its predecessors. Previous studies

31

have considered continuous layers of viscoelastic material covering interface surface areas.
A continuous layer of viscoelastic material at an interface will undoubtedly produce large
amounts of material damping but at the expense of reductions in laminate strength.
Laminate strength is reduced when intimate attachment between neighboring plies is
prohibited by the inclusion of damping material.

An arrangement strategy for the viscoelastic damping strips that received consideration
for the current study specified two-quarter-inch wide strips of damping material be laid
parallel along the length of the beam specimen. This strip arrangement may be
advantageous in the analysis of experimental data because the material damping property is
continuous along the length of the specimen. Continuous damping is desirable for analysis
since the damping characteristic is assumed to be constant throughout the specimen. This
arrangement was not used due to the difficulties associated in fabricating specimens of
repeatable quality. A large potential for error exists when attempting to accurately position
quarter inch wide strips in a parallel fashion across a one foot laminate and machining inch
wide beam specimens with consistent damping strip orientation. Although the parallel
arrangement of the damping strips may be desirable flom an analytical standpoint, the
associated fabrication limitations were deemed too severe.

The damping strip arrangement used for the optimaland general damping strategies has
one-half-inch-wide viscoelastic strips positioned perpendicular to the length axis. This
ensures a simple and highly repeatable specimen fabrication process. The discontinuous
damping arrangement at the interior interfaces is not consistent with the analytical model that
was constructed under the assumption of constant damping throughout the material.
However, one may think of the resulting analytical model of the dynamics as the result of
applying the method of assumed modes. Although this discontinuous arrangement of the
damping strips has shortcomings, the ability to sustain the production of specimens with

consistent location of the damping material was considered to be of greater importance.

32

4.2.2 Fabrication Equipment

The prepreg laminates were cured under elevated temperature, 120'C, and pressure, 620
kPa, in an autoclave. A concern arose that the elevated temperature and pressure of the cure
cycle would have an adverse effect upon the viscoelastic property of the Scotchdamp
material causing a reduction in its energy dissipation capabilities. Consultation with a 3M
representative provided assurance that the Scotchdamp would not suffer permanent material
property degradation. The material retains its viscoelastic damping capabilities to a
temperature of 180'C. The 3M representative advised that Scotchdamp will become less
viscous at the temperatures and pressures of the cure cycle and will slowly flow within the
laminate [28]. Upon laminate cooling, the viscoelastic material will solidify and return to its
original position. Such flow of the viscoelastic material during the cure cycle initiated a
concern that the desired damping strategies could be altered. Upon visual inspection of the
laminate surfaces it was noted that the position of the damping strips remained unchanged.
This inspection was performed visually because the damping strips "read through" to the

laminate’s surface.

4.2.3 Fabrication Methods

The unidirectional, continuous roll of prepreg was wound on a spool and stored in a
freezer to guard against material aging. The five and a quarter inch wide prepreg roll was
cut to make the unidirectional and crossply laminates. Unidirectional laminates were created
by stacking eight layers of prepreg upon one another; extreme care was used to ensure that
the fibers of adjoining plys were oriented parallel to one another so as to maintain the
integrity of the unidirectional laminate. Crossply specimens required four plys sized 5.25 in
by 12 in to produce the 0’ layers and twelve 5.25 in square plys to make the four 90' layers.
Three square plys were needed to compose each 90' ply. Proper positioning of the
Scotchdamp material was assured by employing a wooden, rectangular fixture.

33

Two layers of prepreg were rolled after stacking to improve the degree of interfacial
contact and to remove air trapped at the interface. The direction of rolling was consistent
with the fiber orientation of the top prepreg layer to avoid fiber damage by exerting lateral
loads upon the fibers. Latex gloves were worn during the layup process to prevent body
contact with the carbon/epoxy prepreg or the viscoelastic damping material. Gloves are
essential in protecting the prepregfrom body oils and perspiration, which produce voids and
delamination in a laminate. The necessity of wearing gloves is paramount in the process of
positioning the damping strips on the prepreg. Force is exerted upon the strips with one's
fingers to remove entrapped air and to improve the contact between the prepreg and
viscoelastic materials. If gloves were not worn, one‘s fingers would come into contact with
the exposed surface of the carbon/epoxy prepreg. Prior to the laying out of the damping
strips (when the interface surface is contacted by the latex gloves) the fingertip areas of the
gloves were rinsed with acetone to remove unwanted foreign particles that might have

settled there timing the fabrication process.

4.2.4 Problems Unique To Fabrication

The viscoelastic damping material is adhered to a backing that improves its handling.
This backing material was removed after the damping strips were laid out at the laminate
interface. Ideally the backing peels away flom the damping material without incident.
Occasionally fragments of the backing remain adhered to the damping material and/or the
prepreg. Although very small in size, these remnants must be removed otherwise a void or
delamination may occur in the laminate. Backing remnants are best removed via light
scraping of the prepreg with a scalpel. Such scraping can induce local disruptions of fiber
orientation or sever previously continuous carbon fibers. Disorientation and cutting of
fibers are undesirable occurrences, but necessary to elude the possibility of voids and
delamination since the presence of the latter has a more pronounced effect upon enhancing

laminate damping characteristics.

34

W

The forced vibration method was selected as the experimental means to compare the
damping characteristics of the composite specimens. To assure that the measured damping
was only indicative of material damping, the response of the specimen was measured with
apparatus selected and positioned so as not to violate the assumptions of the Bernoulli-Euler
beam model. Linearity of the laminate specimens and the sensors was verified

experimentally for the operating conditions of interest.

5.1 Experimental Set-up

The experimental equipment was arranged as depicted in Figure 5-1. The forcing signal
driving the clamped-free specimen originated as a random voltage output with specified
frequency content flom a programmable source on a Hewlett-Packard (HP) signal analyzer.
This voltage signal was amplified using the Briiel & Kjaar (B&K) power amplifier prior to
introduction into the B&K Vibration Exciter. The exciter transformed the current signal into
a vertical displacement. Random displacement of the clamped end was used to stimulate the
dynamics of the beam specimen over the specified frequency range. Measurements of the
vertical displacement at the flee end were made using a B&K Laser Displacement/Velocity
Transducer. The clamped end vertical displacement was measured by using a non-
contacting Kaman Measuring Systems (KMS) eddy current sensor. Measured displacement
data were collected using a National Instruments data acquisition board and a Macintosh IIx
computer. The software package LabVIEW was used to create a virtual instrument to aid in
this task. Both the rate of data collection and the number of samples collected were

controlled, and the measurements were automatically stored in a file for subsequent
processing.
5.2 Equipment descriptions

Random displacement at the clamped end was generated by using the programmable
source from a HP 35660A Dynamic Signal Analyzer and amplified by means of a B&K

35

 

 

    
   
    
  

 

Briiel szer
transducer

 

 

Beam Specimen

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hewlett Packard iii”;
Signal analyzer 5:1..2.._;,, . v N 7 '
Briiel Kjazr Kaman signal
7 type/output conditioning module
Briiel Kjarr I T ¢ L
amplifier I ~
National Instruments
virtual instrument

 

 

 

Figure 52: Experimental Set-up

Power Amplifier Type 2706. When using the HP analyzer as a signal generator, the type,
magnitude level, and flequency span of the source output are selected by the user. The
amplifier is beneficial in the start-up experimental process in that it allows for simple and
instantaneous changes of the input signal magnitude, which alters the peak to peak flee end
displacement. The ability to change and monitor the flee end displacement is important
when aerodynamic damping is considered and a minimum flee end amplitude with
acceptable dynamic stimulation is sought. '

The amplified signal was introduced into a B&K Type 4809 Vibration Exciter. This
vibration exciter transfers the current input signal into a vertical displacement. Beam

specimens are rigidly mounted onto the shaker device via an aluminum clamping

36

mechanism. The aluminum clamping device was designed to distribute an equal force
across the laminate while maintaining a minimal weight so as not to impede the performance
of the shaker device. The vibration exciter has a bare table flequency range to 20 kHz. The
acceleration capabilities of the vibration exciter are dependant upon the payload weight it is
required to displace. The clamping device and specimen weigh approximately 130 g which
equates to maximum acceleration levels of 246 "v.2 (25 g), as stated in the B&K Vibration
Exciter manual.

Displacements of the clamped end are measured using a non-contact proximity
measuring device developed by KMS. The eddy current measuring system mensurates
changes in displacement of the aluminum clamping device that is rigidly attached to the
laminate specimen.

The B&K Type 3544 Laser Transducer is non-contacting and quantifies displacement
with an accuracy to 10 run over a flequency span of 0.3 Hz to 20 kHz. The transducer uses a
Helium-Neon laser to measure the velocity of a target surface; displacement values are
obtained through an integration of the velocity signal. Velocity values are acquired by
splitting the laser light into two beams: one of which is directed at the external target, and the
other remains within the unit to serve as a reference beam. Integration of a velocity
measurement of zero flequency yields a bias component in the displacement measrnement.
Removal of this displacement bias is addressed in the Data Manipulation section. Small
pieces of retroreflective tape are adhered on the target surface to reflect the light beam back
into the transducer where it heterodynes with the reference beam. Movement of the target
induces a Doppler Frequency shift between the two light beams. This shift is converted into
a voltage signal which is proportional to the velocity of the target.

The experiment was arranged such that the laser was positioned 25 cm above the

retroreflective tape. This separation guaranteed that the laser was within its optimum

operating distance.

37

A virtual instrument was designed to collect and organize voltage displacement signals
on a Macintosh IIx using the National Instruments LabVIEW 2.0 scientific software
package. The data sampling rate and number of data sets sampled per test run were

specified by modifying the flont panel of the virtual instrument.

5.3 Excitation Method

Specimen excitation must stimulate the vibration modes under evaluation as well as
provide meaningful displacement information. The forced vibration method was selected
over the flee vibration method for its wide band flequency input capabilities and the ease in
control of the magnitude of the input signal, which is essential in regulating the flee end
displacement.

When used on a cantilever beam the forced vibration method has advantages over the
free vibration method in that it allows for convenient adjustment of a specimen’s dynamic
response via selection of beam length as well as the input signal magnitude and flequency
content. Further, the input signal magnitude is efl‘ortlessly varied allowing for direct control
. of the flee end displacement in the clamped-flee test arrangement. Ready regulation of the
flee end displacement is a necessity when one wishes to avoid system damping. The
frequency span of the input signal is easily modified in the forced vibration method, which
allows for control of the frequency content that excites the specimen. This is of importance
when attempting to excite selected vibration modes.

In an attempt to excite the third and forn'th vibration modes, the input band was centered
about the natural flequency of the fourth mode. Such input should adequately stimulate the
mode at the input flequency as well as all lower vibration modes.

5.4 Specimen Linearity
For small vibration displacements within undamaged polymer laminates, one can

assume that the dynamic response is linear if the stiffness and damping are independent of

38

the displacement amplitude [29]. This statement regarding linearity of polymer laminates is
verified by using a process in which peak to peak flee end displacements are monitored for
various known sinusodial excitation signals. As shown in Table 5-1, peak to peak
displacement vary by the same factor as input signal levels thus indicating the polymer
composite laminates have a basically linear response. Data in Table 5-1 represents a
sinusodial input of 18 Hz, the fundamental natural flequency for the crossply- generally
damped laminate used to perform the specimen linearity check. The linearity between
specimen input and peak to peak flee end displacement is also valid when the system
sinusodial excitation signals are increased to the natural frequencies of the second and third
modes. System excitation of flequencies of the second and third modes revealed a weak

coupling between the first two modes and between the first and third modes respectively.

5.5 System Damping

During evaluation of a specimen’s damping capabilities it is essential to assure that the
recorded damping values are only indicative of the energies dissipated by the material.
Extraneous energy can be lost to the experimental apparatus and surroundings. This
undesirable energy loss, termed “parasitic damping”, inflates material damping values.
Energy dissipated to the experimental equipment is termed “apparatus damping” while the
energy lost to the surroundings as a result of the oscillatory beam motion is termed
“aerodynamic damping”. Parasitic damping is recognizable because it typically is non-
linear. This non-linear characteristic of parasitic damping distinguishes it flom the generally
linear damping of an undamaged composite laminate.

W
50 mV 0.4 mm
100 mV 0.8 mm
150 mV 1.2 mm
200 mV 1.6 mm
250 mV 1.9 mm

Table 5-1: Specimen hnearity

39

5.5.1 Apparatus Damping

Apparatus damping accounts for the energy lost to the constraining devices, motion
sensing devices, and lead wire that may contact the specimen during testing. Energy loss
associated with mounting accelerometers or strain guages on the beam specimen was
avoided through the use of non-contact displacement measuring transducers. Frictional
damping accounts for energy lost at the interface between the clamping device and
specimen. Clamping devices that have smaller stiffness values than that of the specimen
will experience deformation in testing resulting in extraneous energy loss. Deformation

losses increase with greater flee end displacement.

5.5.2 Aerodynamic Damping

Aerodynamic damping accounts for vibrational energy losses flom the specimen to the
surrounding air. Aerodynamic damping is dependent upon specimen geometry and the flee
end displacement. The larger the free end displacement and beam surface area hindering the
path of motion, the greater the amount of air the specimen is required to move as it
oscillates. In general, the energy required to remove air flom the specimen’s path is the
energy dissipated due to aerodynamic damping.

Aerodynamic damping is dependent upon the amplitude of the vibrating beam and the
pressure of the ambient air. For a unidirectional carbon fiber reinforced plastic Adams and
Bacon found that a linear relationship existed between aerodynamic damping and beam
amplitude [30]. Results flom Gibson and Plunkett indicate that the loss factor increases
with increases in ambient pressure and beam amplitude [31].

Performing forced vibration cantilever beam experiments in a vacuum is one method that
allows the assumption of negligible aerodynamic damping. A second method to minimize
the effects of aerodynamic damping is to restrict flee end peak to peak displacements to be
less than the specimen thickness. Gibson stated that imposing such a constraint allows the
assumption of negligible aerodynamic damping [29].

5.6 Input Displacement Content

The dynamic response for the clamped-flee specimens was approximated using a four
mode model. In order to evaluate the dynamic response associated with a vibration mode,
that vibration mode needs to be adequately excited by the system input. Modes are
activated when the flequency content of the displacement input has a noticeable magnitude at
a mode’s natrnal flequency. A noticeable magnitude is one which is distinguishable flom
the magnitude of the background noise.

The voltage output signal flom the source on the signal analyzer was adjusted such that a
desirable flequency band was attained. The natural flequencies for the first four vibration
modes of the clamped-flee specimens are less than 800 Hz. In an initial observation it was
assumed that an excitation signal of 0 - 800 Hz would be sufficient to excite the first four
modes. It was later noted that the dynamics of the B&K amplifier and vibration exciter have
a 20 db per decade roll-ofl' and this signal roll-ofl’ results in only adequate stimulation of the
first two modes. Input levels at the third and fourth mode natural frequencies were
indistinguishable flom that of the background noise.

Damping parameter values for the one mode, two mode-one damping parameter, and
two mode-two damping parameter approximations were obtained using the aforestated
excitation signal. Experimental results indicated that modal damping did a better job
predicting flee end displacements than the assumed Bernoulli-Euler beam model.
Henceforth we will be concerned predominantly with models assuming modal damping.
Since the third and fourth vibration modes were not adequately stimulated, accurate
assessments of the associated damping parameters was not obtained.

A second acquisition of experimental data was conducted flom two specimens indicative
of extremes in damping for structrn'al materials. The pair of specimens were excited over a
flequency band of 1100 Hz about the natural flequency of their fourth vibration mode.

Specimen excitation about the natural frequency of the highest mode of interest was

41

necessary to provide stimulation of the lower modes to be investigated. Values for damping
parameters were computed for approximations including one, two, three, and four modes.
Multiple parameters were used to numerate the damping value in the multi-mode
approximations, for example three damping parameters were used in the three mode
approximation. The specimen representing the highest damping was the crossply laminate
equipped with the general damping strategy. Data for this specimen was compared to a
representative monolithic material, aluminum 6061, machined to geometric dimensions

identical to those of the composite specimens.

5.7 Specimen Quality

In experimental determination of material damping values, it is of paramount concern
that specimens are not excited to a level that exceeds their maximum strain amplitude.
When this limiting strain level is surpassed, permanent strain induced internal damage
occurs within the laminate. This microscopic damage will produce significant increases in
material damping values without noticeable losses in stiffness properties [32]. Internal
strain damage was avoided by maintaining an excitation signal small enough so that the
minimal free end amplitudes occurred. Also, specimens were only excited in the
experimental set-up when calibration and test measurements were being recorded. System

set-up and trial scenarios were preformed using laminates other than the experimental beam
specimens.

42

W

6.1 Data Collection

Displacement information was collected using a virtual instrument as described above.
A total of six data sets were accumulated for each test specimen. Three data sets comprised
the calibration data group, used for sealing subsequent data, and three data sets constituted
the actual test data group. A data set consisted of displacement measurements for both the

laser and eddy current systems recorded over one second

6.1.1 Sampling Rate

Two considerations influenced the rate at which the displacement information was
sampled by the virtual instrument. The first concern applied to the accmate reproduction of
the system’s dynamics and the second to the ensuing mathematical post-processin g of data
sets. Dynamics of a system are best captured when data is sampled at a rate that is at least
ten times faster than the highest flequency of concern. For an eight-ply clamped-flee
specimen this frequency of concern is the natural flequency of the fourth mode,
approximately 550 Hz. Upon the completion of the collection process, data sets were
mathematically processed to extract meaningful information regarding the dimensionless
damping and natural flequency characteristics. In light of these concerns and recognizing
the memory capacity and processing rates of a Macintosh fix, the system used to manipulate
the data sets, displacement values were sampled at a rate of 4096 Hz for one second. This
sampling rate was a minimum of seven times the rate of the largest flequency of concern.
Test and calibration data consisted of two sets of 4096 entries; data was written and stored
in two columns. A sampling period of one second was sufficient to capture the random

dynamic trends of the cantilever specimens considered.

6.1.2 Sampling Process
As stated previously, clamped-end displacements were measured using a KMS eddy

current sensor. In order to observe the full range of motion of the clamped end, the KMS

43

potentiometer gain was increased to levels that introduced a bias into its output signal.
When a test specimen was clamped into the experimental set-up, three sets of displacement
measurements were sampled for calibration purposes. The setup for this data set consisted
of positioning both the B&K laser transducer and the eddy current sensor at the clamped end
to measure the motion of the aluminum clamping device under random excitation. After
gathering this data, the laser transducer was repositioned at the flee end at which time three
specimen test data sets were collected.

Since the gain for the Kaman system is highly sensitive, it was of great concern that the
testing equipment not be disturbed (for example by bumping the table) during the sampling

of calibration data and test data or during movement of the laser transducer.

6.2 Data Manipulation

In an effort to ascertain physically meaningful information from the displacement
information, manipulation of the data sets was required. Post processing of data allowed
for recognition and removal of bias components flom both the laser and eddy current
measuring systems and identification and optimization of parameters that yielded values for

specimen damping and stiffness terms.

6.2.1 Laser Calibration
The HP Laser Velocity-transduca displacement measurements were afflicted with a DC
bias. This bias signal occurred because the laser transducer recognizes and measures
motion as a velocity, then integrates this velocity value to obtain a displacement
measurement. The laser bias was removed by subtracting an average value flom individual

laser measurements; this average value was computed for each calibration data group or test
data group.
6.2.2 Kaman Calibration

A calibration group consisting of three data sets was recorded each time a new specimen
was clamped into the experimental set-up. Calibration data sets consisted of laser and eddy

44

current displacements measured at the clamped end. This data was used to compute scale
factors that convert the eddy position change information into a displacement signal
measured in millimeters.

The virtual instrument was arranged such that displacement signals were sampled in an
alternating order separated by equal intervals of time. To accurately compute scale factors,
laser and eddy current data points need to occur at identical times. The linear interpolation
of laser displacement values in equation (6-1) was performed on the two laser measurements
that sandwich an eddy current data point to obtain a locally averaged laser displacement
measurement that occur simultaneously in time with the eddy current measurement. The
process of aligning the time locations of the data values eliminated one measurement flom
the calibration data set. Scale factors, a0 and a1 were computed for each calibration data set
according to the equation below. Factors for the three calibration sets were averaged to
obtain the scale factors for a test group.

y(k) ___ y(k)+:(k -1) (61)

Yr = a0 + arYx (6-2)
The laser and eddy current displacement measrnements are denoted by using y1 and yx

A respectively. The factor a1 represents the Kaman system’s “guage” factor.

6.3 Parameter Identification

Specimen dimensionless damping and natural flequency values were obtained via
optimization of the modal response equation. A modal response equation is an approximate
function whose exactness is improved by increasing the number of vibration modes
included in its representation. New modes are added to the modal response equation in a
block like fashion through the addition of general forms, equation (6-3), of the previously
developed transfer function. A general form of the transfer function was attained by

dividing the density-cross sectional area term, paL‘ , flom the numerator and denominator

45

on the right hand side of the transfer function and substitution of three parameters as detailed

below.

K.
.(1 Sufism < >

 

 

 

1 I
K. a <1>.(1) J ¢.(x)dx
0
_ B11
pl = p 3144
> (6-4)
_ 321
p2: pal-J4
= EI
P3 - paL’ J

 

A value was calculated flom the modal information and assigned to parameter K, for
each mode. This value decreases as the mode number increases indicating that higher
modes have a smaller impact upon system dynamics than lower frequency modes.
Parameters p13 and p3 are proportional to the modal dimensionless damping and the natural
frequency terms respectively. These parameters were found using the optimization
algorithm and were used to generate approximate solutions for the modal response equation.

In the development of the modal response equation, the resulting degree of accuracy is
determined by the number of modes used in its construction. Inclusion of more modes
improves the modal response approximation at the expense of computation time. The modal
response relation can be amended to evaluate an additional mode by including the right hand
side of equation (6-3) into the existing modal approximation. A steady state component
indicative of the zeroth mode was included in the modal response. This component has a
value of one. Equation (6-5) denotes the form of the one mode and two mode

approximations.

46

 

K
On Mod A roximation: Ts =1+ ‘
° ° PP “ [64.11“me

 

 

K K
T oMode-OneDam: T.s =1+ ‘ + 2 * 6-5
“’ P “ Is’+p.>~‘.s+p.?~tI Is2+pars+parI ‘ ’

 

 

 

Two Mode- Two Damp: T}(s)=1+[sz+pl§;+ p311]+[82+ p2£:+ P3142]

Addition of a mode to the modal response relation increases the number of parameters
the optimization routine must determine. An increase in the number of parameters reduces
the function approximation error with an increase in computation time. Modes are
accompanied by two initial condition parameters. The damping term may be easily
represented using one damping parameter as shown in the “Two Mode-One Damp”
approximation of equation (65). A more exact method of representing material damping for
modes other than the fundamental is to assign a damping parameter to each mode as shown
in the “Two Mode-Two Damp” approximation of equation (6-5). A single natural flequency
parameter was defined for the system and remained constant regardless of the number of
modes included in the approximation.

The optimization algorithm used to generate parameter values is in the MATLAB
Optimization Toolbox distributed by The MATH WORKS, Inc. The routine employs the
Sequential Quadratic Programming method to determine the minimum of a non-linear
function and allows constraints to be imposed upon parameters. Since the damping and
flequency parameters represent non-negative physical quantities, their associated parameters
were constrained to be equal to or greater than zero. Limitations of the optimization routine
require the function and constraints to be continuous and no assurance is provided that the

final parameter ‘values represent the desired global extremum; they may instead represent

47

local extrema. The function to be minimized, J, is the time integral of the square of the
difference between the measured flee end displacement, ym, and the predicted flee end

displacement, yp, as shown below.

2
J = :fly,n — yp| dt (6-6)
0

Converging upon local minimums was avoided by careful selection of initial parameter
values. Values determined during the minimization of experimental data included:
dimensionless damping parameters, the natural flequency, and the initial conditions. The
dimensionless damping and natural flequency parameters were of the greatest interest, their
initial values were selected using apriori information in an effort to guard against the
convergence to false minimums.

Since the initial conditions were small in magnitude compared to the damping and
flequency parameters, their initial values were prescribed to be zero for each new data set.
Initial values for the damping and flequency parameters for the one mode approximation
were selected to be close to their analytically predicted values. The initial damping
parameter was chosen as zero and the flequency parameter was specified as one thousand.
Initial parameter values for modes other than the fundamental were specified by using the
final parameters of the next lowest mode.

6.4 Results

Parameter values were used to predict the motion of the specimen’s flee end. For each
test case, predicted traces were superimposed onto the corresponding experimental data.
Plots corresponding to excitation signals of 0 - 800 Hz are located in Appendix E. Appendix
F includes the plots for the 6061 Aluminum and crossply-generally damped specimens with
excitation signals having a 200 Hz band centered about the fourth mode resonant flequency.

48

Sealed flee end displacements of the crossply specimens exhibited a beat phenomena
within their generally oscillatory motion. The beat pulse occurred at approximately two
Hertz. Beat effects were not present in traces of the unidirectional specimen’s flee end
displacements. The beat motion is best displayed in the crossply-undamped test case ‘B’
and the crossply-generally damped test case ‘A’.

6.4.1 Parameter Values

Dimensionless damping parameter values and the natural flequencies for specimens
excited over the 0 - 800 Hz band are located in Tables 6-1 through 6-6. Similar results for
specimens excited about their fourth mode natural flequency are listed in Table 6-8 tlnough
6-15. A value of “0” indicates the minimization algorithm predicts zero dimensionless
damping for the specimen. Values of “0.0000” indicates an amount dimensionless damping
is specified by the minimization algorithm, but the non-zero parameter digits are lost to

truncation.

One mode approx. two mode-one damp. approx. two mode-two damp. approx.
Case mm first. medal medal freq. marl model hm.
test A: 0.0082 31.67 0.0079 0.0493 31.67 0.0082 0.01 15 31.65
test B: 0.0076 31.70 0.0072 0.0452 31.70 0.0076 0.01 15 31.69
test C: 0.0083 31.69 0.0081 0.0508 31.69 0.0083 0.0113 31.69
Total: 0.0080 31.69 0.0077 0.0485 31.69 0.0080 0.0118 31.69
Table 6-1: Crossply-generally damped dimensionless darrrping parameter and natural
frequency values values.

One mode approx. two mode-one damp. approx. two mode-two damp. approx.
Case mad/9.1 freq. medal mndcz freq. medal medal free.
test A: 0.0016 37.52 0.0023 0.0142 37.51 0.0016 0.0143 37.51
test B: 0.0025 37.56 0.0027 0.0171 37.56 0.0025 0.0319 37.56
test C: 0.0032 37.43 0.0043 0.0269 37.40 0.0034 0.0324 37.42
Total: 0.0021 37.54 0.0029 0.0181 37.56 0.0021 0.0353 37.54
Table 6-2: Unidirectional-generally damped dimensionless damping parameter and naunal
flequency values values.

One mode approx. two mode-one damp. approx. two mode-two damp. approx.
Case medal flea. model made]. flea. menu nodal L.
test A: 0.0031 31.84 0.0009 0.0057 31.84 0.0009 0.0057 31.84
test B: 0 31.89 0.0005 0.0029 31.87 0 0.0030 31.87
test C: 0.0008 31.95 0.0007 0.0041 31.94 0.0008 0.0040 31.94
Total: 0.0013 31.91 0.0007 0.0041 31.89 0.0013 0.0083 31.89

Table 6-3: Crossply-optimally damped dimensionless damping parameter and natural
flequency values values.

 

 

49

One mode approx. two mode-one damp. approx. two mode-two damp. approx.

use medal hm. medal medal medal m2 1.
test A: 0 37.89 0 0 37.89 0.0011 0.0028 37.89
test B: 0.0000 37.93 0.0008 0.0050 37.95 0.0000 0.0050 37.95
test C: 0.0009 37.82 0.0011 0.0068 37.82 0.0010 0.0150 37.82
Total: 0.0000 37.88 0.0007 0.0043 37.88 0.0000 0.0103 37.88
Table 6-4: Unidirectional-optimally damped dimensionless damping parameter and natural

flequency values values.

 

One mode approx. two mode-one damp. approx. two mode-two damp. approx.
model madcz

Case medal freq. medal medal first.
test A: 0.0015 30.53 0.0003 0.0021 30.52 0.0015 0.0007 30.52
testB: 0.0006 30.52 0.0003 0.0017 30.52 0.0006 0.0017 30.52
test C: 0.0006 30.55 0.0003 0.0019 30.54 0.0006 0.0017 30.54
Total: 0.0010 30.54 0.0003 0.0018 30.52 0.0010 0.0014 30.53
Table 6-5: Crossply-undamped damped dimensionless damping parameter and natrnal
flequency values values.

One mode approx. two mode-one damp. approx. two mode-two damp. approx.
medez

Case medal freq. medal mcdez flu.
testA: 0.0012 36.73 0.0013 0.0079 36.73 0.0012 0.0088 36.73

test B: 0.0005 36.74 0.0011 0.0067 36.74 0.0005 0.0090 36.74
test C: 0.0007 36.76 0.0012 0.0076 36.74 0.0007 0.0089 36.74
Total: 0.0011 36.73 0.0012 0.0076 36.73 0.0010 0.0089 36.73
Table 6-6: Unidirectional-undamped damped dimensionless damping parameter and natru'al
flequency values values.

In general, the computed natural frequency values were very consistent for both
excitation methods. The largest standard deviation of the 0 - 800 Hz specimens was 0.0057
for the two mode-one damping parameter approximation of the unidirectional- generally
damped specimen. The crossply specimens had larger natural flequencies than
unidirectional specimens for the three damping strategies considered and the all
carbon/epoxy specimens had larger natural flequency values than the aluminum 6061
specimen.

The dimensionless damping parameter values of the optimally damped specimens were
not consistent in value. This lack of commonality was noted for the unidirectional-optimally
damped dimensionless damping parameters that vary by two orders of magnitude for
different test samples of a damping strategy, and optimally damped parameter values of zero

which indicate the physically impossible condition of no material damping. The inconsistent

50

dimensionless damping parameter values for the optimally damped strategy may be
attributed to modelling a discontinuous arrangement of damping strips using a model which
assumes damping is continuous along the length of the specimen. Since the dimensionless
damping parameter values for the optimally damped strategy were scattered, they will not be
considered in the ensuing discussion of experimental results.

Comparison of generally damped and undamped specimen’s dimensionless damping
parameter values indicates the first mode damping of the two mode-two damping parameter
approximation is nearly equal to the dimensionless damping parameter value of the one
mode approximation; these similar values are different flom the dimensionless damping
parameter value of the two mode-one damping parameter approximation. This difference is
anticipated since the two mode-one damping parameter approximation accounts for the
effects of the first and second mode damping whereas the other parameters exclusively
consider only the damping of the fundamental. Unidirectional second mode dimensionless
dampin g parameter of the two mode-two damping parameter approximation had larger
values than the corresponding parameter for the crossply specimens.

An increase in the dimensionless damping parameter value was recognized between the
undamped specimens and the generally damped specimens. This enhancement of material
damping was noted for each test of a specimen and the consistency of damping
improvement is displayed in Table 6-7. The damping enhancement ratios listed are the
quotient of dividing the average generally damped value by an average undamped value.
Average values are the mean of the three test sets and the total case for each damping
strategy. Dimensionless damping parameters of the crossply laminates achieved a factor of

eight improvement while the unidirectional laminates recognized approximately a 2.75 factor

of improvement.
one mode approx. two mode approx.
rimmed: rimmed:
Crossply: 8.63 8.66 8.47
Unidirectional: 2.65 2.84 3.20 ,

Table 6-7: Damping enhancement ratios.

51

Specimens excited about their fourth mode natural flequencies in general had consistent
dimensionless damping parameter values for the one, two, and three mode approximations.
Values corresponding to fourth mode damping of the four mode approximation were
scattered for aluminum and larger than anticipated for the crossply-generally damped
specimen. The inconsistency of the foruth mode values may be attributed to mathematical
difficulties the minimization algorithm encounter during the optimization process or an
insufficient amount of input energy occurs at that frequency. Dimensionless damping
parameter results for the one mode and two mode approximations for the composite
specimen are close in value to the values obtained when the same specimen was excited
using the 0 - 800 Hz band. The first mode dimensionless damping parameter values for the
aluminum specimen were scattered, this scatter may result flom their small magnitudes.

Amnmmmnmcdeldamn modem modem medeidamn Mm
One Mode 0.0000 27. 25
Two Mode 0.0001 0. 00376 - - 27. 21
Three Mode 0.0001 0. 00401 0.00045 - 27.22
Four Mode 0.0000 0.00401 0.00046 2.7280 27 .22

Table 6-8: Aluminum 6061 modal dimensionless damping parameter values and natural
flequency values for one, two three, and four mode approximations; test

case A.
Wm madam mum mcdeidamn Menu
One Mode 0.0005 27.24
Two Mode 0.0007 0. 0041 - 27.20
Three Mode 0.0007 0. 0041 0.0039 - 27.20
Forn' Mode 0.0006 0.0041 0.0039 21.295 27.20

Table 6-9: Aluminum 6061 modal dimensionless damping parameter values and natural
frequency values for one, two three, and four mode approximations; test
case B.

mem McAdam

One Mode 0 27.30
Two Mode 0 0. 0008 - 27.30
Three Mode 0 0.0037 0.0017 - 27.30
Forn' Mode 0.0000 0.0040 0.0019 0.1326 27.27

Table 6-10: Aluminum 6061 modal dimensionless damping parameter values and natural
flequéncy values for one, two three, and four mode approximations; test
case .

52

AmnmanonmedeLdamn modem madeldamn madeAdamn 1112111211911
One Mode 0.0000 27.25
Two Mode 0.0000 0. 0040 - 27.21
Three Mode 0.0003 0. 0040 0.0020 - 27.22
Four Mode 0.0002 0.0040 0.0021 0.0061 27.23

Table 6-11: Aluminum 6061 modal dimensionless damping parameter values and natural
flequency values for one, two three, and four mode approximations; total test
case.

AmrimatinnmdLLdamn madame medeldamn medeidanm

One Mode 0.0082 - - - 31.66
Two Mode 0.0082 0.0124 - - 31.64
Three Mode 0.0073 0.0145 0.0044 - 31.82
Four Mode 0.0073 0.0145 0.0044 0.2526 31.81

Table 6-12: Crossply-generally damped modal dimensionless damping parameter values
and natural flequency values for one, two three, and four mode
approximations; test Case A.

Ammnmafinnmujamn madam mm mm

One Mode 0.0073 31.69
Two Mode 0.0073 0.0123 - - 31.68
Three Mode 0.0073 0. 0169 0.0044 - 31.81
Forn' Mode 0.0073 0.0169 0.0044 0.2526 31.81

Table 6-13: Crossply-generally damped modal dimensionless damping parameter values
and natural flequency values for one, two three, and four mode
approximations; test Case B

Mammary mum mum madam fmnm
One Mode 0. 0081 31.69
Two Mode 0. 0081 0.0120 - 31.68
Three Mode 0.0079 0. 0112 0.0044 - 31.76
Four Mode 0.0079 0.0112 0.0044 0.2526 31.76

Table 6-14: Crossply-generally damped modal dimensionless damping parameter values and
natural flequency values for one, two three, and four mode approximations;

test Case C.
mainstream modem madeidamn mcdeidamn Mum
One Mode 0.0079 31. 68
Two Mode 0.0079 0.0122 - 31. 67
Three Mode 0.0075 0.0143 0.0044 - 31.79
Four Mode 0.0075 0.0143 0.0045 0.1902 31.79

Table 6-15: Crossply- generally damped modal dimensionless damping parameter values
and natural flequency values for one, two three, and four mode
approximations; total test case.

6.4.2 Parameter Magnitude
The optimization algorithm is designed to obtain a minimum for the modal
approximation relation by pertrnbing the parameters such that the modal approximation error
is reduced. The algorithm’s ability to correctly identify parameter values is related to the

53

magnitude of the individual parameter. It was observed the larger the damping parameter’s
magnitude the less effect a perturbation, necessary to reduce modal approximation error
values, had upon the time response. This is evident when comparing the damping
parameter values listed in Tables 6-8 through 6-15 for the crossply- generally damped and
aluminum 6061 specimens excited about their fourth modes. The crossply- generally
damped damping parameters are typically two orders of magnitude larger than those for
aluminum. Damping parameters for the crossply-generally damped specimens have
consistent magnitudes whereas the aluminum damping parameter magnitudes are quite
scattered. This trend of better consistency in identification of damping parameter values for
specimens having larger parameter magnitudes is also demonstrated in the six specimens

excited over the 0-800 Hz band.

6.4.3 Analysis of Time Responses
Minimal deviation between the measured and predicted traces creates difficulty in
discerning the quality of the predicted responses. Qualification of predicted traces is
measured via computation of the square of the 2-norm of the difference between the
measured and predicted data, Be, for the modal approximations. These values were
computed via integration over the normalized beam length according to the equations below.

1
E6 = {I}!!!I - yplzdt (6.7)

|(y..)|’dt (48)

En:

°_.~

The values recorded in Tables 6-16 through 6-19 correspond to the specimens excited
over the 0 - 800 Hz input band. The 2-norm of the error for the two mode approximations
defined with two damping parameters was always less than the norm of the error for the two

mode-one damping parameter approximations excluding the total cases of specimens:

54

crossply- generally damped, unidirectional-generally-damped, and crossply-optimally
damped. The smaller error magnitudes for the two mode-two damping parameter case
indicates specimen damping is best modeled using modal damping. It is also an indication
that the presumed damping mechanism used in the transfer function derivation is not very
representative for multimode approximations. For the multimode models an independent
damping parameter was determined for each mode and. excellent agreement with the
experimental measurements was observed. In general, error signal magnitudes are

approximately 3% of the total measured signal.

Saarinen mm 1:513. terrLC Incl
Crossply-general damp. 0.2793 0.3507 0.4133 1.0433
Unidirect-general damp. 0.3696 0.4153 0.0861 0.8710
Crossply—optimal damp 0.7240 0.3553 1.1397 2.2190
Unidirect-optimal damp. 0.5895 0.4570 0.5441 1.5906
Crossply-undamped 1.5818 0.5925 1.5865 3.7608
Unidirect-undamped 0.6063 0.2101 0.3141 1.5906
Table 6-16: 2-norm of measured displacement signal.
8mm testA term tests: Total
Crossply-general damp. 0.0049 0.0049 0.0043 0.0143
Unidirect-general damp. 0.0051 0.0031 0.0066 0.0158
Crossply-optimal damp 0.0192 0.0216 0.0184 0.0642
Unidirect timal 0.0075 0.0034 0.0149 0.0297
Crossply-undamped 0.0251 0.0262 0.0496 0.1018
Unidirect— 0.0051 0.0045 0.0064 0.0164
Table 6-17: 2—norm of one mode approximation error signal.
Saarinen term can rests: Imal
Crossply-general damp. 0.0029 0.0026 0.0027 0.0082
Unidirect-general damp. 0.0048 0.0025 0.0053 0.0151
Crossply-optimal damp. 0.0056 0.0106 0.0057 0.0251
Unidirect-optimal damp 0.0066 0.0020 0.0142 0.0269
Crossply-undamped 0.0059 0.0036 0.0275 0.0378
Unidirect- 0.0021 0.0030 0.0017 0.0069
Table 6-18: 2-norm of two mode-one damping parameter approximation error signal.
Specimen term 1251.3. rests Incl
Crossply-general damp. 0.0013 0.0011 0.0016 0.2178
Unidirect-general damp. 0.0045 0.0022 0.0052 0.0125
Crossply-optimal damp 0.0056 0.0105 0.0057 0.0243
Unidirect-optimal damp. 0.0066 0.0016 0.0141 0.0252
Crossply-undamped 0.0035 0.0036 0.0274 0.0358
Unidirect-undamped 0.0021 0.0029 0.0015 0.0474

Table 6-19: 2-norm of two mode-two damping parameter approximation error signal.

55

The values recorded in Tables 6-20 and 6-21 are the 2-norm of the measured flee end
displacement and the error between the predicted and measured displacements for the
specimens excited about their fourth mode natural frequencies. The 2-norm magnitudes for
the crossply-generally damped specimens have a consistently larger measured signal than
the aluminum specimens; these greater magnitude values are consistent with the larger peak
to peak displacements on the crossply-generally damped traces of Appendix F.

The 2-norm error between the measured and predicted signals for the two mode, three
mode, and four mode approximations of the crossply-generally damped specimen were all
less than 1% of the measured flee end displacement. The small error percentages indicate
the modal approximations predicted the free end displacement with a high degree of
accuracy. One mode approximations for the crossply- generally damped specimen had 2-
norm error values that were approximately 65% of the measured flee end signal. These
large error signal percentages were evident in the traces for the one mode approximation.
The inability of the one mode approximation to predict flee end displacement may be the
result of insufficient stimulation of the fundamental. The aluminum specimen also
experienced large 2-norm error values for the one mode approximation. The percentage of
enor was approximately 20% of the measured flee end displacement.

Approximation Error (Predicted - Measured) measured

memede unmade mm mm signal
testA: 0.03430 0.00048 0.00048 0.00048 0.05343
test B: 0.05733 0.00038 0.00039 0.00038 0.08472
test C: 0.06566 ~ 0.00078 0.00077 0.00077 0.09101
Table 6-20: 2-norm of measured flee end displacement and the enor between predicted
flee end displacements and measrned flee end displacements, crossply-

generally damped specimen.

Approximation Error (Predicted - Measrned) measured

canned: married: dimmed: featured: signal
test A: 0.00100 0.00021 0.00028 0.00033 0.00869
test B: 0.00176 0.00049 0.00051 0.00051 0.00902
test C: 0.00136 0.00071 0.00072 0.00105 0.00553
Table 6-21: 2-norm of measured flee end displacement and the error between predicted
flee end displacements and measured flee end displacements, aluminum 6061

specrmen.

56

6.4.4 Damping Parameter Convergence To Zero

The damping parameter converged to a value of zero for two test cases. The physical
impossibility of damping parameters being equal to zero indicates either an error in the
minimization process or a problem with the data set. Successive iterations of these test
cases with varying initial conditions each converged to damping parameter values of zero.
Both test cases afflicted with this condition were optimally damped specimens. The
discontinuous nature of the optimal damping strategy may cause the laminate damping to
vary greatly flom the continuous material damping assumption of the modal response
equation. Zero values were converged upon for test case ‘B’ of the crossply-optimally
damped specimen and test case ‘A’ of the unidirectional-optimally damped specimen.
Damping parameters for the two mode crossply—optimally damped approximations did not
converge to zero. ’

Traces of scaled flee end displacements for the two specimens having zero valued
damping parameters were evaluated. Displacements for the unidirectional-optimally damped
specimen appear to be consistent for its three test cases. Displacements had similar peak to
peak displacements and minimal visual variations in their higher mode effects. Analysis of
the three crossply-optimally damped data sets exhibited a unique tendency for the test case
whose first mode damping value converged to zero. Data sets ‘A’ and ‘C’ had
approximately one half of their displacement signals dominated by the oscillatory motion of
the fundamental mode. Data set ‘B’ did not have a region of large first mode oscillations.
This lack of first mode oscillation may have caused the first mode damping parameter value

for test data set ‘B ’ to converge to zero while both of its two mode approximations had

positive damping parameters.

57

W

7.1 Conclusions

1) A general arrangement of viscoelastic material at three ply interfaces symmetric to the
midplane increases the dimensionless damping parameter values of a carbon/epoxy laminate
by a factor of eight for a crossply layup and an approximate factor of three for a

unidirectional layup.

2) Crossply carbon/epoxy laminates exhibit larger dimensionless damping parameter
values than unidirectional carbon/epoxy laminates of similar geometric dimensions.

3) Prediction consistency of dimensionless damping parameter can be improved by
increasing the number of modes included in the modal response relation.

4) Prediction of damping characteristic is improved by using multiple damping
parameters to define modal damping for modes other than the fundamental mode.

5) Parameter values for the optimally damped strategy were not consistent in
magnitude, this may be a result of the discontinuous arrangement of their damping material

and the use of a model that assumes continuous damping.

7.2 Recommendations

The study at hand serves as only an introduction into the realm of enhancing the internal
damping characteristics of composite laminates. It can also serve as a springboard flom
which numerous areas of study may be explored. Additional areas of study can expand
upon the amount and placement of the viscoelastic damping material, variation in the type or
thickness of the damping material, combination of different ply layup orientations to

determine the maximum damping associated with a ply layup strategy.

58

Additionally, investigation of the laminate mechanical properties of toughness, tensile
strength, and fatigue life as a function of the amount and location of viscoelastic material at
ply interfaces would provide information crucial for determining the practical utility of this
model.

Ultrasonic analysis of laminates can be performed to determine if voids, inclusions or
delamination occur within a laminate; such fabrication induced items increase material
dimensionless damping parameter value. Ultrasonic analysis was not performed on the
specimens used in this experimental analysis. Laminates can also be fabricated with voids
or regions of delamination purposely included to quantify their effects upon laminate

damping and strength properties.

59

REEERENQES

l: Magolan, A.M., “An Investigation on the Retrofitment of an Industrial Robot
Manipulator with a Composite Arm: Design, Analysis and Fabrication”, Master of Science
Thesis, Department of Mechanical Engineering, Michigan State University, East Lansing,
MI, 1989.

2: Rivin, E.I., “A Laminated Material for Impact Noise Abatement and its
Applications”, Proceedings of the Sixth Annual ASM/ESD Advanced Composites
Conference, Detroit, Mi, October 8-11, 1990, pp. 491-495.

3: Adams, R.D., “The Dynamic Longitudinal Shear Modulus and Damping of Carbon
Fibers”. WWW 8. PP- 733-743 (1975).

4: Suarez, 8., Gibson, R., Sun, C. and Chaturvedi, S., “The Influence of Fiber
Length and Fiber Orientation On Damping and Stiffness of Polymer Composite Materials”,

W 26 (2). pp. 175-184 (1986)-

5: Way, 8., Ashton, 1N. and El-Sobky, H., “An investigation of the Influence of ,
Anisotropy and Frequency on Damping in Short Glass Fibre Reinforced Polypropylene”,

W 15 (1). pp. 43-60 (1990).

6: Adams, RD. and Bacon, D.G., “Effect of Fibre Orientation and Laminate

Geometry on the Dynamic Properties of CFRP”, WW5, 7 (10),
pp. 402-428 (1973).

7: Andersen, S.M., Jackson, G B. and Vinson, J.R., “Effects of Fiber Orientation on
Material Damping In Beams of Composite Materials”, Proceedings of the Sixth Annual
ASM/ESD Advanced Composites Conference, Detroit, Mi, October 8-11, 1990, pp. 471-
475.

8: Liao, D.X., Sung, GK. and Thompson, BS, “The Optimal Design of Symmetric

Laminated Beams Considering Damping”, W 20 (9), pp.
485-500 (1986).

9: Moser, K. and Lunassegger, M., “Increasing the Damping of Flexural Vibrations of
Laminated FPC Structures by Incorporation of Soft Intermediate Plies with Minimum

Reduction of Stiffness”, Wm 10(4), pp. 321-334 (1985).

10: Sun, C.T., Sankar, B.V. and Roa, V.S., “Damping and Vibration Control of
Unidirectional Composite Laminates Using Add-on Viscoelastic Materials”, J W

Seundandlihmticn. 139(2). pp. 277-287 (1990)-

ll: Lifshitz, J. and Leibowitz, M.,”Optimal Sandwich Beam Design for Maximum

Viscoelastic Darnping”. WW 23 (7). PP- 1027-
1034 (1987).

12: Lall, A. K., Asnani, N. T. and Nakra, B. C., “Vibration and Damping Analysis of
Rectangular Plate With Partially Covered Constrained Viscoelastic Layer”, W

W 109 (7). PP 241-247 (1987)

13: Mantena, PR. and Gibson, R.F., “Constrained Layer Damping Treatments for
Vibration Control of Composite Structural Elements”, Proceedings of the Sixth Annual
ASM/ESD Advanced Composites Conference, Detroit, Mi, October 8-11, 1990, pp. 3-14.

14: Plunkett, R. and Lee, C.T., “Length Optimization for Constrained Viscoelastic
Layer Darnping”. WWW 48. pp- 150-161 (1970).

15: Q: Sun, C.T., Sankar, B.V. and Rao, V.S., “Damping and Vibration Control of
Unidirectional Composite Laminates Using Add-On Viscoelastic Materials”, W

Saundnnflihrancn. 139 (2). pp. 277-287 (1990).

16: Sun, C., Wu, J. and Gibson, R., “Preditcion of Material Damping of Laminated
Polymer Composite”. IcumalnLMatenalaScience 22. pp. 1006-1012 (1987).

17: Lin, D.X., Ni, RC. and Adams, R.D., “Predictions and Measurement of the
Vibrational Damping Parameters of Carbon and Glass Fibre-Reinforced Plastics Plates”,

Walt. 18 (3). pp. 132-152 (1984).

18: Ni, R. and Adams, R., “A Rational Method for Obtaining the Dynamic Mechanical
Properties of Laminae for Predicting the Stiffness and Damping of Laminated Plates and

BeamS”. Commuter. 15 (3). 99- 193-199 (1984).

19: Ni, R. and Adams, R., “The Damping and Dynamic Moduli of Symmetric
Laminated Composite Beams-Thoeretical and Experimental Results”, W
M, 18 (3). Pp. 104-121 (1984).

61

20: Lee, C.Y., Pfeifer, M., Thompson, B. and Gandhi, M., “The Characterization of
Elastic Moduli and Damping Capacities of Graphite [epoxy Composite Laminated Beams in
Hygrothermal Environments”, MW. 23 (8), pp. 819-845
(1989).

21: Ni, R.G., Lin, D.X. and Adams, R.D., “The Dynamic Properties of Carbon/Glass
Fibre Sandwich-Laminated Composites: Theoretical, Experimental and Economic
Considerations”, qupgsites, 15 (4), pp. 297-304 (1984).

22: Crane, RM. and Gillespie Jr., J .W., “Effect of Vibration Amplitude and Frequency
on the Vibration Damping Loss Factor of Glass/Epoxy Composites”, Proceedings of the
Sixth Annual ASM/ESD Advanced Composites Conference, Detroit, Mi, October 8-11,
1990, pp.461-470.

23: Adams, R. D, “Damping Properties Analysis of Composites”, ASMJnmman'Qnal
-- - - . . - pp.,206-217 1988.

 

24: Sun, C.T., Sankar, B.V. and Rao, V.S., “Damping and Vibration Control of
Unidirectional Composite Laminates Using Add-On Viscoelastic Materials”, W
Scuniandlrbrancn 139(2) 99- 277-237 (1990).

25: Rudin, A., - - - ‘ ,_ .
W Academic Press. Inc.. 1982

26: Sanjana, S.N., “Average Indications for Prepreg Products”, W,
25(1), pp. 5 - 11 (1980).

 

27: Akay, M., “Effects of Prepreg Ageing and Post Cure Hygrothermal Conditioning
on Mechanical Behaviour of Carbon-Fibcr/Epoxy Laminates”, W

manning. 38(4). pp- 359-370 (1990).

28: Johnson, R., 3M Corporation, telephone conversation June 28, 1990.

29: Gibson, Ronald E, “Damping Characteristics of Composite Materials and
Structures”, Proceedings of the Sixth Annual ASM/ESD Advanced Composites Conference,
Detroit, Mi, October 8-11, 1990, pp. 441-449.

62

30: Adams, R. D. and Bacon, D. C. “Measurement of the flexural damping capacity
and dynamic Young’s modulus of metals and reinforced plastics’, W

131212181211er 6. pp- 27-41 (1972).

31: Gibson, RF. and Plunkett, R., “A Forced-vibration Technique for Measurement of
Material Dimming”. merimentaLMechanics 17(8) PP- 297-302 (1977).

32: Gibson, R., “Dynamic Mechanical Behavior of Fiber-Reinforced Composites:
Measurement and Analysis”, W, 10 (10), pp. 325-341 (1976).

63

Appendix A

General eigensolution matrix.

The following two relations will be used to solve (3-19).

ea + ’1’ = e"(cosb + jsinb) and e' - ’1’ = e‘(cosb — jsinb)

Rewrite (3-18):

Det [F] = 0 = 4+e'a+m+e'(l"j)"+ea'j)x+ea+’0)"
Combination of (A-1) and (A-2) produce:
4 + e"(cos(—b) + jsin(-b)) + e"(cos(b) — jsin(b)) +
e‘(cos(-b) - jsin(—b)) + e‘(cos(b) + jsin(b)) = 0
note: sin(b) = —sin(-b) and cos(b) = cos(-b)
Equation (A-4) allows consolidation of (A-3) as follows:

4 + e’“(2 cosb) + e"(2 cos b) = 0
4 + Zeosb[e_a + ea] = 0

e" + e“

recall: = cosha

 

4+4cosbcosha = 0
recall: a = b = 3.
4+4coshcosh71 = 0

Thus: coslicosh 1., = —l

(A-l)

(A-2)

(A-3)

(A-4)

(A-5)

Appendix B

Values of variables used in chapters are listed below:

51. =3”, = saw-(0.0148...) =0.1128m

r = 2.9738 x10'12m4

P(w'h) 1 = 1.232 x 10-2 kg

9,-(1)
-1.960669
+1.863479

-1.776306

W
specimen length a l = 28 an
specimen base a b = 2.54 cm
specimen height
material density a P = 1.55 Slcm3
laminae thickness 5 t1 = 0.014cm
Young’s modulus E E = 1.2609 x 10+11k3/s2m
area moment of inertia E
specimen mass 5 m =
PER MQDE games
mode eigen- normalized
am 28111:. £21 £22
1 1.875100 1.000007 -0.734105
2 4.694090 0.999985 -1.018454
3 7 .854760 0.999959 -0.999183
4 10.995541 1.023346 -1.023349

+0.927019

21.
0.765639
0.468193
0.230822

0.289813

Ki
-1.501166
+0.872468
-0.410010

+0.268662

65

Appendix C

Solving for the non-dimensional damping parameter in terms of material and geometric
properties.
Compare the following two equations:

i1 + P139191 + Paxiq = 0 (C‘l)

'y' + (25%),» + (w:)y = o (02)
Relations (C-3a) and (C-3b) can be obtained flom (C—1) and (C-2) by equating the

coefficients for the terms with similar derivatives.

2W. = 11.27»? (a) w: = p.15. (b) (03)
Substitute [H-3b] into [Ii-3a]:
2: P3)”: = 131.291 (C'4)

We will solve (G4) to obtain a relation for the damping ratio, §, based upon the known
values: eigenvalue, 1... Young’s modulus, E, area moment of inertia, I, laminate density, p,

and specimen cross sectional area, a.

 

12.
£3. = 2:71): (CI-5)
3

Appendix D

Magnitude and Phase angle plots for varying values of the dimensionless damping

parameter.

50 1 I I rtvvvv I 1 f

 

 

 

magnrtude (dB)
o
l

 

 

 

_50 1 1 1 1A 1 1 1 1 .1
100 101 102 103

fmquency (HZ)

O 1 r r vvt1tr X1 VjUIIW U U vrr

 

 

 

phase (degrees)

 

m 1 1 1 1 1 1 1 11 1 1 1 1
100 101 102 103
flequency (Hz)

Figure D-l: Dimensionless damping parameter of the first mode = 0.025.

67

 

VIVVUI V V 7 '1 IV 1 I VIII!
A _.

-50 . ......A . .....
100 101 102 103
flequency (Hz)

 

 

magnitude (dB)
0

 

 

 

 

 

 

 

 

’1"? O IUI\TVIUIIYI f

fi-2oo- _

0

E

g._400 . ........ . . .-.-..
100 101 102 103

flequency (Hz)
Figure D-2: Dimensionless damping parameter of the first mode = 0.050:

 

50 T T f 7711—77—77 T Y r Vfitfirji V U fi'

 

 

 

magmtude (dB)
:3
h f

 

_50 1L1. 1L1. m A
100 101 102 103

 

 

 

4m 1 . 1 14-1111
100 101 102 103
flequency (Hz)
Figure D-3: Dimensionless damping parameter of the first mode = 0.075.

 

phase (degrees)
1'6
8

 

68

 

 

 

magnitude (dB)

 

 

 

 

100 101 102
fi'equency (Hz)

103

 

 

 

phase (degrees)

 

 

 

100 I I ‘ ' ‘ I 161 I I 102
frequency (Hz)
Figure D-4: Dimensionless damping parameter of the first mode = 0.100.

103

 

20 t 1 Yivvvvr t fiIW'I'II

o f\

 

 

-20

I

magnitude (dB)

 

 

 

_40 . 1 1 g111..11
100 101 102

fmquency (HZ)

103

 

 

phase (degrees)

 

 

 

100 I I ‘11‘011 ' ”H.162
frequency (Hz)
Figure D—5: Dimensionless damping parameter of the first mode = 0.200.

69

Appendix E

Traces of predicted and measured free end displacement for the six clamped-free specimens
for the one mode approximation, the two mode-one damping parameter approximation and
the two mode-two damping parameter approximation. All data is for the 0 - 800 Hz band of

random excitation at the clamped end.

Crossply ' General Damping (0116 mode approx. - test "a")

 

 

 

 

3
A 2 '- q
% 1 .- d
if 0‘ -
5
E
§ '1 ' ‘
.9?
“c _2 _ _

_3 . . 1 _1

0 0.2 0.4 0.6 0.8 1

time (seconds)

displacement (millimeters)

displacement (mllllme' ' ters)

70

Crossply - General Damping (one mode approx. - test "b")

 

 

 

02 04 0:6 0.8
time (seconds)

Crossply - General Damping (one mode approx. - test "c")

 

 

 

02 0.4 06 0.8
time (seconds)

 

 

displacement (millimeters)

displacement (millimeters)

71

Crossply - General Damping (two mode-one damp approx. - test "a")

 

 

 

 

 

0 0:2 0:4 0:6 0:8
time (seconds)

Crossply - General Damping (two mode—one damp approx. - test "b")

 

 

 

 

 

0 02 (f4 0.6 0.8
time (seconds)

displacement (millimeters)

displacement (millimeters)

72

 

CTOSSPIY ' General Damping (two mode-one damp approx. - test "c")

 

 

 

0 012 0.4 0:6 0:8
time (seconds)

Crossply ' General Damping (tWO mode-two damp approx. - test "a")

 

 

 

 

0 0:2 04 0:6 0:8
time (seconds)

displacement (millimeters)

displacement (millimeters)

73

Crossply - General Damping (two mode-two damp approx. - test "b")

 

 

 

 

 

0 02 04 0.6 0.8
time (seconds)

Crossply ' General Damping (two mode-two damp approx. - test "c")

 

 

 

 

0 0.2 04 0.6 03
time (seconds)

displacement (millimeters)

displacement (millimeters)

74

Unidirect - General Damping (one mode approx. - test "a")

 

 

 

 

02 04 0.6 0:8 1
time (seconds) '

Unidirect - General Damping (one mode approx. - test "b")

 

 

 

 

0.2 0.4 0.6 0:8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

75

Unidirect - General Damping (one mode approx. - test "c")

 

 

 

 

0 0:2 0:4 0:6 0:8 1
I time (seconds)

 

Unidirect - General Damping (one mode-one damp approx. - test "a")

 

 

 

0 0:2 0.4 0:6 0:8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

76

Unidirect - General Damping (one mode-one damp approx. - test "b")

 

 

 

 

0 0:2 04 0.6 0,3 1
time (seconds)

Unidirect - General Damping (one mode-one damp approx. - test "c")

 

 

 

 

0 0.2 0.4 0.6 03; 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

77

Unidirect - General Damping (one mode-two damp approx. - test "a")

 

 

 

 

0 0:2 0:4 0:6 0:8 1

time (seconds)

Unidirect - General Damping (one mode-two damp approx. - test "b")

 

 

 

 

0 0.2 0:4 0:6 0:8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

78

Unidirect - General Damping (one mode-two damp approx. - test "c")

 

 

 

0 0:2 0:4 0.?
time (seconds)

0.8

Crossply ' Optimal Damping (one mode approx. - test "a")

 

 

 

0 0:2 0,4 0:6
time (seconds)

08

 

 

displacement (millimeters)

displacement (millimeters)

79

Crossply - Optimal Damping (one mode approx. - test "b")

 

 

 

0 0.2 0T4 0:6 0.8 1
time (seconds)

Crossply - Optimal Damping (one mode approx. - test "c")

 

 

 

 

 

 

 

 

 

0 0:2 04 06 0:8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

3 ,

 

Crossply - Optimal Damping (two mode-one damp approx. - test "a")

 

80

 

 

Crossply - Optimal Damping (two mode-one damp approx. - test "b")

02

0.4

06

time (seconds)

0.8

 

0.2

04

0:6

time (seconds)

0:8

 

 

 

displacement (millimeters)

displacement (millimeters)

81

Crossply - Optimal Damping (two mode-one damp approx. - test "c")

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.2 04 0:6 0:8
time (seconds)

3 Crossply - Optimal Damping (two mode-two damp approx. — test "a")

 

 

 

 

 

0 02 0:4 06 0.8
time (seconds)

displacement (millimeters)

displacement (millimeters)

82

Crossply - Optimal Damping (two mode-two damp approx. - test "b")

 

 

 

 

 

 

 

 

 

 

 

 

3
2 _
1 -
O " ‘ ‘ ‘ I '4
1 r . '
-1 _ ..
-2 _ 1
-3 1 m m 1
0 0.2 0.4 0.6 0.8 1
time (seconds)
3 Crossply - Optimal Damping (two mode-two damp approx. - test "c")
2 - 1
1 - 1
o .
-1 I .
-2 _ -
-3 . . 1 1
0 0.2 0.4 0.6 0.8 1

time (seconds)

displacement (millimeters)

displacement (millimeters)

83

Unidirect - Optimal Damping (one mode approx. - test "a")

 

 

 

 

 

0 012 0.4 (K 0:8

time (seconds)

Unidirect - Optimal Damping (one mode approx. - test "b")

 

 

 

 

 

0 0.2 0.4 06 08
time (seconds)

displacement (millimeters)

displacement (millimeters)

84

Unidirect - Optimal Damping (one mode approx. - test "c")

 

 

 

 

 

 

 

 

 

0 0:2 04 0.6 0:8
time (seconds)

Unidirect - Optimal Damping (two mode-one damp approx. - test "a")

 

 

 

 

0 0.2 0.4 06 08
time (seconds)

displacement (millimeters)

displacement (millimeters)

85

Unidirect - Optimal Damping (two mode-one damp approx. - test "b")

 

 

 

 

 

0 02 0.4 0:6 03
time (seconds)

Unidirect - Optimal Damping (two mode-one damp approx. - test "c")

 

 

 

 

 

 

0 02 0.4 0.6 0.8
time (seconds)

86

Unidirect - Optimal Damping (two mode-two damp approx. - test "a")

 

 

diSplacement (millimeters)
o

 

 

 

-1 _. -
-2 1. '1
-3 1 . 1 .
0 0.2 0.4 0.6 0.8 1
time (seconds)

Unidirect - Optimal Damping (two mode-two damp approx. - test "b")

 

 

diSplacement (millimeters)
o

 

 

 

-1 _ -
-2 _ -
-3 . 1 1 ,

0 0.2 0.4 0.6 0.8 1

time (seconds)

displacement (millimeters)

diSplacement (millimeters)

87

 

 

 

 

 

Unidirect - Optimal Damping (two mode-two damp approx. - test "c")

 

 

0 0: 2 0.4 0T6
time (seconds)

Crossply - Undamped (one mode approx.

08

- test "a")

 

 

 

 

 

 

 

 

 

 

0 02 0.4 0:6
time (seconds)

0.8

p—n

displacement (millimeters)

displacement (millimeters)

88

Crossply - Undamped (one mode approx. - test "b")

 

 

 

 

 

0 0.2 0.4 06 0.8 1
time (seconds)

Crossply - Undamped (one mode approx. - test "c")

1 .. 11W-

 

 

 

O
r
a.-
“is
l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0:2 0:4 0:6 0:8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

89

Crossply - Undamped (two mode approx. - test "a")

 

 

“ 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0:2 04 0.6 08 1
time (seconds)

Crossply — Undamped (two mode approx. - test "b")

 

 

 

 

0.2 014 0.6 0.8 1
time (seconds)

 

displacement (millimeters)

displacement (millimeters)

90

Crossply - Undamped (two mode approx. - test "0")

 

 

 

1M1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2 0.4 0.6 0.8
time (seconds)

Crossply - Undamped (two mode-two damp approx. - test "a")

 

 

 

 

 

 

 

 

 

 

 

L

 

 

0.2 0.4 0.6 0.8
time (seconds)

 

displacement (millimeters)

displacement (millimeters)

91

Crossply - Undamped (two mode-two damp approx. - test "b")

 

 

 

 

0 032 0.4 06 0:8 1
time (seconds)

 

Crossply - Undamped (two mode-two damp approx. - test "c")

1- all

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 012 04 0:6 0:8
time (seconds)

displacement (millimeters)

displacement (millimeters)

92

 

 

Unidirect - Undamped (one mode approx. - test "a")

 

 

 

0.2 0.4 0.6 0.8 1
time (seconds)

Unidirect - Undamped (one mode approx. - test "b")

 

 

 

 

0.2 04 06 0.8 1
time (seconds)

displacement (millimeters)

displacement (millimeters)

93

Unidirect - Undamped (one mode approx. - test "c")

 

 

 

 

0 02 04 0.6 0:8
time (seconds)

 

Unidirect - Undamped (two mode-one damp approx. - test "a")

 

 

 

0 0:2 04 0.6 0.3
time (seconds)

displacement (millimeters)

displacement (millimeters)

94

Unidirect - Undamped (two mode-one damp approx. - test "b")

 

 

 

 

0 02 0.4 0.6 0.8
time (seconds)

Unidirect - Undamped (two mode-one damp approx. - test "c")

 

 

 

 

0 0.2 0.4 06 0.8

time (seconds)

displacement (millimeters)

displacement (millimeters)

95

Unidirect - Undamped (two mode-two damp approx. - test "a")

 

 

 

0.2 04 0:6 018
time (seconds)

Unidirect - Undamped (two mode—two damp approx. - test "b")

 

 

 

0.2 04 0:6 08
time (seconds)

 

 

displacement (millimeters)

96

 

Unidirect - Undamped (two mode-two damp approx. - test "c")

 

 

 

0 0:2 0.4 0:6 0:8
time (seconds)

97

Appendix F

Time versus displacement traces for the crossply-generally damped and the aluminum 6061
specimens excited about their fourth mode resonant frequencies.

 

 

 

 

 

l r
E 0.5 - 1
33'
% § 1 I h. I, ' I | ..
t: O 1 f b ‘1. ’ ' \ t n I I ‘ J l I
g I 1‘ ‘3 J ‘ at ' ‘ l1 1 1’
0
E
O:
m ~0.5- -
‘5

-1 . 1 . 1

0 0.2 0.4 0.6 0.8 1

time (seconds)

Figure F- 1: Crossply-generally damped one mode approximation, test A.

98

 

0.5 "' I I

-0.5 -

displacement (millimeters)
o

 

 

 

0 02 0.4 0.6 0:8
time (seconds)

Figure F-2: Crossply- generally damped two mode approximation, test A.

 

.9
{It
I
.-

displacement (millimeters)
.<'>
u. o

 

 

 

 

0 0.2 0.4 0:6 08
time (seconds)

Figure F-3: Crossply- generally damped three mode approximation, test A.

99

 

 

 

 

 

 

 

 

 

1 m
E 0.5 ~ 1 ~
13’
.5
E
.5 0
§ 1
E r
at
m -0.5- —
=8
-1 . 1 1 1
0 0.2 0.4 0.6 0.8 1
time (seconds)
Figure F-4: Crossply-generally damped four mode approximation, test A.
1 1
g 0.5 -
E
E "1 a k‘ ‘1 I I1 ‘ 1‘ l' \l
E 01- ,1‘”: “' ‘1 1"“). {'1 Raina", ' t! ‘1‘ ‘lu’:
a ~ ' .,
0 I
.8.
81 -0.5 ~ -
=8
_1 1 1 1 1
0 0.2 0.4 0.6 0.8
time (seconds)

Figure F-5: Crossply- generally damped one mode approximation, test B.

100

 

0.5 -

 

 

displacement (millimeters)
o

 

 

 

0 02 04 0.6 0.8 1
time (seconds)

Figure F-6: Crossply- generally damped two mode approximation, test B.

 

0.5 - a

displacement (millimeters)
o

 

 

 

0 02 0.71 0:6 03 1
time (seconds)

Figure F-7: Crossply- generally damped three mode approximation, test B.

101

 

0.5 .

 

displacement (millimeters)
o

 

 

 

 

 

 

 

0 0:2 04 0:6 0.8 1
time (seconds)
Figure F-8: Crossply- generally damped four mode approximation, test B.

1
g; 0.51 .1
t: O I 1 ‘k r; I f “ ‘s ‘ ’ ’ s 'r I I) 1‘
5 1 J \ J s. s 1‘ \ , a,
g 1
.22
1% -O.5 — -
=6

_1 1 1 1 1
0 0.2 0.4 0.6 0.8 1

time (seconds)

Figure F-9: Crossply-generally damped one mode approximation, test C.

102

 

0.5 —

-0.5 ~

displacement (millimeters)
o

 

J

 

 

0 0:2 0:4 0:6 ' 0.8
time (seconds)

Figure F-10: Crossply-generally damped two mode approximation, test C.

 

0.5 ~—

displacement (millimeters)
o

 

 

 

0 0.2 0.4 0.6 0:8
time (seconds)

Figure F—l 1: Crossply- generally damped three mode approximation, test C.

103

 

0.5 r-

displacement (millimeters)
o

 

 

 

0 0.2 04 0:6 0:8
time (seconds)

Figure F-12: Crossply-generally damped four mode approximation, test C.

 

l

0.5

-0.5 —

displacement (millimeters)
o

 

 

 

0 02 0:4 06 0.8
time (seconds)

Figure F-l3:A1uminum 6061 one mode approximation, test A.

104

 

0.5

l

displacement (millimeters)
o

 

 

 

 

0 02 0:4 0:6 0:8
time (seconds)

Figure F—14:Aluminum 6061 two mode approximation, test A.

 

l

0.5

displacement (millimeters)
o

 

 

 

0 02 04 06 08
time (seconds)

Figure F-15:Aluminum 6061 three mode approximation, test A.

105

 

displacement (millimeters)
o

 

 

 

 

 

 

 

-1 . 1 .
0 0.2 0.4 0.6 0.8 1
time (seconds)
Figure F—16:Aluminum 6061 four mode approximation, test A.
1
g 0.5 - -
E .. I L, I ' I ‘ .l
T: 0 - ' '3
t: _ ' .
1%
E
9" - 1— -l
‘6 0.5
_1 . 1 a 1
0 0.2 0.4 0.6 0.8

time (seconds)

Figure F-17:Aluminum 6061 one mode approximation, test B.

 

106

 

0.5 .

displacement (millimeters)
o

 

 

 

0 0:2 0:4 0:6 0.8
time (seconds)

Figure F-18:Alurninum 6061 two mode approximation, test B.

 

0.5

T

displacement (millimeters)
o

 

 

 

0 02 0:4 0:6 08
time (seconds)

Figure F-19:Aluminum 6061 three mode approximation, test B.

107

 

displacement (millimeters)
o

 

 

 

0 0:2 04 06 0:8
time (seconds)

Figure F-20:Aluminum 6061 four mode approximation, test B.

 

displacement (millimeters)
o

 

 

 

0 012 0.4 0:6 0.8
time (seconds)

Figure F-leAluminum 6061 one mode approximation, test C.

108

 

0.5 -

displacement (millimeters)
o

 

 

0 02 0.4 (T6

time (seconds)

Figure F—22:Aluminum 6061 two mode approximation, test C.

0.8

 

I

0.5

displacement (millimeters)
o

 

 

0 012 04 06
time (seconds)

Figure F-23:Aluminum 6061 three mode approximation, test C.

08

 

 

 

109

 

0.5 —

displacement (millimeters)
o

 

 

 

 

0 02 0:4 0:6 08

time (seconds)

Figure F-24:Aluminum 6061 four mode approximation, test C.

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