FREE VIBRATION OF LAMINATED RECTANGULAR PLATES EMPLOYING
FINITE ELEMENT ANALYSIS
by
James Tyler Roach
B.S., University of Colorado at Denver, 1991
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
1995
1995 by James Tyler Roach
All rights reserved.
This thesis for the Master of Science
degree by
James Tyler Roach
has been approved for the
Graduate School
by
/kotM % ffl.T
( ' Date
Roach, James Tyler (M.S., Mechanical Engineering)
Free Vibrations of Laminated Rectangular Plates Employing
Finite Element Analysis
Thesis directed by Professor James C. Gerdeen
ABSTRACT
The development of equations for the free vibration of
laminated rectangular plates is discussed. An iterative
numerical method is used to determine the frequencies
predicted by closedform solutions.
Additionally, a subspace modal analysis using the Finite
Element Method (FEM) is performed for various boundary
conditions. FEM results are compared against those of
the closedform solutions. A convergence analysis
illustrating loss of accuracy with different numbers of
elements was done. Shell element theory used in the FEM
is provided to yield a higher level of insight to the
nature of the problem.
Several boundary conditions are addressed for homogeneous
isotropic and anisotropic laminated plates.
Further, a box shell is analyzed using FEM. The local
vibrational modes of the box faces (breathing modes) are
IV
addressed as they relate to the plate vibration analysis.
This abstract accurately represents the content of the
candidate's thesis. I recommend its publication.
Signed
James C. Gerdeen
v
I would like to dedicate this thesis to my mother,
kind and patient way has led me to a successful
conclusion.
Her
CONTENTS
Chapter
1. Introduction .................................... 1
1.1 Prologue..................................... 1
1.2 Problem Statement............................ 3
1.3 Nomenclature................................. 3
2. Free Vibration of a Laminated Plate.......... 8
2.1 Effective Properties for a Single Lamina ... 10
2.2 Plate Flexural Rigidities ...................... 14
2.3 Vibration Equation for a Four Layer Laminate . 16
2.4 Vibration Equation: Simply Supported Plate . 23
3. Homogeneous Isotropic Plates ................... 25
3.1 SSSS Boundary Condition...................... 29
3.1.1 Finite Element Analysis Results ................ 30
3.1.2 Comparison of Closedform and FEA Results 39
3.2 SFSF Boundary Condition...................... 40
3.2.1 Finite Element Analysis Results ................ 41
3.2.2 Comparison of Closedform and FEA Results . . 45
3.3 CCCC Boundary Condition...................... 48
3.3.1 Finite Element Analysis Results ................ 49
4. Anisotropic, Inhomogeneous (Laminated) Plates 52
4.1 Effect of Lamina Orientations................ 53
4.1.1 SSSS Laminated Plate........................ 54
vii
4.1.2 SFSF Laminated Plate........................ 58
4.1.3 CCCC Laminated Plate........................ 61
4.2 Effect of Aspect Ratios........................ 63
5. Box Shell Finite Element Analysis.............. 71
5.1 Box Shell Analysis Results ..................... 72
6. Conclusions and Recommendations................. 80
Appendix
A. Boundary Conditions ................................ 82
B. Frequency and Mode Shape Equations................ 85
C. Finite Element Analysis Models ..................... 95
List of References..................................... 106
viii
The list of those that have helped me through this paper
could quite possibly rival the writing itself. However,
in particular I would like to acknowledge the members of
the staff and faculty of the Mechanical Engineering
office at CUDenver, from whose advise I have benefitted
many times.
Additionally, Joseph Anderer, a fellow struggling
student, has been very kind in spurring me on towards
completion of this project. May he find much happiness
in his new marriage.
Thank you all.
IX
1.
Introduction
1.1 Prologue
The solution of the free vibration of a simply supported,
isotropic, and homogeneous plate is usually presented in
most texts. In most cases, the solution to a circular
plate is provided as well [3,5,6,11,20,22,23,27]. This
solution is similar to the simply supported rectangular
plate, which is readily obtained, and provides a
fundamental insight to the general analysis of plates.
The appearance of closedform solutions for other
boundary conditions is disturbingly lacking in the
literature, due to the difficulty of solving them
analytically. Approximations, such as the RayleighRitz
energy method, are usually employed.
To provide free vibration equations for various boundary
conditions, equations for a homogeneous isotropic plate
deflection under a statically applied load were used.
Additionally, to account for the behavior of a laminated
plate, work by Timoshinko [23] concerning the deflection
of orthotropic plates was combined with work done by
Gerdeen [10] concerning the development of the effective
properties of composites. Additionally, analysis follows
in similar fashion to that done by Gorman [11]. This
1
allowed incorporation of the composite nature of plates
into vibrational motion.
To some extent the ability to use analytical theory in
addition to the use of the Finite Element Method (FEM),
yields another tool available for verification of the
results obtained by FEM.
One must be cautious not to assume that closedform
equations provide the entire scope of solution. They
simply add to the pool of knowledge available to
designers and engineers. In order to provide a strong
foundation for the analysis, equations were tested using
homogeneous isotropic plates with different aspect ratios
and boundary conditions. Analysis was repeated using
laminated anisotropic plates. A convergence analysis
(using increasingly larger mesh sizes) was performed on
the same models using a Finite Element Analysis (FEA)
package. The analysis then proceeded to a more general
laminate to ascertain the degree of certainty for the use
of different mesh sizes with varying lamina orientations.
A box shell assembly was analyzed to provide additional
insight to the vibration of plates.
2
1.2
Problem Statement
As industry increases the use of laminated composites in
design and application, finite element modeling of the
vibration of such structures gains importance. This
thesis provides information on the finite element
analysis of the free vibration of rectangular laminated
plates. Additionally, the text should provide assistance
to those wishing to solve problems of a similar nature.
1.3 Nomenclature
The boundary condition notation used for the rectangular
plate begins on the left side of the plate and lists the
boundary conditions in a counterclockwise fashion,
as shown in Figure 1.1.
It is further assumed that there is no change in the
boundary condition on any side (i.e. the left side of the
plate does not change from a simple support to a clamped
condition halfway along the plate's edge). The letter
notation used is as follows: "S" indicates a simple
3
4
S
F
Figure 1.1 Method used to describe boundary conditions
support (translation in a direction out of the plane of
the plate is constrained, inplane translation and
rotation are allowed), "F" designates a free condition
(that is, one devoid of any restrictions on forces and
moments) and "C" indicates that the edge is clamped (all
translations and rotations are constrained). In order to
describe the boundary conditions of the plate, a four
letter designation is given, beginning at the left of the
plate and rotating in a counterclockwise fashion about
the plate. In the case where all four sides are simply
supported, we would designate the plate as: SSSS; where
opposing sides are simply supported and free,
respectively, we would describe the boundary conditions
as: SFSF.
For purposes of describing the Finite Element Models, the
4
elements for each model were symmetric about the mid
point of the plate. As a result, for purposes of brevity
the models are referred to by the number of elements on
one side. For example, a sixteen element model would be
referred to as model S4, and a 1024 element model would
be referred to as model S32 (4x4= 16 total elements, and
32x32= 1024 total elements).
Due to the volume of the modal data, only the
displacements for six points on each plate are presented.
The locations of these points are provided in Figure 1.2.
Figure 1.2: Location of
points used to described
modal analysis.
5
A list of variables and their definitions follows:
E Young's modulus for isotropic materials
Ex Longitudinal Young's modulus of unidirectional
components
Ey Transverse Young's modulus of unidirectional
components
Ez Young's modulus through the thickness of the plate
G Shear modulus for isotropic materials
G, Shear modulus about the zaxis
*Jr
Gyz Shear modulus about the xaxis
Gxz Shear modulus about the yaxis
u Displacement along the xaxis
v Displacement along the yaxis
w Displacement along the zaxis
v Poisson's ratio for isotropic materials
v^ Major (Longitudinal) Poisson's ratio for anisotropic
materials
vyz Minor (Transverse) Poisson's ratio for anisotropic
materials
vxz Poisson's ratio for anisotropic materials
0 Angle of ply orientation; counterclockwise rotation
is positive
h Total thickness of a laminate
h Thickness of a single lamina
6
n Total number of plies in a laminate
Dx Flexural rigidity of a laminate about the xaxis
Dy Flexural rigidity of a laminate about the yaxis
Dx Flexural rigidity of a laminate encompassing
curvature
H Combined rigidity
I Moment of inertia (second moment of area)
M Moment
p Mass density per unit area
x Longitudinal direction of plate
y Transverse direction of plate
z Direction normal to plate (before deformation)
L Longitudinal direction of a single lamina
T Transverse direction of a single lamina
% Unit mass for a single lamina
A Surface area of a plate
t Time
7
2.
Free Vibration of a Laminated Plate
The vibration equations used in this writing are
specified for a laminated plate made up of four
orthotropic layers. Each layer is homogeneous throughout
its thickness. By incorporating relationships developed
from equations based on work by Timoshinko [23] and
Gerdeen [10] the composite nature of the plate was taken
into account. The plate remains within the elastic limit
for the material and conforms to the rules for the
deflection of plates used in plate theory. The form of
the equations describing the frequencies and mode shapes
reduces properly to the homogeneous case when all of the
lamina are the same (isotropic) material.
Firstly, the effective material properties for a single
lamina were developed. Secondly, the effective
properties of each layer were used to obtain the
relationships needed to encompass the overall properties
of a general laminated plate, assuming that each lamina
remains homogeneous throughout its thickness and that the
orthotropic characteristics remain intact over the plate.
In order to obtain the effective properties of interest,
in this case the flexural rigidities, pure bending is
assumed. After distillation of these equations, the
8
effective flexural rigidities encompass the laminated
nature of the composite.
Vibration equations are obtained by introducing a dynamic
load into a deflection equation for a rectangular plate
under a statically applied load. The resulting bi
harmonic equation is solved explicitly for each of the
boundary conditions of interest. In order to demonstrate
a method of solution of the biharmonic vibration
equation, the simply supported case is examined. It
should be noted that the solution to the simply supported
case is well known. For this reason, it provides for an
excellent way to demonstrate the validity of the method
of solution. For purposes of simplicity, inplane
displacements are ignored. This implies that vibrational
modes that lie in the plane of the plate will not be
predicted by the equations used in this writing. For a
more detailed look at vibrational modes, it is
recommended that the more complete form of the equation
be utilized (see Appendix A for more information).
9
2.1
Effective Properties for a Single Lamina
First, the effective properties for an orthotropic lamina
are obtained for use in the vibration equation for a
general laminated plate. Begin with a lamina as found in
the laminated plate shown in Figure 2.1.
L
X
By examining a single lamina possessing orthotropic
properties, the strain of the material may be described.
We have, from Gerdeen [10] :
r  ' r
*x ^11 S12 S16 xx
eyy = ^12 S22 S26 yy
e*y. *^16 ^26 *^66 xy.
(2.1)
10
where,
S^= C^49+Â§ig4e+^2
ET
LT
B,
cos20 sin20
x;
7T_ sin40 ^ cos40 +( 1
*^22
ET
E"
\ GLT
^)cos2 0 sin2 0
JL )
512 =
+ )cos20sin20
^ Et GltJ
vlt(cos40 + sin40)
~ E~r
566= 4
1 + 2v
LT + )cos20sin20
Et I
+ (cos20 sin20)
LT
^ ( 2 (1 +vLT)
516 =
cos30sin0
JLT)
V.T 2 +2 1 1 \
El Et 5LTt
cos 0 sin30
for any given angle 6 from the principal material
(2.2a)
(2.2b)
(2.2c)
(2.2d)
(2.2e)
axes.
11
&L GLt,
(2.2f)
Each Sifj is a compliance coefficient indicating the
relationship between the stress and strain. It should be
noted that Gerdeen's [10] development is similar to that
taken by Stephen Tsai [24]. Further, each of these
coefficients are obtained by taking into account the
material properties of the lamina in the lamina's
principal direction and the orientation, or rotation,
from that direction. It should also be noted that a
further step may be taken to account for the
micromechanical properties of a composite lamina (such as
fiberglass). Methods for computing the effective
macromechanical properties from the constituent materials
are covered by Gerdeen [10] and Tsai [24] and are beyond
the scope of this writing.
By assuming a uniaxial tensile load P and a constant
crosssectional area A, we have for the stresses that
(2.3)
12
Substituting this into equation (2.1) and solving for the
strains leads to
S1XP
S12P
yy
 S'*P
(2.4)
The effective Young's modulus in the longitudinal
direction is the ratio of the stress to the strain in
that direction and is given by
Ex'
PA 1
(2.5)
For a transverse load the effective Young's modulus is
given by
V
(2.6)
and for pure torsion we have
G =
(2.7)
13
2.2 Plate Flexural Rigidities
The effective material properties of each ply are used in
the laminate to describe the flexural rigidities
(resistance to bending) of the composite plate. Perfect
bonding between lamina is assumed; therefore, no
slippage between lamina occurs. Additionally, although
an interlaminar stress arises, the displacement at each
contact point between lamina will be identical, thus
allowing the use of simple plate theory. To simplify the
analysis, the laminate was limited to four layers,
although the equations may be readily expanded to
encompass more lamina, if desired. The vibration
equations only see the flexural rigidities in their
final, constructed form. In order to address different
laminates, such as a crossply configuration,
modifications to the flexural rigidities can be
introduced as constants within the vibration equations.
All assumptions are consistent with classical plate
theory.
The plate rigidities are determined by applying pure
14
bending to the plate and describing the bending moment
through the laminate thickness, h (see Figure 2.2).
Figure 2.2: Plate under pure bending
Because the lamina material does not vary through its
thickness, they are constant and the effective moduli may
be pulled out of the integrals. The resulting invariants
are the plate flexural rigidities.
/
h
2
h
4
0
h
4
h
2
h
2
h
2
h
4
0
h
4
'yi dx2 1 dy2
32P/ ip d2W
r ~r
2
4
h
h
4
15
where,
D
D.
A3(7gy1' +EyJ +ev3' + 7jO
192
h3 (7E,1" + E2" + Â£3" + 7E4")
192
(2.9)
The forms for Dx and are arrived at in similar
fashion:
II
D.
xy
h3 (7JSL + Ex + Ex + 7EX ')
192
h3 (7Gi + G2" + G3" + 7G4")
192
(2.10)
The D values are the effective flexural rigidities of
the plate.
2.3 Vibration Equation for a Four Layer Laminate
From Timoshinko [23] the equation for pure bending of an
anisotropic plate is given by
16
dx4 dx2dy2 y 3y4
2H Dv~ = q
(2.11)
where,
(2.12)
For purposes of illustration, a fourply laminated
rectangular plate, consisting of orthotropic lamina
possessing arbitrary orientations within the plane of the
plate, is considered. In addition, there is perfect
bonding between each lamina.
In the bending equation above, q is a statically applied
pressure normal to the plate, and is a function of x and
y, the length and width coordinates of the plate. For
free vibrational analysis, by applying d'Alembert's
principle, this load was replaced by an inertial force
described by
where p is the density per unit area, and mt is the unit
(2.13)
17
mass for the ith ply in the laminate. The governing
differential equation becomes
D,
d4W
dx4
+ 2H
dx2dy2
+ D.
a4 w
dy4
d2W
at2
= 0
(2.14)
By using separation of variables, a solution of the above
equation is derived in terms of unknown constants.
W(x,y,t) = W(x,y)T(t) (2.15a)
' dx4
dx2dy2
(2.15b)
D T
x dx4
+ 2HT
dx2dy7
+ D T^
y dy4
d2T
p W2== 0
H at2
(2.15c)
Dx d4W + 2H d4W + Dy d4W
pW dx4 pW dx2dy2 pW dy4
i a2r
t at2
(2.15d)
The left and right sides of equation (2.15d) can only be
equal if both sides are equal to the same constant, co2.
18
After some reduction, each equation may be expressed as
dzT
dt2
+ cj2t
o
d*W + 2H &W + Dy &W pG)2W
dx4 Dx dxzdy2 Dx dy4 Dx
(2.16)
The solution of the second order differential equation in
T is well known. In order to arrive at the mode shape
relationship for the plate, the fourth order differential
equation in W is utilized. The displacement equation for
W(x,y) is nondimensionalized in order to achieve a
generality in the solution [11]. For this purpose, the
following variables are defined:
Â£ =
x
Opiate
T =
Opiate
X2 = CO
Opiate
Opiate
(2.17)
This transformation results in the following for the mode
shape equation in differential form:
d4vni,n) + 2H$Z a4w(5,n) + DyV WE,tp
0n4 DX dr\2dl2 Dx ae
 o
^x
(2.18)
19
Since a complex sinusoidal form for the free vibration is
expected, assume a Levy type solution.
IV( ^, T)) = Â£ Ym{r\) sin {nml) (2.19)
m=1
where k is some value chosen to obtain reasonable
accuracy in the series.
Substituting the Levy solution into equations
(2.17), (2.18), and simplifying, results in the relation:
drj4
2{rm) 2 d2Ym 4
dx dn2
Dy(im)4
A4
(2.20)
This is a fourth order homogeneous equation in Ym with
constant coefficients. The method of solution chosen was
to take the equation into state space. Define the
following:
d2Y
5t2
&Y
0ti3
= W
= U3>'
(2.21)
Solving for the system's eigenvalues Xi results in
20
[*] =
Ry
Â£>
0 1 0 0
0 0 1 0
0 0 0 1
(irm) 4A4) 0 2H$2 (mn.) 2 0
Z>
Xi.2 = (fc
N D
(J777t) 2 +
H
\2
\Dx)
D
(irm) 4 + A4
= K
and
X3.4 = <1>
\ D
(imz) 2 +
V
\
JL
K DXj
Â£>
(im)4 + A4
= Y
(2.22)
(2.23)
(2.24)
21
The form for Yra varies depending on whether or not
complex values are present. These forms are:
CASE 1: If X4 > ^(imt)4, then
D
_________________ y____________
=^coshPm1l +SmsinhPmTi +CjnsinYmii ^cosy^
(2.25)
CASE 2:
If X4 < inrn)4
___________
=^coshPmri +BmsinhPmTi +CmsinhYmTl +Â£>JBcoshYJBTi
The constants A,,,, Bm/ Cm/ Dm are to be determined by
solving equations (2.17), (2.18) for the specific
boundary conditions of the plate. These constants are a
function of the effective flexural rigidities and will
change as the laminate's layup changes. The constants
beta and gamma provide us with a ready means by which we
may include the effects of a laminated plate and are
referred to as transition constants in the remainder of
the text.
22
2.4 Vibration Equation: Simply Supported Plate
At this point, the specific case of the vibration of a
simply supported plate can be evaluated. In the general
vibration equation, there are two different possible
solutions depending on whether or not imaginary values
are present.
At the boundaries of a simply supported plate the
displacement and induced moment are zero. By applying
these conditions to equation (2.27) we can state that
1 0 0 1 A
coshp sinhp siny cosy B
P2 0 0 y2 C
P2coshp P2sinhP y2siny y2cosy D
(2.26)
where the subscripts have been dropped for clarity.
In a modal analysis, only the relative magnitude of
displacements is of interest. Therefore, A is
arbitrarily set equal to unity. After substituting A
into the system of equations described by equation
(2.26), constant values are obtained:
23
A
1
B
1
tanhp
C = D  1
tany
(2.27)
For the first case, when the constants are introduced
into the Levy form, the following mode shape relation
results:
coshpr 
sintiPn
tanhp
^ sinyri
tany
COSYTl
sinmTi^
(2.28)
For the second case, imaginary values are introduced.
Again, the displacement and induced moment are zero at
the boundaries. When introduced into the Levy form, the
following mode shape relation results:
coshpri 
sinhPii
tanhp
sinhyg
tanhy
coshyri
sir
(2.29)
Through trigonometric manipulation, the forms for both
cases can be shown to reduce to
P/(I[,T) = A^sint/mc?) sin(miri) (2.30)
which is consistent with the work presented by Gorman
[11] .
24
3. Homogeneous Isotropic Plates
Free vibration of a homogeneous and isotropic rectangular
plate with three different boundary conditions is used to
demonstrate the Finite Element Analysis technique. This
project was undertaken in order to describe a clear
procedure for interpretation of finite element analysis
of plates. The goal of performing analysis on
homogeneous, isotropic plates was to establish a number
of elements, or mesh size, to be used for future study.
In order to develop confidence that the results for the
laminated case were valid, mesh sizes indicating
convergence were determined using well known constraints.
As the text demonstrates, a surprisingly large number of
elements must be employed to obtain stable numbers from
the model.
Two parameters were used to define convergence: the
excitation frequencies (in Hertz) and mode shape data.
The FEA package used performed a subspace modal analysis,
using Sturm numbers [13] as an indicator that all modes
of vibration were present.
An hconvergence analysis study continually decreases the
size of the elements used in a model, and thereby
25
increases the overall mesh size, until an acceptable
state of convergence has occurred. This is similar to a
pconvergence study, in which the fundamental equations
(basis functions) of the elements themselves are
modified. This type of analysis was a modified h
convergence; modified in the sense that nodes were
manually relocated for purposes of comparison between
models. The initial points were decided upon by the
generation of a sixteenelement model (square, four
elements on each side).
Finite Element Model construction involved a combination
of solid modeling techniques and direct construction
techniques. "Solid modeling" is a process of generating
a body by employing three dimensional entities. These
entities are divided into a grid called an element mesh
used for analysis. For these problems, the generation of
an area was sufficient to describe the model. "Direct
generation" is the process of generating a body to be
analyzed by directly specifying where the mesh was to be
placed. When the mesh was generated using solid
modeling, a link was made between the "solid" entities
and the "direct" entities. In this case, the areas that
described each plate were intricately linked to the FEA
nodes and elements. In order to demonstrate clear
26
convergence of each mode shape, nodes were defined that
coincided with specific locations on the mesh. The mesh
generator in the FEA package used assumed a balanced
distribution of the mesh relative to the plate surface
area. The nodes of interest were relocated into position
manually. When each node was moved, the Jacobian of each
element related to the node was checked for potential
numerical error. If the value of the Jacobian was
unsatisfactory (i.e. negative or very small, indicating
that excessive distortion was present in the element),
the condition was intercepted and corrected.
Due to the sensitivity of the FEA method to incorrectly
placed nodes, it became necessary to adjust complete rows
and column values related to a relocated node. In order
to alleviate conflict between the element data and the
solid model information the model data base was cleared
of all solids model data, leaving only nodal coordinate
and connectivity information (element definitions)
behind.
The package generated excessive overhead: large files
(often larger than ten Megabytes) containing all
information pertinent to the analysis were generated.
Additionally, a great deal of memory was demanded from
27
the computer in order to solve several of the models.
Although access to generous resources was granted, this
additional overhead still pushed the limits of the
available space (during the course of this thesis, the
space required for analysis doubled twice).
After the analysis had been performed, most of this
information was no longer required for the convergence
analysis. Through trial and error, a system was devised
to optimize the available resources. As each model was
solved, the results were obtained in the form of graphics
files, excitation frequencies and modes shapes. The key
information was compiled into tape archive files and
compressed. The process was repeated for each of the one
hundred and twenty models observed.
The structure of the analysis followed a straightforward
approach: determine the first ten modes of analysis for
a given boundary condition, plate aspect ratio and mesh
size. The mesh size was improved and the analysis
repeated, until the model had apparently converged. A
spreadsheet was utilized to combine the various results
for different mesh sizes (all other factors remained
constant) and graphs were generated that clearly
demonstrated the trends of convergence (Â§ 3.1.1).
28
3.1
SSSS Boundary Condition
For simple support along all four edges, displacement out
of the plane of the plate is restricted along all edges.
Two edges are constrained so that inplane motion
transverse to each edge is restricted. The other two
edges are free of such constraints. To negate rigid body
motion, translations and rotations of the node at the
origin was constrained in all three cartesian coordinate
directions (Figure 3.1).
M
SSSS
Figure 3.1: Simplysupported boundary
condition
29
3.1.1 Finite Element Analysis Results
This section covers the vibration of a homogeneous and
isotropic square plate supported on all four edges by
three different boundary conditions. In short, a general
discussion of the convergence of the solution of a modal
(vibrational) analysis is presented here.
In order to present the information as clearly and
succinctly as possible, the discussion is broken into two
parts.
The vibration of a simply supported plate is approached
first in order to provide a strong comparison to a very
well known analytical solution. Even this stage is
divided into two substeps in order to provide for the
best possible representation of the information. First
the convergence of the excitation frequencies is examined
in order to determine an appropriate mesh size for use in
further analyses.
No vibration analysis is complete without the
introduction of the vibrational mode shapes (the
displacement of the body at each excitation frequency).
Specific points are selected on the plate and are
30
referenced in all subsequent analysis results. The
displacement of each of these points is tracked for each
mesh size and an appropriate point of convergence is
determined. Again, it must be emphasized that the
initial analysis is performed for a simplysupported
plate and that the mode shapes varied when the boundary
conditions were altered.
After an appropriate mesh size was determined for the
simplysupported case, two other boundary conditions were
examined in order to provide for a broader base of
understanding for the problem at hand. The other
boundary conditions used were those of a
simplefreesimplefree case (SFSF: opposing sides have
identical constraints) and the fully clamped case (CCCC:
the motion of all four edges are fully restricted). The
reason for using these other two models is simple. The
SFSF case is a moderately wellknown solution determined
through the use of the Rayleigh energy method, and
therefore provides for a good comparison with regards to
the present convergence analysis. The CCCC case may be
determined only for precisely defined cases and therefore
provides a strong comparison for a situation that is not
easily determined.
31
As can be seen from the figure, the mode shapes for a
plate tend to be either symmetrical or antisymmetrical
across the body. For most cases, particularly the
simplysupported boundary condition, this type of
behavior has been taken to advantage in that a basic
sinusoidal combination is used to approximate the
displacements of the plate (see chapter 2) In general,
these types of shapes were very well supported by the
results of FEA (eightnode quadrilateral elements using
cubic basis functions were employed in all models). An
interesting anomaly at the second mode of vibration would
randomly appear in the results. This random behavior is
discussed later.
The excitation frequencies converge at a surprisingly
fast rate, so that even very low mesh sizes seem to
indicate reasonable convergence. These values compare
favorably to the wellknown closed form solution, as they
approach asymptotically to closed form values.
The convergence of the mode shapes varies slightly
depending on the modal frequency sought. At lower
excitation frequencies the convergence appears quite good
for mesh sizes larger than nine elements. Resolution is
lost rapidly as higher modes of vibration are sought.
32
Essentially, with the use of quadrilateral elements, the
size of the mesh is directly related with the number of
degrees of freedom in the model. At extremely small mesh
sizes (those around four elements) resolution is
practically nonexistent for all mode shapes greater than
that of the fundamental frequency.
One must take care in the determination of the points
used to describe the mode shapes. In a Finite Element
Analysis, the data obtained from a vibrational analysis
may be extensive. Practically speaking, this volume of
information is probably unnecessary for most engineering
applications. Of more interest is the location of
maxima and minima and the frequency at which they occur.
As stated previously, in order to demonstrate
convergence, specific locations were chosen on the plate.
Armed with the knowledge that the mode shapes would most
likely be either symmetric or antisymmetric, points were
chosen to lie on even divisions of the plate.
Particularly, the points lie on the onequarter, halfway,
and threequarter fractions of the plate, essentially
defining a rectangle evenly dividing the plate's surface
area.
Points lying on maxima or minima were relatively stable
33
across all mesh sizes greater than one hundred elements.
Points stationed at positions of high gradients (perhaps
between varying maxima and minima) were likely to change
displacements depending upon the mesh size. In order to
account for this possibility of error, a full
hconvergence analysis was performed. Even at relatively
high mesh sizes (thirtytwo elements on a side, or one
thousand and twentyfour elements), the magnitude of the
displacements of these points had a tendency to vary
slightly (Figure 3.2 and Figure 3.3). Therefore, caution
would be advised when interpreting the results of the
Finite Element Analysis technique when plate elements are
used.
Displacement vs Mesh Size
HI, Aspect^ 1, BC: SSSS, Mode= 1
26
25
24
t 23
I 22
0
1 21
o
20
19
18
17
0 5 10 15 20 25 30
Number o(Bemerts on SWe o? Mesh
Figure 3.2: Mode shape of
selected nodes for first
mode (SSSS case).
34
Normalized Displacement vs Mesh Size
HI, Aspect= 1. BC: SSSS, Mode= 1
Figure 3.3: Mode shape of
selected nodes for first
mode (SSSS casenormalized
data).
Throughout the course of the analysis, it was determined
that a model with a mesh size of one hundred elements was
sufficient to properly describe a range of mode shapes
extending to the tenth mode of vibration. Resolution was
lost for some points in models containing high gradients
(Figure 3.4), so the decision to use a four hundred
element mesh for subsequent comparisons was made.
35
Normalized Displacement vs Mesh Size
HI, Aspect= 1, BC: SSSS, Mode= 3
Figure 3.4: Mode shape of
selected nodes for third
mode (SSSS casenormalized
data).
Figure 3.5: Possible
anomaly at second mode for
SSSS case.
36
An interesting anomaly occurred at the second mode of
vibration. The mode shapes found by the SSSS model
indicated a fortyfive degree rotation with respect to
the axes of the plate (Figure 3.5). Initially, an
examination of the closedform equations did not reveal
the source of the behavior. Upon examination of further
models, however, a theory emerged. One of the
characteristics of an anisotropic plate (a plate with
material properties that are allowed to vary with
direction) is that when the effective Young's moduli are
dominant in a fortyfive degree direction relative to the
axes of the plate the mode shapes shift to that direction
as well. The plate flexural rigidities are such that the
contributions of the sinusoidal modes of vibration are
dominant in the rotation orientation. Although the model
was did not incorporate a rotation of material properties
(in fact, the FEA package used was specifically
instructed to analyze isotropic materials) the results
corresponded to those of a laminated plate with material
properties dominant along rotated axes. The behavior may
have been related to an increasing numerical error in the
subspace solution technique employed. Although much
effort was spent in verifying the boundary conditions,
another possibility might have been that the model was
slightly overconstrained. The behavior was not repeated
37
in any higher mode of vibration or other boundary
condition and only appeared in the case of a square
plate. Discrete elements may result in a pseudo
anisotropy, however this effect should have decreased
when more elements were used in the model. In order to
ensure that a proper mode shape has been determined, a
solution using a reduced mass matrix (similar to lumped
mass) might be employed. The first and third modes of
vibration are illustrated in the following figures
(Figure 3.6 and Figure 3.7). The first mode is
reminiscent of the fundamental mode of a beam, and is
related. The third mode of vibration was chosen
arbitrarily in order to demonstrate the behavior of a
slightly higher mode of vibration.
Figure 3.6: First mode of
vibration for SSSS case.
38
4
2 Fr* vibration (HI. CIcatAts* 3 02< Aspect* 1. 8C* SSSS!
Figure 3.7: Third mode of
vibration for SSSS case.
3.1.2 Comparison of Closedform and FEA Results
Although a derivation is provided in chapter 2, the
equations describing the vibration of a simply supported
plate are listed here.
Excitation frequencies (eigenvalues) for a rectangular
plate simply supported on all edges (SSSS).
X2 = (/nn) 2+(rm) 2/<\>2 (3.1)
39
Mode shapes for a rectangular plate simply supported on
all edges (SSSS).
= A^sinfjmtl;) sin(rmr\) (3.2)
As stated previously, the solution for this case is very
well known. For mesh sizes larger than one hundred
elements, the range of valid solutions extends smoothly
into the tenth mode of vibration.
3.2 SFSF Boundary Condition
To demonstrate a model of slightly increased difficulty
level, a plate with two opposing simply supported edges
and two opposing edges free of constraints was examined
(Figure 3.8). A description of the constraints for a
simply supported edge may be found in section 3.1. To
negate rigid body motion, translations and rotations of
an additional point on the boundary of the plate was
constrained in all three cartesian coordinate directions.
40
A
00
///Mlv/m/
Figure 3.8: SimpleFreeSimpleFree
boundary condition
3.2.1 Finite Element Analysis Results
For the SFSF case, the mode shapes were similar to those
found for the simply supported case. The pattern of
convergence closely followed that of the SSSS boundary
condition in that at very low mesh sizes (less than nine
elements), the mode shapes showed high variation, and the
frequencies appeared to converge at relatively low mesh
sizes. At mesh sizes around one hundred elements the
mode shapes appeared to converge to reasonably constant
values (Figure 3.9 and Figure 3.10), and at mesh sizes
approaching four hundred elements, all displacements
seemed to have settled down. All values compared
favorably to those found by the closed form solutions.
41
Normalized Displacement vs Mesh Size
HI, Aspect^ 1, BC: SFSF, Mode= t
Figure 3.9: First mode for
selected nodes, SFSF,
normalized data.
42
Normalized Displacement vs Mesh Size
HI, Aspect= 1, BC: SFSF, Mode= 3
Figure 3.10: Third mode for
selected nodes, SFSF,
normalized data.
43
The first and third mode are shown graphically in
Figure 3.11 and Figure 3.12.
Figure 3.11: First mode of
vibration for SFSF case.
Figure 3.12: Third mode of
vibration for SFSF case.
44
3.2.2 Comparison of Closedform and FEA Results
The solution for a plate with opposing simply supported
and opposing free edges is moderately complex, depending
upon whether symmetric or antisymmetric modes are
present. Additionally, due to the presence of a radical,
the solution must be broken down further in order to
accommodate the possible introduction of complex numbers
[11] .
In the interests of simplifying this writing, the
equations are provided for an anisotropic plate. The
equations reduce to the isotropic case when the flexural
rigidities are equal (see chapter 2).
The equations are summarized as:
Excitation frequencies (eigenvalues) for a rectangular
plate with boundary condition SFSF: X4 > (Dy/Dx) (nm)4
(symmetric).
Y (P2 v (
P (P2v* (
45
Mode shapes for a rectangular plate with boundary
condition SFSF: X4 > (Dy/Dx) (nnr)4 (symmetric) .
W(S,ti) =
COSYT) +
(Y2 + v (4>jn~n:) 2) cos (y/2) cohQ
(P2 v ((Jwnir) 2) cosh (P/2)
(3.4)
sirnnnc
Excitation frequencies (eigenvalues) for a rectangular
plate with boundary condition SFSF: X4 < (Dy/Dx) (m7r)4
(symmetric).
Y (y2v* (
P (P2 v* (tyimz)2) (y2 v (
Mode shapes for a rectangular plate with boundary
condition SFSF: X4 < (Dy/Dx) (mir)4 (symmetric) .
WU,ti) =
coshTH W2v (cj>imr)2) cosh (y/2) coshpi1
(P2v (/r77C) 2) cosh(P/2)
SirunTT
p
6)
Excitation frequencies (eigenvalues) for a rectangular
plate with boundary condition SFSF: X4 > (Dy/Dx) (imt)4
(antisymmetric) .
Y (y2+v* (
P (P2v* (mit) 2) (y2+v (<{mm)2) sin (y/2) cosh (p/2) =0
46
Mode shapes for a rectangular plate with boundary
condition SFSF: X4 > (Dy/Dx) (nnr)4 (antisymmetric) .
IV(^, T)) = sinyri +
(P2v (<()7777r) 2) sinh(P/2)
(y2+v [fyimz)2) sin (y/2)
sinhpr) sinmTi
Excitation frequencies (eigenvalues) for a rectangular
plate with boundary condition SFSF: X4 < (Dy/Dx) (mir)4
(aptisymmetric).
Y (y2v* (fymn)2) (P2v (fymz)2) sinh (P/2) cosh (y/2) (3.9)
P (P2v* (/mt) 2) (y2v (<)imt) 2) sinh (y/2) cosh(P/2) =0
Mode shapes for a rectangular plate with boundary
condition SFSF: X4 < (Dy/Dx) (imr)4 (antisymmetric) .
For mesh sizes over sixtyfour elements, the closed form
solutions compared favorably with the FEA results. The
estimated error for each case proved to be well within
reasonable limits. For mesh sizes smaller than sixty
four, the comparison was approximate. It was difficult
to determine whether or not the error rested with the
IV( , T) = sinhyri
(y2v (
(P2v {
sinhpr sinm
47
analytical model or with the discretized finite element
model. In order to clarify this discrepancy, data could
be generated empirically and compared to each of these
models. A far better solution would be to simply
increase the model size (in the FEA case) or settle with
the approximation of the analytical model. Unless
exceptionally high accuracy is desired, anything else
would be a waste of resources.
Some equations for other boundary conditions may be found
in Appendix B.
3.3 CCCC Boundary Condition
The two previous solutions are fairly well known in
literature. To demonstrate a model not as readily
solved, a plate fully clamped around its boundaries was
examined (Figure 3.13). All translations and rotations
are fully constrained about the edge of the plate. There
was no need to fix an additional point as rigid body
motion is not possible in this configuration.
48
Figure 3.13:
condition
(b)
Fully clamped boundary
3.3.1 Finite Element Analysis Results
Interestingly, the fully clamped (CCCC) case was very
similar to the simply supported boundary condition. All
frequencies were higher, indicating the presence of the
stiffer boundary condition, but appeared to follow the
same pattern as that found in the simply supported case.
The mode shapes held a strong similarity to the simply
supported case when away from the plate's edge. In fact,
the mode shape patterns appeared to be practically
identical to those shown by the SSSS case if the plate
edges were ignored. This is demonstrated by Figure 3.14
and Figure 3.15.
49
Figure 3.14: First mode of
vibration for CCCC case.
Figure 3.15: Third mode of
vibration for CCCC case.
In general, although there is a strong similarity in
appearance to the simply supported case, the
displacements for the modes showed stronger sensitivity
50
to mesh size.
Normalized Displacement vs Mesh Size
HI, Aspect= 1, BC: CCCC, Mode= 1
Figure 3.16: First mode for
selected nodes, CCCC,
normalized data.
51
Normalized Displacement vs Mesh Size
HI, Aspect= 1, BC: CCCC, Mode= 3
Figure 3.17: Third mode for
selected nodes, CCCC,
normalized data.
52
4.
Anisotropic. Inhomogeneous (Laminated) Plates
The goal of performing analysis on homogeneous and
isotropic plates was to establish a number of elements,
or mesh size, to be used as a baseline for comparison.
In order to develop confidence that the results for the
laminated case were valid, mesh sizes indicating
convergence were determined using well known constraints.
As the previous text indicates, a large number of
elements must be employed to obtain stable numbers from
the model.
The first ten modes were determined for each of the three
boundary conditions examined for the homogeneous and
isotropic plates covered in chapter 3. The plate aspect
ratio and mesh size were modified in a manner consistent
with previous analysis. A spreadsheet was utilized to
combine the various results for different mesh sizes (all
other factors remained constant) and graphs were
generated that clearly demonstrated the trends of
convergence.
53
4.1
Effect of Lamina Orientations
To examine the effects of different laminates with
varying material properties, a four layer laminate
consisting of two different materials each possessing
anisotropic properties was used. Three classic lamina
orientations were used in the analysis to demonstrate the
impact of lamina orientation on mode shapes and
excitation frequencies. The layups employed in the
analysis were 1) all four lamina were oriented with the
Longitudinal Young's modulus at zero degrees to the
longitudinal axis of the plate (material properties were
aligned: [0,0,0,0]) 2) the lamina were at forty
five degree increments from the longitudinal axis of the
plate (angleply: [+45,45,+45,45]), and 3) the
lamina were rotated at ninety degree increments from the
longitudinal axis of the plate (crossply:
[0,90,0,90]). For purposes of discussion, the
longitudinal axis is defined at the axis lying along the
longer of the length or width of the plate.
The two materials used were: 1) Scotch Ply 1002
(Glass/Epoxy) and 2) AS/3501:(Graphite/Epoxy). The
longitudinal modulus for AS/3501 is four times larger
than that for the Scotch Ply (which has a modulus of
54
5.6xl06 psi). The transverse moduli are similar at
around 1.25xl06 psi. The major Poisson ratios for the
Scotch Ply and AS/3501 are 0.26 and 0.30, and the
densities are 0.065 and 0.058 lbf/in3, respectively.
These materials were chosen for their contrasting
longitudinal moduli and the fact that their other
material properties are similar.
4.1.1 SSSS Laminated Plate
The first mode of vibration for a homogeneous and
isotropic plate is shown in Figure 4.1. This shape was
used as a baseline for the following discussion.
Similarities or deviations will be highlighted where
appropriate.
In the interest of brevity, only the mode shapes at the
fundamental frequency (the first excitation frequency)
have been presented here.
55
Figure 4.1: First mode of
vibration for a homogeneous
and isotropic simply
supported plate.
For the first laminar orientation ([0,0,0,0]) there
seemed to be no significant change from the homogeneous,
isotropic case. The excitation frequency changed due to
the fact that different materials have been used (see
chapter 2 for a discussion on moduli and their effects on
flexural rigidities). In all other respects, the mode
shape appears as it should for the homogeneous case. A
check using a uniform material with equivalent moduli
showed that the frequency provided by the laminate model
was reasonable.
56
Figure 4.2: Anisotropic,
inhomogeneous simply
supported plate
[0,0o,0o,0].
As the lamina were rotated into the next orientation,
however, the dominance of the moduli became apparent
(Figure 4.3). The mode shape rotates by 45, so that the
circular deflections are elongated in that direction.
For this boundary condition, there is no shift in the
position of the maxima, however. Equation (2.21)
supports this rotation of mode shapes by its combination
of sinusoidal terms through the transition constants
provided.
57
Figure 4.3: Anisotropic,
inhomogeneous simply
supported plate
[+45,45,+45,45].
As the lamina continued to rotate, the original mode
shape (and frequency) were regained (Figure 4.4).
58
Free Vieration til. Elements
a. Aspect I. 6C SSSS)
Figure 4.4: Anisotropic,
inhomogeneous simply
supported plate
[0,90,0,90].
4.1.2 SFSF Laminated Plate
The SFSF case, with its nonsymmetrical boundary
condition, showed far more interesting results. At the
aligned laminar orientation, the mode shape appeared
similar to that obtained by the homogeneous, isotropic
material (Figure 4.5).
59
Figure 4.5: Anisotropic,
inhomogeneous SFSF plate
[0,0o,0o,0] .
As the lamina were rotated into the angleply
configuration, the stiffer moduli had less of an effect
on the mode shape. The general shape appeared similar to
that of the previous laminar orientation (Figure 4.6).
As the rotation reached the crossply configuration, the
effective flexural rigidities were such that the
resistance to bending was at a minimum as two sides of
the plate were not reinforced by constraints. Again, the
general mode shape appeared similar (Figure 4.7), but the
excitation frequency dropped dramatically (from 28 Hz for
the aligned case to 12 Hz for the crossply
configuration).
60
m
M
mJSLm
free Vieration HI, EIcent [QO; Aspect* 1, BC~ SfSFj
Figure 4.6: Anisotropic,
inhomogeneous SFSF plate
[+45,45,+45,45].'
Figure 4.7: Anisotropic,
inhomogeneous SFSF plate
[0,90,0,90].
61
4.1.3
CCCC Laminated Plate
At the aligned laminar orientation, the mode shape
appeared similar to that obtained by the homogeneous,
isotropic material (Figure 4.8).
Figure 4.8: Anisotropic,
inhomogeneous CCCC plate
[0o,0,0o,0] .
When the lamina were rotated into the angleply
configuration, the shape once again was reminiscent of
the homogeneous case. The results were similar to those
obtained for the simply supported case (Figure 4.9).
62
Figure 4.9: Anisotropic,
inhomogeneous CCCC plate
[+45,45,+45,45] .
The final configuration's solution (Figure 4.10) reverted
back to the same mode shape and frequency combination
found for the aligned laminar orientation. In short, for
a symmetrical boundary condition constraining a square
plate, the solutions for both the aligned and crossply
configurations are identical. The relationship between
the solutions is equivalent to a rigid body rotation
about the center of the plate.
63
Figure 4.10: Anisotropic,
inhomogeneous CCCC plate
[0,90,0,90].
4.2 Effect of Aspect Ratios
The sensitivity of the vibrational modes with respect to
a changing aspect ratio is sought. The same boundary
conditions applied to previous models are used in this
section. Where appropriate, the similarities and
discrepancies between boundary conditions are addressed.
64
It is well known, the vibrations of the plate will, in
the limit, approach those of a beam under the same
boundary conditions (all of the texts under the list of
references demonstrate this, with the exception of [13]).
The goal of the following analysis was not to determine.
the final values at which socalled "beam modes"
dominate, but rather to demonstrate the trend and the
sensitivity of the elements to a changing aspect ratio.
As seen in Figure 4.11, the rate of convergence for a
simply supported plate was abrupt. The displacement as a
function of aspect ratio appeared to converge moderately
quickly to values of approximately twenty and twentyfive
units (as this was a modal analysis, the value has no
physical significance. Only the relative displacement is
of importance), for different points on the plate. It
was originally thought that an aspect ratio would lead to
reasonable beam modes, and the initial data did suggest a
very steep trend in that direction. As can be seen from
the graph, however, this is not the case.
65
Displacement vs Aspect Ratio
AI, Aspect= varies, BC: SSSS, Mode= 1
Figure 4.11: Anisotropic,
inhomogeneous SSSS plate.
Sensitivity to aspect ratio
(raw data).
Normalized Displacement vs Aspect Ratio
. AI, Aspect= varies, BC: SSSS, Mode= 1
Figure 4.12: Anisotropic,
inhomogeneous SSSS plate.
Sensitivity to aspect ratio
(normalized data).
66
1 4
2 3
Free Violation UI. EIcmau* . 00. Aspect % 0C SSSS)
Figure 4.13: First mode for
SSSS case (aspect ratio= 5).
The deflection for a simplefreesimplefree boundary
condition showed a more severe sensitivity to a changing
aspect ratio. Due to fact that more degrees of freedom
exist, the model showed a distinct change as the plate
grew in length. It should be noted that the simply
supported boundary conditions are placed along the
shorter edges of the plate, so that in the limit the
plate eventually assumed the characteristics of a simply
supported beam. As demonstrated by Figure 4.14, beam
modes did not yet converge at a plate aspect ratio of
five (it was originally thought that the beam modes would
be present at an aspect ratio of approximately three).
67
Displacement vs Aspect Ratio
AI, Aspect= varies, BC: SFSF, Mode= 1
Figure 4.14: Anisotropic,
inhomogeneous SFSF plate.
Sensitivity to aspect ratio
(raw data).
Normalized Displacement vs Aspect Ratio
AI, Aspect= varies, BC: SFSF, Mode= 1
Figure 4.15: Anisotropic,
inhomogeneous SFSF plate.
Sensitivity to aspect ratio
(normalized data).
68
1 4
2 at 3
Free Vitretion ttl. Eleaents* 00. Asoect S. BC SFSF)
Figure 4.16: First mode for
SFSF case (aspect ratio= 5) .
Not surprisingly, the sensitivity to a changing aspect
ratio for the fully clamped case (CCCC) was less than
that for either the simply supported or the simplefree
simplefree cases. The pattern was again similar to that
for a simply supported case, and beam modes appeared to
be reached near an aspect ratio of five (Figure 4.17) .
As demonstrated in Figure 4.18, the normalized data was
essentially identical to that of the simply supported
case for the same laminate (Figure 4.12) This was not
surprising as the fully clamped case appeared to be
merely a stiffer version of the simply supported boundary
condition.
69
Normalized Displacement vs Aspect Ratio
AI, Aspect^ varies, BC: CCCC, Mode= 1
Figure 4.17: Anisotropic,
inhomogeneous CCCC plate.
Effect of aspect ratio on
mode shape.
Figure 4.18: Anisotropic,
inhomogeneous CCCC plate.
Aspect ratio= 5.
70
Figure 4.19: First mode for
CCCC case (aspect ratio= 5).
71
5.
Box Shell Finite Element Analysis
The fourhundred element model was extruded into the
third dimension in order to generate a box shell. One
side of the box was fully clamped over its surface so
that the box was cantilevered. It was thought that the
box shell was an advance over the plate problems in that
in dealing with three dimensional entities there was a
probability of generating translations and rotations in
all three coordinate directions.
The shell used for the analysis is constructed of a
homogeneous and isotropic material (Aluminum). As
demonstrated in previous chapters, the introduction of a
laminated shell would have necessitated indepth
discussion as to the effects of the different layups and
lamina orientations. By keeping with a simple material
we are able to examine the effects of geometry and
determine the vibrational modes under controlled
conditions.
The first ten modes of vibration are examined and
similarities and differences are addressed where
appropriate.
72
Finally, the sides of the box shell are examined with the
intent of relating each side to a plate with a similar
boundary condition.
5.1 Box Shell Analysis Results
The first mode of vibration (Figure 5.1) exhibited a
motion similar to the first vibrational mode of a simply
supported or clamped plate. In fact, given that each
side of the box is rigidly attached to the other side,
with the exception of the excitation frequency the motion
on one side was very similar to that of a fully clamped
plate.
The other sides of the box balanced displacement in a
symmetrical fashion so that as one side flexed outward,
all four remaining sides moved inward. This type of
behavior was consistent across all ten modes examined.
The excitation frequency at this mode was significantly
lower than that of a fully clamped plate (approximately
80 Hz as opposed to over 450 Hz). Apparently, the
severity of the constraints are relaxed by the.presence
of the other four sides of the box.
73
Figure 5.1: Cantilevered
box shell (first mode)
The second mode of vibration again showed a symmetric
displacement (Figure 5.2). The side opposite the clamped
surface showed virtually no motion. The remaining four
sides showed a mode shape similar to the first mode of a
fully clamped plate. Although the frequency was higher
for this mode, it was lower than that for a fully clamped
plate. A point of interest was that each side still
oscillated as if its edges were fully clamped. In fact,
the original geometry of the box shell was still
predominant (there was no "swaying" action present at
these modes). The primary displacement present was a
"breathing" action of the box.
74
Figure 5.2: Cantilevered
box shell (second mode)
The sides of the box in the third mode of vibration
(Figure 5.3) still exhibited shapes reminiscent of those
of a fully clamped plate. In this case, however, it
appeared that only two sides of the box were excited.
The vibrational mode was antisymmetric and all edges
remained rigidly fixed. It is possible that at higher
frequencies, swaying actions would be present, but the
stiffness of each side prohibits this action in the lower
modes of vibration. A relaxed model was evaluated, not
present in this writing, so that two opposing sides were
removed. The resulting loss of stiffness introduced
swaying action at relatively low frequencies.
75
Figure 5.3: Cantilevered
box shell (first mode)
The fourth mode (Figure 5.4) of vibration demonstrated
symmetric displacements. Again, the edges of the box
remained fixed and acted as if they has been rigidly
clamped in place.
Also, although all sides of the box acted in unison, each
side exhibited motion not unlike that of a single, fully
clamped plate.
76
Figure 5.4: Cantilevered
box shell (second mode)
The fifth mode of vibration (Figure 5.5) was the first to
demonstrate behavior similar to that of the second mode
for a fully clamped plate. Taken as a whole, the mode
was antisymmetric. The side opposing the clamped
surface and all edges appeared rigid.
Although the analysis had not been performed, based on
the behavior of the plates taken individually and that of
the sides of the box when looked upon as plates, it is
reasonable to assume that if each were hinged rather than
firmly attached, the vibrational modes would be similar.
77
Figure 5.5: Cantilevered
box shell (first mode)
The remaining modes are shown in figures 5.5 through 5.9.
It can be readily seen that in all cases, each side of
the box acts, in essence, as if it were an isolated
plate.
In all cases, the vibrational modes are symmetric or
antisymmetric across the box and, due to relaxed
boundary conditions, the excitation frequencies are much
lower than those of an individually clamped plate.
78
Figure 5.6: Cantilevered
box shell (second mode)
Figure 5.7: Cantilevered
box shell (first mode)
79
Figure 5.8: Cantilevered
box shell (second mode)
Figure 5.9: Cantilevered
box shell (first mode)
80
6.
Conclusions and Recommendations
The models that were run exhibited consistent
characteristics throughout the analysis, in that a
surprisingly large number of elements needed to be
included in the model in order to obtain consistently
accurate results.
It is highly recommended that the analyst using FEA be
familiar with as many of the characteristics of the
system being examined. This will allow for a substantial
increase in the efficiency of the process of solution.
Additionally, the use of submodel analysis may be
recommended if the system is moderately complex. It was
found that if the translations were restrained in the
SSSS case the FEA solution resulted in a higher frequency
than expected for a small deflection solution. Inplane
translations had to be relaxed in the analysis in order
to obtain results in agreement with the closedform
results.
An anomoly was found that was not apparent in the
classical literature. The Finite Element Model analysis
appears to indicate a new mode of vibration. This is
81
worthy of additional investigation.
The vibrational frequencies were close to convergence at
relatively low mesh sizes, however the examination of
mode shapes must be done with caution.
In essence, the higher the mode being sought, the finer
the mesh should be in order to properly capture the
correct contours present during vibration. Even with the
eightnode quadrilateral elements used in this
examination, substantial accuracy was often lost at lower
mesh sizes.
In addition, an increase in the order of the basis
functions used for each element might be increased (a p
convergence analysis). This would result in higher
computational requirements, however a significant
increase in accuracy may be obtained.
82
Appendix A. Boundary Conditions
83
The following equations have been nondimensionalized in
order to allow for broader application [11, 23].
v= Poisson ratio
v* =2v
(A. 1)
Simply Supported Edges: The displacement and induced
moment are zero at the edges.
W(Â£,,r\) =0
(A.2)
d2w(^,n) _0
dt2
Clamped Edges: The displacement and slope are zero along
the edges.
T)) =0
(A.3)
dW{^ T) _n
dl
84
Free Edges:
There are no bending and twisting moments
and there are no vertical shearing forces.
edge \ =1:
d2W{l,r\) ^ v d2W{Â£, ti) =Q
dÂ£2
d3ftr(g,ii) ^ v* d3W(1jf ri) _Q
dV <])2 dW
edge n=l:
d2^(Â£,Tl) +vd)2 & =Q
dr)2 dt,2
+V*(K2 =0
3t)3 dr\d^2
(A.4)
85
Appendix B. Frequency and Mode Shape Equations
86
Transition Constants
The transition constants that take into account the
plate's composite properties are
and
Pm=
H
_ Dy
(B.l)
(flm)2 +
Ry
Dv
(nrn) 4 + XA
(B.2)
In the homogeneous case, the flexural rigidities Dx, Dy,
and H are equivalent to D (the flexural rigidity of a
homogeneous plate) and the values for /3 and y reduce to:
(t>>/ (77m)2 + X2 
(B. 3)
87
Simplysupported plate (SSSS)
Eigenvalues for a rectangular plate simply supported on
all edges (SSSS).
A2 = (/7m) 2+ (rm) 2/2 (B.4)
Mode shapes for a rectangular plate simply supported on
all edges (SSSS).
= A^sininm^) sin(tmri) (B.5)
88
Boundary condition: SCSS
Eigenvalues for a rectangular plate with boundary
condition SCSS: X4 > (Dy/Dx) (m7r)4.
ysinhPcosy PcoshPsiny = 0
Mode shapes for a rectangular plate with boundary
condition SCSS: X4 > (Dy/Dx)(m7r)4.
W(Â£,t)) = sinyT) 
(B.6)
(B.7)
89
Boundary condition: SCSC
Eigenvalues for a rectangular plate with boundary
condition SCSC (symmetric modes).
ycosh^sin^ + Bsinh^cos^ = 0
1 2 2 K 2 2
Mode shapes for a rectangular plate with boundary
condition SCSC (symmetric modes).
eW(Â£, T))
cosyti
cos(y/2)
cosh(P/2)
jcoshp^
sirumt^
Eigenvalues for a rectangular plate with boundary
condition SCSC (antisymmetric modes).
Mode shapes for a rectangular plate with boundary
condition SCSC (antisymmetric modes).
Itf(S,Ti)
sinyri
sin(y/2)
sinh(p/2)
jsinhprj
sinmni;
(B. 8)
(B.9)
(B.10)
(B.ll)
90
Boundary condition: SFSS
Eigenvalues for a rectangular plate with boundary
condition SFSS: X4 > (Dy/Dx) (imr)4.
y (y2 + v* (
P (P2 v* (
(B.12)
Mode shapes for a rectangular plate with boundary
condition SFSS: X4 > (Dy/Dx) (nm)4.
=
smYT]
(Y2 +v ((fr/nrc)2) siny
(P2 v (
sinhpT]
simrnt^
(B.13)
Eigenvalues for a rectangular plate with boundary
condition SFSS: X4 < (Dy/Dx) (imr)4.
Y (y2 v* (4>ijnt) 2) (P2 v (/7m)2) sinhpcoshY 
P(P2 v*($nm)2) (y2 v (4>inu)2) coshpsinhY = 0
Mode shapes for a rectangular plate with boundary
condition SFSS: X4 < (Dy/Dx) (nnr)4.
Tl)
sinhyTi 
(Y2v((l)inu)2_)_sinhYsinhp
(P2 v 2) sinhp
CB.15)
simrrni;
91
