Research A rticle
Frequency Domain Spectral Element Model for the Vibration
Analysis of a Thin Plate with Arbitrary Boundary Conditions
Ilwook Park, Taehyun Kim, and Usik Lee
Department of Mechanical Engineerin g, In ha Un i versity, 100 Inha-ro, Na m-gu, Incheon 402-751, Republic of Kor ea
Correspondence should be addressed to Usik Lee; ulee@inha.ac.kr
Received June ; Revised August ; Accepted September
Academic Editor: Giovanni Garcea
Copyright © Ilwook Park et al. is is an open access article distributed under the Creative Commons At tribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a new spectral element model for nite rectangular plate elements with arbitrary boundary conditions. e new spectral
element model is developed by modifying the boundary splitting method used in our previous study so that the four corner nodes of
a nite rectangular plate element become active. us, the new spectral element model can be ap plied to any nite rectangular plate
element with arbitrary boundary conditions, while the spectral element model introduced in the our previous study is valid only
for nite rectangular plate elements with four xed corner nodes. e new spectral element model can be used as a generic nite
element model because it can be assembled in any plate direction. e accuracy and computational eciency of the new spectral
element model are validated by a comparison w ith exact solutions, solutions obtained by the standard nite element method, and
solutions from the commercial nite element analysis p ackage ANSYS.
1. Introduction
e plate is a representative structural element that is widely
used in many engineering elds such as mechanical, civil,
aerospace, shipbuilding, and structural engineering. Severe
or unwanted vibration of a plate is a very important engi-
neering problem. us, it is required to accurately predict
the vibration characteristics of a plate during the design
phase. Exact solutions are available only for Levy-type plates
[, ]. us, numerous computational methods have been
developed for the vibrations of plates during the last two
centuries.
e nite element method (FEM) is one of the most
widely used computational methods that can be applied to
various complex structures including the plates. e FEM
in general provides reliable solutions in the low frequency
range, but poor solutions in the high frequency range. us,
to improve the solution accuracy in the high frequency range,
a nite element must be divided in to many smaller nite
elements so that their sizes are smaller than the wavelengths
of the highest vibration mode of interest. However, this will
result in a signicant increase in computation cost. us, as
an alternative to FEM, we can consider the spectral element
method (SEM) for the vibration analysis of plates.
e SEM considered in this study is the fast Fourier
transform- (FFT-) based frequency domain analysis method
[, ]. e spectral element matrix (or exact dynamic stiness
matrix) used in the SEM is formulated from free wave solu-
tions that satisfy governing dierential equations of motion
in the frequency domain. us, compared with FEM, the
SEM can provide exact solutions by representing a uniform
structure as a single nite element, regardless of the size of
the uniform structure. Accordingly the SEM is known as an
exact solution method that has the exibility of FEM and the
exactness of continuum elements [].
Despite the outstanding features of the SEM, it is mostly
used in one-dimensional (D) structures [, ]. e SEM
application to two-dimensional (D) structures such as plates
has been limited to very specic geometries and boundar y
conditions, for example, Levy-type plates [–] and innite
or semi-innite plates [–]. Some researchers [, ] have
introduced the spectral super element method (SSEM) for
rectangular plates with prespecied boundary conditions on
two parallel edges in one direction (say, the -direction).
However, as their spectral element models can be assembled
only in another direction (the -direction), their applica-
tions must be limited to very specic boundary condi-
tions. Recently, Park et al. [, , ] developed spectral
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2016, Article ID 9475397, 20 pages
http://dx.doi.org/10.1155/2016/9475397
Mathematical Problems in Engineering
element models for rectangular membrane, isotropic plate,
andorthotropiclaminatedcompositeplateelementsby
using two methods in combination: the boundary splitting
method [] and the spectral super element method (SSEM)
[]. ey derived frequency-dependent shape functions by
applying a Kantorovich method-based nite strip element
method in one direction and the SEM in another direction
in combination, and vice versa. Accordingly, their spectral
element models can be assembled in both the -and-
directions. However, the spectral elemen t models by Park
etal.[,,]stillhavesomelimitationsbecausetheyare
valid only for nite rectangular membranes or plate elements
whose four corner nodes are xed. To the authors’ best
knowledge, there have been no reports on a generic type of
spectral element model that can be assembled in any direction
of a plate subjected to arbitrary boundary conditions.
us, the purpose of this study is to develop a new spectral
element model for nite rectangular plate elements that can
be a pplied to any plate subjected to arbitrary boundary
conditions. e new spectral element model is developed
by modifying the boundary splitting technique used in our
previous study [] so that the four corner nodes of a nite
rectangular plate element become active. e p erformance of
the new spectral element model is evaluated by comparison
with exact solutions, FEM solutions, and solutions using the
commercial nite element analysis p ackage ANSYS [].
2. Spectral Element Model for
a Finite Plate Element
2.1. Governing Equations in the Frequency Domain. e time
domain equation of motion and the boundary conditions of
plate structures with transverse vibrations are described in
[]. e time domain equation of motion of a plate can be
transformed into a frequency domain equation of motion of
the plate by using the FFT as follows []:
4
4
+2
4
2
2
+
4
4
−
2
=,,
()
where (,)is the transverse displacement in spectral form,
(,)is the external force in spectral form, is the mass
per unit area of the plate, and =
3
/[12(1−]
2
)]is the
exural bending r igidity of the plate where is the modulus
of elasticity, ] is Poisson’s ratio, and h is the plate thickness.
Similarly, the time domain boundary conditions can be
transformed into frequency domain boundary conditions as
follows:
𝑦
,−
1
2
𝑦
=−
𝑦1
(
)
or
𝑦
,−
1
2
𝑦
=
𝑦1
(
)
,
𝑦
,−
1
2
𝑦
=−
𝑦1
(
)
or ,−
1
2
𝑦
=
1
(
)
,
𝑦
,
1
2
𝑦
=
𝑦2
(
)
or
𝑦
,
1
2
𝑦
=
𝑦2
(
)
,
𝑦
,
1
2
𝑦
=
𝑦2
(
)
or ,
1
2
𝑦
=
2
(
)
,
𝑥
−
1
2
𝑥
,=−
𝑥1
or
𝑥
−
1
2
𝑥
,=
𝑥1
,
𝑥
−
1
2
𝑥
,=−
𝑥1
or −
1
2
𝑥
,=
3
,
𝑥
1
2
𝑥
,=
𝑥2
or
𝑥
1
2
𝑥
,=
𝑥2
,
𝑥
1
2
𝑥
,=
𝑥2
or
1
2
𝑥
,=
4
,
()
where
𝑥
and
𝑦
are the dimensions of a nite plate in the -
and -directions, respectively;
𝑥
and
𝑦
are the resultant
moments; and
𝑥
and
𝑦
are the resultant transverse shear
forces dened by
𝑥
=−
2
2
+]
2
2
,
𝑥
=−
2
2
+]
2
2
−2
(
1−]
)
3
2
𝑦
=−
2
2
+]
2
2
,
𝑦
=−
2
2
+]
2
2
−2
(
1−]
)
3
2
.
()
And
𝑥
and
𝑦
are the slopes dened by
𝑥
=
,
𝑦
=
.
()
We need to obtain frequenc y domain free wave solutions
for a homogeneous equation of motion in order to formulate
Mathematical Problems in Engineering
=
x
y
+
=
w
B
(x, y)
w(x, y)
)
w
A
(x, y
(a) Original problem (b) Problem A
(c) Problem B (d) SEM model
x, L
y
/2 =
2
(x)
−L
x
/2, y =
1
(y)
L
x
/2, y =
2
(y)
y
x, −L
y
/2 =
1
(x)
A
x, L
y
/2 =0
A
x, −L
y
/2 =0
A
−L
x
/2, y =
1
(y)
xA
L
x
/2, y =
x2
(y)
B
x, L
y
/2 =
2
(x)
B
x, −L
y
/2 =
1
(x)
B
−L
x
/2, y =0
B
L
x
/2, y =0
w
y
w
y
w
y w
y
w
y
w
y
w
y
w
y
w
y
w
x
w
x
w
x
w
x
w
x
w
w
x
w
x
w
w
x
w
F : Boundary splitting method used in the previous study [] to derive (,)for a rectangular plate element subjected to arbitrary
boundary conditions (e:activenodes;O:xednodes).
w
B
(x, y)
w(x
x
y
,y)
)
w
A
(x, y
w
y2
(x)
w
y
w
y
w
y
1
(x)
w
y2
(x) − w
y2
(x)
1
(x) − w
y1
(x)
+==
(a) Original problem (b) Problem A
(c) Problem B
(d) SEM model
−L
x
/2, y =
1
(y)
L
x
/2, y =
2
(y)
A
x, L
y
/2 =
A
−L
x
/2, y =
1
(y)
A
L
x
/2, y =
2
(y)
𝐰
yB
x, L
y
/2 =
B
−L
x
/2, y =0
B
L
x
/2, y =0
x, −L
y
/2 =
1
(x)
A
x, −L
y
/2 =
B
x, −L
y
/2 =
w
y
w
y
x
w
x
w
x
w
x
w
x
w
x
w
x
w
x
w
x
w
x
w
x, L
y
/2 =
2
(x)w
y
w
y
w
y
w
y
F : Boundary splitting method used in this study to derive (,)for a rectangular plate element subjected to arbitrary boundary
conditions (e:activenodes).
the spectral element model for a nite plate element. To real-
ize this, the homogeneous equation of motion is considered
by removing external force (,)in () as follows:
4
4
+2
4
2
2
+
4
4
−
2
=0.
()
e weak form of () can be obtained in the following
form:
𝑥
𝑦
2
2
+]
2
2
2
2
+2
(
1−]
)
2
2
+
2
2
+]
2
2
2
2
−
2
=0.
()
A free vibration solution satisfying the weak form given
in () can be obtained approximately by using two combined
methods: the boundary splitting method [] and the spec-
tral super element method (SSEM) []. e SSEM uses a
combination of the Kantorovich method (based on the nite
strip element method) and the frequency domain D spectral
element method.
e concept of the boundary splitting method is illus-
trated in Figures and . Figure indicates the concept used
in our previous study []. Figure indicates the concept used
in the present study. e original problems, shown in Figures
(a) and (a), are represented by the sum of two partial
problems, Problem and Problem .InFiguresand,the
geometric boundary conditions of the original problems are
presented in simple forms by using the following denitions:
w
𝑥
,=,,
𝑥
,=
𝑇
w
𝑦
,=,,
𝑦
,=
𝑇
w
𝑥𝐴
,=
𝐴
,,
𝑥𝐴
,=
𝐴
𝑇
w
𝑦𝐴
,=
𝐴
,,
𝑦𝐴
,=
𝐴
𝑇
w
𝑥𝐵
,=
𝐵
,,
𝑥𝐵
,=
𝐵
𝑇
w
𝑦𝐵
,=
𝐵
,,
𝑦𝐵
,=
𝐵
𝑇
.
()
In our previous study [], Problem ,showninFig-
ure (b), has xed (null) boundary conditions on the parallel
Mathematical Problems in Engineering
edges at =−
𝑦
/2 and
𝑦
/2.Problem,shownin
Figure (c), has xed (null) boundary conditions on the
parallel edges at =−
𝑥
/2 and
𝑥
/2.Asaresult,the
spectral element model developed in [] is valid only for
nite rectangular plate elements whose four corner nodes are
xed, as shown in Figure (d). Accordingly, an applicat ion of
this approach should be limited to very specic problems as
considered in [].
We propose a new boundary splitting method by mod-
ifying the boundary splitting method used in [] such that
the four corner nodes of a nite plate element become active.
Problem , shown in Figure (b), has arbitrary boundary
conditions rather than xed boundary conditions on the
parallel edges at =−
𝑦
/2and
𝑦
/2,anditssolutionis
represented by
𝐴
(,).Problem, shown in Figure (c),
has xed (null) b oundary conditions on the parallel edges
at =−
𝑥
/2and
𝑥
/2.However,theboundaryconditions
at =−
𝑦
/2 and
𝑦
/2 in Problem must be specied
such that the sum of the boundary conditions at =−
𝑦
/2
and
𝑦
/2in Problem and those in Problem is identical
to the boundary conditions at =−
𝑦
/2and
𝑦
/2in the
original problem. e solution of Problem is represented by
𝐵
(,).en,thesolution(,)of the original problem
can be obtained by summing the solutions to Problem and
Problem as follows:
,=
𝐴
,+
𝐵
,.
()
Accordingly, compared to the spectral element model devel-
oped in our previous study [] based on the boundary
splitting shown in Figure , the present spectral element
model that was developed based on the boundary splitting
showninFigurehasfouractivecornernodes.us,itcanbe
used as a generic nite element model that can be assembled
in both the -and-directions of a plate with arbitrary
boundary conditions.
2.2. Derivation of
𝐴
(,). To obtain the solution
𝐴
(,)
for Problem by using the SSEM, a rectangular nite plate
element is divided into
𝑦
nite strip elements in the -
direction,asshowninFigure(a).eth nite strip element,
which has a width of
(𝑗)
𝑦
=
𝑗
−
𝑗−1
in the -direction, is
shown in Figure (b).
e displacement eld in the th nite strip element can
be represented by
(𝑗)
𝐴
,=Y
(𝑗)
𝐴
w
(𝑗)
𝐴
(
)
𝑗−1
≤≤
𝑗
,
()
where Y
(𝑗)
𝐴
()is a one-by-four interpolation function matrix
and w
(𝑗)
𝐴
() arethenodallinedegreeoffreedom(DOF)
functions dened by
Y
(𝑗)
𝐴
=Y
(𝑗)
𝐴1
,Y
(𝑗)
𝐴2
w
(𝑗)
𝐴
(
)
=
w
(𝑗−1)
𝑦𝐴
(
)
w
(𝑗)
𝑦𝐴
(
)
,
()
x
y
(1)
(2)
(3)
(j)
d
U
d
L
d
R
A
w
(N
𝑦
)
yA
(x) = w
y2
(x)
w
(N
𝑦
−1)
yA
(x)
w
(j)
yA
(x)
w
(j−1)
yA
(x)
y
j−1
y
j
w
(2)
yA
(x)
w
(1)
yA
(x)
w
(0)
yA
(x) = w
y1
(x)
.
.
.
.
.
.
L
y
N
y
L
x
𝐝
B
A
(a)
(j)
w
(j)
yA
(x) = w
y
w
(j−1)
yA
(x) = w
y
w
yA
(x, y) = (x, y), 𝜃
yA
(x, y)
T
l
(j)
y
w
(j)
A
(x, y)
A
x, y
j
A
x, y
j−1
w
A
(b)
F : Finite strip element representation of a rectangular nite
plate element subjected to arbitrary boundary conditions at =
−
𝑦
/2and
𝑦
/2(e: nodes; grey circles: no dal values).
where
Y
(𝑗)
𝐴1
=
(𝑗)−3
𝑦
−
𝑗
2
⋅2+
𝑗
−3
𝑗−1
,
(𝑗)−2
𝑦
−
𝑗
2
−
𝑗−1
Y
(𝑗)
𝐴2
=−
(𝑗)−3
𝑦
−
𝑗−1
2
⋅2+
𝑗−1
−3
𝑗
,
(𝑗)−2
𝑦
−
𝑗−1
2
−
𝑗
,
()
w
(𝑗−1)
𝑦𝐴
(
)
=w
𝑦𝐴
,
𝑗−1
=
𝐴
,
𝑗−1
,
𝑦𝐴
,
𝑗−1
𝑇
w
(𝑗)
𝑦𝐴
(
)
=w
𝑦𝐴
,
𝑗
=
𝐴
,
𝑗
,
𝑦𝐴
,
𝑗
𝑇
.
()
By using (), the displacement eld
𝐴
(,) over the
entire domain of the nite plate element can be represented
as
𝐴
,=Y
𝐴
w
𝐴
(
)
−
1
2
𝑦
≤≤
1
2
𝑦
,
()
Mathematical Problems in Engineering
where
w
𝐴
(
)
=w
(0)𝑇
𝑦𝐴
(
)
,w
(1)𝑇
𝑦𝐴
(
)
,...,w
(𝑗)𝑇
𝑦𝐴
(
)
,...,
w
(𝑁
𝑦
−1)𝑇
𝑦𝐴
(
)
,w
(𝑁
𝑦
)𝑇
𝑦𝐴
(
)
𝑇
,
()
Y
𝐴
=L
(0)
𝐴
,L
(1)
𝐴
,...,L
(𝑗)
𝐴
,...,L
(𝑁
𝑦
−1)
𝐴
,
L
(𝑁
𝑦
)
𝐴
()
with
L
(0)
𝐴
=
(1)
𝐴
Y
(1)
𝐴1
L
(𝑗)
𝐴
=
(𝑗)
𝐴
Y
(𝑗)
𝐴2
+
(𝑗+1)
𝐴
Y
(𝑗+1)
𝐴1
=1,2,...,
𝑦
−1
L
(𝑁
𝑦
)
𝐴
=
(𝑁
𝑦
)
𝐴
Y
(𝑁
𝑦
)
𝐴2
.
()
In (),
(𝑗)
𝐴
()are functions dened by
(𝑗)
𝐴
=−
𝑗−1
−−
𝑗
,
()
where ()is the Heaviside unit step function.
Substituting () into () yields
A
4
4
w
𝐴
4
+A
2
2
w
𝐴
2
+A
0
−
2
M
𝐴
w
𝐴
=0,
()
where
=
,
A
0
=Λ
𝐴4
,
A
2
=] Λ
𝐴3
+Λ
𝑇
𝐴3
−2
(
1−]
)
Λ
𝐴2
A
4
=Λ
𝐴1
,
M
𝐴
=Λ
𝐴1
()
with the following denitions:
Λ
𝐴1
=
+(1/2)𝐿
𝑦
−(1/2)𝐿
𝑦
Y
𝑇
𝐴
Y
𝐴
,
Λ
𝐴2
=
+(1/2)𝐿
𝑦
−(1/2)𝐿
𝑦
Y
𝑇
𝐴
Y
𝐴
Λ
𝐴3
=
+(1/2)𝐿
𝑦
−(1/2)𝐿
𝑦
Y
𝑇
𝐴
2
Y
𝐴
2
,
Λ
𝐴4
=
+(1/2)𝐿
𝑦
−(1/2)𝐿
𝑦
2
Y
𝑇
𝐴
2
2
Y
𝐴
2
.
()
e constant matrices Λ
𝐴1
, Λ
𝐴2
, Λ
𝐴3
,andΛ
𝐴4
are provided
in Appendix A.
Next, we assume solutions of () to be in the following
form:
w
𝐴
(
)
=
1
(2)
𝐴
.
.
.
(2(𝑁
𝑦
+1))
𝐴
𝐴
+𝑘
𝑥
𝑥−(1/2)
𝑘
𝑥
𝐿
𝑥
=r
𝐴
𝐴
+𝑘
𝑥
𝑥−(1/2)
𝑘
𝑥
𝐿
𝑥
,
()
where
𝐴
is a constant,
𝑥
is the wavenumber in the -
direction, and
𝑥
=
+
𝑥
if Re
𝑥
>0
−
𝑥
if Re
𝑥
<0
0 if Re
𝑥
=0.
()
Substituting () into () gives the following eigenvalue
problem:
A
4
4
𝑥
+A
2
2
𝑥
+A
0
−
2
M
𝐴
r
𝐴
=0 ()
or
A
𝐴
2
𝑥
+A
2
𝑥
+A
0
−
2
M
𝐴
r
𝐴
=0
𝑥
=
2
𝑥
.
()
e dispersion relation (i.e., the frequency-wavenumber
relationship) can be obtained from () as follows:
det A
4
2
𝑥
+A
2
𝑥
+A
0
−
2
M
𝐴
=0. ()
From (), the wavenumbers can be computed as
𝑥(𝑗)
=+
𝑥(𝑗)
𝑥(4(𝑁
𝑦
+1)+𝑗)
=−
𝑥(𝑗)
=−
𝑥(𝑗)
=1,2,...,4
𝑦
+1.
()
By using the wavenumbers
𝑥(𝑗)
given by (), we can
write the general solution of () in the following form:
w
𝐴
(
)
=R
𝐴
E
𝐴
(
;
)
a
𝐴
,
()