COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
In-Plane Vibrations of Rectangular Plates with Rectangular Cutouts
I. Shufrin 1, M. Eisenberger 1, 2*
1
2
Faculty of Civil Engineering, Technion – Israel Institute of Technology, Haifa, 32000, Israel
Department of Building and Construction, City University of Hong Kong, Hong Kong
Email: [email protected], [email protected]
Abstract This work presents accurate numerical values of the natural frequencies and mode shapes for the in plane
vibrations of rectangular plates with rectangular cutouts. The solutions are obtained using the multi term extended
Kantorovich method. In this method a solution is assumed in one direction of the plate, and this enables to transform
the coupled partial differential equations of the plate equilibrium into an a set of ordinary differential equation. These
equation are solved exactly by the exact element method, and an approximate natural frequency is obtained. In the
second step, the derived solution is taken as the assumed solution in one direction, and the process is repeated to find an
improved approximation of the natural frequency. This process converges with a small number of solution cycles. For
plates with cutouts this process yields very accurate values even with 1 term expansion, and in some cases one can
improve these values by adding additional functions in the expansion. Several examples are given and compared with
solutions from finite element analysis.
Key words: in-plane vibration, Extended Kantorovich Method, rectangular plates, cutouts
INTRODUCTION
Plate elements are used in civil, mechanical, aeronautical and marine structures. The consideration of vibration
frequencies and modes for such plates is essential to have an efficient and reliable design. Modern engineering practice
requires also an account of higher natural frequencies of vibrations, and thus the in-plane vibrations, which are higher
then the out of plane modes become more significant in the analysis. Exact solutions for the in-plane dynamic
characteristics of rectangular plates are available only for a limited number of boundary conditions on the four edges of
the plate. In this paper the solution is obtained using the Kantorovich method. The method proposes to use for the
solution the sum of multiplications of functions in one direction by functions in the second direction. Then, one
assumes a solution in one direction, and upon substitution of this solution into the partial differential equations of the
in-plane vibrations of the plate, a set of ordinary differential equations is obtained. Solving these equations the natural
frequencies and modes are found. This solution is approximate, and the accuracy is dependent on the assumed solution.
In the extended Kantorovich method [1] this solution is a starting point for another cycle, where the derived solution is
used as the assumed solution in the second direction, and a solution is sought in the first direction using the same
procedure. After repeating this process several times convergence is obtained, and the solution is still an approximate
solution, but with very small relative error.
The extended Kantorovich method was used for the out of plane vibration of variable thickness rectangular plates by
Eisenberger and Moyal [2], using the exact beam solution as an assumed solution in the first iteration. Shufrin and
Eisenberger [3,4] have extended the solution to thick plates with constant and variable thickness using first and higher
order shear deformation theories.
For in-plane vibrations of plates Bardell et al [6] used the h-p finite element method. They solved completely free and
clamped plates with different aspect ratios. Farag and Pan [7] solved for completely clamped plate using assumed
⎯ 1185 ⎯
summation of sine functions in the two directions, and later on [8] expanded their solution to plates with two opposite
edges clamped and the other two either clamped or free. Gang and Wereley [9] used the extended Kantorovich method
for the analysis of in-plane vibrations of rectangular plates. They used only one function in the expansion and
compared their results with finite element results of a highly densed mesh. Gorman [10,11] in his work used the
superposition method to obtain high accuracy vibration frequencies and modes for plates with various combinations of
boundary conditions. Singh and Muhammad [12] presented a modified form of the Rayliegh-Ritz method for the
in-plane vibration analysis of plates and compared their solution with those in [7] and [9]. Results for the in-plane
vibration analysis of rectangular plates with cutouts were not found in the open literature.
In this paper the solution using the extended Kantorovich method is derived as multi function expansion using several
terms, and highly accurate results are obtained for the in-plane free vibration frequencies of plates with rectangular
cutouts. Results are given for several plates, and compared with FE results.
VIBRATIONS OF RECTANGULAR PLATES
1. Basic Equations The total energy of the in-plane harmonic vibrations of the plate is given by [5]
2
2
⎤
1 ⎡ ⎛ ∂v ∂u ⎞
2Gh ⎛ ∂u ∂v ⎞
∂u ∂v
Π = ∫∫ ⎢Gh⎜ + ⎟ +
− ρω 2 h(u 2 + v 2 )⎥ dxdy
⎜ + ⎟ − 4Gh
2 A ⎣⎢ ⎝ ∂x ∂y ⎠
(1 − ν ) ⎝ ∂x ∂y ⎠
∂x ∂y
⎦⎥
(1)
where u(x,y) and v(x,y) are the inplane displacements in the x and y directions, respectively, G is the shear modulus,
h ( x, y ) is the thickness, ω is the frequency of harmonic vibrations, ρ is the mass density, and ν is Poisson’s ratio.
According to the principle of minimum energy, the first variation of the functional should be equal to zero. Thus,
taking the variation
δΠ = ∫∫
A
1 −ν
⎡
⎤
(
)
(
h
u
+
h
v
u
+
h u , y + h v, x )δu , y + ⎥
ν
δ
,
,
,
x
y
x
⎢
2
⎢
⎥
⎢+ 1 − ν (h v + h u )δv + (h v + νh u )δv + ⎥ dxdy = 0
,x
,y
,x
,y
,x
,y
⎢ 2
⎥
⎢
⎥
− Ω 2 h uδu − Ω 2 h vδv
⎢
⎥
⎢⎣
⎥⎦
(2)
with
Ω 2 = ρω 2
(1 − ν )
2G
= ρω 2
(1 − ν 2 )
(3)
E
According to the Kantorovich solution procedure [2], the solution is assumed as
N
u ( x, y ) = ∑ u i ( x )U i ( y ) = {u} {U }
T
(4a)
i =1
N
v( x, y ) = ∑ vi (x )Vi ( y ) = {v} {V }
T
(4b)
i =1
where N is the number of functions in the expansion. The plate thickness is taken as follows
M
M
i =1
i =1
h ( x, y ) = ∑ h0 hi ( x )H i ( y ) = ∑ h0 hi H i
(5)
and in the current work there is no variation in the thickness in both the x and y directions.
2. Derivation of the Equations of Motion In the extended Kantorovich method the functions in the y direction are
assumed known. Then substitution of Eqs. (4) and (5) into Eqn. (2), yields
⎯ 1186 ⎯
M
⎡
⎤
⎞
⎛M
T
T
{
}
h
u
h
H
U
U
hi {v}T νh0 H i {V, y }{U }T ⎟{δu , x } +
{
}{
}
+
⎜
∑
∑
i ,x
0 i
⎢
⎥
⎠
⎝ i =1
i =1
⎢
⎥
M
⎢ ⎛M
⎥
1 −ν
⎞
T 1 −ν
h0 H i {U , y }{U , y }T + ∑ hi {v, x }T
h0 H i {V }{U , y }T ⎟{δu} + ⎥
⎢+ ⎜ ∑ hi {u}
2
2
⎠
i =1
⎢ ⎝ i =1
⎥
⎢ ⎛M
⎥
M
1 −ν
T
T 1 −ν
T ⎞
⎢ + ⎜ ∑ hi {v, x }T
h0 H i {V }{V } + ∑ hi {u}
h0 H i {U , y }{V } ⎟{δv, x } + ⎥ dxdy = 0
2
2
⎢ ⎝ i =1
⎠
⎥
i =1
⎢
⎥
M
M
⎛
⎞
⎢
⎥
+ ⎜ ∑ hi {v}T h0 H i {V, y }{V, y }T + ∑ hi {u , x }T νvH i {U }{V, y }T ⎟{δv} +
⎢
⎥
⎝ i =1
⎠
i =1
⎢
⎥
M
M
T
T
T
T
2
2
⎢
⎥
− Ω ∑ hi {u} h0 H i {U }{U } {δu} − Ω ∑ hi {v} h0 H i {V }{V } {δv}
⎢⎣
⎥⎦
i =1
i =1
δΠ = ∫∫
A
(6)
And in a more compoact form, after factoring the terms in the y direction,
[ ]
[
]
M
T ⎞
⎡
⎤
⎛M
T
(1)
{
}
h
u
S
+
hi {v}T S i(5 ) ⎟{δu , x } +
⎜
∑
∑
i ,x
i
⎢
⎥
⎠
⎝ i =1
i =1
⎢
⎥
M
M
⎢
⎥
⎛
⎞
+ ⎜ ∑ hi {u}T S i(3) + ∑ hi {v, x }T S i(6 ) ⎟{δu} +
⎢
⎥
⎝ i =1
⎠
i =1
⎢
⎥
⎢
⎥
M
M
T
⎛
⎞
T
T
⎢
⎥ dx = 0
+ ⎜ ∑ hi {v, x } S i(2 ) + ∑ hi {u} S i(6 ) ⎟{δv, x } +
⎢
⎝ i =1
⎠
⎥
i =1
⎢
⎥
M
M
⎛
⎞
⎢
⎥
+ ⎜ ∑ hi {v}T S i(4 ) + ∑ hi {u , x }T S i(5 ) ⎟{δv} +
⎢
⎥
⎝ i =1
⎠
i =1
⎢
⎥
M
M
⎢− Ω 2 ∑ h {u}T S (1) {δu} − Ω 2 2 ∑ h {v}T S (2 ) {δv}⎥
i
i
i
i
⎢⎣
⎥⎦
1 − ν i =1
i =1
[ ]
Lx
δΠ =
∫
0
[
]
[
]
[
]
[
]
[
(7)
]
[ ]
[
]
where the coefficients S(1) through S(6) are defined as
[S ]= h H
Ly
(1)
0
i
∫ {U }{U }
T
i
dy
(8a)
0
[S ]
(2 )
i
1 −ν
h0 H i
=
2
Ly
∫ {V }{V }
T
dy
(8b)
0
Ly
[ S ( ) ] = 1 −2ν h H ∫ {U }{U } dy
(8c)
[ S ( ) ] = h H ∫ {V }{V } dy
(8d)
T
3
i
0
i
,y
,y
0
Ly
T
4
0
i
,y
i
,y
0
Ly
[ S ] = ν h H ∫ {U }{V } dy
(5 )
T
i
i
0
(8e)
,y
0
[S ]
(6 )
i
Ly
1 −ν
h0 H i ∫ {V
=
2
0
}{U , y }T dy
(8f)
Integration by parts leads to a system of one-dimensional equations of motion
M
∑
i =1
M
∑
i =1
(h [ S
i
[
(1)
i
]{u }+ h ([ S ( ) ] − [ S ( ) ] ){v }− h ([ S ( ) ] − Ω [ S ( ) ]){u}) = {0}
6 T
5
, xx
]
i
i
([
3
i
,x
] [ ]
⎛
⎜⎜ hi S i(2 ) {v, xx } + hi S i(6 ) − S i(5 )
⎝
T
)
i
2
i
[
]
1
i
[
]
2
{u , x }− hi ⎛⎜ S i(4 ) − 2Ω S i(2 ) ⎞⎟{v}⎞⎟⎟ = {0}
1 −ν
⎝
⎠ ⎠
⎯ 1187 ⎯
(9)
and natural boundary conditions
[ ]
[ ]
M
M
{N x } = ⎛⎜ ∑ hi S i(1) {u , x }+ ∑ hi S i(5) {v}⎞⎟
⎝ i =1
⎠
i =1
{Q y } = ⎛⎜ ∑ hi [ S i(2 ) ]{v, x }+ ∑ hi [ S i(6 ) ]{u}⎞⎟
M
M
⎝ i =1
⎠
i =1
Lx
0
(10)
Lx
0
where Nx is the resultant force normal to the edge, and Qy is the in-plane shear force parallel to the edge (and in the y
direction). Therefore, we can impose four types of boundary conditions on each edge: (1) FF - completely free, i.e. Nx
and Qy both equal zero, (2) FC - free to move normal to the edge and restrained parallel to the edge, i.e. Nx = 0 and Qy
≠ 0, (3) CF - restraind for normal displacement and free to slide along the edge, i.e. Nx ≠ 0 and Qy =0, and (4) CC fully restrained, i.e. Nx ≠ 0 and Qy ≠ 0.
3. The Solution Procedure The solution will be performed using the exact element method [2,3,4]. For the solution,
we use the following dimensionless coordinates: ξ=x/Lx and η=y/Ly, and Eqn. (9) and Eqn. (10) can be written as
{u(ξ ) }+ [A ]{v(ξ ) }+ [A ]{u(ξ ) } + Ω [A ]{u(ξ ) } = { 0 }
{v(ξ ) }+ [A ]{u(ξ ) }+ [A ]{ v(ξ ) } + Ω [A ]{ v(ξ ) } = { 0 }
,ξξ
,ξξ
2
,ξ
1
2
,ξ
4
3
2
5
with
([ ] [ ] )⎞⎟⎠L
−1
[ ]
M
M
[A1 ] = ⎛⎜ ∑ hi S i(1) ⎞⎟ × ⎛⎜ ∑ hi S i(5 ) − S i(6 )
⎝ i =1
⎠
⎝ i =1
−1
[ ]
[
T
(12)
x
]
M
M
[A2 ] = ⎛⎜ ∑ hi S i(1) ⎞⎟ × ⎛⎜ − ∑ hi S i(3) ⎞⎟ L x 2
⎠
⎝ i =1
⎠
⎝ i =1
−1
[ ]
(13)
[ ]
M
M
[A3 ] = ⎛⎜ ∑ hi S i(1) ⎞⎟ × ⎛⎜ ∑ hi S i(1) ⎞⎟ Lx 2
⎠
⎝ i =1
⎠
⎝ i =1
[ ]
(14)
([ ] [ ] )⎞⎟⎠L
−1
M
M
[A4 ] = ⎛⎜ ∑ hi S i(2 ) ⎞⎟ × ⎛⎜ ∑ hi S i(6 ) − S i(5 )
⎝ i =1
⎠
⎝ i =1
[A5 ] = ⎛⎜ ∑ hi [ S i(2 ) ]⎞⎟
−1
T
(15)
x
[ ]
⎞ 2
⎛ M
× ⎜ − ∑ hi S i(4 ) ⎟ Lx
⎠
⎝ i =1
⎠
M
⎝ i =1
[ ]
(11)
6
−1
M
[A6 ] = ⎛⎜ ∑ hi S i(2 ) ⎞⎟ × ⎛⎜ 2
⎝ i =1
⎠
⎝ 1 −ν
∑ h [ S ( ) ]⎟⎠ L
M
2
i =1
i
i
(16)
⎞
2
(17)
x
and the resultant edge forces
{N x } = ([ A7 ]{u , x }+ [ A8 ]{v})
{Q } = ([ A ]{v }+ [ A ]{u})
9
y
,x
Lx
(18)
0
Lx
(19)
10
0
with
[A7 ] =
1
Lx
∑ h [S( )]
M
1
i =1
i
(20)
i
[A8 ] = ∑ hi [ S i(5 ) ]
M
(21)
i =1
[A9 ] =
1
Lx
∑ h [S( )]
M
2
i =1
i
(22)
i
[A10 ] = ∑ hi [ S i(6 ) ]
M
(23)
i =1
⎯ 1188 ⎯
Assuming that the unknowns u and v are infinite power series, as
∞
{ u } = ∑ { u }i ξ i
i =0
∞
{ v } = ∑ { v }i ξ i
i =0
the solution is obtained using the exact element method [3,4], which enables to derive the exact dynamic stiffness
matrix of the one dimensional element. The complete procedure is not given here. The natural frequencies of
vibrations are the values of ω that will cause the assembled dynamic stiffness matrix of the structure, after imposing
the particular boundary conditions, to become singular.
4. Treatment of cutouts The cutouts are realized by proper representation of the plate as composed of several
subdomains, and in each of these the thickness is represented by a sum of two terms,
h = h0 h1 H 1 + h0 h2 H 2
(24)
such that in the cutout areas these sum up to zero. As an example, three plates with cutouts are shown in Fig. 1:
(a)
centrally located square cutout in a square plate (b) rectangular cutout in a rectangular plate and (c) square corner
cutout in a square plate. In the same way several cutouts with various geometric setups can be analyzed in a plate.
Figure 1: Three plates with internal and corner cutouts
EXAMPLES
The first example is of a square plate with a cenrally located square cutout as shown in Fig. 2. The plate is fully
rstrained (CC) on all edges. The edge size is given by c=0.4a, where a is the edge length. In all calculations Poisson’s
ratio ν is taken as 0.3. The natural frequencies are presented in non-dimensional form as Ω = ω a 2 ρ (1 − ν 2 ) E .
Figure 2: Example 1 – fully restrained square plate with a square cutout
⎯ 1189 ⎯
In Table 1 comparison is made between of the results of the extended Kantorovich solution and results calculated using
the ANSYS FE program with 8520 DOF. The number of functions in the expansion (N) is taken between 1 and 3. In
Table 2 the first 5 mode shapes are given for the calculated frequencies and compared with the results from ANSYS.
From the results in Tables 1 and 2 it can be seen that the convergence of the frequency values and the mode shapes is
very rapid. For the fourth mode the single function result is already converged and below the finite element result. For
the fifth mode, the addition of the second function results in the converged result.
The second example is for a plate where the cut out is along the edges and termed corner cutout, as shown in Fig. 3. The
upper quarter part of the plate is cut out, and on these boundaries the plate is completely free (FF), while all the other
boundaries are fully restrained (CC). In Table 3 comparisson is given with the results from Ansys FE analysis using
7400 DOF. It is seen that with 3 function expansion almost converged results are obtained, even for this complex
structure. In Table 4 the mode shapes are compared, and very close similarity is observed.
Table 1 Example 1 - Non-dimensional frequencies for a square with a square cutout
Present
ANSYS
8520 DOF
ANSYS
35000 DOF
1 term
2 terms
3 terms
4.369
4.367
4.609
4.403
4.378
4.467
4.466
4.545
4.480
4.471
4.467
4.466
4.545
4.481
4.472
4.546
4.545
4.541
5.426
5.426
5.443
5.426
Table 2 Example 1 – Mode shapes for a square with a square cutout
First inplane mode
U
V
Second inplane mode
U
V
Third inplane mode
U
V
ANSYS
8520 DOF
4.369
4.467
4.467
4.609
4.545
4.545
4.403
4.480
4.481
4.378
4.471
4.472
One term
solution
Two terms
solution
Three
term
solution
⎯ 1190 ⎯
Table 2 Example 1 – Mode shapes for a square with a square cutout (continued)
Fourth inplane mode
U
Fifth inplane mode
V
U
V
ANSYS
8520 DOF
4.546
5.426
4.541
5.443
One term
solution
Two terms
solution
5.426
Figure 3: Example 2 – fully restrained square plate with a corner cutout
Table 3 Example 2 - Non-dimensional frequencies for a square plate with a square cut at the corner
ANSYS
7400 DOF
1 Term
2 Terms
3 Terms
3.349
3.505
3.360
3.351
3.591
3.740
3.598
3.592
3.941
4.167
3.957
3.943
⎯ 1191 ⎯
Table 4 Example 2 – Mode shapes for a square plate with a square cut at the corner
First inplane mode
U
V
Second inplane mode
U
V
Third inplane mode
U
V
ANSYS
7400 DOF
3.349
3.591
3.941
3.351
3.592
3.943
3 Term
solution
SUMMARY
The vibration frequencies and mode shapes for a rectangular plate with rectangular cutouts were found using the
extended Kantorovich method. In this work multi-term expansion was used to enable fast convergence of the results. It
was shown that 3 term expansion gave almost complete convergence.
REFERENCES
1. Kerr AD. An extended Kantorovich method for the solution of eigenvalue problem. Int. Jour. Solids Struc., 1969;
15: 559-572.
2. Eisenberger M, Moyal H. Vibration frequencies of variable thickness plates. Proc. Advances in Structural
Dynamics 2000, Hong Kong, China, 2000; 1: 645-650.
3. Shufrin I, Eisenberger M. Stability and vibration of shear deformable plates – first order and higher order
analyses. Int. Jour. Solids Struc., 2005; 42: 1225-1251.
4. Shufrin I, Eisenberger M. Vibration of shear deformable plates with variable thickness – first order and higher
order analyses. Jour. Sound Vibr., 2006; 290: 465-489.
5. Reddy, JN. Theory and Analysis of Elastic Plates. Taylor & Francis, Philadelphia, PA, USA, 1999.
6. Bardell NS, Langley RS, Dundson JM. On the free in-plane vibration of isotropic rectangular plates. Jour. Sound
Vibr., 1996; 191: 459-467.
7. Farag NH, Pan J. Free and forced in-plane vibration rectangular plates. Jour. Acoust. Soc. Am., 1998; 103:
408-413.
8. Farag NH, Pan J. Modal characterstics of in-plane vibration rectangular plates. Jour. Acoust. Soc. Am., 1999;
105: 3295-3310.
9. Wang G, Wereley NM. Free in-plane vibration of rectangular plates. AIAA Jour., 2002, 40: 953-959.
10. Gorman DJ. Free in-plane vibration analysis of rectangular plates by the method of superposition. Jour. Sound
Vibr., 2004; 272: 831-851.
11. Gorman DJ. Accurate analytical type solutions for the free in-plane vibration of clamped and simply supported
rectangular plates. Jour. Sound Vibr., 2004; 276: 311-333.
12. Singh AV, Muhammad T. Free in-plane vibration of isotropic non-rectangular plates. Jour. Sound Vibr., 2004;
273: 219-231.
⎯ 1192 ⎯