ABSTRACT: The development of the dynamic stiffness matrix for rectangular plate element undergoing in-plane vibration is presented in this paper. Gorman’s superposition method is exploited to obtain the solution of the governing equations of motion. The dynamic stiffness matrix of completely free rectangular plate element is derived using the Projection method. Free vibration analysis of plate assemblies with arbitrary boundary conditions is carried out. The results obtained using the proposed method, are in very good agreement with the results obtained using exact solutions and finite element method. Modeling of plate assemblies using the spectral element method demonstrate high precision, accuracy and low memory cost in comparison with the finite element method.

KEY WORDS: In-plane vibration; Dynamic stiffness matrix; Spectral element method.

1 INTRODUCTION

Unlike the free transverse vibration, only few studies have been dedicated to in-plane free vibration of rectangular plates, since it is more likely to excite the transverse vibration of plates by the external excitations. Bardell et.al. [1] exploited the Rayleigh-Ritz method to obtain free vibration characteristics of rectangular plates with simply supported, clamped and free boundaries. Xing and Liu [2] used the direct separation of variables to derive all possible exact solutions for free in-plane vibration of rectangular plates for two opposite edges simply supported. Instead of the widely used Finite element method (FEM) [3] the Spectral element method (SEM) (or the dynamic stiffness method) can be used in order to solve various types of dynamic problems. The SEM is based on the spectral representation of the displacement field and on the exact solution of the governing equations of motion defined in the frequency domain. Consequently, the SEM is especially efficient for structures consisting of one-dimensional elements (beams and bars). For two-dimensional elements - plates exact solutions of the governing equations of motion for both transverse and in-plane vibration exist only for rectangular plates with special boundary conditions: two opposite edges simply supported [4-6] and two-edge infinite plates [7-9]. Gorman [10] developed an analytical-type solution using the superposition method to obtain free in-plane natural frequencies of completely free rectangular plate. This method cannot be easily applied in the free vibration analysis of complex structures like plate assemblies consisting of plates of different geometrical and material properties. Therefore, the method is efficient for calculation the free in-plane vibrations of an individual rectangular plate.

A general method for free in-plane vibration analysis of rectangular plates having arbitrary boundary conditions, geometrical properties and non-uniform distribution of structural materials is presented in this paper. The development of the dynamic stiffness matrix of completely

free rectangular plate is grounded on Gorman’s superposition method [10] and the Projection method [11]. The global dynamic stiffness matrix of plate assemblies with arbitrarily assigned boundary conditions has been constructed using the same assemblage procedure as in the FEM. Verification of the developed dynamic stiffness matrix has been conducted through several numerical simulations of free in-plane vibrations of plates with different types of boundary conditions. The results have been compared with the results computed using exact solutions from the literature [2], [10], the results based on the Rayleigh-Ritz method [1], and also the results obtained using the finite element software SAP2000 [12].

2 DYNAMIC STIFFNESS FORMULATION FOR IN-PLANE VIBRATION OF RECTANGULAR PLATE ELEMENT

2.1 General solution

Equations of motion of rectangular plate element presented in Figure 1 can be written as:

2 2 2 2

1 22 2 2

2 2 2 2

1 22 2 2

0

0

u u v h ua a

x y Dx y t

v v u h va a

x y Dy x t

(1)

where

1 2 2

1 1, ,

2 2 1

Eha a D

,

h is plate thickness, ρ is mass density, E is the Young’s modulus, is the Poisson’s ratio of the plate material. If in-plane displacements u and v are given in the spectral representation:

Free in-plane vibration of rectangular plates using spectral element method

M. Nefovska-Danilović1, M. Petronijević1

1Department of Engineering Mechanics and Theory of Structures, Faculty of Civil Eng., University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

email: marija@grf.bg.ac.rs, pmira@grf.bg.ac.rs

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

3869

ˆ( , , ) ( , , )

ˆ( , , ) ( , , )

i t

i t

u x y t u x y e

v x y t v x y e

(2)

Figure 1. Rectangular plate element and boundary conditions

where ω is angular frequency, ˆ( , , )u x y and ˆ( , , )v x y are

the amplitudes of displacements in the frequency domain, the Fourier transform of Eq. (1) can be expressed as:

2 2 2 2

1 22 2 2

2 2 2 2

1 22 2 2

ˆ ˆ ˆˆ 0

ˆ ˆ ˆˆ 0

p

p

u u va a u

x yx y c

v v ua a v

x yy x c

(3)

where 21

p

Ec

represents the longitudinal wave

speed. Introducing Gorman’s superposition method [10], the in-plane displacements of a rectangular plate can be split up into four contributions: symmetric-symmetric (SS), anti-symmetric-anti-symmetric (AA), symmetric-anti-symmetric (SA) and anti-symmetric-symmetric (AS):

ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , )

ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , )SS AA SA AS

SS AA SA AS

u x y u x y u x y u x y u x y

v x y v x y v x y v x y v x y

(4)

The free vibration mode is symmetric about an axis if displacement normal to this axis has a symmetric distribution about it, while the vibration mode is anti-symmetric about an axis if displacement normal to this axis has an anti-symmetric distribution about it. In order to find the solution of equations (3), the in-plane displacements are presented in infinite series form, which are truncated to a point M for practical purposes. Each symmetry contribution can be written in the following form:

1 1 2 1

1 2 2 2

ˆ ( , )

ˆ ( , )

Mm m

ij ij m ij mm

Mm m

ij ij m ij mm

u x y U y h x U x h y

v x y V y h x V x h y

(5)

where i, j = S, A, 1mh x , 2

mh x , 1mh y and 2

mh y are

trigonometric functions, which depend on the type of

symmetry contribution. Functions 1 mijU y , 2 m

ijU x ,

1 mijV y and 2 m

ijV x , given in Eq.(5) for each contribution,

represent the solutions of the system of equations (3). They correspond to the Levy-type solutions for plate with the following boundary conditions assigned to parallel edges:

Figure 2. One quarter of rectangular plate element

ˆˆ , 0 , 0 for

ˆˆ , 0 , 0 for

xy

xy

u a y N a y x a

v x b N x b y b

(6)

These functions are defined in the Appendix.

2.2 Projection method for the double symmetry contribution

Gorman’s superposition method presented in the previous section enables the analysis of one quarter of the rectangular plate element for each symmetry contribution, Figure 2. The development of the dynamic stiffness matrix for the double symmetry contribution will be presented in the following. The dynamic stiffness matrices of the other three contributions can be obtained likewise. The in-plane displacement vector SSq and the corresponding

force vector SSQ for the SS contribution for each frequency

ω, along the boundary x = a and y = b of the quarter segment of rectangular plate element are defined as:

1 2 3 4

1 2 3 4

ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )

ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )SS SS SS SS

TSS SS SS SS SS

SS SS SS SS

TSS SS SS SS SS

x xy xy y

q q q q

u a y v a y u x b v x b

Q Q Q Q

N a y N a y N x b N x b

q

Q (7)

where

1

ˆ ˆˆ ,

ˆ ˆˆ ,

ˆ ˆˆ ,

x

y

xy

u vN x y D

x y

u vN x y D

x y

u vN x y Da

y x

(8)

The displacement and force vectors SSq and SSQ are

continuous functions of spatial variables s (s=x, y). Consequently, it is not possible to define the relation between

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the displacement and force vector on the boundary as in the case of one-dimensional elements. In order to solve the problem, Kevorkian and Pascal [11] defined the Projection method for free transverse vibration of rectangular plate. It is based on the projection of the displacement and force vector onto a set of projection functions h(s):

1

1

( ) , , 1,..., 4

( ) , , 1,..., 4

i i

i i

M

SS SS m mm

M

SS SS m mm

q s q h h s i

Q s Q h h s i

(9)

where the projection of displacement and force functions along the plate boundary s is defined as:

2, ( )

2, ( )

i i

i i

SS m SS ms

SS m SS ms

q h q s h s dsL

Q h Q s h s dsL

(10)

Projection functions for in-plane vibration and each symmetry contribution are given in the Appendix. Substituting Eq.(5)

into Eq.(9) and using Eq.(10), the projections SSq and SSQ

of the vectors SSq and SSQ are obtained in the following

form:

SS SS

SS SS

q D C

Q F C

(11)

where SSD and SSF are frequency dependent matrices that

relate the projections of the displacement and force vectors and the vector of integration constants C , respectively. Elimination of the vector C from equations (11) leads to the dynamic stiffness matrix of rectangular plate element for the double symmetry contribution:

1

SS

SS

SS SS

SS SS

D

D

Q K q

K F D

(12)

Equations (11) and (12) have the same form as the corresponding expressions for one-dimensional spectral elements, [13]. However, the displacement and force vectors

SSq and SSQ in this case comprise the projections of the

displacements and forces along the plate boundary. In addition, the size of these vectors is influenced by the number of terms M in the general solution. Consequently, the size of

the dynamic stiffness matrix SSDK is 4M. Sizes of the

dynamic stiffness matricesSADK (

ASDK ) and AADK are

4M+1 and 4M+2, respectively.

2.3 Development of the dynamic stiffness matrix of completely free plate element

The dynamic stiffness matrix of completely free plate element can be derived from the dynamic stiffness matrices of each symmetry contribution using the transformation matrix T. The

transformation matrix relates the projections of the displacement vector oq defined for one quarter of plate

element and the projection of the displacement vector q of

the whole completely free plate element, in the following way:

1

2oq Tq (13)

where

(16 4) (16 4)

,

SS SS

SA SA

AS AS

AA AA M x M

o

q T

q Tq T

q T

q T

 

A detailed development of the transformation matrix T can be found in reference [14]. Similarly, the reverse procedure is carried out for the corresponding force vectors and the following relation is obtained:

T oQ T Q (14)

where Q is the projection of the force vector of the whole

completely free plate element and oQ comprises the

projections of forces defined for one quarter of plate element: T

SS SA AS AA oQ Q Q Q Q

Since

0 0 0

0 0 0

0 0 0

0 0 0

SS

SA

AS

AA

D

Do o o o

D

D

K

KQ K q q

K

K

(15)

the dynamic stiffness matrix of completely free rectangular plate element is obtained from equations (13) – (15) in the form:

1

2TD oK T K Tq (16)

3 NUMERICAL ANALYSIS AND RESULTS

The presented approach has been used for the free vibration analysis of plates with different boundary conditions. For this reason, a computer program for in-plane free vibration analysis of plate assemblies with arbitrary boundary conditions was developed using Matlab [15]. The global dynamic stiffness matrix of plate assemblies was obtained applying the same assemblage procedure as in the FEM.

In order to verify the accuracy and convergence of the proposed method two plates with aspect ratios b/a = 1 and b/a = 1.2 were analyzed. Obtained results were compared with the results available in the literature [1], [10], [2] and [6], as well with the results obtained using SAP2000 [12]. Four different Dimensionless natural frequencies pa c have been

calculated for different number of terms in the general solution - M and for the following boundary conditions:

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simply supported edge (S1): u ≠0, v = 0 for x =±a and u = 0, v ≠0 for y =±b,

simply supported edge (S2): u = 0 , v ≠0 for x =±a and u ≠0 , v = 0 for y =±b,

clamped edge (C): u = 0, v = 0,

free edge (F): u ≠0, v ≠0.

The natural frequencies are determined as maxima of the following expression, [16]:

1log

det

nnDK

(17)

where nnDK is the dynamic stiffness sub-matrix of plate

assemblies related to the unknown generalized displacement projections. Dimensionless natural frequencies for square and rectangular plate are presented in Table 1 and Table 2, respectively. The convergence of the proposed method has been achieved for only 3 – 5 terms in the general solution. In addition, the obtained results are in very good agreement with the results obtained using the exact solutions [6] and [10], as well as the results obtained by Bardell [1] and SAP2000 [12]. The efficiency and accuracy of the proposed method is shown on the example of plate assembly with non-uniform geometrical properties and different types of boundary conditions, Figure 3. The natural frequencies, summarized in Table 3 are compared only with the FEM results, since there are no other available results in the literature. As the number of finite elements increases, the natural frequencies computed using the FEM converge to the present solutions. In the present method the discretization is influenced only by the plate geometry. Therefore, the assembly is discretized only by using two spectral elements. It significantly increased the accuracy and decreased the computational time in comparison with the FEM.

Table 1. Dimensionless frequencies of square plate (b/a = 1).

BC Freq. num.

Present solution Ref. [10]

Ref. [1] M=3 M=5 M=7 M=11

F –

F –

F -

F

1 2 3 4 5 6 7 8

1.160 1.241 1.317 1.501 1.726 1.868 2.177 2.499

1.160 1.237 1.315 1.497 1.726 1.864 2.161 2.490

1.160 1.237 1.315 1.495 1.726 1.863 2.157 2.487

1.160 1.236 1.314 1.494 1.726 1.862 2.153 2.486

1.160 1.236 1.314 1.494 1.726 1.862 2.153 2.485

1.160 1.236 1.314 1.493 1.726 1.868 2.177 2.499

S1

- S

1 -

S1

- S

1 1 2 3 4 5 6 7

0.923 1.317 1.865 2.074 2.210 2.636 2.764

0.927 1.315 1.858 2.076 2.217 2.631 2.780

0.928 1.315 1.860 2.077 2.219 2.630 2.784

0.929 1.315 1.857 2.078 2.221 2.629 2.786

-

0.929 1.314 1.858 2.078 2.222 2.628 2.788

C –

C –

C–

C 1

2 3 4 5

1.778 2.122 2.593 2.929 2.947

1.778 2.120 2.593 2.930 2.948

1.778 2.118 2.592 2.930 2.948

1.778 2.118 2.592 2.930 2.948

-

1.778 2.118 2.593 2.930 2.948

Table 2. Dimensionless frequencies of rectangular plate (b/a = 1.2).

BC Freq. num.

Present solution

Ref. [6]

Ref. [2]

FEM (SAP2000)

M=11 50x60

elements 100x120 elements

S1

- S

1 -

S1

- S

1

1 2 3 4 5 6 7 8 9 10

0.774 0.929 1.210 1.548 1.806 1.857 2.012 2.044 2.322 2.420

0.774 0.929 1.210 1.549 1.806 1.859 2.013 2.044 2.323 2.419

0.774 0.929 1.210 1.549 1.806 1.859 2.013 2.044 2.323 2.419

0.774 0.929 1.209 1.548 1.805 1.857 2.012 2.044 2.321

2.416

0.774 0.929 1.209 1.548 1.806 1.858 2.013 2.044 2.322

2.418

S2 -

F -

S1

- C

1 2 3 4 5 6 7 8 9 10

0.685 0.838 1.301 1.420 1.818 1.895 2.132 2.240 2.381 2.442

-

0.684 0.839 1.298 1.417 1.821 1.896 2.130 2.242 2.384 2.445

0.684 0.839 1.298 1.417 1.820 1.894 2.128 2.240 2.382 2.443

0.684 0.839 1.298 1.417 1.820 1.895 2.129 2.241 2.383 2.444

Figure 3. Geometry of plate assembly

4 CONCLUSIONS

The general method for the development of the dynamic stiffness matrix of rectangular plate element undergoing in-plane vibration has been presented in this paper, using Gorman’s superposition method and the Projection method. It allows free vibration analysis of any rectangular plate assemblies with arbitrary boundary conditions and different geometrical and material properties that cannot be solved using the available solutions in the literature. The present solution has been validated against the exact results available in the literature and the results obtained from the SAP2000 finite element code. The results indicated high precision of the proposed solution in comparison with the FEM. The proposed method has used the same assemblage procedure as in the FEM to obtain the global dynamic stiffness matrix of plate assemblies. However, the discretization has been minimized, which significantly increases the accuracy and decreases the computational time.

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Table 3. Natural frequencies (in Hz) of plate assembly

BC Freq. num.

Present solution

FEM (SAP2000)

M=7 M=9 24x20

elements 48x40

elements 96x80

elements

C –

F –

F–

F

1 2 3 4 5 6 7 8 9

10

305.6 835.1 1007 2142 2228 2494 2747 3167 3542 3569

305.4 835.2 1007 2141 2228 2493 2746 3166 3542 3569

305.3 835.3

1005.2 2126.5 2228.4 2487.7 2737.7 3145.2 3532.9 3550.5

305.2 835.2

1006.2 2136.5 2228.2 2491.3 2743.1 3160.0 3539.8 3564.5

305.1 835.1

1006.4 2139

2227.9 2492.2 2744.3 3163.5 3541.4 3567.8

C –

C –

F–

F

1 2 3 4 5 6 7 8 9

10

976.4 1381 1671 2648 2956 3313 3460 3853 4017 4127

976.4 1381 1670 2648 2955 3312 3460 3852 4017 4126

977.4 1379.5 1663.7 2637.6 2933.9 3292.8 3446.4 3849.4 3989.5 4098.8

977.8 1379.8 1666.8 2641.3 2943.3 3301.5 3456.6 3874.5 4004.8 4118.3

977.9 1379.8 1667.5 2642.1 2945.5 3303.6 3459.1 3880.5 4008.5 4123.2

ACKNOWLEDGMENTS

The authors are grateful to the Ministry of Education, Science and Technology, Republic of Serbia, for the financial support of this research within the Project TR 36046.

REFERENCES [1] N. S. Bardell, R. S. Langley, J. M. Dunsdon, On the free in-plane

vibration of isotropic rectangular plates. Journal of Sound and Vibration 1996; 191(3): 459-467.

[2] Y. F. Xing, B. Liu, Exact solutions for the free in-plane vibrations of rectangular plates. International Journal of Mechanical Sciences 2009; 51: 246-255.

[3] K. J. Bathe, E. Wilson, Numerical method in finite element analysis. Prentice-Hall, 1976.

[4] U. Lee, J. Lee, Spectral element method for Levy-type plates subject to dynamic loads. Journal of Engineering and Mechanics 1999; 125: 243-247.

[5] M. Boscolo, J. R.Banerjee, Dynamic stiffness elements and their application for plates using first order shear deformation theory. Computers and Structures 2011; 89: 395-410.

[6] M. Boscolo, J. R.Banerjee, Dynamic stiffness method for exact inplane free vibration analysis of plates and plate assemblies. Journal of Sound and Vibration 2011; 330: 2928-2936.

[7] M. S. Anderson, F. W. Williams, C. J. Wright, Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supporting structure. International Journal of Mechanical Sciences 1983; 25: 585-596.

[8] A. N. Danial, J. F. Doyle, S. A. Rizzi , Dynamic analysis of folded plate structures. Journal of Vibration and Acoustics 1996; 118: 591-598.

[9] A. N.Bercin , Eigenfrequencies of rectangular plate assemblies. Computers and Structures 1997; 65: 703-711.

[10] D. J. Gorman , Free in-plane vibration analysis of rectangular plates by the method of superposition. Journal of Sound and Vibration 2004; 272: 831-851.

[11] S . Kevorkian, M. Pascal , An accurate method for free vibration analysis of structures with application to plates. Journal of Sound and Vibration 2001; 246(5): 795-814.

[12] SAP2000. Integrated finite element analysis and design of structures. Berkeley, California: Computers and Structures Inc., 1996.

[13] J . R. Banerjee, Dynamic stiffness formulation for structural elements: A general approach. Computers and Structures 1997; 63(1): 101-103.

[14] M. Nefovska-Danilovic Dynamic analysis of soil-structure system using spectral element method, Phd Thesis, Faculty of Civil Engineering, University of Belgrade, 2013

[15] Matlab, MathWorks Inc. The Language of technical computing. MATLAB 2011b. 2011.

[16] J. F. Doyle, Wave propagation in structures. New York: Springer-Verlag, 1997.

APPENDIX

In-plane displacements and projection functions for the SS contribution

12 1 4 1

22 2

11 1

21 2 3 2

sin sin

cos cos

cos cos

sin sin

m m m m

m m

m m

m m m m

SSm m m

SSm m m

SSm m m

SSm m m

U y C y D y

U x A x B x

V y C y D y

V x A x B x

 

 

1

2

1

2

2 1cos

22 1

sin2

2 1sin

22 1

cos2

m

m

m

m

m xh x

am x

h xa

m yh y

bm y

h yb

In-plane displacements and projection functions for the AA

contribution

12 1 4 1

22 2

11 1

21 2 3 2

cos cos

sin sin

sin sin

cos cos

m m m m

m m

m m

m m m m

AAm m m

AAm m m

AAm m m

AAm m m

U y C y D y

U x A x B x

V y C y D y

V x A x B x

 

 

1

2

1

2

sin

cos

cos

sin

m

m

m

m

m xh x

am x

h xa

m yh y

bm y

h yb

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In-plane displacements and projection functions for the SA contribution

12 1 4 1

22 2

11 1

21 2 3 2

sin sin

sin sin

cos cos

cos cos

m m m m

m m

m m

m m m m

SAm m m

SAm m m

SAm m m

SAm m m

U y C y D y

U x A x B x

V y C y D y

V x A x B x

 

 

1

2

1

2

sin

cos

2 1sin

22 1

cos2

m

m

m

m

m xh x

am x

h xam y

h yb

m yh y

b

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