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Free and Forced Vibration of Rectangular Plates Using the Finite

Difference Method

Yousef S. Al Rjouba* and Osama Abdeljaber b aCivil Engineering Department, Jordan University of Science and Technology, P. O. 3030, Irbid,

Jordan.

bGraduate Student, Civil Engineering Department, Jordan University of Science and Technology, P. O.

3030, Irbid, Jordan.

ABSTRACT

This paper presents a finite difference method to solve free and forced

vibration problems of rectangular plates with differing boundary conditions. The

natural frequencies are obtained from the peaks of the free vibration response in the

frequency domain by exciting the plate with an initial displacement. The free

vibration response in the time domain is calculated using the finite difference method.

This is then converted to the frequency domain using Fourier transform. In this paper,

the plate is subjected to various dynamic loadings, namely, a step function,

rectangular and triangular loads, and a sinusoidal harmonic loading. The present

results are compared to analytical and numerical solutions available in the literature.

The results obtained are in good agreement with those of exact and numerical results

available in the literature.

Keywords—Dynamic loadings, Finite difference method, Natural frequency,

Plate.

___________________________________________

*Corresponding author. E-mail: [email protected], Tel: +962 795204911, Fax: +962 2 70950185

mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Introduction and literature review

Plate elements are being used increasingly in various fields of engineering,

especially in aerospace, mechanical, civil, and hydraulic structures. The analysis of

free and forced vibration of rectangular plates is highly significant in the performance

of such structures. As a result, much research has been conducted to determine the

dynamic plate behavior. Leissa (1969, 1973) presents comprehensive and accurate

analytical results of the free vibration of rectangular plates with differing boundary

conditions. Twenty-two cases are considered, based on the possible combinations of

clamped, simply-supported, and free-edge conditions. An exact solution is obtained

for six cases having two opposite sides simply-supported. The remaining 15 cases are

analyzed using the Ritz method, with 36 terms containing the products of beam

functions. An approximate solution for the free vibration of rectangular plates using

the Rayleigh-Ritz method has been obtained by several researchers (Leissa, 2005;

Ilanko, 2009; Young, 1950; Bassily and Dickinson, 1975; Bhat, 1985; Liew and Lam,

1991; Lim and Liew, 1995). Other researchers (Leung and Chan, 1998; Zienkiewicz

and Taylor, 1989; Reddy, 1993) used a finite element method. Eftehari and Jafari

(2012) presented a combined application of the Ritz and differential quadrature (DQ)

methods to solve the free and forced vibration of rectangular plates with differing

boundary conditions. Yang, et al. (2012) used the finite difference method to obtain

the complex natural frequencies for linear free vibration, bifurcation, and chaos for

forced nonlinear vibration of an axially moving plate. Gupta, et al. (2012) presented

the forced vibration of a non-homogeneous simply-supported-free-simply-supported-

free rectangular plate of variable thickness. An approximate formula is obtained to

estimate the maximum deflection of a rectangular plate subject to uniform harmonic

loading. Wang and Xu (2010) used discrete singular convolution (DSC) to analyze thefree vibration of beams, annular plates, and rectangular plates with free boundary

edges. This method is examined using eight examples of free vibration of beams,

annular plates, and rectangular plates with differing boundary conditions. Duccechi, et

al. (2014) used the Von Karman equations to analyze the nonlinear vibration of thin

rectangular plates. The eigenmodes of a fully clamped plate are calculated based on a

fast converging solution strategy. The nonlinear dynamic of the first two modes, both

in the free and forced regimes, is taken into consideration. A closed form solution of

the natural frequencies and the corresponding mode shapes of thin plates with any


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type of boundary condition is obtained by Xing and Liu (2009), using a dual method.

A closed form solution for the free vibrations of rectangular Mindlin plates with any

boundary condition is proposed by Xing and Liu (2009), using a new two-

eigenfunctions theory. The proposed theory is obtained by reformulating the three

classical eigenvalue differential equations of a Mindlin plate. Then, separation of

variables is used to solve the two differential equations, which are identical to those of

Kirchhoff plate theory, Timoshenko and Krieger (1959). Werfalli and Karoud (2012)

studied the free vibration of thin, isotropic rectangular plates with various edge

conditions using a Galerkin-based finite element method. A closed form solution for

the free vibration of rectangular thin plates with three edge conditions, namely fully

simply-supported, fully clamped, and two opposite edges simply supported and the

other two edges clamped, is obtained by Wu, et al (2007), using Bessel functions.

Njoku, et al (2013) used the peculiar shape functions of the Taylor series, along with

Galerkin's method, to determine the natural frequencies of a fully clamped, isotropic,

thin rectangular plate. An analytical solution of the free vibration of a completely

simply supported rectangular Kirchhoff plate is obtained by Bahrami, et al. (2008),

using a wave propagation approach. Jain, et al. (1973) studied the free vibration of

rectangular plates having parabolically varying thickness, with two simply-supported

parallel edge conditions, based on the classical theory of plates. The Frobenius

method is used to solve the equation of motion. As a result of the product of an

infinite series and a function satisfying the boundary conditions at two simply

supported parallel edges, the deflection of the plate is obtained accordingly. Numayr,

et al (2004) used the finite difference method to solve the free vibration of composite

plates with differing boundary conditions. In their study, the effects of shear

deformation and rotary inertia are included. Yeh, et al. (2006) analyzed the free

vibrations of clamped and simply- supported rectangular thin plates using the finite

difference and differential transformation methods. The order of the differential

transformation, the number of sub-domain spaces, the variable conditions, and the

type of initial condition are used as investigative parameters.

It is clear form the previously outlined literature that the dynamic behavior of

rectangular plate subjected to impact loading is very important. In this paper, an

attempt to understand the dynamic behavior of rectangular plates with differing

boundary conditions using the finite difference method to perform dynamic analysis is


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made. In this paper, Fourier transform is used to convert the free vibration response

from the time domain, obtained by the finite difference method, to the frequency

domain, thereby obtaining the natural frequencies. In the forced vibration case, the

rectangular plate is subjected to various impact loadings, such as, constant force,

rectangular, triangular, and sinusoidal harmonic. The natural frequencies are obtained

for plates with differing boundary conditions and are compared to exact and

numerical results available in the literature. The dynamic response in the forced

vibration regime is studied.

Mathematical Model of a Plate

Consider an isotropic, elastic, rectangular Kirchhoff plate of thickness h, lengths a

and b in the x- and y-directions, respectively, as shown in Fig. 1. The plate has mass

density , Young's modulus E, Poisson's ratio , and flexural rigidity) ν12(1

EhD

2

3

.

The differential equation of motion for forced vibration is given by [20] as:

t)y, p(x,t

t)y,w(x,

ρhy

t)y,w(x,

yx

t)y,w(x,

2x

t)y,w(x,

D 2

2

4

4

22

4

4

4

(1)

where w(x,y,t), and p(x,y,t) are the transverse deflection of the plate and the

dynamic loadings applied, respectively.

Four types of dynamic loadings are considered, namely, a step function of infinite

duration, a rectangular load of finite (td=0.1 sec) duration, a triangular load of finite

(td=0.1 sec) duration, and a sinusoidal harmonic loading of finite duration (t d=0.1 sec),

Fig. 2.

In this study, a plate with six different boundary conditions are considered,

namely, SSSS, SCSS, SCSC, CCCC, CCCS, and CCSS, where S refer to a simply-

supported edge and C to clamped edge. The boundary conditions for simply-

supported edge can be written as,

0y

t)y,w(x, υ

x

t)y,w(x,t)y,w(x,

2

2

2

2

at x=0 or a,


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0x

t)y,w(x, υ

y

t)y,w(x,t)y,w(x,

2

2

2

2

at y=0 or b.

and for a clamped edge as,

0x

t)y,w(x,t)y,w(x,

at x=0 or a,

0y

t)y,w(x,t)y,w(x,

at y=0 or b.

Equation (1) can be written in a finite difference form, Fig. 3, as:

2)t j,w(i,Δt

ρh1)t j,w(i,

Δt

2ρt) j, p(i,

D

Δt) j,2,w(i

t) j,2,w(it)2, jw(i,t)2, jw(i,t)1, j1,2w(i

t)1, j1,2w(it)1, j1,2w(it)1, j1,2w(it) j,1,8w(i

t) j,1,8w(it)1, j8w(i,t)1, j8w(i,t) j,w(i,

DΔΔ

ρhΔ20

22

4

2

4

h

(2)

where is the size of the square mesh which is assumed to be equal to 0.02m in

this study, Δt is the time interval (0.001 sec), t is the time in sec, and i and j are the

node numbers in the x- and y-directions, respectively.

The free vibration of the plate is obtained by setting the external loads to zero. For

all plates considered in this study, the plate is excited by an initial displacement at

point (a/4, b/4), as shown in Fig. 4. The deflection at point (3a/4, 3b/4) is calculated

using the finite difference method. Fourier transform is then used to convert the

response from the time domain to the frequency domain. The natural frequencies are

measured from the peaks of the curve, see Fig. 5. To demonstrate the forced vibration

cases, equation (2) is used to calculate the deflection at the central point (i, j) of the

plate due to the various dynamic loadings, by representing that equation as:

RHS

LHSt) j,w(i, (3)

where LHS is the left hand side of equation (2), and RHS is right hand side of the

same equation.

For each time step, 0.001sec, the deflection is computed by equation (3), and the

corresponding response due to each dynamic loading is obtained. It should be


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mentioned here that a Matlab program titled "Free and forced vibration of plates" was

written to demonstrate the present results.

Results and discussion

To demonstrate the free and forced vibrations, the material properties used in

this study are as follows: concrete Young's modulus, 29 N/m1030E ; Poisson's

ratio, 0.3 ν ; mass density, 32800kg/mρ ; and plate thickness, 0.1mh . Table

1-3 shows the first four non-dimensional frequency parameters, ρ/Dωaλ 2 , for

rectangular plates with different values of a/b, namely, 2/3, 1, and 1.5, and various

boundary conditions, namely, SSSS, SCSS, and SCSC, respectively, calculated by the

present finite difference method and compared to the exact solution of Leissa (1969;

1973) and the numerical solution of Eftehari and Jafari (2012). It is clear from the

results that the present finite difference method agrees well with those of the exact

solution, Leissa (1969; 1973) and the numerical solution, Eftehari and Jafari (2012).

Table 4-6 also shows the first four non-dimensional frequency parameters,

ρ/Dωaλ 2 , for CCCC, CCCS, and CCSS rectangular plates, respectively, with

aspect ratio values of 2/3, 1, and 1.5, calculated by the present finite difference

method and compared to the numerical solutions of Leissa (1969; 1973) and those of

Eftehari and Jafari (2012). Again, the present finite difference method results agree

well with those of the numerical solution obtained by Leissa (1969; 1973) and those

of Eftehari and Jafari (2012). It should be mentioned here that the non-dimensional

frequency parameter for a/b values of 2/3, and 1.5 is not available in Eftehari and

Jafari (2012). The proposed finite difference method is examined for the forced

vibration analysis of rectangular plates subjected to various dynamic loadings,

namely, constant force, rectangular force with time of duration, t d=0.1 sec, triangular

force with time of duration, td=0.1 sec, and sinusoidal pulse with time of duration, td=

0.1 sec, and frequency of 20 rad/sec. The amplitude of all dynamic loadings is

assumed to be 20 kN with time step, 0.001secΔt . Figures 6-11 show the central

displacement of a square plate with various boundary conditions, namely, SSSS,

SCSS, SCSC, CCCC, CCCS, and CCSS, respectively. It is clear from the results that

the central displacement has a sudden jump at td= 0.1 sec for rectangular, triangular,

and sinusoidal loadings. The central displacements for the case of a rectangular loadare higher than those due to sinusoidal and triangular loadings at all times, regardless


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of the boundary conditions. The response for all plates will go to zero, except for the

case of constant load. By comparing the results of Fig. 6 with those of Fig. 9, one sees

that the rate of convergence of the response to zero for SSSS plates is slower than that

of CCCC plates. The central displacements of plates having two opposite simply

supported edges, namely, SSSS, SCSS, and SCSC is higher than those of plates not

having two opposite simply supported edges, namely, CCCC, CCCS, and CCSS,

compare Figures 6-8 with Figures 9-11.

Conclusions

In this paper, the free and forced vibration of rectangular plates is studied

using the finite difference method. Six different cases of rectangular plates are

considered based on their boundary conditions. Fourier transform is used to convert

the time response obtained by the finite difference method to the frequency domain

for an initially displaced plate. The corresponding natural frequencies of the

rectangular plates are obtained. The dynamic behavior of rectangular plates is studied

in this paper by subjecting them to various dynamic loadings. The results obtained in

this study are in good agreement with exact and numerical results available in the

literature.

REFERENCES

[1] A.W. Leissa, Vibration of Plates, Washington, NASA SP-160, US Government

Printing Office, 1969.

[2] A.W. Leissa, The free vibration of rectangular plates, Journal of Sound and

Vibration 31 (3) (1973) 257 –293.

[3] A.W. Leissa, The historical bases of the Rayleigh and Ritz methods, J. Sound Vib.

287 (2005) 961 –978.

[4] S. Ilanko, Comments on the historical bases of the Rayleigh and Ritz methods, J.

Sound Vib. 319 (2009) 731 –733.[5] D. Young, Vibration of rectangular plates by the Ritz method, ASME J. Appl.

Mech. 17 (1950) 448 –453.

[6] S.F. Bassily, S.M. Dickinson, On the use of beam functions for problems of plates

involving free edges, ASME J. Appl. Mech. 42 (1975) 858 –864.

[7] R.B. Bhat, Natural frequencies of rectangular plates using characteristic

orthogonal polynomials in Rayleigh-Ritz method, J. Sound Vib. 102 (1985) 493 – 499.

[8] K.M. Liew, K.Y. Lam, A set of orthogonal plate functions for vibration analysis of

regular polygonal plates, ASME J. Vib. Acoust. 113 (1991) 182 –186.


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[9] C.W. Lim, K.M. Liew, Vibrations of perforated plates with rounded corners, J.

Eng. Mech. 121 (2) (1995) 203 –213.

[10] A.Y.T. Leung, J.K.W. Chan, Fourier p-element for analysis of beams and plates,

J. Sound Vib. 212 (1) (1998) 179 –185.

[11] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1, McGraw-

Hill, New York, 1989.

[12] J.N. Reddy, An Introduction to the Finite Element Method, second ed., McGraw-Hill, New York, 1993.

[13] S.A. Eftekhari, A.A. Jafari, A mixed method for free and forced vibration ofrectangular plates, Applied Mathematical Modelling, 36 (2012), 2814 –2831.

[14] X. D. Yang, W. Zhang, L. Q. Chen, M. H. Yao, Dynamical analysis of axially

moving plate by finite difference method, Nonlinear Dynamics, 67 (2012), 997 –1006.

[15] A. K. Gupta, M. Saini, S. Singh and R. Kumar, Forced vibrations of non-

homogeneous rectangular plate of linearly varying thickness, Journal of Vibration and

Control, 12 (2012) 1 –9.

[16] X. Wang, S. Xu, Free vibration analysis of beams and rectangular plates withfree edges by the discrete singular convolution, Journal of Sound and Vibration 329

(2010) 1780 –1792.

[17] M. Ducceschi, C. Touzé, S. Bilbao, and C. J. Webb, Nonlinear dynamics of

rectangular plates: investigation of modal interaction in free and forced vibrations,

Acta Mechnics 225 (2014), 213 –232.[18] Y. Xing, B. Liu, New exact solutions for free vibrations of rectangular thin plates

by symplectic dual method, Acta Mech Sin (2009) 25:265 –270.

[19] Y. Xing, B. Liu, Closed form solutions for free vibrations of rectangular Mindlin plates, Acta Mech Sin (2009) 25:689 –698.

[20] S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-

Hill, New York, 1959.[21] N. M. Werfalli, A. A. Karoud, Free Vibration Analysis of Rectangular Plates

Using Galerkin-Based Finite Element Method, International Journal of Mechanical

Engineering, 2,2, 59-67.

[22] J. H. Wu, A. Q. Liu, and H. L. Chen, Exact Solutions for Free-Vibration

Analysis of Rectangular Plates Using Bessel Functions, Journal of Applied

Mechanics, (2007), 74, 1247-1251.

[23] Njoku K. O., Ezeh J. C., Ibearugbulem O. M., Ettu L. O., and Anyaogu L.

(2013). “Free vibration analysis of thin rectangular isotropic CCCC plate using

Taylor series formulated shape function in Galerkin's method,” Academic Research

International, vol. 4, no. 4, pp. 126-132.

[24] M. N. Bahrami, M. Loghmani, and M. Pooyanfar, Analytical Solution for FreeVibration of Rectangular Kirchhoff Plate from Wave Approach, World Academy of

Science, Engineering and Technology, (2008), 39, 221-223.

[25] Jain, R. Kbishan, and S. R. Soni. "Free vibrations of rectangular plates of

parabolically varying thickness." Indi. J. Pure Appl. Math 4 (1973): 267-277.

[26] K. S. Numayr, R. H. Haddad, and M. A. Haddad. "Free vibration of composite

plates using the finite difference method." Thin-walled structures 42.3 (2004): 399-

414.

[27] Y. L Yeh, M. J Jang, and C. C. Wang, Analyzing the free vibrations of a plate

using finite difference and differential transformation method. Applied mathematics

and computation, 178(2), (2006), 493-501.


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Table 1 Natural frequency parameter for SSSS plates.

Ritz and DQ

Eftekhari, Jafari Exact

Leissa Present SolutionMode numberAspect ratio

a/b

N.A.

14.25614.3371

2/327.41627.3712

43.86544.3153

49.34848.2264

19.73919.73919.3591 1 49.34849.34849.0432

49.34849.34849.0433

78.95778.95778.2074

N.A. 32.07632.2581

1.5 61.68561.5842

98.69699.7093

111.033108.5094

Table 2 Natural frequency parameter for SCSS plates.

Ritz and DQ

Eftekhari, Jafari Exact


a/b

N.A.

15.57815.6411

2/331.07230.6122

44.56444.4153

55.39353.4384

23.64623.64623.4611 1 51.67451.67450.8322

58.64658.64658.6523 86.13586.13584.7194

N.A. 42.52842.6971

1.5 69.00367.9782

116.267115.6133

120.996118.7264

Table 3 Natural frequency parameter for SCSC plates.

Ritz and DQ

Eftekhari, Jafari

Exact

Leissa

Present SolutionMode numberAspect ratioa/b

N.A.

17.37316.4281

2/335.34534.3902

45.42944.4663

62.05459.8574

28.95128.95127.3711

1 54.74354.74353.2662

69.32769.32767.8573

94.58594.58591.2354

N.A. 56.34854.2731

1.5 78.98476.7222

123.172121.5603

146.268141.0784


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Table 4 Natural frequency parameter for CCCC plates. Ritz and DQ

Eftekhari, Jafari Ritz


a/b

N.A.

27.01027.3711

2/341.71641.7072

66.14362.5623

66.55263.8654

35.98535.99236.4941 1 73.39373.41372.9892

73.393 73.413 72.9893

108.217108.270105.5744

N.A. 60.77261.5851

1.5 93.86093.8412

148.820140.7653

149.740143.6964

Table 5 Natural frequency parameter for CCCS plates. Ritz and DQ

Eftekhari, Jafari Ritz


a/b

N.A.

25.86125.4121

2/338.10237.7972

60.32561.2553

65.51664.3664

31.82631.82931.2811

1 63.33163.34762.5522

71.07671.08471.6863

100.792100.83099.0564

N.A. 48.16747.3391

1.5 85.50784.4372

123.990125.0363

143.990145.2304

Table 6 Natural frequency parameter for CCSS plates. Ritz and DQ

Eftekhari, Jafari

Ritz

Leissa

Present SolutionMode numberAspect ratioa/b

N.A.

19.95219.5501

2/3

34.02433.8872

54.37054.7423

57.51756.0454

27.05427.05627.3711

1 60.54460.54459.9552

60.79460.79161.2583

92.85392.86591.2374

N.A. 44.89343.9891

1.5 76.55476.2462

122.330123.1673

129.410126.0984


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Figure 1 Thin rectangular plate geometry and coordinate system

Figure 2 Dynamic loading functions (a) constant, (b) rectangular, (c) triangular,

(d) sinusoidal.

x

y

a

b

h

t

t t

t

F(t)F(t)

F(t)F(t)

(a) (b)

(c) (d)

0F 0F

0F 0F

dt

dt dt


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Figure 3 Plate mesh numbering in x-, y-, and t-coordinates.


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Figure 4 Initially displaced plate.


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Figure 5 Calculation of the non-dimensional frequency using Fourier transform,

for an SSSS plate.


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Figure 6 Central displacement versus time for an SSSS plate for various

dynamic loadings.


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Figure 7 Central displacement versus time for an SCSS plate for various

dynamic loadings.


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Figure 8 Central displacement versus time for an SCSC plate for various

dynamic loadings.


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Figure 9 Central displacement versus time for an CCCC plate for various

dynamic loadings.


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Figure 10 Central displacement versus time for an CCCS plate for various

dynamic loadings.


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Figure 11 Central displacement versus time for an CCSS plate for various

dynamic loadings.

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