Shock and Vibration

Volume 2014 (2014), Article ID 286710, 11 pages

http://dx.doi.org/10.1155/2014/286710

Research Article

An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

Xue Kai, Wang Jiufa, Li Qiuhong, Wang Weiyuan, and Wang Ping

College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China

Received 4 March 2013; Accepted 19 September 2013; Published 25 February 2014

Academic Editor: Nuno Maia

Copyright © 2014 Xue Kai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An analysis method is proposed for the vibration analysis of the Mindlin rectangular plates with general elastically restrained edges, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier cosine series and four one-dimensional Fourier series. The use of these supplementary functions is to solve the possible discontinuities with first derivatives at each edge. So this method can be applied to get the exact solution for vibration of plates with general elastic boundary conditions. The matrix eigenvalue equation which is equivalent to governing differential equations of the plate can be derived through using the boundary conditions and the governing equations based on Mindlin plate theory. The natural frequencies can be got through solving the matrix equation. Finally the numerical results are presented to validate the accuracy of the method.

1. Introduction

Rectangular plates are important structural elements and the analysis of the vibration is very important for the design of plate-type structures in aerospace, electronic, mechanical, marine, nuclear, and structural engineering. Thus, many researchers have done much work and got a lot of results. However, this research is based on the classical Kirchhoff hypothesis. This theory neglects the effect of shear deformation and rotary inertia which result in the over-estimation of vibration frequencies. This deviation will increase with increasing plate thickness. To improve the results, the Mindlin first-order plate can be employed. So the vibration of Mindlin rectangular plates has begun to gain attention.

The effective methods used to analyze the vibration of Mindlin rectangular plates include Rayleigh-Ritz method [1–4] and some numerical methods, such as differential quadrature method (DQM) [5, 6] and discrete singular convolution method (DSC) [7, 8]. Several analysis methods were also proposed by some researchers. Hashemi et al. derived the exact close form characteristic equations and their associated eigenfunctions for the thick rectangular plates with two opposite sides simply supported [9, 10]. Gorman used the superposition method to obtain a solution for the Mindlin plates [11, 12].

Most existing studies are limited to the classical homogeneous boundary conditions. Recently, Li proposed a Fourier series method for the vibration analysis of arbitrarily supported beam [13]. The flexural displacement of the beam is sought as the linear combination of a Fourier series and an auxiliary polynomial function. Subsequently, this method is extended to the flexural and in-plane vibration of rectangular plates under general boundary conditions. The flexural and in-plane displacement of the plate is sought as the linear combination of a double Fourier cosine series and auxiliary series functions [14–16]. It has been shown that this solution method works very well for various edge supports.

In this paper, an improved Fourier series method is employed to analyze the free vibration of Mindlin rectangular plates with general elastic boundary supports, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier cosine series and four one-dimensional Fourier series. The possible discontinuities problems, which maybe encountered in the displacement and rotations partial differentials along the edges, can be solved by these supplementary functions. Then, an exact solution for Mindlin rectangular plates with arbitrary elastically restrained edges can be obtained. Finally several numerical examples and the comparisons with those results reported in the literature are presented to validate the accuracy of the present approach.

2. Mathematical Modeling and Solution Methodology

Consider a rectangular Mindlin plate elastically restrained along all edges, as shown in Figure 1. The boundary conditions are physically realized in terms of three kinds of restraining springs (translational, rotational, and torsional springs) attached to each edge. Different boundary conditions can be directly obtained by changing the stiffness of springs. The governing differential equations for free vibration of a Mindlin plate are given by

where is the transverse displacement, and are the slope due to bending alone in the respective planes, is the shear correction factor to account for the fact, is the shear modulus, is the Poisson’s ratio, is the flexural rigidity, is the mass density, and is the thickness of the plate.

Figure 1: A Mindlin plate with general elastic boundary support.

In terms of transverse displacements and slope, the bending and twisting moments and the transverse shearing forces in plates can be expressed as

There are three forces along every edge and they are the bending moment, the twisting moment, and the shearing forces. Three kinds of restraining springs (rotational, torsional, and translational springs) along every edge can be corresponding to these three forces. The boundary conditions for an elastically restrained rectangular plate are as follows:

where and ( and ) are the translation spring constants, and ( and ) are the rotational spring constants, and and ( and ) are the torsional spring constants at and ( and ), respectively. All classical homogeneous boundary conditions can be easily derived by simply setting each of the spring constants to be infinite or zero.

According to the Mindlin plate theory, the transverse displacement of the plate median surface and the rotations of the cross-section, respectively, along the direction and the direction are utilized. In this study, these quantities are expressed in form of improved Fourier series expansions [16]:

where , , , , , , , , and  are the expansion coefficients, , , and are the length and width, respectively, and

Theoretically, there is an infinite number of these supplementary functions. However, one needs to ensure that the selected functions will not nullify any of the boundary conditions. It is easy to verify that , , , ,  similar conditions exist for the supplementary function in -direction. Though these conditions are not necessary, they can simplify the subsequent mathematical expressions and the corresponding solution procedures.

One will notice from (4) that beside the standard double Fourier series, four single Fourier series are also included. The potential discontinuity associated with the -derivative and -derivative of the original function along the four edges can be transferred onto these auxiliary series functions. Then, the Fourier series would be smooth enough in the whole solving domain. Therefore, not only is this Fourier series representation of solution applicable to any boundary conditions but also the convergence of the series expansion can be improved.

Substituting (4) into the boundary conditions, for example, , at ; one can have

In order to derive the constraint equations for the unknown coefficients, all the sine terms and the auxiliary series functions will be expanded into Fourier cosine series. The related formulas are provided in Appendix A. Then, by equating the coefficients for the like terms on both sides, one can obtain the following equations:

Similarly, the substitution of (4) into the remaining boundary conditions will lead to eleven equations that can be obtained from (3):

When all the series expansions are truncated to and in numerical calculations, the twelve equations can be rewritten in a matrix form as

where

The elements of the matrices and are defined in Appendix B.

By substituting (4) into the governing differential equation (1), as mentioned earlier, all the sine terms and the auxiliary series functions are expanded into Fourier cosine series. Then, by equating the coefficients for the like terms on both sides, one can obtain the following equations:

Writing in matrix form, we have

Substituting (9), the final system equations can be obtained as

where

where and . The element of the matrices , , , and are defined in Appendix B. The natural frequencies and eigenvectors can be obtained through solving (13). Then, the physical mode shapes also can be got using (4) and (9).

3. Result and Discussion

Several examples involving various boundary conditions will be discussed in this section. To avoid any comparison of the roundoff results, which might be unrealistic, the nondimensional frequency is used. For the analysis, Poisson’s ratio and shear correction factor are used.

First, convergence studies are carried out. Table 1 gives the frequencies calculated by using different number of terms in the series expansion for F-F-F-F Mindlin rectangular plates (for and ). It shows that the results are very accurate when and are small numbers. When and are larger than 10, results are almost invariant. So this method has a very good convergence characteristic.

Table 1: The first seven frequency parameters 1/2 for F-F-F-F Mindlin rectangular plates.

1
2
3
4
5
6
7

5
1.2978
1.9202
2.3644
3.2556
3.2556
5.6148
5.6148

7
1.2924
1.9196
2.3637
3.2438
3.2438
5.6105
5.6105

9
1.2904
1.9195
2.3635
3.2388
3.2388
5.6092
5.6092

10
1.2903
1.9194
2.3634
3.2371
3.2371
5.6089
5.6089

11
1.2895
1.9194
2.3634
3.2366
3.2366
5.6087
5.6087

12
1.2895
1.9194
2.3633
3.2358
3.2358
5.6085
5.6085

13
1.2892
1.9194
2.3633
3.2355
3.2355
5.6085
5.6085

14
1.2892
1.9194
2.3633
3.2355
3.2355
5.6084
5.6084

In order to evaluate the accuracy of the present method, the comparisons with those results reported in the literature are carried out. First of all, considering a plate with two opposite simple edges and the other two edges free (S-F-S-F). A simple edge can be obtained by setting the translational and torsional spring constants to be infinite (here the infinite is represented by a very large number, , and is the flexural rigidity) and the rotational spring constants to be zero. A free edge can be obtained when all the spring constants are zero. In Table 2, the first seven nondimensional frequency parameters, , are given with different aspect ratios and thickness ratios. At the same time, a comparison of the exact solution in [9] is also presented. The results show that the calculated frequencies are excellent. As mentioned earlier, the series expansion will have to be truncated in numerical calculations. In this example and all the subsequent calculations, the setting is used. Specially, in order to compare with the results in [9], shear correction factor is used.

Table 2: The first seven frequency parameters 1/2 for S-F-S-F Mindlin rectangular plates.

Method
1
2
3
4
5
6
7

0.4
0.1
Present
9.5829
10.7868
14.5793
20.7593
29.3643
36.7449
37.8085

[9]
9.5814
10.7809
14.5672
20.7245
29.3338
36.736
37.7879

0.2
Present
9.1316
10.1983
13.5314
18.7785
25.8073
31.7024
32.4788

[9]
9.1313
10.1968
13.5287
18.7728
25.8016
31.7000
32.4730

1
0.1
Present
9.4461
15.4109
33.9251
36.4266
42.8986
62.3508
66.3889

[9]
9.4458
15.4054
33.9160
36.4246
42.8870
62.3304
66.3720

0.2
Present
8.9997
14.1349
29.2566
31.434
36.1663
49.8976
52.8026

[9]
8.9997
14.1341
29.2558
31.4338
36.1646
49.8953
52.8012

2.5
0.1
Present
9.3065
29.6407
35.8486
64.0871
75.9262
105.417
125.3433

[9]
9.3065
29.6389
35.8486
64.0850
75.9234
105.4121
125.3334

0.2
Present
8.8835
24.2058
30.9976
49.7612
59.4412
77.0664
90.361

[9]
8.8835
24.2054
30.9976
49.7609
59.4404
77.0654
90.3658

Three more classical cases (C-F-F-F, C-F-S-F, and C-S-S-F) were considered, and the corresponding frequency parameters are listed in Tables 3, 4, and 5. A clamped edge can be viewed as all the spring constants set to be infinite. The frequency parameters solved by Ritz method and DSC method also are given as a comparison. A good agreement is also observed among these solutions.

Table 3: The first seven frequency parameters 1/2 for C-F-F-F Mindlin rectangular plates.

Method
1
2
3
4
5
6
7

0.4
0.1
Present
2.1006
2.7297
4.3933
7.2648
10.4212
11.1567
11.4449

[1]
2.1006
2.7294
4.3921
7.2622
10.4206
11.1554
11.4433

0.2
Present
1.8579
2.2911
3.4911
5.5123
6.9021
7.3873
8.077

[1]
1.8579
2.2910
3.4909
5.5119
6.9020
7.3871
8.0767

1
0.1
Present
0.3476
0.8171
2.0357
2.584
2.8633
4.8186
5.4836

[1]
0.3476
0.8168
2.0356
2.5836
2.8620
4.8162
5.4834

0.2
Present
0.3384
0.7446
1.7806
2.2766
2.4207
3.8853
4.3156

[1]
0.3384
0.7445
1.7806
2.2765
2.4205
3.8851
4.3168

2.5
0.1
Present
0.0554
0.2798
0.3432
0.8865
0.9517
1.6251
1.8326

[1]
0.0554
0.2795
0.3431
0.8854
0.9516
1.6228
1.8320

0.2
Present
0.0550
0.2626
0.3337
0.8196
0.8958
1.4639
1.6566

[1]
0.0550
0.2625
0.3337
0.8192
0.8958
1.4632
1.6564

Table 4: The first seven frequency parameters 1/2 for C-F-S-F Mindlin rectangular plates.

Method
1
2
3
4
5
6
7

0.4
0.1
Present
7.8209
8.2262
9.6485
12.1337
15.6899
20.1409
20.358

[8]
7.8285
8.2339
9.6566
12.1373
15.6917
20.1394
20.3607

0.2
Present
5.5099
5.7895
6.7405
8.3478
10.4967
12.6125
12.8205

[8]
5.5128
5.7926
6.7436
8.3495
10.4978
12.6141
12.8219

1
0.1
Present
1.4735
1.9497
3.6462
4.502
5.0407
6.7828
6.8161

[7]
1.4738
1.9536
3.6499
4.5031
5.0483
6.7909
—

0.2
Present
1.3254
1.7019
3.0527
3.6262
4.0033
5.2067
5.3485

[7]
1.3255
1.7030
3.0534
3.6265
4.0054
5.2084
—

2.5
0.1
Present
0.2413
0.5789
0.776
1.2825
1.5996
2.1811
2.4288

[7]
0.2413
0.5817
0.7761
1.2859
1.5995
2.1852
—

0.2
Present
0.2362
0.5402
0.7365
1.1704
1.4595
1.9285
2.1794

[7]
0.2362
0.5409
0.7366
1.1714
1.4594
1.9300
—

Table 5: The first seven frequency parameters 1/2 for C-S-S-F Mindlin rectangular plates.

Method
1
2
3
4
5
6
7

0.4
0.1
Present
7.9414
8.9713
11.137
14.4651
18.7651
20.4622
21.36

[8]
7.9491
8.9792
11.1430
14.4679
18.7554
20.4646
21.3629

0.2
Present
5.5943
6.3055
7.752
9.8284
12.295
12.6792
13.2365

[8]
5.5972
6.3085
7.7544
9.8298
12.2958
12.6807
13.2381

1
0.1
Present
1.6196
2.9171
4.6617
5.7683
5.9723
8.5764
8.8548

[8]
1.6281
2.9271
4.6650
5.7741
5.9769
8.5795
9.8555

0.2
Present
1.4451
2.4997
3.7407
4.6356
4.677
6.4325
6.4907

[8]
1.4476
2.5027
3.7419
4.6375
4.6787
6.4338
6.4910

2.5
0.1
Present
0.3658
0.9415
1.746
1.7792
2.3445
2.8701
3.217

[8]
0.3832
0.9487
1.7596
1.7828
2.3543
2.8721
3.2233

0.2
Present
0.3526
0.8834
1.6006
1.6092
2.0839
2.4914
2.7667

[8]
0.3574
0.8854
1.6043
1.6102
2.0867
2.4919
2.7685

This method not only can solve the Mindlin rectangular plates with classical boundary conditions but also can solve plates with the elastical supports. Xiang et al. [3] and Gorman [11] and Zhou [4] also researched Mindlin plates with elastically restrained edges, but they both used two kinds of springs (rotational and translational springs) to realize the elastical supports along every edge. As mentioned earlier, three kinds of springs along every edge are needed to truly realize the general elastic supports, including all classical homogeneous boundary conditions. Table 6 gives the first seven frequency parameters for Mindlin rectangular plates (for and ) with two opposite free edges, the other two edges only elastically restrained against translation, and Table 7 gives the first seven frequency parameters for the plates with two opposite free edges, the other two edges symmetrically elastically restrained. Table 8 lists the frequency parameters when the translational, rotational, and torsional spring constants are all varying at . For simplicity, all the restraining springs are assumed to have the same stiffness in the three examples. Table 8 shows that the results are invariant when is larger than 10⁷, so it is appropriate to set infinite to be in this paper.

Table 6: The first seven frequency parameters 1/2 for Mindlin rectangular plates with two opposite free edges, the other two edges only elastically restrained against translation .

Method
1
2
3
4
5
6
7

5
Present
0.3051
0.3071
0.5397
1.2922
1.8248
2.2009
2.8356

[4]
0.3052
0.3075
0.5396
1.3117
1.8447
2.2089
2.8913

50
Present
0.6926
0.8072
1.5785
1.9220
2.1374
2.7775
3.3443

[4]
0.6941
0.8225
1.5797
1.9268
2.1634
2.7919
3.3823

500
Present
0.8782
1.2203
2.6177
2.8662
3.1461
4.2179
5.0805

[4]
0.8831
1.3081
2.7583
2.8767
3.2089
4.3584
5.0980

5000
Present
0.905
1.306
2.7851
3.1319
3.4845
4.7069
5.1891

[4]
0.9105
1.4254
2.9541
3.1485
3.6157
4.9856
5.3230

Table 7: The first seven frequency parameters 1/2 for Mindlin rectangular plates with two opposite free edges, the other two edges symmetrically elastically restrained and .

Method
1
2
3
4
5
6
7

5
10
Present
0.3105
0.3148
0.9155
1.4411
2.006
2.8874
2.9385

[4]
0.3104
0.3148
0.9208
1.4505
2.0122
2.8989
2.9659

50
50
Present
0.8936
0.901
1.6281
1.9545
2.1865
3.2206
3.412

[4]
0.8944
0.9032
1.6305
1.9599
2.1912
3.2374
3.4255

5000
5000
Present
1.7537
1.958
3.0634
3.9445
4.1976
5.1453
5.2879

[4]
1.7607
1.9901
3.1318
3.9578
4.2553
5.2883
5.3548

Table 8: The first seven frequency parameters 1/2 for C-C-F-C Mindlin rectangular plates with translational, rotational, and torsional restraints at , .

1
2
3
4
5
6
7

0
2.2388
3.6105
5.4944
6.6135
6.8022
9.4188
9.7667

0.1
2.2476
3.6233
5.5034
6.6282
6.811
9.4295
9.775

1
2.3132
3.7213
5.5721
6.7447
6.8826
9.5158
9.8391

10²
2.878
4.562
6.0184
7.641
7.6707
10.2284
10.3535

10⁴
3.2854
6.2286
6.2735
8.7475
10.2438
10.4305
12.3813

10⁶
3.2954
6.2857
6.2861
8.8102
10.3788
10.4778
12.5528

10⁷
3.2955
6.2863
6.2863
8.8109
10.3797
10.4786
12.5545

10⁹
3.2955
6.2863
6.2863
8.8109
10.3797
10.4786
12.5546

3.2955
6.2863
6.2863
8.8109
10.3797
10.4786
12.5546

4. Conclusions

An improved Fourier series method is proposed to analyze the free vibration of Mindlin rectangular plates with general elastic boundary supports. The general boundary conditions are physically realized with the uniform distribution of springs on each boundary edge, and different boundary conditions can be directly obtained by changing the stiffness of the springs. The vibration displacements and the cross-sectional rotations of the mid-plane are sought as the linear combination of a double Fourier cosine series and four single auxiliary series functions, respectively. The use of these supplementary functions is to solve the discontinuity problems which were encountered in the displacement and rotations partial differentials along the edges. The unknown expansion coefficients can be solved through using the boundary conditions and the governing equations. In this method, analytical solution is derived for the vibrations of Mindlin rectangular plates with general elastic boundary support. Finally, the numerical results and the comparisons with those reported in the literature are presented to validate the accuracy of the method.

Appendices

A. Supplementary Series

We have

B. Definitions of Matrices

We have

where , ,   ,  ,  ,  and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge the support of this work by the National Natural Science Foundation of China (no. 51105087) and the Fundamental Research Funds for the Central Universities (HEUCF110701). The authors would like to express their sincere thanks to the reviewers for the constructive and positive comments.

References

K. M. Liew, Y. Xiang, and S. Kitipornchai, “Transverse vibration of thick rectangular plates—I. Comprehensive sets of boundary conditions,” Computers and Structures, vol. 49, no. 1, pp. 1–29, 1993.

K. M. Liew, K. C. Hung, and M. K. Lim, “Vibration of mindlin plates using boundary characteristic orthogonal polynomials,” Journal of Sound and Vibration, vol. 182, no. 1, pp. 77–90, 1995.

Y. Xiang, K. M. Liew, and S. Kitipornchai, “Vibration analysis of rectangular mindlin plates resting on elastic edge supports,” Journal of Sound and Vibration, vol. 204, no. 1, pp. 1–16, 1997.

D. Zhou, “Vibrations of Mindlin rectangular plates with elastically restrained edges using static Timoshenko beam functions with the Rayleigh-Ritz method,” International Journal of Solids and Structures, vol. 38, no. 32-33, pp. 5565–5580, 2001.

F.-L. Liu and K. M. Liew, “Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method,” Journal of Sound and Vibration, vol. 225, no. 5, pp. 915–934, 1999.

P. Malekzadeh and S. A. Shahpari, “Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM,” Thin-Walled Structures, vol. 43, no. 7, pp. 1037–1050, 2005.

Y. Hou, G. W. Wei, and Y. Xiang, “DSC-Ritz method for the free vibration analysis of Mindlin plates,” International Journal for Numerical Methods in Engineering, vol. 62, no. 2, pp. 262–288, 2005.

Y. Xiang, S. K. Lai, and L. Zhou, “DSC-element method for free vibration analysis of rectangular Mindlin plates,” International Journal of Mechanical Sciences, vol. 52, no. 4, pp. 548–560, 2010.

S. H. Hashemi and M. Arsanjani, “Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates,” International Journal of Solids and Structures, vol. 42, no. 3-4, pp. 819–853, 2005.

S. H. Hashemi, K. Khorshidi, and H. Rokni Damavandi Taher, “Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory,” Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 883–900, 2009.

D. J. Gorman, “Free vibration analysis of Mindlin plates with uniform elastic edge support by the superposition method,” Journal of Sound and Vibration, vol. 207, no. 3, pp. 335–350, 1997.

F. Ohya, M. Ueda, T. Uchiyama, and M. Kikuchi, “Free vibration analysis by the superposition method of rectangular Mindlin plates with internal columns resting on uniform elastic edge supports,” Journal of Sound and Vibration, vol. 289, no. 1-2, pp. 1–24, 2006.

W. L. Li, “Free vibrations of beams with general boundary conditions,” Journal of Sound and Vibration, vol. 237, no. 4, pp. 709–725, 2000.

W. L. Li, “Vibration analysis of rectangular plates with general elastic boundary supports,” Journal of Sound and Vibration, vol. 273, no. 3, pp. 619–635, 2004.

J. Du, W. L. Li, G. Jin, T. Yang, and Z. Liu, “An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges,” Journal of Sound and Vibration, vol. 306, no. 3–5, pp. 908–927, 2007.

W. L. Li, X. Zhang, J. Du, and Z. Liu, “An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports,” Journal of Sound and Vibration, vol. 321, no. 1-2, pp. 254–269, 2009.