Research Papers

Modal Analysis of General Plate Structures

[+] Author and Article Information
Hongan Xu

Department of Mechanical Engineering,
Wayne State University,
5050 Anthony Wayne Drive,
Detroit, MI 48202

W. L. Li

Department of Mechanical Engineering,
Wayne State University,
5050 Anthony Wayne Drive,
Detroit, MI 48202
e-mail: wli@wayne.edu

Jingtao Du

College of Power and Energy Engineering,
Harbin Engineering University,
Harbin 150001, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 19, 2011; final manuscript received June 27, 2013; published online November 26, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 136(2), 021002 (Nov 26, 2013) (11 pages) Paper No: VIB-11-1248; doi: 10.1115/1.4025876 History: Received October 19, 2011; Revised June 27, 2013

A general analytical method, referred to as the Fourier spectral element method, is presented for the dynamic analysis of plate structures consisting of any number of arbitrarily oriented rectangular plates. The compatibility conditions between any two adjacent plates are generally described in terms of three-dimensional elastic couplers with both translational and rotational stiffnesses. More importantly, all plates involved can be arbitrarily restrained along any edges in contrast to the commonly imposed condition: each plate has to be simply supported along, at least, one pair of parallel edges. Thus, plate structures here are not limited to Levy-type plates as typically assumed in other techniques. The flexural and in-plane displacement fields on each plate are analytically expressed as accelerated Fourier series expansions and the expansion coefficients are considered as the generalized coordinates to be determined using the familiar Rayleigh–Ritz technique. The accuracy and reliability of the present method are validated by both finite element analysis (FEA) and experimental data for box structures under various boundary conditions.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Langley, R. S., and Cotoni, V., 2004, “Response Variance Prediction in the Statistical Energy Analysis of Built-Up Systems,” J. Acoust. Soc. Am., 115(2), pp. 706–718. [CrossRef] [PubMed]
Popplewell, N., 1971, “The Vibration of Box-Like Structure I: Natural Frequencies and Normal Modes,” J. Sound Vib., 13(4), pp. 357–365. [CrossRef]
Popplewell, N., 1975, “The Response of Box-Like Structures to Weak Explosions,” J. Sound Vib., 42(1), pp. 65–84. [CrossRef]
Dickinson, S. M., and Warburton, G. A., 1967, “Vibration of Box-Type Structures,” J. Mech. Eng. Sci., 9(4), pp. 325–335. [CrossRef]
Kim, C. S., and Dickinson, S. M., 2008, “The Flexural Vibration of Line Supported Rectangular Plate Systems,” J. Sound Vib., 114(1), pp. 129–142. [CrossRef]
Yuan, J., and Dickinson, S. M., 1992, “The Flexural Vibration of Rectangular Plate Systems Approached by Using Artificial Springs in the Rayleigh-Ritz Method,” J. Sound Vib., 159(1), pp. 39–55. [CrossRef]
Yuan, J., and Dickinson, S. M., 1993, “On the Continuity Conditions Enforced When Using Artificial Springs in the Rayleigh-Ritz Method,” J. Sound Vib., 161(3), pp. 538–544. [CrossRef]
Langley, R. S., 1991, “An Elastic Wave Technique for the Free Vibration Analysis of Plate Assemblies,” J. Sound Vib., 145(2), pp. 261–277. [CrossRef]
Rebillard, E., and Guyader, J. L., 1995, “Vibration Behavior of a Population of Coupled Plates: Hypersensitivity to the Connection Angle,” J. Sound Vib., 188(3), pp. 435–454. [CrossRef]
Park, D. H., and Hong, S. Y., 2001, “Power Flow Models and Analysis of In-Plane Waves in Finite Coupled Thin Plates,” J. Sound Vib., 244(4), pp. 651–668. [CrossRef]
Fulford, R. A., and Peterson, A. A. T., 2000, “Estimation of Vibration Power in Built-Up Systems Involving Box-Like Structures, Part 1: Uniform Force Distribution,” J. Sound Vib., 232(5), pp. 877–895. [CrossRef]
Langley, R. S., 1989, “Application of the Dynamic Stiffness Method to the Free and Forced Vibrations of Aircraft Panels,” J. Sound Vib., 135(2), pp. 319–331. [CrossRef]
Khumbah, F. M., and Langley, R. S., 1998, “Efficient Dynamic Modeling of Aerospace Box-Type Structures,” The 27th international Congress on Noise Control Engineering (InterNoise'98), Christchurch, New Zealand, November 16-18, Vol. 3.
Bercin, A. N., and Langley, R. S., 1966, “Application of the Dynamics Stiffness Technique to the In-Plane Vibration of Plate Structure,” Comput. Struct., 59(5), pp. 869–875. [CrossRef]
Bercin, A. N., 1997, “Eigenfrequencies of Rectangular Plate Assemblies,” Comput. Struct., 65(5), pp. 703–711. [CrossRef]
Azimi, S., Hamilton, J. F., and Soedel, W., 1984, “The Receptance Method Applied to the Free Vibration of Continuous Rectangular Plates,” J. Sound Vib., 93, pp. 9–29. [CrossRef]
Beshara, M., and Keane, A. J., 1998, “Vibrational Energy Flows Between Plates With Compliant and Dissipative Couplings,” J. Sound Vib., 213(3), pp. 511–535. [CrossRef]
Kim, H. S., Kang, H. J., and Kim, J. S., 1994, “Transmission of Bending Waves in Inter-Connected Rectangular Plates,” J. Acoust. Soc. Am., 96, pp. 1557–1562. [CrossRef]
Cuschieri, J. M., 1990, “Structural Power-Flow Analysis Using a Mobility Approach of an L-Shaped Plate,” J. Acoust. Soc. Am., 87(3), pp. 1159–1165. [CrossRef]
Grice, R. M., and Pinnington, R. J., 2000, “Vibration Analysis of a Thin-Plate Box Using a Finite Element Model Which Accommodates Only In-Plane Motion,” J. Sound Vib., 232(2), pp. 449–471. [CrossRef]
Grice, R. M., and Pinnington, R. J., 2002, “Analysis of the Flexural Vibration of a Thin-Plate Box Using a Combination of Finite Element Analysis and Analytical Impedances,” J. Sound Vib., 249(3), pp. 499–527. [CrossRef]
Patera, A. T., 1984, “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion,” J. Comput. Phys., 54, pp. 468–488. [CrossRef]
Zak, A., Krawczuk, M., and Ostachowicz, W., 2006, “Propagation of In-Plane Elastic Waves in a Composite Panel,” Finite Elements in Analysis and Design, 43, pp. 145–154. [CrossRef]
J.P.Boyd, 1989, Chebyshev and Fourier Spectral Methods, Springer, Berlin.
Kudela, P., Żak, A., Krawczuk, M., and Ostachowicz, W., 2007, “Modelling of Wave Propagation in Composite Plates Using the Time Domain Spectral Element Method,” J. Sound Vib., 302, pp. 728–745. [CrossRef]
Gopalakrishnan, S., and Mitra, M., 2010, Wavelet Methods for Dynamical Problems: With Application to Metallic, Composite, and Nano-Composite Structures, CRC Press, Boca Raton, FL.
Doyle, J. F., 1997, Wave Propagation in Structures, Springer, New York.
Ahmida, K. M., and Arruda, J. R. F., 2001, “Spectral Element-Based Prediction of Active Power Flow in Timoshenko Beams,” Int. J. Solids Struct., 38, pp. 1669–1679. [CrossRef]
Igawa, H., Komatru, K., Yamaguchi, I., and Kasai, T., 2004, “Wave Propagation Analysis of Frame Structures Using the Spectral Element Method,” J. Sound Vib., 277, pp. 1071–1081. [CrossRef]
Lee, U., 2009, Spectral Element Method in Structural Dynamics, Wiley, New York.
Birgersson, F., Ferguson, N. S., and Finnveden, S., 2003, “Application of the Spectral Finite Element Method to Turbulent Boundary Layer Induced Vibration of Plates,” J. Sound Vib., 259(4), pp. 873–891. [CrossRef]
Lee, U., Kim, J., and Leung, A. Y. T., 2000, “The Spectral Element Method in Structural Dynamics,” Shock Vib., 32(6), pp. 451–465. [CrossRef]
Leissa, A., 1969, “Vibration of Plates,” NASA Paper No. SP-160.
Blevin, R. D., 1979, Formulas for Natural Frequency and Mode Shape, VanNostrand, New York.
Gorman, D. J., 1999, Vibration Analysis of Plates by the Superposition Method, World Scientific, Singapore.
Bhat, R. B., Singh, J., and Mundkur, G., 1993, “Plate Characteristic Functions and Natural Frequencies of Vibration of Plates by Iterative Reduction of Partial Differential Equation,” ASME J. Vib. Acoust., 115, pp. 177–181. [CrossRef]
Cortinez, V. H., and Laura, P. A. A., 1990, “Analysis of Vibrating Rectangular Plates of Discontinuously Varying Thickness by Means of the Kantorovich Extended Method,” J. Sound Vib., 137, pp. 457–461. [CrossRef]
Wang, G., and Wereley, N. M., 2002, “Free In-Plane Vibration of Rectangular Plates,” AIAA J., 40, pp. 953–959. [CrossRef]
Li, W. L., 2000, “Free Vibrations of Beams With General Boundary Conditions,” J. Sound Vib., 237, pp. 709–725. [CrossRef]
Li, W. L., Bonilha, M. W., and Xiao, J., 2007, “Vibrations and Power Flows in a Coupled Beam System,” ASME J. Vib. Acoust., 129, pp. 616–622. [CrossRef]
Li, W. L., and Xu, H. A., 2009, “An Exact Fourier Series Method for the Vibration Analysis of Multi-Span Beam Systems,” ASME J. Computational Nonlinear Dyn., 4, pp. 1–9. [CrossRef]
Du, J. T., Li, W. L., Li, W. Y., Zhang, X. F., and Liu, Z. G., 2010, “Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges,” ASME J. Vib. Acoust., 132, p. 031002. [CrossRef]
Zhang, X. F., and Li, W. L., 2009, “Vibrations of Rectangular Plates With Arbitrary Non-Uniform Elastic Edge Restraints,” J. Sound Vib., 326(2009), pp. 221–234. [CrossRef]
Dai, L., Yang, T. J., Li, W. L., Du, J. T., and Jin, G. Y., 2012, “Dynamic Analysis of Circular Cylindrical Shells With General Boundary Condition,” ASME J. Vib. Acoust., 134, p. 041004. [CrossRef]
Xu, H. A., Du, J. T., and Li, W. L., 2010, “Vibrations of Rectangular Plates Reinforced by any Number of Beams of Arbitrary Lengths and Placement Angles,” J. Sound Vib., 329, pp. 3759–3779. [CrossRef]
Tolstov, G. P., 1965, Fourier Series, Englewood Cliffs, NJ.
Zygmund, A., 1968, Trigonometric Series, Cambridge University, Cambridge, UK, Vol. I.
Gottlieb, D., and Orszag, S. A., 1977, Numerical Analysis of Spectral Methods. SIAM, Philadelphia, PA.
Du, J. T., Li, W. L., Liu, Z., Jin, G., and Yang, T. J., 2011, “Free Vibration of Two Elastically Coupled Rectangular Plates With Uniform Elastic Boundary Restraints,” J. Sound Vib., 330, pp. 788–804. [CrossRef]


Grahic Jump Location
Fig. 3

Coordinate systems defining the coupling or restraining springs

Grahic Jump Location
Fig. 2

The coordinate transformation between two coupled plates

Grahic Jump Location
Fig. 1

Schematic of a plate structure consisting of arbitrarily oriented plates

Grahic Jump Location
Fig. 4

The construction of a box structure

Grahic Jump Location
Fig. 5

Convergence of the natural frequencies predicted based on different truncation numbers: M = N = 5, 6, 7, 8, 9

Grahic Jump Location
Fig. 6

The mode shapes for the free open box: the (a) 1st mode; (b) 2nd mode; (c) 3rd mode; (d) 4th mode; (e) 5th mode; (f) 6th mode; (f) 7th mode; and (f) 8th mode. Panel (a): calculated; panel and (b): measured.

Grahic Jump Location
Fig. 7

The mode shapes for the open box under CSFF boundary conditions: the (a) 1st mode; (b) 2nd mode; (c) 3rd mode; (d) 4th mode; (e) 5th mode; (f) 6th mode; (f) 7th mode; and (f) 8th mode. Panel (a):Left—calculated; panel and (b): FEM.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In