Nonlinear partial differential equations for vibration of an orthotropic thick plate on a two-parameter (Pasternak type) elastic foundation subjected to a nonuniform initial stress are derived. Both rotary inertia and transverse stress are considered in the derivation. The Galerkin method is used to transform the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge–Kutta method is used to obtain the ratio of nonlinear to linear frequency. Numerical examples are presented for the combination of pure bending stresses and extensional stresses are taken to be the initial stresses in the plane of the plate. These equations are then applied to a simply supported orthotropic plate on a Pasternak foundation to solve its motion of nonlinear vibration. The effects of the material properties, initial stress, amplitude of vibration, and foundation stiffness on frequency ratio are discussed.
Chauhan, H. N. and Herrmann, G. (1956). Influence of Large Amplitudes on Free Flexural Vibrations of Elastic Plates , Journal of Applied Mechanics, 23: 532–540 .
2.
Chauhan, A. P. and Ashwell, D. G. (1969). Small and Large Amplitude Free Vibrations of Square Shallow Shells , International Journal of Mechanical Sciences, 11: 337–350 .
3.
Berger, H. M. (1955). New Approach to the Analysis of Large Deflections of Plates , Journal of Applied Mechanics, 22: 465–472 .
4.
Sathyamoorthy, M. (1979). Effects of Large Amplitude Shear and Rotary Inertia on Vibration of Rectangular Plates , Journal of Sound and Vibration, 63(2): 161–167 .
5.
Wu, C. I. and Vinson, J. R. (1969). Influences of Large Amplitude Transversely Shear Deformation and Rotatory Inertia on Lateral Vibrations of Transversely Isotropic Plates , Journal of Applied Mechanics, 36: 254–260 .
6.
Wu, C. I. and Vinson, J. R. (1969). On the Non-linear Oscillations of Plates Composed of Composite Materials , Journal of Computer Mathematic, 3: 548–561 .
7.
Wu, C. I. and Vinson, J. R. (1971). Non-linear Oscillations of Laminated Specially Orthotropic Plates with Clamped and Simply Supported Edges , Journal of Acoustic Society America, 49: 561–567 .
8.
Bennett, J. A. (1971). Nonlinear Vibrations of Simply Supported Angle-ply Laminated Plates , AIAA Journal, 9: 1997–2003 .
9.
Chandra, R. and Raju, B. B. (1975). Large Deflection Vibration of Angle Ply Laminated Plate , Journal of Sound and Vibration, 40(3): 393–408 .
10.
Chandra, R. (1976). Large Deflection Vibration of Cross-ply Laminated Plates with Certain Edge Condition , Journal of Sound and Vibration, 48(4): 509–514 .
11.
Chia, C. Y. and Prabhakara, M. K. (1978). A General Mode Approach to Nonlinear Flexural Vibrations of Laminated Rectangular Plates , Journal of Applied Mechanics, 45: 623–628 .
12.
Reddy, J. N. and Chao, W. C. (1982). Nonlinear Oscillations of Laminated, Anisotropic, Rectangular Plates , Transaction of ASME, 49: 396–402 .
13.
Ganapathi, M. , Varadan, T. K. and Sarma, B. S. (1991). Nonlinear Flexural Vibrations of Laminated Orthotropic Plates , Computers and Structures, 39(6): 685–688 .
14.
Yeh, F. H. and Liu, W. H. (1991). Nonlinear Analysis of Rectangular Orthotropic Plates , International Journal of Mechanical Sciences, 33(7): 563–578 .
15.
Sathyamoorthy, M. (1996). Nonlinear Vibrations of Plates: An Update of Recent Research Developments , Applied Mechanics Reviews, 49(10): 55–62 .
16.
Eratll, N. and Akoz, A. Y. (1997). The Mixed Finite Element Formulation for the Thick Plates on Elastic Foundations , Computers and Structures, 65(4): 515–529 .
17.
Omurtag, M. H. and Kadioglu, F. (1998). Free Vibration Analysis of Orthotropic Plates Resting on Pasternak Foundation by Mixed Finite Element Formulation , Computers and Structures, 67(4): 253–265 .
18.
Aiello, M. A. and Ombres, L. (1999). Buckling and Vibrations of Unsymmetric Laminates Resting on Elastic Foundations under In-plane and Shear Forces , Computers and Structures, 44(1): 31–41 .
19.
Chen, C. N. (2000). Vibration of Prismatic Beam on an Elastic Foundation by the Differential Quadrature Element Method , Computers and Structures, 77(1): 1–9 .
20.
Filipich, G. P. and Rosales, M. B. (2002). A Further Study about the Behaviour of Foundation Piles and Beams in a Winkler-Pasternak Soil , International Journal of Mechanical Sciences, 44(1): 21–36 .
21.
Xu, C. S. and Chia, C. Y. (1993). Large-amplitude Vibrations of Unsymmetrically Laminated Imperfect Thick Circular Plates on an Elastic Foundation , Composites Science and Technology, 49(2): 131–137 .
22.
Shih, Y. S. and Blotter, P. T. (1993). Non-linear Vibration Analysis of Arbitrarily Laminated Thin Rectangular Plates on Elastic Foundations , Journal of Sound and Vibration, 167(3): 433–459 .
23.
Patel, B. P. , Ganapathi, M. and Touratier, M. (1999). Nonlinear Free Flexural Vibrations/Post-buckling Analysis of Laminated Orthotropic Beams/Columns on a Two Parameter Elastic Foundation , Computers and Structures, 46(2): 189–196 .
24.
Brunelle, E. J. and Robertson, S. R. (1974). Initially Stressed Mindlin Plates , AIAA Journal, 12(3): 1036–1045 .
25.
Brunelle, E. J. and Robertson, S. R. (1976). Vibrations of an Initially Stressed Thick Plate , Journal of Sound and Vibration, 45(3): 405–416 .
26.
Yamaki, N. and Chiba, M. (1983). Nonlinear Vibrations of a Clamped Rectangular Plate with Initial Deflection and Initial Edge Displacement – Part I: Theory , Thin-Walled Structures, 1(1): 3–29 .
27.
Chen, L. W. and Doong, J. L. (1983). Large Amplitude Vibration of an Initially Stressed Moderately Thick Plate , Journal of Sound and Vibration, 89(4): 499–508 .
28.
Chen, L. W. and Doong, J. L. (1983). Large Amplitude Vibration of an Initially Stressed Thick Circular Plate , AIAA Journal, 21: 1317–1324 .
29.
Cheung, Y. K. , Zhu, D. S. and Iu, V. P. (1998). Nonlinear Vibration of Thin Plates with Initial Stress by Spline Finite Strip Method , Thin-Walled Structures, 32(4): 275–287 .
30.
Chen, C. S. , Cheng, W. S. , Chien, R. D. and Doong, J. L. (2002). Large Amplitude Vibration of an Initially Stressed Cross Ply Laminated Plates , Applied Acoustics, 63(9): 939–956 .
31.
Chen, C. S. and Fung, C. P. (2004). Nonlinear Vibration of an Initially Stressed Hybrid Composite Plates , Journal of Sound and Vibration, 274(3–5): 1013–1029 .
32.
Chen, C. S. (2005). Large Amplitude Vibration of a Shear Deformable Functionally Graded Plate , Composite Structures, 68(3): 295–302 .
33.
Bolotin, V. V. (1963). Nonconservative Problems of the Theory of Elastic Stability, Macmillan Comp., New York .
34.
Singh, P. N. , Sundarajan, V. and Das, Y. C. (1974). Large Amplitude Vibration of Same Moderately Thick Structural Elements , Journal of Sound and Vibration, 36(3): 375–387 .
35.
Singh, G. , Raju, K. K. and Rao, G. V. (1990). Nonlinear Vibrations of Simply Supported Rectangular Cross Ply Plates , Journal of Sound and Vibration, 142(2): 213–226 .
36.
Jones, R. M. (1975). Mechanics of Composite Materials, McGraw-Hill, New York .
