Large amplitude free vibration analyses of thin-to-moderately thick angle-ply laminated rectangular plates with edges rotationally restrained against rotation are investigated using the differential quadrature method (DQM). The governing equations are based on two different theories: thin or classical plate theory and the first order shear deformation plate theory. In both theories, Green's strain in conjunction with the von Karman assumption is employed. The boundary conditions are implemented precisely at boundary grid points and are built into the equations of motion without any extra efforts. After transforming the governing equations from temporal to frequency domain, a direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and its accuracy is demonstrated by comparing the results with those for limit cases, i.e., laminated plates with classical boundary conditions and also for linear free vibration analysis of laminated plates with restrained edges. Finally, the effects of different parameters on the ratio of nonlinear to linear natural frequency of plates are studied.
Sathyamoorthy, M. (1987). Nonlinear Vibration Analysis of Plates: A Review and Survey of the Current Developments, Applied Mechanics Review, 40: 1553—1561.
2.
Chiang, C.K. , Mei, C. and Gray, C.E. (1991). Finite Element Large-amplitude Free and Forced Vibrations of Rectangular Thin Composite Plates, Trans ASME Journal of Vibration Acoustics, 113: 309—315.
3.
Han, W. and Petyt, M. (1997). Geometrically Nonlinear Vibration Analysis of Thin, Rectangular Plates using the Hierarchical Finite Element Method-II: 1st Mode of laminated Plates and Higher Modes of Isotropic and Laminated Plates , Computers and Structures, 63: 309—318.
4.
Ribeiro, P. and Petyt, M. (1999). Non-linear Vibration of Composite Laminated Plates by the Hierarchical Finite Element, Composite Structures, 46: 197—208.
5.
Ribeiro, P. (2005). First-Shear Deformation, p-version, Finite Element for Laminated Plate Nonlinear Vibrations, American Institute of Aeronautical and Astronautical Journal, 43: 1371—1379.
6.
Ribeiro, P. and Petyt, M. (1999). Nonlinear Vibration of Plates by Hierarchical Finite Element and Continuation Methods, International Journal of Mechanical Science, 41: 437—459.
7.
Harras, B. and Benamar, R.G. (2002). Geometrically Non-linear Free Vibration of Fully Clamped Symmetrically Laminated Rectangular Composite Plates, Journal of Sound and Vibration, 251: 579—619.
8.
Chien, R.D. and Chen, C.S. (2005). Nonlinear Vibration of Laminated Plates on Nonlinear Elastic Foundation, Composite Structures, 70: 90—99.
9.
Chen, C.S., Fung, C.P. and Chien, R.D. (2006). Nonlinear Vibration of Orthotropic Plates with Initial Stresses on a Two Parameter Elastic Foundation, Journal of Reinforced Plastics and Composites, 25: 283—301.
10.
Chen, C.S., Fung, C.P. and Chien, R.D. (2007). Nonlinear Vibration of an Initially Stressed Laminated Plate According to a Higher-order Theory, Composite Structures , 77: 521—532.
11.
Li, J.J. and Cheng, C.J. (2005). Differential Quadrature Method for Nonlinear Free Vibration of Orthotropic Plates with Finite Deformation and Transverse Shear Effect, Journal of Sound and Vibration , 281: 295—309.
12.
Ramachandran, J. and Bhaskar, K. (1985). Large Amplitude Vibrations of Rectangular Plates with Nonuniform Elastic Edge Supports, Journal of Sound and Vibration, 103: 153—158.
13.
Bhattacharya, A.P. and Sivakumaran , K.S. (1982). Nonlinear Flexural Vibrations of Initially Deflected Cross-ply Laminated Plates with Elastically Restrained Edges, Fiber Science Technology , 17: 157—168.
14.
Liu, W.H. and Yeh, F.H. (1993). Non-linear Vibrations of Initially Imperfect, Orthotropic, Moderately Thick Plates with Edge Restraints , Journal of Sound and Vibration, 165: 101—122.
15.
Singha, M.K. and Ganapathi, M. (2004). Large Amplitude Free Flexural Vibrations of Laminated Composite Skew Plates, International Journal of Non-Linear Mechanic, 39: 1709—1720.
16.
Liew, K.M., Xiang, Y. and Kitipornchai, S. (1997). Vibration of Laminated Plates Having Elastic Edge Flexibility, ASCE Journal of Mechnical Engineering, 123: 1012—9.
17.
Karami, G., Malekzadeh, P. and Mohebpour, S.R. (2006). DQM Free Vibration Analysis of Moderately Thick Symmetric Laminated Plates with Elastically Restrained Edges, Composite Structures, 74: 115—125.
18.
Nallim, L.G. and Grossi, R.O.(November 2007). Vibration of Angle-ply Symmetric Laminated Composite Plates with Edges Elastically Restrained, Composite Structures, 81(1): 80—83.
19.
Ashour, A.S. (2006). Vibration of Angle-ply Symmetric Laminated Composite Plates with Edges Elastically Restrained, Composite Structures , 74: 294—302.
20.
Bert, C.W., Jang, S.K. and Striz, A.G. (1988). Two New Approximate Methods for Analyzing Free Vibration of Structural Components, American Institute of Aeronautical and Astronautical 1964 Journal, 26: 612—618.
21.
Bert, C.W. and Malik, M. (1996). Differential Quadrature Method in Computational Mechanics: A review, Applied Mechanics Review, 49: 1—27.
22.
Karami, G. and Malekzadeh, P. (2002). A New Differential Quadrature Methodology for Beam Analysis and the Associated DQEM, Computer Method in Applied Mechanics and Engineering, 191: 3509—3526.
23.
Karami, G. and Malekzadeh, P. (2003). Application of a New Differential Quadrature Methodology for Free Vibration Analysis of Plates, International Journal of Numerical Method in Engineering, 56: 847—867.
24.
Malekzadeh, P. and Shahpari, S.A. (2005). Free Vibration Analysis of Variable Thickness Thin and Moderately Thick Plates with Elastically Restrained Edges by DQM, Thin-Walled Structures, 43: 1037—1050.
25.
Striz, A.G. , Jang, S.K. and Bert, C.W. (1988). Nonlinear Bending Analysis of Thin Circular by Differential Quadrature, Thin-Walled Structures , 6: 51—62.
26.
Bert, C.W.Jang, S.K. and Striz, A.G. (1989). Nonlinear Bending Analysis of Orthotropic Rectangular Plates by the Method of Differential Quadrature, Computational Mechanics, 5: 217—226.
27.
Lin, R.M. , Lim, K.M. and Du, H. (1994). Large deformation analysis of plates under thermal loading, Computational Method in Applied Mechanics Engineering, 117: 381—390.
28.
Chen, W., Shu, C., He, W. and Zhong, T. (2000). The Application of Special Matrix Product to Differential Quadrature Solution of Geometrically Nonlinear Bending of Orthotropic Rectangular Plates, Computers and Structures, 74: 65—76.
29.
Malekzadeh, P. and Karami, G. (2006). Differential Quadrature Nonlinear Analysis of Skew Composite Plates Based on FSDT, Engineering Structure, 28: 1307—1318.
30.
Malekzadeh, P. and Setoodeh, A.R. (2007). Large Deformation Analysis of Moderately Thick Laminated Plates on Nonlinear Elastic Foundations by DQM , Composite Structures, 80: 569—579.
31.
Malekzadeh, P. and Farid, M. (2007). A DQ Large Deformation Analysis of Composite Plates on Nonlinear Elastic Foundations, Composite Structures, 79: 251—260.
32.
Malekzadeh, P. and Fiouz, A.R. (2007). Large Deformation Analysis of Orthotropic Skew Plates with Nonlinear Rotationally Restrained Edges Using DQM, Composite Structures, 80: 196—206.
33.
Malekzadeh, P. (2007). A Differential Quadrature Nonlinear Free Vibration Analysis of Laminated Composite Skew Thin Plates , Thin-Walled Structures, 45: 237—250.
34.
Malekzadeh, P. (2007). Differential Quadrature Large Amplitude Free Vibration Analysis of Laminated Skew Plates Based on FSDT, Composite Structures, Proof, Available online 18 April 2007.
35.
Feng, Y. and Bert, W. (1991). Application of the Quadrature Method to Flexural Vibration Analysis of a Geometrically Nonlinear Beam, Nonlinear Dynamics, 3: 13—18.
36.
Guo, Q. and Zhong, H. (2004). Non-linear Vibration Analysis of Beams by a Spline-based Differential Quadrature Method, Journal of Sound and Vibration, 269: 413—420.
37.
Zhong, H. and Guo, Q. (2003). Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method, Nonlinear Dynamics , 32: 223—234.
38.
Reddy, J.N. (1997). Mechanics of Laminated Composite Plates Theory and AnalysisCRC, Boca Raton.
39.
Venkateswara Rao, G., Raju, I.S. and Kanaka Raju , K. (1976). A Finite Element Formulation for Large Amplitude Flexural Vibrations of Thin Rectangular Plates, Computers and Structures, 6: 163—167.
