DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges

Abstract This paper presents an efficient and accurate differential quadrature (DQ) large amplitude free vibration analysis of laminated composite thin beams on nonlinear elastic foundation. Beams under consideration have elastically restrained against rotation and in-plane immovable edges. Elastic foundation has cubic nonlinearity with shearing layer. We impose the boundary conditions directly into the governing equations in spite of the conventional DQ method and without any extra efforts. A direct iterative method is used to solve the nonlinear eigenvalue system of equations after transforming the governing equations into the frequency domain. The fast rate of convergence of the method is shown and their accuracy is demonstrated by comparing the results with those for limit cases, i.e. beams with classical boundary conditions, available in the literature. Besides, we develop a finite element program to verify the results of the presented DQ approach and to show its high computational efficiency. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams are studied.

[1]  Ghodrat Karami,et al.  A new differential quadrature methodology for beam analysis and the associated differential quadrature element method , 2002 .

[2]  C. Bert,et al.  Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam , 1992 .

[3]  T. K. Varadan,et al.  Lagrange-type formulation for finite element analysis of non-linear beam vibrations , 1983 .

[4]  P. Malekzadeh,et al.  Large deformation analysis of orthotropic skew plates with nonlinear rotationally restrained edges using DQM , 2007 .

[5]  Ghodrat Karami,et al.  In-plane free vibration analysis of circular arches with varying cross-sections using differential quadrature method , 2004 .

[6]  T. K. Varadan,et al.  On various formulations of large amplitude free vibrations of beams , 1988 .

[7]  B. P. Patel,et al.  Application of spline element for large-amplitude free vibrations of laminated orthotropic straight/curved beams , 1998 .

[8]  Hongzhi Zhong,et al.  Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method , 2003 .

[10]  G. Karami,et al.  Application of a new differential quadrature methodology for free vibration analysis of plates , 2003 .

[11]  G. Venkateswara Rao,et al.  Large-amplitude free vibrations of beams—A discussion on various formulations and assumptions , 1990 .

[12]  Gangan Prathap,et al.  Galerkin finite element method for non-linear beam vibrations , 1980 .

[13]  T. K. Varadan,et al.  The large amplitude vibration of hinged beams , 1978 .

[14]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[15]  J. Reddy.  Mechanics of laminated composite plates : theory and analysis , 1997 .

[16]  M. Touratier,et al.  Nonlinear free flexural vibrations/post-buckling analysis of laminated orthotropic beams/columns on a two parameter elastic foundation , 1999 .

[17]  Pedro Ribeiro,et al.  HIERARCHICAL FINITE ELEMENT ANALYSES OF GEOMETRICALLY NON-LINEAR VIBRATION OF BEAMS AND PLANE FRAMES , 2001 .

[18]  G. Karami,et al.  Static and stability analyses of arbitrary straight-sided quadrilateral thin plates by DQM , 2002 .

[19]  J.-J. Li,et al.  Differential quadrature method for nonlinear vibration of orthotropic plates with finite deformation and transverse shear effect , 2005 .

[20]  T. K. Varadan,et al.  Ritz finite element approach to nonlinear vibrations of beams , 1984 .

[21]  P. Malekzadeh,et al.  Out-of-plane static analysis of circular arches by DQM , 2003 .

[22]  Rakesh K. Kapania,et al.  Nonlinear vibrations of unsymmetrically laminated beams , 1989 .

[23]  A. R. Setoodeh,et al.  Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM , 2007 .

[24]  Hongzhi Zhong,et al.  Non-linear vibration analysis of beams by a spline-based differential quadrature method , 2004 .

[25]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[26]  Ghodrat Karami,et al.  Differential quadrature nonlinear analysis of skew composite plates based on FSDT , 2006 .