Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

In this article, the nonlinear equations of motion for anti-symmetric angle-ply composite rectangular plates have been derived using the first-order shear deformation theory, including shear deformation and rotary inertia. By using the Galerkin method, five coupled nonlinear partial differential equations of motion are reduced to a nonlinear ordinary differential equation. Then, the multiple time scales method is used to solve the obtained equation and to derive an analytical relation for the nonlinear frequency. Results are compared with the literature and good agreement is achieved. After proving the validity of this study, nonlinear free and forced vibration of a fiber metal laminated rectangular plate have been studied and the effects of some system parameters on the nonlinear behavior of the FML rectangular plate have been investigated.

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“…Finally, after substituting equations ( 39) and ( 41) into equation (38), to have a periodic solution…”

Section: Free Vibrationmentioning

confidence: 99%

Nonlinear free and forced vibrations of anti-symmetric angle-ply hybrid laminated rectangular plates

Shooshtari

Razavi

2013

Journal of Composite Materials

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“…Large amplitude free vibration behavior of plates [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] have received considerable attention by various researchers due to its relative importance to the field of structural engineering. Leung and Mao [7] applied an Hamilton's formulation and symplectic integration methods to investigate the non-linear vibration behavior of beams and plates.…”

Section: Introductionmentioning

confidence: 99%

Large Amplitude Free Vibration Analysis of Thin Rectangular Plates: Simple Closed-form Solutions

Gunda¹

2013

Nonlinear Engineering

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Large amplitude free vibration behavior of thin, isotropic rectangular plate configurations are expressed in the form of simple closed-form solutions by using an application of the Ritz method based on coupled displacement fields. Influence of plate aspect ratio a b and Poisson ratio . / on the behavior of back-bone curves is briefly discussed for various boundary configurations of the rectangular plate. Proposed closed-form solutions are corrected for the simple harmonic motion (SHM) assumption using the well established harmonic balance method which is applicable for the homogeneous form of cubic non-linear Duffing equation.Keywords. Large amplitude free vibration, Thin rectangular plates, Ritz method, Closed-form solutions, Coupled displacement fields, von-Kármán type of geometric nonlinearity, Non-linear frequency. Nomenclature a = length of the rectangular plate b = width of the rectangular plate C = axial rigidity of the plate E h .1 2 / Á D = flexural rigidity of the plate E h 3 12.1 2 / Á E = Young's modulus h= thickness of the plate N x , N y and N xy = in-plane stress resultants M x , M y and M xy = moment resultants u = displacement in x-direction v = displacement in y-direction U = strain energy of the plate w = transverse displacement T = kinetic energy of the plate x, y = in-plane co-ordinates z 1 = maximum reference amplitude of the rectangular plate z 1 h = maximum reference amplitude to thickness ratio x , y and xy = in-plane strain terms L = linear frequency parameter h! L 2 a 4 4 D Á NL H = non-linear frequency parameter obtained on the basis of SHM assumption Â h! NL H 2 a 4 4 D Ã NL = non-linear frequency parameter h! NL 2 a 4 4 D Á = Poisson ratiox , y and xy = curvature terms Subscripts L = linear NL = non-linear NL H = non-linear based on harmonic motion assumption x = partial derivative with respect to x xx = second partial derivative with respect to x y = partial derivative with respect to y yy = second partial derivative with respect to y xy = partial derivative with respect to x and y 1 Introduction

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“…The hierarchal polynomials have been used as shape functions of transverse vibrating membrane [15] and plate [16][17][18] elements. In the present research work, the same polynomials are employed to establish the shape functions associated with the super-convergent element models.…”

Section: Introductionmentioning

confidence: 99%

Shape functions associated with inverse element formulations

Faroughi

Ahmadian

2010

Proceedings of the Institution of Mechanical Engineers, Part C:

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Super-convergent element formulations in local co-ordinates are obtained using inverse strategies. In the inverse approach discretization errors of the element formulation are minimized leading to super-convergent solutions. In the development of the inverse element model, no shape functions are introduced and therefore the task of element transformation from local to global co-ordinates system remains a challenge. In this paper, a procedure is proposed to produce shape functions associated with the inverse element formulations via hierarchical polynomials. A membrane element formulation is developed using inverse strategy as an example and its associated shape functions are determined using hierarchical polynomials. Numerical results indicate higher accuracy of the developed model in global co-ordinates compared to the reported models in the literature for the same element.

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Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

Cited by 9 publications

References 14 publications

Nonlinear free and forced vibrations of anti-symmetric angle-ply hybrid laminated rectangular plates

Nonlinear free and forced vibrations of anti-symmetric angle-ply hybrid laminated rectangular plates

Large Amplitude Free Vibration Analysis of Thin Rectangular Plates: Simple Closed-form Solutions

Shape functions associated with inverse element formulations

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