Non-linear flexural vibration of orthotropic skew plates
TL;DR: In this article, the large amplitude (non-linear) free flexural vibration of thin, elastic, orthotropic skew plates clamped along all four edges was analyzed using the Galerkin's method.
Abstract: This report deals with the large amplitude (non-linear) free flexural vibration of thin, elastic, orthotropic skew plates clamped along all four edges. The analysis is based on the non-linear dynamic equations applicable for rectilinearly orthotropic skew plates, derived in terms of the stress function, F , and the lateral displacement, W . Solutions obtained on the basis of assumed vibration modes make use of Galerkin's method. Curves of amplitude versus period for clamped skew plates have been obtained for two types of orthotropy and, in each case, for different aspect ratios and sweep angles of the plate. The corresponding relationship for the isotropic case has also been obtained. The results when specialized for the cases of isotropic skew plates and orthotropic rectangular plates agree well with those in the literature. The results show that the non-linearity is of the “hardening” type, that is, the period of non-linear vibration decreases with increasing amplitude.
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Abstract: Here, the large amplitude free flexural vibration behavior of symmetrically laminated composite skew plates is investigated using the finite element method. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The nonlinear matrix amplitude equation obtained by employing Galerkin's method is solved by direct iteration technique. Time history for the nonlinear free vibration of composite skew plate is also obtained using Newmark's time integration technique to examine the accuracy of matrix amplitude equation. The variation of nonlinear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, fiber orientation and boundary condition.
76 citations
Cites background from "Non-linear flexural vibration of or..."
...Sathyamoorthy and Pandalai [37] analyzed single layer orthotropic skew plates for movable in-plane edges and Berger approximation for immovable in-plane edge conditions....
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TL;DR: In this paper, a nonlinear free vibration analysis of thin-to-moderately thick laminated composite skew plates is presented based on the first order shear deformation theory (FSDT) using differential quadrature method (DQM).
Abstract: Nonlinear free vibration analysis of thin-to-moderately thick laminated composite skew plates is presented based on the first order shear deformation theory (FSDT) using differential quadrature method (DQM). The geometrical nonlinearity is modeled using Green’s strain and von Karman assumptions in conjunction with the FSDT of plates. After transforming and discretizing the governing equations, which includes the effects of rotary inertia, direct iteration technique as well as harmonic balance method is used to solve the resulting discretized system of equations. The effects of skew angle, thickness-to-length ratio, aspect ratio and also the impact due to different types of boundary conditions on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Numerical solutions for nonlinear frequency laminated skew plates under different geometrical parameters and mixed boundary conditions are presented.
72 citations
TL;DR: Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented in this paper, where the governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions.
Abstract: Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented The governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions To cause the impact due to nonlinear terms more significant, in-plane immovable simply supported, clamped and different combinations of them are considered The effects of different parameters on the convergence and accuracy of the method are studied The resulted solutions are compared to those from other numerical methods to show the accuracy of the method Some new results for laminated composite skew plates with different mixed boundary conditions are presented and are compared with those obtained using the first order shear deformation theory based DQ (FSDT-DQ) method Excellent agreements exist between the solutions of the two approaches but with much lower computational efforts of the present DQ methodology with respect to FSDT-DQ method
60 citations
TL;DR: In this paper, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach, which includes the effects of shear deformation, in-plane and rotary inertia.
Abstract: Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.
54 citations
Cites background from "Non-linear flexural vibration of or..."
...Sathyamoorthy and Pandalai [21] have analyzed single layer orthotropic skew plates for movable in-plane edges and Berger approximation for immovable in-plane edge conditions....
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TL;DR: In this paper, the dynamic instability and nonlinear response of rectangular and skew laminated plates subjected to periodic in-plane load are studied, and the region of dynamic instability associated with the effect of the stacking sequence of lamination and the skew angle of plate are also investigated and discussed.
Abstract: The dynamic instability and nonlinear response of rectangular and skew laminated plates subjected to periodic in-plane load are studied. Based on von Karman plate theory, the large amplitude dynamic equations of thin laminated plates are derived by applying the approach of generalized double Fourier series. On the assumed mode shape, the governing equations are reduced to the Mathieu equation using Galerkin's method. The incremental harmonic balance (IHB) method is applied to solve the nonlinear temporal equation of motion, and the region of dynamic instability is determined in this work. Calculations are carried out for isotropic, angle-ply and arbitrarily laminated plates under two cases of boundary conditions. The principal region of dynamic instability associated with the effect of the stacking sequence of lamination and the skew angle of plate are also investigated and discussed. The results obtained indicated that the instability behavior of the system is determined by the several parameters, such as the boundary condition, number of the layers, stacking sequence, in-plane load, aspect ratio, amplitude and the skew angle of plate.
26 citations
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TL;DR: In this article, the Von Karman field equations for flexible oblique plates with an initial curvature are extended to a dynamical case using series of initial and additional deflections and Galerkin's procedure, the governing equation for an admissible mode time function is established using this single assumed modal deflection, and assuming built-in edge fiee to move in the inplane directions.
Abstract: Von Karman field equations for flexible oblique plates with an initial curvature are extended to a dynamical case Using series iepresentation of initial and additional deflections and Galerkin's procedure, the governing equation for an admissible mode time function is established Using this single assumed modal deflection, and assuming built-in edge fiee to move in the inplane directions, the following particular cases are discussed: buckling of an oblique plate under uniaxial compressive load, free linear vibrations of a square plate, large deflections of a uniformly loaded square plate, snap-through phenomena of a curved oblique plate under uniform transverse load, and free nonlinear vibrations A numeiical example concerning a rhombic plate is discussed in more detail The well-known fact of a decrease of the period of nonlinear vibrations with an increasing amplitude is corroborated, this relation being less pronounced for larger sweep angles
20 citations