Finite Element Large-Amplitude Free and Forced Vibrations of Rectangular Thin Composite Plates

Abstract

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration responses of arbitrarily laminated anisotropic composite rectangular thin plates. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite rectangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The finite element results are compared with available approximate continuum solutions. The amplitude-frequency relations for convergence with gridwork refinement, different boundary conditions, aspect ratios, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Non-Linear Forced Vibrations of Plates by an Asymptotic–numerical Method

“…The non-linearity associated with rectangular plates is more pronounced than that associated with square plates. The same results were presented and discussed by Mei and co-workers using FEM and the iterative process for isotropic and composite plates [14,15]. Non-linear deflection shapes of a fully clamped square plate under a harmonic distributed force (P d 0 ¼ 0Á2) corresponding to various frequencies are presented in Figure 6.…”

“…In order to make comparisons between numerical results available based on the finite element method or on analytic procedures, the same formulation of the excitation forces as that presented by Mei et al [14,15] is considered, as explained in section 2.3. Frequency ratios for non-linear vibration of fully clamped and simply supported isotropic square plates under a harmonic uniformly distributed force are presented in Tables 1 and 2.…”

“…Free and steady state vibrations of plates using FEM have been studied by Wellford et al [33]. Intensive and instructive studies of forced vibrations of structures based on the finite element method have been presented by Mei et al [13,14], Chiang et al [15] Shi et al [19,21] and Zhou et al [20]. Based on Galerkin's method and the harmonicbalance method applied to a single-degree-of-freedom system, some frequency-amplitude relationships were presented by Dumir and Bhaskar [34] for beams and plates and by Sherif [35] for unsymmetric sandwich circular plates.…”

“…Attention was particularly focused on the various analytical, semi-analytical and numerical methods used in various studies related to the subject. Various works assume the dependence in time to be harmonic and combine the FEM with linearizing techniques [12][13][14][15] or with iterativeincremental procedures [16,17]. A reduced basis technique based on FEM was developed and applied to non-linear vibrations by Noor et al [18].…”

Finite element analysis of nonlinear flutter of composite panels

“…The aerodynamic load Pa is described by the first-order The external virtual work is expressed in terms of the element mass and aerodynamic matrices Equations (20) and (24) are substituted into Equation (16) in order to obtain the elemental equations of motion…”