Large amplitude flexural vibration of simply supported skew plates.

Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement— Part II: Experiment

Abstract: Theoretical analyses are presented for nonlinear vibrations of a clamped rectangular plate under a uniformly distributed periodic load, with the effect of both initial deflection and initial edge displacement taken into consideration The dynamic analog of the Marguerre equations is used and the steady-state solutions are obtained by first applying the Galerkin method and then the harmonic balance method Actual calculations are carried out for the square plate under the assumption of the three degrees-of-freedom system and the frequency response characteristics and typical waveforms are determined under various initial edge displacements including initially buckled cases Effects of both initial deflection and initial edge displacement on the static deflection as well as the lower natural frequencies are also clarified

Non-linear vibrations of a clamped circular plate with initial deflection and initial edge displacement, part I: Theory

Abstract: Theoretical analyses are presented for axisymmetric non-linear vibrations of a clamped circular plate of isotropic materials, under uniformly distributed lateral loading, with the effect of both initial deflection and initial edge displacement taken into consideration. The dynamic analog of the Marguerre equations is used and the steady state solutions are obtained by first applying the Galerkin method and then the harmonic balance method. Actual calculations are carried out for the three degree of freedom Galerkin system and the frequency response characteristics and typical waveforms are determined under various initial edge displacements including initially buckled cases. Effects of both initial deflection and initial edge displacement on the static deflection as well as the lower natural frequencies are also clarified.

Analysis of dynamic instability for arbitrarily laminated skew plates

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.

Non-linear flexural vibrations of anisotropic skew plates

Abstract: The large amplitude free flexural vibrations of thin, elastic anisotropic skew plates are studied by using the von Karman field equations in which the governing non-linear dynamic equations are derived in terms of the stress function and the lateral displacement. Clamped boundary conditions are chosen and the in-plane edge conditions considered are either immovable or movable. Solutions are obtained by the Galerkin method on the basis of a one-term assumed vibration mode. The degree of non-linearity is obtained as a function of skew angle, aspect ratio and types of orthotropy. The results, on specializing for an isotropic skew plate and an orthotropic rectangular plate, agree well with those found in the literature. The use of the Berger approximation to study a skew plate with in-plane immovable edges is shown to lead to errors of both a quantitative and qualitative nature.

On the berger approximation: A critical re-examination

Abstract: Since the first paper by Berger some two decades ago, the simplification known as the Berger approximation has been invoked by the authors of several score papers in spite of the fact that no rational mechanical basis for the approximation could be found. Many recent papers have raised doubts on its applicability. In this paper, using certain well known results from the two dimensional theory of elasticity, a plausible explanation for the origin of the Berger method is suggested. The arguments can be developed further to show that other specious Berger-like approximations can be developed, all of them leading to uncoupled non-linear equations yielding different overall results. Further, it is shown that such methods fail to predict the non-linear behaviour with respect to important parameters and that whatever accuracy is obtained in the solution of a particular problem can at best be attributed to fortuity.

A new approach to the analysis of large deflections of plates

Abstract: As a result of the assumption that the strain energy due to the second invariant of the middle surface strains can be neglected when deriving the differential equations for a flat plate with large deflections, simplified equations are derived that can be solved readily. Computations using the solution of these simplified equations are carried out for the deflection of uniformly loaded circular and rectangular plates with various boundary conditions. Comparisons are made with available numerical solutions of the exact equations. The deflections found by this approach are then used to obtain the stresses, and the resulting stresses are compared with existing solutions. In all the cases where comparisons could be made, the deflections and stresses agree with the exact solutions within the accuracy required for engineering purposes.

Influence of Large Amplitudes on Flexural Vibrations of Elastic Plates

Abstract: Approximate solutions for the nonlinear bending vibrations of thin plates are presented for the cases of rectangular and circular plates subjected to various boundary conditions, and the effects of large amplitudes on both the free and forced vibrations are clarified

Large amplitude flexural vibration of rectangular plates

Abstract: By using an approximate formulation due to Berger it is shown that the vibration of rectangular plates with large amplitudes may be treated in a simple and unified manner. Numerical results are given for various boundary conditions.

Large-amplitude oscillations of oblique panels with an initial curvature

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

Non-linear flexural vibration of orthotropic skew plates

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.