Forced Vibrations Of Plates Research Articles

Nonlinear vibration of circular plate with variable thickness

This paper is concerned with the nonlinear vibration problems of circular plates with variable thickness. The nonlinear equations of plates with variable thickness are extended to the dynamic case. The resulting equations can be solved by using an iterative method, a Galerkin’s approach and a perturbation method. Detailed solutions and numerical results are given for two kinds of boundary conditions, the clamped edge and the supported edge. The results show that the solutions for the case of the plates with uniform thickness can be included in the solution herin as a special case. The effect of various thickness parameters is investigated in detail. Also, a Runge-Kutta method is used to solve the free and forced vibrations of plates with variable thickness, and the results are obtained firstly. It has shown that the adoption of variable thickness plate would be useful in engineering design.

A reduction method for the analysis of non-linear forced vibration of plates

The non-linear forced vibration of plates is investigated by using an implicit time integration scheme and a reduction method. The plate is first discretized into generalized finite strips with the non-linear strains of von Karman type included. A set of basis vectors is generated and transformation on to the subspace spanned by these vectors is then performed in such a way that the time-stepping dynamic analysis is totally carried out within the subspace. Because of the reduction of number of unknowns, a significant reduction in computing time can be achieved as compared with that for the analysis of the complete system. An example of a plate with different time-varying loads is given to demonstrate the potential of the method.

Comments on ‘‘A general formulation for the sound radiation from rectangular, baffled plates with arbitrary boundary conditions’’ [Berry e t a l., J. Acoust. Soc. Am. 8 8, 2792–2802 (1990)

The recent, excellent article ‘‘A general formulation for the sound radiation from rectangular, baffled plates with arbitrary boundary conditions,’’ by A. Berry, J. L. Guyader, and J. Nicolas [J. Acoust. Soc. Am. 88, 2792–2802 (1990)] calls for some remarks and clarifications concerning use of polynomial functions when dealing with free and forced vibrations of plates.

Free and forced vibrations of plates by boundary elements

A direct boundary element methodology is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the frequency domain dynamic fundamental solution of the problem and this essentially creates only boundary integrals. Thus only the plate perimeter has to be discretized and this is accomplished with the aid of quadratic isoparametric boundary elements for increased accuracy. Both free and forced vibration problems are considered. The free vibration problem is reduced to a matrix eigenvalue problem wherein the matrix coefficients are complex Bessel functions of the frequency parameter. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. Numerical examples are presented to illustrate the proposed methodology and demonstrate its advantages.

Free and forced vibrations of plates by boundary and interior elements

AbstractA direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.

A Mixed Method in Elasticity

A mixed method for three-dimensional elasto-dynamic problems has been formulated which gives a complete choice in prescribing the boundary conditions in terms of either stresses, or displacements, or partly stresses and partly displacements. The general expressions for the responses of the elastic body have been derived in the form of transcendental partial differential equations of a set of initial functions, which can be evaluated in terms of the prescribed boundary conditions. The method so-formulated has been illustrated by applying it to the theory of plates. Only plates subjected to antisymmetric loads have been considered for illustration. Some examples of free and forced vibration of plates have been presented. Results are compared with solutions from existing theories.