A simple formula to study the large amplitude free vibration behavior of structural members, such as beams and plates, is developed. The nonlinearity considered is of von Karman type, and after eliminating the space variable(s), the corresponding temporal equation is a homogeneous Duffing equation. The simple formula uses the tension(s) developed in the structural members due to large deflections along with the corresponding buckling load obtained when the structural members are subjected to the end axial or edge compressive load(s) and are equal in magnitude of the tension(s). The ratios of the nonlinear to the linear radian frequencies for beams and the nonlinear to linear time periods for plates are obtained as a junction of the maximum amplitude ratio. The numerical results, for the first mode of free vibration obtained from the present simple formula compare very well to those available in the literature obtained by applying the standard analytical or numerical methods with relatively complex formulations.

Simple Formula to Study the Large Amplitude Free Vibrations of Beams and Plates

In this article, the prediction of free vibration characteristics of various initially compressively loaded structural members is given in the form of a master formula. These structural members are optimised for the strength criteria and are subjected to high initial compressive loads (or stresses). These initial compressive loads have to be considered when the free vibration characteristics are evaluated. The standard method is to solve the differential equation considering the term corresponding to these loads. Though this is possible for simple structural members, it is often difficult while dealing with complex structural members with complicated effects. Under these circumstances, the popular approximate continuum and numerical methods like the Rayleigh–Ritz and finite element methods are used to obtain reliable and accurate solutions to the initially loaded free vibration characteristics of these structural members. However, these approximate methods are also having their inherent difficulties. To a...

Master formula for evaluating vibration frequencies of structural members under compressive loads

In this study a simple displacement field is proposed for finite element vibration analysis of circular plates. The displacement field is a combination of a conventional field, which is very popular for three-dimensional vibration analysis of circular plates or cylinders, and another field, which is most general for axisymmetric vibration. The investigation here is intended, through mathematical formulation and examples, to verify the existence of coupled axisymmetric vibration modes. These modes cannot be found by conventional displacement field, but can easily be revealed by the present model. Validity and accuracy of the proposed displacement field to show all the prominent vibration modes of circular plates or cylinder-like structures is also demonstrated by comparing results using the present method with those of a three-dimensional finite element analysis and experiments.

A simple and unified displacement field for three-dimensional vibration analysis of prestressed circular plates

The present study presents the effect of deformation and stress due to preloading on vibration of circular plates. The preloadings may be applied in the radial direction on edge or in the transverse direction on plate surface. The research features that, first, a unified displacement field for circular plates is employed; second, the formulation is based on elasticity and has fewer approximations or assumptions than plate theories; third and most noticeable, the analysis takes into account the deformation due to preloading, in addition to the stresses and shows that such deformation has significant effect on vibration of circular plates for some cases.

G. Venkateswara Rao

Papers

Prediction of fundamental frequency of initially in-plane-loaded moderately thick circular plates