Frequency Equations for the In‐Plane Vibration of Circular Annular Disks

S Bashmal,S Rakheja,R Bhat

doi:10.1155/2010/501902

Abstract

This paper deals with the in‐plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in‐plane free vibration of an elastic and isotropic disk is studied on the basis of the two‐dimensional linear plane stress theory of elasticity. The exact solution of the in‐plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson′s ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.

Highlights

The out-of-plane vibration properties of circular disks subjected to a variety of boundary conditions have been extensively investigated (e.g., [1,2,3,4])
This study aims at generalized formulation for in-plane vibration analyses of circular annular disks under different combinations of clamped, free, or flexible boundary conditions at the inner and outer edges
The exact frequency parameters for the annular disks were subsequently obtained under different combinations of boundary conditions at the inner and outer edges

Summary

Introduction

The out-of-plane vibration properties of circular disks subjected to a variety of boundary conditions have been extensively investigated (e.g., [1,2,3,4]). The in-plane vibration of circular disks was first attempted by Love [8] who formulated the equations of motion for a thin solid circular disk with free outer edge together with the general solution. The equations of motion were subsequently solved by Onoe [9] to obtain the exact frequency equations corresponding to different modes of a solid disk with free outer edge. The in-plane vibration characteristics of solid disks clamped at the outer edge have been investigated in a few recent studies. Farag and Pan [11] evaluated the frequency parameters and the mode shapes of in-plane vibration of solid disks clamped at the outer edge using assumed deflection modes in terms of trigonometric and Bessel functions. Park [12] studied the exact frequency equation for the solid disk clamped at the outer edge

Objectives

Results

Conclusion