Three-dimensional vibration analysis of circular and annular plates via the Chebyshev–Ritz method

Abstract

A three-dimensional free vibration analysis of circular and annular plates is presented via the Chebyshev–Ritz method. The solution procedure is based on the linear, small strain, three-dimensional elasticity theory. Selecting Chebyshev polynomials which can be expressed in terms of cosine functions as the admissible functions, a convenient governing eigenvalue equation can be derived through the Ritz method. According to the geometric properties of circular and annular plates, the vibration is divided into three distinct categories: axisymmetric vibration, torsional vibration and circumferential vibration. Each vibration category can be further subdivided into the antisymmetric and symmetric ones in the thickness direction. Convergence and comparison study demonstrated the high accuracy and efficiency of the present method. The present approach shows a distinct advantage over some other Ritz solutions in that stable numerical operation can be guaranteed even when a large number of admissible functions is employed. Therefore, not only lower-order but also higher-order eigenfrequencies can be obtained by using sufficient terms of the Chebyshev polynomials. Finally, some valuable results for annular plates with one or both edges clamped are given and discussed in detail.