3D vibration analysis of solid and hollow circular cylinders via Chebyshev–Ritz method

Abstract

A general approach is presented for solving the free vibration of solid and hollow circular cylinders. The analysis procedure is based on the small-strain, linear and exact elasticity theory. By taking the Chebyshev polynomial series multiplied by a boundary function to satisfy the geometric boundary conditions as the admissible functions, the Ritz method is applied to derive the frequency equation of the cylinder. According to the axisymmetric geometrical property of a circular cylinder, the vibration modes are divided into three distinct categories: axisymmetric vibration, torsional vibration and circumferential vibration. Moreover, for a cylinder with the same boundary conditions at the two ends, the vibration modes can be further divided into antisymmetric and symmetric ones in the length direction. Convergence and comparison studies demonstrate the high accuracy and small computational cost of the present method. A significant advantage over other Ritz solutions is that the present method can guarantee stable numerical operation even when a large number of terms of admissible functions are used. Not only the lower-order but also the higher-order frequencies can be obtained by using a few terms of the Chebyshev polynomials. Finally, the first several frequencies of circular cylinders with different boundary conditions, with respect to various parameters such as the length–radius ratio and the inside–outside radius ratio, are given.