Rayleigh-Ritz Method

Abstract

This is the first of two chapters devoted to the approximate analysis of continuous systems. In this chapter, a global method of approximation is considered: the Rayleigh-Ritz method. It is based on the definition of a set of global assumed modes defined on the entire domain and satisfying the kinematic boundary conditions. The method is based on Hamilton’s principle: the strain energy and the kinetic energy are expressed in terms of the generalized coordinates and the Lagrange equations are used to derive a set of differential equations which constitute an approximation of the partial differential equation governing the continuous system. The quality of the approximation depends on the capability of the assumed modes to represent the actual deformations of the system. The method is illustrated with the axial vibration of a bar and the planar vibration of a beam, including with axial prestress (e.g. building with gravity loads). This chapter also introduces the Rayleigh quotient and the principle of stationarity, which justifies that the Rayleigh-Ritz method tends to overestimate the natural frequencies of the system. A set of problems is proposed at the end of the chapter.

La sagesse a ses excès et n’a pas moins besoin de modération que la folie. Montaigne, Essais (1572–1588)