Rayleigh Quotient Algorithm for Modal Analysis of Structural Models

Abstract

An algorithm for the computation of natural frequencies in continuum models based on classical separation of variables and the Rayleigh quotient is presented. The iterative method uses frequency-dependent shape functions derived from conventional modal analysis to assemble the boundary condition matrix resulting from the application of separation of variables for modal analysis, and the corresponding mass and stiffness matrices. An order reduction to a single generalized coordinate allows the application of the Rayleigh quotient for the estimation of a particular natural frequency. An iterative procedure is followed which starts from an initial estimate of a desired natural frequency. The evaluation of the Rayleigh quotient provides an improved estimate of the squared natural frequency and the corresponding improvement of the shape functions that estimate a mode shape. The recursive application of the proposed algorithm allows the estimation of any natural frequency and mode shape of the continuum model. The application of the proposed technique is illustrated using continuum-parameter rod and beam elements in longitudinal and flexural vibrations, respectively. Some characteristics of the speed of convergence and regions of convergence are explored. The algorithm shows excellent convergence characteristics.