PERIODIC RESPONSE OF MECHANICAL SYSTEMS WITH LOCAL NON-LINEARITIES USING AN ENHANCED GALERKIN TECHNIQUE

Abstract

A new semi-analytical framework for the study of mechanical systems with local non-linearities is presented. It recognizes that many practical built-up structures consist of non-linearities, typically at joints or junctions, with a few degrees-of-freedom, coupled with many linear degrees-of-freedom of the adjoining components. Unlike linear systems, sinusoidal excitation produces a periodic response, including super and subharmonics. A Galerkin based computational method for the solution of the steady state periodic response of mechanical systems with local non-linearities, defined in the time and/or frequency domains is proposed. The method incorporates a form of order reduction and numerical continuation with distinct benefits. Order reduction enables inclusion of the extensive and necessary, but often linear, assembled component dynamics with minimal computational cost. Additionally, the proposed form of reduction allows non-linearities explicitly defined in the frequency and the time domain to be handled simultaneously. The continuation scheme, based on the QR decomposition, facilities parametric studies for design by using the system solution for one set of parameters to optimally predict the steady state periodic solution for a similar set of parameters. Two specific examples have been chosen to illustrate the key concepts and methodology of the dual domain method. In the first example, a rigid body connected to a simply supported elastic beam via a non-linear spring is considered. The hydraulic engine mounting system is presented as the second example; a practical representative of the issues discussed in this article. Results of digital and analog computational studies verify the accuracy of the proposed method and highlight its unique capabilities.