Extension of the direct parametrization method of invariant manifolds for systems with variable mass matrices: application to the Ziegler column

Abstract

Models of mechanical systems with variable mass matrices are present in many physical and engineering applications, most notably in fluid–structure interaction and in multibody systems such as articulated machines and mechanisms. At the same time, the presence of nonlinearities makes the simulations all the more costly, as they often need to be repeated for a large number of parameters. Thus, model order reduction techniques are highly important from a practical standpoint, from the design phase to the analysis. Among the available methods for vibrating systems, the computation of nonlinear normal modes, defined as invariant manifolds, is a direct, simulation-free approach that has demonstrated its potential in a variety of contexts. In this paper, an extension of the direct parametrization method for invariant manifolds is proposed to handle the case of systems with a variable mass matrix. To illustrate the capability of the reduction technique, the nonlinear dynamics of a Ziegler column is investigated under static and pulsating loads.