Nonlinear Normal Modes of Nonconservative Systems

Abstract

Linear modal analysis is a mature tool enjoying various applications ranging from bridges to satellites. Nevertheless, modal analysis fails in the presence of nonlinear dynamical phenomena and the development of a practical nonlinear analog of modal analysis is a current research topic. Recently, numerical techniques (e.g., harmonic balance, continuation of periodic solutions) were developed for the computation of nonlinear normal modes (NNMs). Because these methods are limited to conservative systems, the present study targets the computation of NNMs for nonconservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, a new finite element technique is proposed to solve the set of partial differential equations governing the manifold geometry.

Nonlinear normal mode continuation through a Proper Generalized Decomposition approach with modal enrichment

“…After a spatial discretisation, using for example the Finite Elements method, the differential equations governing the motion of a nonlinear dynamical system can usually take the following form [17,7], where the nonlinear efforts are separated from the linear ones:…”

“…A NNM is then defined as a two-dimensional invariant manifold in phase space. The reader can find more information about NNMs definitions in the works of Kerschen and Renson [19,17,5] for instance.…”

“…Although they do not have orthogonality properties like LNMs and do not decouple the equations, NNMs can be useful to investigate the modal interaction between widely spaced modes or modal bifurcations [17,5]. One of the major features of NNMs is that frequencies depends on the response amplitude on the contrary to LNMs.…”

On the computation of the slow dynamics of nonlinear modes of mechanical systems

“…The invariant manifold approach was extended to account for the effect of harmonic excitation [3] and viscous damping [5]. It has been applied to various problems including piecewise linear systems [6], internally resonant nonlinear modes [4] and generic nonlinearly damped systems [7].…”

“…These computations can involve several thousands of nonlinearly coupled algebraic equations [3]. Moreover, the development of numerically robust algorithms for the treatment of generic, in particular nonconservative nonlinearities, seems to be an unresolved problem, see e. g. [7]. Furthermore, since the time-dependency is lost in the problem definition, the characteristic frequencies cannot directly be obtained from the computed manifold, but has to be identified from simulation results [6,7].…”