Abstract

A nonlinear modal analysis procedure is presented for the forced response of nonlinear structural systems. It utilizes the notion of invariant manifolds in the phase space, which was recently used to define nonlinear normal modes and the corresponding nonlinear modal analysis for unforced vibratory systems. For harmonic forcing, a similar procedure could be formulated, simply by augmenting the size of the free vibration problem. However, in order to accommodate general, nonharmonic external excitations, the invariant manifolds associated with the unforced system are used herein for the forced response analysis. The procedure allows one to generate reduced-order models for the forced analysis of structural systems. Although strictly speaking the invariance property is violated, good results are obtained for the case study considered. In particular, it is found that fewer nonlinear modes than linear modes are needed to perform a forced modal analysis with the same accuracy. For systems with small and/or diagonal damping, approximate invariant manifolds are determined, which are shown to yield good results for both the unforced and forced responses. (Author)

Highlights

  • The analysis of the free and forced responses of linear dynamic systems is a well established field, with many analytical and numerical tools available.1'3 In particular, modal analysis allows one to break a problem into smaller, more solved sub-problems, and to consider the solution of the original problem as, in some sense, a post-processing product, using the theorem of superposition

  • Because they typically seek the solutions in time of all the differential equations of motion simultaneously, perturbation techniques are not usable for systems with many degrees of freedom, and do not provide a model reduction technique

  • Again, this method is cumbersome for large systems, does not provide a model reduction technique, and is limited to special kinds of external forcing (the normal form theory, as applied in reference 7 for forced response problems, requires that the forcing functions be considered as the solution of a suitable ordinary differential equation, which essentially reduces its use to problems with harmonic forcing or no forcing)

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Summary

Introduction

The analysis of the free and forced responses of linear dynamic systems is a well established field, with many analytical and numerical tools available.1'3 In particular, modal analysis allows one to break a problem into smaller, more solved sub-problems, and to consider the solution of the original problem as, in some sense, a post-processing product, using the theorem of superposition. The dynamics on the invariant manifold can if desired, be simplified by use of the normal form theory on the (reduced) set of modal oscillators, in view of an analysis by perturbation methods (see reference 13 for a case with an internal resonance) This invariant manifold procedure is geometric in nature, and is theoretically applicable to many non-linear structural systems, including gyroscopic and/or non-proportionally damped ones. This article investigates the possibility of neglecting the time variations of the manifolds and, of utilizing the invariant single- or multimode manifolds of the unforced system to perform a non-linear modal analysis of the forced response of the system This allows one to develop an efficient and systematic model reduction procedure, where only the modes that are mainly excited by the external forces need to be modeled.

Non-Linear Modes and Invariant Manifolds
Overview
Preliminary Comments
Conclusions
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