Definition and Fundamental Properties of Nonlinear Normal Modes

Abstract

The concept of a normal mode is central in the theory of linear vibrating systems. Besides their obvious physical interpretation, the linear normal modes (LNMs) have interesting mathematical properties. They can be used to decouple the governing equations of motion; i.e., a linear system vibrates as if it were made of independent oscillators governed by the eigensolutions. Two important properties that directly result from this decoupling are:

1. Invariance: if the motion is initiated on one specific LNM, the remaining LNMs remain quiescent for all time.

2. Modal superposition: free and forced oscillations can conveniently be expressed as linear combinations of individual LNM motions.

In addition, LNMs are relevant dynamical features that can be exploited for various purposes including model reduction (e.g., substructuring techniques, experimental modal analysis, finite element model updating and structural health monitoring.