Modal Analysis for Nonlinear Vibration

Abstract

Linear vibration theory uses the concept of defining a specific set of modes of vibration for the system under consideration. Physically, each mode relates to a particular geometric configuration in the system, such as two lumped masses oscillating either in- or out-of-phase with each other. For linear systems, the superposition principle means that the complete vibration response can be computed as a summation of the responses from each mode. In general terms, modal analysis has come to mean considering the response of a system by studying its vibration modes; modal decomposition is the process of transforming the system from a physical to a modal representation. This is particularly useful in linear systems, because each mode has an associated resonance, and understanding where resonances could occur in a structure is a key part of analysing vibration problems. In this chapter the use of modal analysis for nonlinear systems is also considered. First, the decomposition of discrete and continuous linear systems into modal form is reviewed and the effect of nonlinear terms on this analysis is discussed. Following this, methods for decomposing nonlinear systems are considered. Initially a brief discussion of nonlinear normal modes is given and a special case system, in which there is nonlinear but no linear coupling between two oscillators, is analysed using the harmonic balance approach. Following this, attention is turned to the main technique for carrying out nonlinear modal decomposition, which is the method of normal forms that was introduced in Sect. 4.5. This is a technique that transforms the system to the simplest form possible. The approach described here uses linear modal decomposition as the first step in the process. The main advantage of using normal forms is that information about nonlinear (also called internal) resonances in the system can be obtained. As a result, a normal form analysis can be used to obtain information about both linear and nonlinear resonances in a nonlinear vibration problem. The modal decomposition techniques are used to find backbone curves that represent the undamped vibration response of the system in the frequency domain. The reason for taking this approach is that modal analysis is most relevant for lightly damped systems, where multiple resonant peaks can occur in the response. Just like linear systems, nonlinear systems with light damping have a forced response that is determined by the underlying undamped characteristics (It’s possible to define systems that don’t have this property, but we will restrict our discussion to systems that do.). In the frequency domain this is captured by the backbone curves. As a result, defining the backbone curves for the system gives a nonlinear modal model. The nonlinear examples considered in this chapter are confined to two degrees-of-freedom, but can be extended to higher degrees-of-freedom, and a short discussion of relevant literature on this topic is given at the end of the chapter.