Modal Analysis of Nonlinear Mechanical Systems
June 25, 2012 — June 29, 2012
- Gaetan Kerschen (University of Liège, Belgium)
Modal analysis and testing of linear structures has been developed over the past 40–50 years, and the techniques available today are really quite sophisticated and advanced. Because nonlinearity is a frequent occurrence in real-life applications, there is a need for efficient nonlinear modal analysis methods. In this context, nonlinear normal modes (NNMs) offer a solid mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear conceptual relation to the classical linear normal modes (LNMs), with which structural engineers are familiar. However, for reasons that will be detailed during the course, most practicing engineers still view NNMs as a concept that is foreign to them, and they do not yet consider them as a practical tool.
This course will first introduce the concept of NNMs and their two main definitions. The undamped definition considers NNMs as a family of periodic orbits in the vicinity of an equilibrium point, whereas the damped definition, which is more general by essence, views an NNM as an invariant manifold in phase space. The fundamental differences between LNMs and NNMs will be highlighted (e.g., the frequency-energy dependence, bifurcations and modal interactions of NNMs) and illustrated using simple examples. The pedagogical treatment of NNMs will be an important objective of the course so that the theory will be accessible to attendees coming from both academic and industrial areas.
Different methods for computing NNMs from a mathematical model will be presented. Participants will be exposed to both advanced analytical and numerical methods. Particular attention will be devoted to the invariant manifold and normal form theories. We will also show that numerical algorithms pave the way for an efficient and practical computation of NNMs. Realizing that a large body of the literature deals with low-order lumped-mass models, complex structures including rotorcraft blades and a full-scale aircraft will be examined.
The course will also discuss experimental modal analysis, which amounts to extracting NNMs directly from experimental data. Two methods will be presented. The first method is based on the concept of slow flow and builds a model based on intrinsic modal oscillators. The second method relies on the generalization of the phase lag quadrature criterion to nonlinear systems. Advanced signal processing, including the wavelet transform, will prove very useful for experimental NNM identification.
Finally, the course will describe several important applications of the NNM theory, including model validation, model reduction, and vibration and acoustic mitigation.
KEYWORDS: Modal Analysis, Nonlilnear Structures, Nonlinear, Normal Modes, Model Validation.