Normal form transformations for structural dynamics: an introduction for linear and nonlinear systems.

The aim of this paper is to provide an introduction to using normal form transformations for linear and nonlinear structural dynamics examples. Starting with linear single-degree-of-freedom systems, a series of examples are presented that eventually lead to the analysis of a system of two coupled nonlinear oscillators. A key part of normal form transformations are the associated coordinate transformations. This review includes topics such as Jordan normal form and modal transformations for linear systems, while for nonlinear systems, near-identity transformations are discussed in detail. For nonlinear oscillators, the classical methods of Poincaré and Birkhoff are covered, alongside more recent approaches to normal form transformations. Other important topics such as nonlinear resonance, bifurcations, frequency detuning and the inclusion of damping are demonstrated using examples. Furthermore, the connection between normal form transformations and Lie series is described for both first and second-order differential equations. The use of normal form transformations to compute backbone curves is described along with an explanation of the relationship to nonlinear normal modes. Lastly, conclusions and possible future directions for research are given.