Nonlinear Modal Decomposition Using Normal Form Transformations

Abstract

In this paper we discuss a technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form. This type of decomposition technique is an established cornerstone of linear modal analysis. Extending this type of technique to nonlinear multi-degree-of-freedom systems has been an important area of research in recent years. The key result in this work is that a theoretical transformation process is used to reveal both the linear and nonlinear system resonances. For each resonance, the parameters which characterise the backbone curves and higher harmonic components of the response, can be obtained. The underlying mathematical technique is based on a near identity normal form transformation for systems of equations written in second-order form. This is a natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. The example is a system with cubic nonlinearities, and shows how the transformed equations can be used to obtain a time independent representation of the system response. It is shown that when the natural frequencies are close to an integer multiple of each other, the backbone curve bifurcates. Examples of the predicted responses are compared to time-stepping simulations to demonstrate the accuracy of the technique.