An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems

This paper presents a technique for the explicit formulation of frequency response functions (FRFs) for nonlinear MDOF systems, yielding closed-form analytical expressions. The technique produces FRFs at selected coordinates only, regardless of the system's size or the type of nonlinearity. The nonlinear formulation in this work is based on first-order describing functions, which represent the nonlinearities by amplitude-dependent coefficients. For the sake of clarity, the paper is focused on cubic stiffness and friction damping nonlinearities, although, given the generality of the formulation, the inclusion of other types should be straightforward. Conducted entirely in the physical domain, the approach is based on manipulating the coefficients stored in the explicitly formulated system matrices, thus the term “explicit”. An enhanced version, which can tackle large systems, is also presented and validated against real measurements taken from a test rig. It was found that the nonlinear behaviour was predicted with reasonable accuracy in all cases studied.