Computing backbone curves for nonlinear oscillators with higher order polynomial stiffness terms

Single-degree-of-freedom (SDOF) nonlinear oscillators are widely used for modelling systems with just one degree-of-freedom in addition to single mode approximations to structural elements such as beams and cables, as well as other multi-degree-of-freedom (MDOF) applications. In this work, an investigation of the behavior of SDOF nonlinear oscillators is carried out using the method of direct normal forms. So far, this method has only been considered as a theoretical technique used for solving limited nonlinear dynamical systems in which low orders of nonlinearities appear, involving quadratic and cubic nonlinearities. In this work, thanks to the implementation of symbolic computations, the method of direct normal forms is generalized for solving nonlinear SDOF systems with any order of polynomial (or geometric) weak nonlinearities. Using this new approach, the effect of any higher order nonlinear term, or any combination of nonlinear terms can be investigated. Backbone curve relations are obtained for a selection of example systems representing both hardening and softening systems, and the results are verified by comparing the approximate analytical solutions to numerical solutions generated using COCO numerical continuation toolbox in Matlab.