Reduced-order modelling in non-linear dynamics: an approach based on non-linear modes

It is largely accepted that non-linear modes of vibration may be particularly suitable for obtaining ‘reduced-order’ models in non-linear dynamics, for their ability to grasp the essential qualitative system information that a much larger number of linear modes are required to. Previous work by the first author on ‘reduced-order’ modelling in non-linear dynamics did not account for the velocity contents within non-linear modes. For many systems, this simplifying assumption does not, in fact, spoil the quality of the ‘reduced-order’ model. Nevertheless, it is not to be generally taken for granted. In this article, a generalised procedure for ‘reduced-order’ modelling in non-linear dynamics that uses the full displacement and velocity contents of non-linear modes is addressed and illustrated. Two case studies are presented and conclusions regarding the relevance of the velocity contents are drawn. Comparison between non-linear dynamic responses of finite-element and ‘reduced-order’ models under different load conditions is made. For both external and parametric resonances, a remarkable agreement between them was achieved, provided the velocity contents within the non-linear modes are retained. In the second case study, damping is essential to help the system settling down in a post-critical periodic attractor, otherwise wave propagation and reflection will have an enduring effect.