Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes

The definition of a non-linear normal mode (NNM) as an invariant manifold in phase space is used. In conservative cases, it is shown that normal form theory allows one to compute all NNMs, as well as the attendant dynamics onto the manifolds, in a single operation. Then, a single-mode motion is studied. The aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results. Using single-non-linear mode motion predicts the correct behaviour. Hence, the non-linear change of co-ordinates allowing one to pass from the linear modal variables to the normal ones, linked to the NNMs, defines a framework to properly truncate non-linear vibration PDEs. Two examples are studied: a discrete system (a mass connected to two springs) and a continuous one (a linear Euler–Bernoulli beam resting on a non-linear elastic foundation). For the latter, a comparison is given between the developed method and previously published results.