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 trunc ature using a single linear mode c an lead to erroneous results. Using single-non-linear mode motion predic ts the c orrec t behaviour. Henc e, the nonlinear c hange of c o-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 disc rete system (a mass c onnec ted to two springs) and a c ontinuous 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.