Model reduction for discrete and elastic structures with inertial quadratic non-linearities

This work discusses model reduction for non-linear structural systems under harmonic excitations. Model reduction can be achieved by different techniques, one of the recent techniques being the non-linear normal modes (NNMs) of the system. The dimension reduction achievable depends on the possibilities of internal resonances. A master–slave separation of degrees of freedom is used, and a non-linear relation between the slave and master coordinates is constructed based on the method of multiple timescales. More specifically, three cases involving external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic resonances for systems with 1:2 internal resonances are considered. Reduced-order models based on the ‘Conservative NNMs’ as well as ‘Damped NNMs’ are constructed. The steady-state periodic responses of the reduced models determined by the method of multiple timescales are compared to exact solutions of the system models computed by the bifurcation analysis and parameter continuation software AUTO. The analysis is specifically applied to a spring-mass-pendulum system with external excitation, and to an elastic three-beam-tip-mass structure, which is first reduced to a high-fidelity non-linear discretized model through a Galerkin approximation. Both systems exhibit essential quadratic non-linearities and couplings between the various generalized coordinates. The NNMs of the two systems are used to perform model reductions when excited by harmonic excitations. It is seen that for systems with essential inertial quadratic non-linearities, the technique for model-order reduction through multiple timescales approximation based on NNMs over-predicts the softening non-linear response in each of the cases studied.