$$N-1$$ N - 1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities

Abstract

In this paper the $$N-1$$ nonlinear modal interactions that occur in a nonlinear three-degree-of-freedom lumped mass system, where $$N=3$$ , are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. $$\omega _{n1}: \omega _{n2}:\omega _{n3} \approx 1:1:1$$ ) is considered. However, due to the symmetries of the system under consideration, only $$N-1$$ modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.