Bifurcations of attractors in forced system with nonlinear energy sink: the effect of mass asymmetry

System under investigation comprises a harmonically forced linear oscillator and a nonlinear energy sink (NES). The NES is a small mass (relative to that of the linear oscillator) which is attached to the primary system via a linear damper and strongly nonlinear spring (pure cubic nonlinearity). Among possible responses there exists one characterized by extremely deep modulation of the oscillations and referred to as a strongly modulated response regime (SMR). Numeric simulations demonstrate that the SMR can exist only for sufficiently small values of the NES mass. Known analytical approximations for description of the SMR deal with the lowest order of the asymptotic approximation and, consequently, work fairly well only for very small values of the NES mass and do not take into account its actual value. In the present study, we develop the analytical tools to investigate the higher-order asymptotic approximation. This enables us to depict the qualitative changes in the regime for the growing values of a NES mass and also to provide a crude estimation for a NES mass threshold. It is also demonstrated that in some cases the mechanisms of loss of stability by SMR (due to the growing values of NES mass) can be illustrated and explained via one-dimensional mapping diagrams. The described novel analytical approach is verified numerically and a fairly good agreement between the numerical and analytical models is observed.