A new way to introduce geometrically nonlinear stiffness and damping with an application to vibration suppression

The single-degree-of-freedom linear oscillator (LO) under the action of harmonic forcing is coupled to a light attachment that acts, in essence, as a nonlinear energy sink (NES). The complex dynamics of this two-degree-of-freedom system is investigated. Strong stiffness and damping nonlinearity in this system is introduced by coupling the LO and NES by two inclined linear spring-damper elements. The inclination of the coupling elements during the motion introduces strongly nonlinear geometric effects in the forced dynamics. A slow/fast partition of the dynamics is introduced by applying a complexification-averaging method, based on which a bifurcation analysis is performed. The effects of the initial angle, i.e., the initial inclination angle of the spring-damper elements, are thoroughly studied. The topology of the trajectories on the slow manifold of the dynamics is studied, and the different steady-state response regimes are predicted analytically. The conditions for existence of strongly modulated responses and of “folding singularities” are studied, and their effects on the nonlinear dynamical responses are revealed. Comparisons between analytical and numerical results indicate a good agreement between the two and provide a means of verification of the analytical findings. The analytical results show that, by increasing the initial angle of inclination, one can shrink, and even completely eliminate, unwanted high-amplitude steady-state responses of the LO that co-exist with desirable low-amplitude forced responses over definitive frequency ranges. This finding has significant practical implications for the vibration mitigation efficiency and robustness of the proposed NES, enhancing drastically the vibration suppression of the forced response of the LO. The presented results and the associated analytical modeling can be used to design and enhance the performance of nonlinear vibration absorbers as vibration mitigating devices.