Research On Dynamical Characteristics Of Vibration Absorber With Geometrically Nonlinear Damping

Mechanical structures suffer from various kinds of unwanted vibrations in complex working environments,and many scholars have conducted extensive research for the purpose of vibration control.Regarding passive vibration mitigation devices,compared to the linear vibration absorber with limited effective frequency range,the introduction of stiffness nonlinearities in the vibration absorber can enlarge the effective bandwidth.The nonlinear energy sink(NES)is a representative nonlinear shock absorber,due to its simple structure and low cost,NES has been widely applied in diverse engineering fields.However,the previous scholars only focus on the research of the NES with stiffness nonlinearities,the effects of nonlinear damping are ignored.The geometric damping nonlinearities are generated by two initially inclined pairs of damper elements with specific position and initial inclination angle.This dissertation aims to investigate the effects of geometrically nonlinear damping on the dynamics and mitigation performance of NES and provides theoretical foundation for potential practical engineering applications.The research contents are listed as:(1)A new kind of nonlinear energy sink is proposed by grounding the nonlinear oscillator by extra cubic nonlinear spring and damper.With the application of complex-averaging method,the steady-state dynamical behavior of the system is investigated by the slow invariant manifold(SIM),folding singularities and equilibrium points.Different scenarios of strongly modulated responses(SMR)are presented based on the geometry of SIM.The incremental harmonic balance method is applied to detect the frequency response curves of the system around the fundamental resonance,and the accuracy of the theoretical analysis is fully verified by the numerical results obtained by direct integration of equations of motion of the system.It is demonstrated that the increase of external forcing amplitude,global nonlinear stiffness and local nonlinear stiffness can drive the frequency response curves move toward the right and widen the frequency bandwidth of the coexistence of multiple steady-state response regimes,while the increase of nonlinear damping the reverse.The numerical simulation results also show that the addition of geometrically nonlinear damping and local potential in the proposed NES can drastically enhance the capacity of the nonlinear vibration absorber to suppress the shock-induced response of the LO,and the proposed NES is effective for a comparatively broad range of applied impulsive energies,particularly for the high impulsive energies.(2)The effects of geometrically nonlinear damping on the multi-degree-of-freedom(MDOF)nonlinear energy sink are studied.The geometry of the SIM of the MDOF system is more complex compared to the single-degree-of-freedom(SDOF)NES,and there exist multiple unstable branches on the SIM,which indicates that different scenarios of SMRs can happen for various initial conditions and system parameters when the system reaches steady state.The theoretical predictions are validated by direct integrations of equations of motion of the original system.By investigating the dynamics at slow time scale,the frequency response curves are obtained around the fundamental resonance of the linear oscillator.It is demonstrated that the addition of geometrically nonlinear damping in the MDOF nonlinear vibration absorber can dramatically reduce the amplitude of the linear oscillator around the fundamental resonance.(3)The mechanical realization of aforementioned form of geometrically nonlinear damping is based on the assumption that the initial angle of two inclined linear damper elements is zero.The strongly nonlinear geometric effects of the initial angle of inclination on the forced dynamics are thoroughly studied.The topology of the trajectories on the SIM of the dynamics is studied,and the different steady-state response regimes are predicted analytically.The conditions for existence of SMRs 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.(4)For the continuous system,the effects of geometrically nonlinear damping on the cantilever beam are studied.The beam is clamped at one of its ends and the other end is pinned to the grounded parallel spring-damper element with an initial inclination angle.The external harmonic excitation is applied to the end of the beam,and the frequency response plots are obtained by finite element method.The geometric effects lead to different dynamical regimes by varying the initial angle of inclination,indicating the complete transition from hardening resonances to softening resonances by changing the geometry parameters.The reduced SDOF model of the continuous system is used to verify the findings from the finite element method.The nonlinear hardening and softening resonances at different initial angles of inclination are validated by investigating the time series of the tip of the beam and the relationship between restoring force and displacement at steady state.