Analytical Analysis Of Performance And Optimization Design Of Parameters For Dampers Considering Nonlinear Stiffness

The occurrence of natural disasters will bring serious damage and large-scale damage to buildings.Therefore,more and more scholars at home and abroad begin to study vibration control technology.This paper studies two kinds of passive control devices:tuned mass damper（TMD）and cubic stiffness nonlinear energy sink（NES）.Due to the large displacement and the use of limit devices,TMDs will show some nonlinear characteristics during vibration.However,in most previous studies,the nonlinearity in TMD is usually ignored.In this paper,it is proved by numerical method that ignoring the nonlinear behavior in TMD will have an adverse impact on the vibration reduction effect.Due to the limitations of narrow damping band and low damping efficiency of linear dampers,some nonlinear dampers such as NES began to attracted attention gradually.However,due to the difficulty of solving strong nonlinear systems and the high sensitivity to initial conditions,there are not many analytical optimization designs of NES independent of initial conditions at present.In view of the above problems,this paper mainly completes the following studies.1.Taking a two-degree-freedom system composed of a linear primary structure and a hardening nonlinear TMD as the dynamic model.The response of the system is deduced by using the analytical method of complex variable average method and multi-scale method,and the approximate frequency response formula of the nonlinear system in steady-state is obtained.This formula is applied to the optimal design of the nonlinear hardening TMD.The natural frequency ratio of nonlinear TMD designed by linear method is modified.The influence of the change of TMD damping on the frequency domain response is simulated and analyzed with MATLAB,and the damping ratio of TMD is optimized.The optimization of TMD natural frequency and damping ratio reduces the steady-state amplitude of the primary structure effectively,and improves the stability of the primary structure response near resonance through the results of Simulink,and the vibration reduction effect has been significantly improved compared with the linear design method.The experimental study of hardening nonlinear TMD shows that there is weak nonlinearity in TMD even if linear connection is adopted.The experimental results show that the numerical model and the analytical derivation in this paper are accurate and reliable.The optimization method in this paper is practical.The nonlinear TMD designed by the optimal frequency ratio formula in this paper has a good damping effect,which is obviously better than the linear method.2.The phenomenon of SN bifurcation and Hopf bifurcation in the hardening nonlinear TMD system and the ranges of parameter where bifurcation can occur are analyzed in detail by combining analytical and numerical methods,some rules and conclusions are summarized.The formula of SIM is obtained by the complex variable average method and the multi-scale method.The"bulge phenomenon"caused by the nonlinear jump in the curves of’（?）-Z₂’obtained by different methods is found for the first time when analyzing the influence of the mass ratio of TMD on the steady-state response mode of the system.The TMD quality ratio interval of the modulation response is obtained through some analytical derivation.Based on the"bulge phenomenon",an analytical method which can directly optimize the mass ratio of hardened nonlinear TMD is proposed.The results of Simulink shows that this method can improve the vibration reduction efficiency,and also provides a new way for the analysis and parameter optimization design of the nonlinear hardening TMD.3.SN bifurcation analysis and Hopf bifurcation analysis for softening nonlinear TMD system are carried out.The effects of parameter on bifurcation results and peak shift of the frequency response curves are studied.The different performances of softening TMD and hardening TMD in nonlinear characteristics are compared,and specific examples are given to show several different bifurcation phenomena and response modes with the change of parameters.When analyzing the invariant manifold of softening nonlinear TMD,the parameter conditions under which the softening nonlinear TMD can jump are derived.The approximate steady-state analytical formula of the softening nonlinear TMD system is obtained by the complex variable average method.Based on this formula and Simulink,the natural frequency ratio of the softening nonlinear TMD is optimized.The numerical results show that this method reduces the amplitude of the primary structure near the resonance and improves the robustness of the primary structure response near the resonance.4.Taking the two-degree-freedom system composed of a single-degree-freedom linear primary structure and a single-degree-freedom cubic stiffness NES with a harmonic excitation as the dynamic model.The slow varying models of NES on the fast and slow time scales are obtained by the analytical method of complex variable average method and multi-scale method.When studying the variation law of the steady-state response with the excitation amplitude,the"platform phenomenon"caused by the modulation response and some relevant laws are proposed for the first time.Based on the"platform phenomenon",the excitation amplitude range for the modulation response when the other parameters are determined is derived.And the NES stiffness range for the modulation response is derived.On this basis,the optimal stiffness ratio of NES independent of the initial conditions is obtained.The results of the numerical analysis show that the optimization method reduces the steady-state amplitude of the primary structure effectively and improves the stability of the primary structure in the frequency domain.From the numerical results,it can be seen that the mode of strong modulation response（SMR）is beneficial to the vibration reduction effect.