| With the increased demand for precision instruments and structural safety on the performance of vibration suppression equipment,the research of efficient and stable vibration absorbers is necessary.Meanwhile,the inerter,as a novel vibration control unit,plays a great role in promoting structural vibration damping.Connecting it between the primary system and the subsystem can change the inertial coupling of the system,and connecting it to the ground can reduce the volume and self-weight of the equipment.However,the improper structure with inerter will reduce the system performance.In addition,high-performance vibration-damping devices are becoming more and more complex.The coupling effects between complex structures and their influence on the response mechanism and dynamics of the system are not clear.The thesis aims to enhance the vibration suppression capability of vibration absorbers with inerter and to explore the effects of different configurations and installation methods for inerter on vibration absorber systems.Various passive/active dynamic vibration absorbers,nonlinear energy sinks,and tuned inerter dampers are presented and investigated in detail.The major research is as follows.(1)The research background and significance of this thesis,the research progress of inerter,linear/nonlinear vibration absorber,fractional order dynamics,and active control are briefly introduced.The major research contents and innovations of this thesis are discussed.(2)Based on the typical Voigt-type dynamic vibration absorber,a novel dynamic vibration absorber model with inerter and grounded stiffness is presented.The analytical solutions of optimal design parameters for this model are studied in detail.Firstly,the motion differential equation of the two-degree-of-freedom system is established through Newton’s second law.The analytical solution of the system is calculated,and the optimal frequency ratio of the system is obtained using fixed-point theory.The best working range of the inerter is derived,and finally the optimal grounded stiffness ratio and the approximate optimal damping ratio are obtained.The working condition when the inerter coefficient is not within the best range is discussed,and suggestions for practical application are given.The correctness of the analytical solution is verified by numerical simulation.It is compared with a variety of existing dynamic vibration absorbers under harmonic and random excitations.Subsequently,the amplification mechanism is introduced into it,and a similar phenomenon is found,as well as further improvements in the damping performance of the absorber.The effects of inerter coefficient and magnification on the response of the primary system are analyzed.(3)Deformed on the basis of the Voigt vibration absorber,a grounded stiffness vibration absorber is presented,and the equivalent tuned inerter damper is given.An improved method based on fixed-point theory is proposed to calculate the closed-form solution of the fixed-point coordinates of the system.Analytical and numerical methods are used to study the maximum amplitude of the primary system response,and global optimization is performed according to the fixed-point amplitude characteristics.The design formulas for the optimal natural frequency ratio and the optimal damping ratio are derived.Numerical simulation is used to verify the correctness of the analysis.Under the excitation of harmonic force,the comparison is made with the local and global optimized Voigt vibration absorber and grounded damper vibration absorber.(4)Under various working conditions,the accuracy differences and applicability conditions of each method are revealed by comparing the differences between approximate solution methods of different nonlinear systems.The complexification-averaging method,multi-scale method,and harmonic balance method are applied to obtain the analytical solutions of single-degree-of-freedom autonomous and non-autonomous systems.The Duffing oscillator is used as an example for numerical verification.Semi-analytical solutions of the steady-state response of a two-degree-of-freedom nonlinear energy sink system are derived.The amplitude and root mean square are used as evaluation indicators to describe the precision of the system response.(5)A kind of vibration absorber with geometrically nonlinear inerter is studied.The generation mechanism of the nonlinear restoring force of the geometrically nonlinear structure is analyzed,and a high-order Taylor series approximation is used to reduce the computational complexity.The analytical solution of the approximate response is obtained using the harmonic balance method,and the amplitude-frequency response equation and the phase-frequency response equation are derived.Sensitivity analysis is performed for each variable,and the parameters are optimized by the grey wolf algorithm.The feasibility and accuracy of the grey wolf algorithm in the parameter optimization problem of absorbers are verified.Finally,the displacement transmissibility,attenuation rate,damping bandwidth,and optimal parameter values are used as indicators to compare with other linear/nonlinear vibration absorbers.(6)The effect of the combined inerter and grounded stiffness structure on the response mechanism and damping effect of nonlinear energy sink is investigated.And the primary system damping,which has been neglected in most studies,is added to the model.The closed-form solution of the system is derived using the complexification-averaging method and verified numerically,based on which the stability analysis equation and the local bifurcation boundary equation are calculated.Combined with the multi-scale method to analyze the slow invariant manifold,sufficient conditions of strongly modulated response are derived.Several sets of typical parameters are selected and analyzed in detail,as well as the correctness of the analytical results is verified.The amplitude-frequency response equation of the primary system is obtained by the harmonic balance method,and its parameter effects and amplitude-frequency characteristics are analyzed.Based on the H_∞optimization criterion,the multi-objective grey wolf algorithm is applied for parameter optimization,and the superiority of the model in vibration reduction performance is illustrated in different working cases.The model is simplified to obtain a cubic stiffness nonlinear energy sink,and the feasibility of eliminating the high-branch response by parameter optimization without changing its structure is discussed.Lastly,it is compared with other linear vibration absorbers and nonlinear energy sinks in terms of damping effect,bandwidth,and system stability.(7)An active tuned inerter damper with delayed fractional-order PID based on the displacement feedback of the primary system is proposed.The displacement amplitude expression and stability condition of the primary system are derived by the averaging method,Lyapunov indirect method,and Routh-Hurwitz criterion.Their correctness is verified by using the power series method.The equivalent mechanical mechanism of the control force is analyzed,where the differential order belongs to the fully parametric domain.The definitions of equivalent stiffness and damping coefficients are given when the differential order belongs to 0-1.The definitions of equivalent stiffness,damping coefficient,and mass are given when the order belongs to 1-2.At last,parameter tuning is performed through the improved grey wolf algorithm.Under different working conditions,a comparative study with the passive,PID,and fractional-order PID-controlled tuned inerter dampers is carried out.Finally,the key results of this thesis are outlined,and the lack of current research and the direction of future research are pointed out. |
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