Parametric Vibration Control Method Of Hanging Cable Using Inerter Dampers

The cable-stayed fastening-hanging cantilever construction method has been prevalently employed in the construction of arch bridges.However,the hanging cables’ parametric vibration frequently appears during the cantilever construction stage of arch rings.Inerter dampers are effective control strategies compared with conventional dampers,and have been the focal point of research for improving vibration mitigation effects.Therefore,to control the hanging cables’ parametric vibration,this article conducts a detailed study on the vibration mitigation effects and influence mechanism of the inerter dampers on hanging cables’ parametric vibration.The research contents have essential theoretical and practical significance.This article investigates the control methods of hanging cables’ parametric vibration based on inerter dampers.The vibration model of hanging cable with inerter dampers installed under end excitation is the object of study,and the following research contents are carried out by combining theoretical analysis and numerical calculations.(1)Based on Newton’s laws and the Galerkin method,the vibration equations of the hanging cable with a parallel inerter damper(PVMD)and the hanging cable with a tuned inerter damper(TID)under support excitation were established.The vibration equations were solved using the Runge-Kutta method and the method of multiple scales(MMS),respectively.This yielded multi-scale analytical approximate solutions for the cable-PVMD system and the cable-TID system under end excitation.The accuracy of the multi-scale analytical approximate solutions was validated by comparing them with the numerical approximate solutions obtained using the Runge-Kutta method.(2)The detailed analysis was conducted on the influence mechanism of the parameters of PVMD and TID on the cable’s parametric vibration.The findings revealed that the damping component of the PVMD increases the minimum excitation amplitude required for the initiation of hanging cable’s parametric vibration,thereby suppressing the parametric vibration.The inerter component of PVMD amplifies the vibration suppression effect of the damping component.Upon the installation of the TID,the amplitude-frequency response curve of the hanging cable’s parametric vibration exhibits multi-peak and differentiation phenomena,significantly reducing the vibration amplitude of the hanging cable.The damping coefficient and inertance of TID influence the multi-peak and differentiation phenomena of the amplitude-frequency response curve by affecting the phase difference between the TID and the cable.Specifically,the damping coefficient of TID suppresses the multi-peak phenomenon,while the inertance of TID promotes it.Moreover,The frequency ratio of TID and hanging cable plays a role in adjusting the ratios of the various peaks.(3)By comparing the amplitude-frequency response curves of the hanging cable equipped with VD,PVMD,and TID,respectively,the vibration control efficacy of these three dampers on the second superharmonic resonance,primary resonance,and principal parametric resonance of the hanging cable are evaluated.The conclusion is that the control efficacy of TID surpasses that of PVMD and VD,and the control efficacy of PVMD outperforms that of VD.Subsequently,the superior control efficacy of inerter dampers was further elucidated from a mechanistic perspective,using the force-displacement hysteresis curves of the three types of dampers.(4)The problem of numerical stability in the approximate numerical solution is prevalent in studies on parametric vibrations.Specifically,the approximate numerical solution does not fully align with the upper branch stable solution of the approximate analytical solution,and this problem persists even after the installation of inerter dampers.In this regard,this study posits the initial condition dependence of the Runge-Kutta solution,based on the comparison results of the Runge-Kutta method and MMS.The cause of the initial condition dependence of the Runge-Kutta solution was investigated by analyzing the phase portraits of multiple-scale solutions.The results indicate that different initial conditions lead the transient amplitude of the Runge-Kutta solution to develop along different phase trajectories and ultimately converge to different steady-state solutions.Then,this study provides a general method for determining reasonable initial conditions of Runge-Kutta method and the improved calculation steps of Runge-Kutta method for the cable-PVMD and cable-TID systems,respectively.Finally,the vibration response of a hanging cable equipped with an inerter damper under random excitation was calculated using the enhanced Runge-Kutta method.The results indicate that the inerter damper significantly suppresses the parametric vibration of the hanging cable under random excitation.Moreover,the TID demonstrating superior suppression effects compared to the PVMD.