Equal-Peak Method And Its Applications Of Time-Delayed Vibration Absorbers In Nonlinear Systems

The anti-resonance response is the minimum point of the frequency response curve(FRC).Usually,this characteristic is applied to design the structural parameters of passive absorbers.By equalizing the natural frequencies of the absorber and the primary system,the resonance peak with high amplitude becomes the anti-resonance point with low amplitude,and the vibrations around the anti-resonance frequency are absorbed.Since additional energy sources are not required,passive absorbers are widely used in engineering.However,passive absorbers are not suitable for the working condition with variable frequency excitations,because their parameters are fixed and cannot be adjusted actively.To tackle this challenge,time-delayed vibration absorbers(TDVAs)were proposed.It is found that time-delayed feedback can actively adjust the equivalent stiffness and damping of the absorber.Thus,the broadband vibration absorption effects can be realized by equalizing the anti-resonance frequency and the excitation frequency.The broadband vibration absorption effects are realized by TDVAs based on the anti-resonance frequency modulation,but multiple resonance peaks with different amplitudes are also triggered.Due to the rapid change of the responses between resonance peaks and anti-resonance points,small drifts of excitation frequency will result in the sharp deterioration of vibration suppression effects.In addition,with the increase of system nonlinearity,the resonance peaks increase rapidly,and some complex phenomena such as the multiple steady-state are induced in the resonance band.These phenomena make the absorber design complex.To sum up,it is necessary to study the design method of TDVA which can suppress multiple peaks of nonlinear systems simultaneously and eliminate multiple steady-state phenomena.To tackle the above challenges,the vibration suppression of nonlinear systems is investigated,and the equal-peak method of TDVA and its applications in different nonlinear structures are explored.Then,a general,systematic and complete design method for TDVA is developed.To achieve this goal,this thesis mainly focuses on the following three scientific problems:(1)how to establish the equal-peak method of TDVA,suppress multiple resonance peaks of the nonlinear system simultaneously,develop and improve the research on the vibration absorption mechanism of TDVA;(2)how to build the optimization method and process for time-delayed parameters,realize the equal-peak property,reveal the relationship between time-delayed parameters and vibration suppression performance;(3)how to extend the equal-peak design method to different nonlinear structures,verify the effectiveness of the proposed method.To tackle the above three scientific problems,the main contents of this thesis are shown as follows:1.For the first scientific problem,this thesis studies the nonlinear primary system coupled with TDVA and its dynamics.We clarify the amplitude modulation mechanism on resonance response and the elimination mechanism on multiple steady-state phenomena,including jumping and Detached Resonance Curve(DRC)for timedelayed parameters.Based on these mechanisms,we establish the equal-peak method of TDVA for nonlinear systems.First,the stability condition of time-delayed parameters is proposed such that the equilibrium of the system is stable.Second,the necessary and sufficient condition of the equal-peak method is proposed,and the eigenvalues of the Jacobian matrix of the resonance response are obtained.Then,the relationship between these eigenvalues,the equal-peak property of FRC and the resonance amplitude is revealed.Third,the minimum peak condition is proposed,and the optimal time-delayed parameters with different external excitation amplitudes are obtained.Finally,the effectiveness of the proposed method is verified numerically for high-order nonlinear systems.It is found that with the proposed equal-peak method,the resonance responses are suppressed simultaneously and the jumping phenomena are eliminated.2.For the second scientific problem,this thesis studies the multiple DOF nonlinear system coupled with multiple TDVAs,and proposes the optimization algorithm for TDVAs.First,the underdetermined nonlinear algebraic equations which describe the equal-peak property are calculated.In the calculation process,the conditions of timedelayed parameters proposed in the equal-peak method are embedded.With the continuous variation of structural parameters of TDVAs and external excitation amplitude,the arc-length method with cubic extrapolation is used to obtain the optimal time-delayed parameters.Then,the relationship between the optimal time-delay parameters and the vibration suppression performance is revealed.Finally,the optimization algorithm is generalized to study the vibration suppression of nonlinear systems with even order nonlinearities.It is found that the resonance responses around different modes are suppressed and the jumping phenomena are eliminated.3.For the third scientific problem,this thesis studies the single-span suspension bridge coupled with multiple TDVAs.First,the vibration suppression of the first-order resonance of the bridge with a single TDVA is studied.Based on the proposed optimization algorithm,the influence of the structural parameters of TDVA on the vibration suppression performance is clarified.Then,with the optimization of the stiffness,mass and damping parameters of TDVA,the vibration suppression effect of the equal-peak method on the resonance response is improved,and the multiple steadystate phenomena of DRC is eliminated.Then,these works are extended to study the vibration suppression of multi-order resonance of suspension bridges.The relationship between the structural parameters of TDVAs and the related optimal time-delayed parameters is established.The effect of multiple TDVAs on suppressing multi-order resonance of suspension bridges is improved.Besides,two kinds of multiple steadystate phenomena,namely the jumping and the DRC,are eliminated,Thus,the adaptability of the equal-peak method to different nonlinear structures is verified.The main innovations of this paper can be summarized as follows:1.The equal-peak method of TDVAs is established,and the tuning effect of timedelayed feedback on multiple resonance amplitude is found.The modulation mechanism of TDVA on resonance amplitude is revealed for nonlinear systems.The broadband vibration suppression effect for multiple DOF nonlinear systems is realized.2.The optimization algorithm for control parameters of single TDVA and multiple TDVAs is proposed,and the relationship between the parameters of TDVA and the broadband vibration suppression performance is established.The numerical simulation results show that the equal-peak method has an excellent vibration suppression effect.3.The time-delayed controller for vibration suppression of nonlinear chain structure and single-span suspension bridge is designed.The equal-peak method and optimization algorithm of time-delayed parameters are verified.The problem of suppressing multiple resonance amplitudes of the suspension bridge is solved.