In this paper, a time delay is coupled to a dynamic vibration absorber with an inerter-based grounded stiffness that contains an amplification mechanism, and a model of the dynamic vibration absorber with time-delay feedback control containing an inerter is obtained. For this model, the vibration differential equation of the system is first established, and the optimal structural parameters of the passive system with an inerter are analyzed under the condition of no time delay. When the time delay is considered, the characteristic equation of the system is a transcendental equation with exponential terms. The CTCR method and the Dixon resultant are used to analyze the stability of the system, and the stable intervals of the feedback gain coefficient and time delay are obtained. Then, according to the anti-resonance peak optimization criteria, the structural parameters of the system are further adjusted under the condition of ensuring the system’s stability, and a set of optimal structural parameters is successfully obtained, which can not only meet the minimum anti-resonance peak criterion but also have a wider vibration reduction frequency band and a good vibration reduction effect. Finally, the optimization results are verified in both the time domain and the frequency domain. Compared with the traditional dynamic vibration absorber, the dynamic vibration absorber designed in this paper has achieved a good vibration reduction effect.
References
[1]
Frahm, A. (1909) Device for Damping Vibrations of Bodies. US Patent 989958.
[2]
Ormondroyd, J. and Den Hartog, J.P. (1928) The Theory of the Dynamic Vibration Absorber. TransactionsoftheAmericanSocietyofMechanicalEngineers, 49, Article ID: 021007. https://doi.org/10.1115/1.4058553
[3]
Den Hartog, J.P. (1974) Mechanical Vibrations. McGraw-Hill Book Company.
[4]
Den Hartog, J.P. (1985) Mechanical Vibrations. Dover Publications.
[5]
Brock, J.E. (1946) A Note on the Damped Vibration Absorber. JournalofAppliedMechanics, 13, A284. https://doi.org/10.1115/1.4009588
[6]
Nishihara, O. and Asami, T. (2002) Closed-Form Solutions to the Exact Optimizations of Dynamic Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors). JournalofVibrationandAcoustics, 124, 576-582. https://doi.org/10.1115/1.1500335
[7]
Asami, T. and Nishihara, O. (2003) Closed-Form Exact Solution to H∞ Optimization of Dynamic Vibration Absorbers (Application to Different Transfer Functions and Damping Systems). JournalofVibrationandAcoustics, 125, 398-405. https://doi.org/10.1115/1.1569514
[8]
Liu, L.L., Ren, B.L. and Zhu, G.D. (2017) Research on Dynamic Characteristics of Bistable Electromagnetic Vibration Absorber Considering Nonlinear Damping. Journal of Vibration and Shock, 36, 91-96. https://doi.org/10.13465/j.cnki.jvs.2017.17.015
[9]
Zhao, Y.Y. and Xu, J. (2008) Damping Mechanism of Time-Delayed Nonlinear Dynamic Vibration Absorber. Chinese Journal of Theoretical and Applied Mechanics, 40, 98-106. https://doi.org/10.6052/0459-1879-2008-1-2007-078
[10]
Zhao, Y.Y. and Xu, J. (2011) Vibration Control of Self-Parameter Vibrating Systems Using Time-Delayed Feedback. Chinese Journal of Theoretical and Applied Mechanics, 43, 894-904. https://doi.org/10.6052/0459-1879-2011-5-lxxb2010-652
[11]
Peng, H.B., Shen, Y.J. and Yang, S.P. (2015) Parameter Optimization of a Novel Dynamic Vibration Absorber with Negative Stiffness Components. Chinese Journal of Theoretical and Applied Mechanics, 47, 320-327. https://doi.org/10.6052/0459-1879-14-275
[12]
Oyelade, A.O., Wang, Z. and Hu, G. (2017) Dynamics of 1D Mass-Spring System with a Negative Stiffness Spring Realized by Magnets: Theoretical and Experimental Study. TheoreticalandAppliedMechanicsLetters, 7, 17-21. https://doi.org/10.1016/j.taml.2016.12.004
[13]
Yi, J., Xu, K. and He, X.H. (2024) Parameter Optimization and Seismic Response Control of Tuned Negative Stiffness-Inertial Mass Damping System. Journal of Vibration Engineering, 37, 1015-1022. https://doi.org/10.16385/j.cnki.issn.1004-4523.2024.06.012
[14]
Gao, H., Xing, C.X. and Wang, H. (2023) Enhancement of Seismic Performance of Isolated Structures Using Tuned Negative Stiffness-Inertial Mass Dampers. Journal of Southeast University (Natural Science Edition), 53, 592-599. https://doi.org/10.3969/j.issn.1001-0505.2023.04.004
[15]
Wang, J., Zhang, Y. and Huang, S. (2023) Analytical Study on Parameter Optimization of Inertial-Mass Damping Systems with Negative Stiffness under Displacement Excitation. Journal of Vibration Engineering, 36, 804-814. https://doi.org/10.16385/j.cnki.issn.1004-4523.2023.03.023
[16]
Zhang, Y., Liu, G.L. and Zhou, P. (2024) Study on the Vibration Reduction Effect of Cable Systems with Negative Stiffness Dampers under Flexible Supports. Highway, No, 5, 142-146.
[17]
Tu, L., Ning, D., Sun, S., Li, W., Huang, H., Dong, M., et al. (2020) A Novel Negative Stiffness Magnetic Spring Design for Vehicle Seat Suspension System. Mechatronics, 68, Article ID: 102370. https://doi.org/10.1016/j.mechatronics.2020.102370
[18]
Benacchio, S., Malher, A., Boisson, J. and Touzé, C. (2016) Design of a Magnetic Vibration Absorber with Tunable Stiffnesses. NonlinearDynamics, 85, 893-911. https://doi.org/10.1007/s11071-016-2731-3
[19]
Hu, F.Y., Liu, Q.H. and Cao, J.Y. (2021) Parameter Identification Method for the Restoring Force Surface of Negative Stiffness Nonlinear Systems. Journal of Xi’an Jiaotong University, 55, 95-106. https://doi.org/10.7652/xjtuxb202104011
[20]
Zang, J., Yuan, T., Lu, Z., Zhang, Y., Ding, H. and Chen, L. (2018) A Lever-Type Nonlinear Energy Sink. JournalofSoundandVibration, 437, 119-134. https://doi.org/10.1016/j.jsv.2018.08.058
[21]
Yang, W.Q., Hua, X.G. and Wen, Q. (2021) Vibration Reduction Optimization of Lever-Type Tuned Mass Dampers for Vortex-Induced Vibrations of Twin Cable Suspensions. Journal of Vibration Engineering, 34, 819-827. https://doi.org/10.16385/j.cnki.issn.1004-4523.2021.04.019
[22]
Xu, X.M., Fu, M.H. and Li, J.X. (2016) Study on a Novel Lever-Type Variable Friction Damper for Heavy-Duty Truck Bogies. Mechanical Engineering and Automation, No. 6, 18-20. https://doi.org/10.3969/j.issn.1672-6413.2016.06.008
[23]
Li, C.X. and Xiong, X.Y. (2003) Dynamic Characteristics of a New Model Strategy for Lever-Type Tuned Mass Damper (LT-TMD). Sichuan Building Science, No. 4, 73-75.
[24]
Wang, X., He, T., Shen, Y., Shan, Y. and Liu, X. (2019) Parameters Optimization and Performance Evaluation for the Novel Inerter-Based Dynamic Vibration Absorbers with Negative Stiffness. JournalofSoundandVibration, 463, Article ID: 114941. https://doi.org/10.1016/j.jsv.2019.114941
[25]
Yang, X.T., Shen, Y.J. and Wang, J.F. (2022) Parameter Optimization of a Dynamic Vibration Absorber with an Amplification Mechanism, Inerter and Grounded Stiffness. Journal of Vibration and Shock, 41, 308-315. https://doi.org/10.13465/j.cnki.jvs.2022.21.035
[26]
Sipahi, R. and Olgac, N. (2006) Stability Robustness of Retarded LTI Systems with Single Delay and Exhaustive Determination of Their Imaginary Spectra. SIAMJournalonControlandOptimization, 45, 1680-1696. https://doi.org/10.1137/050633238
