一种含放大机构的负刚度动力吸振器的参数优化
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摘要
在大多数情况下机械振动是有害的,它不仅产生噪声还会降低设备的工作精度和使用寿命.采用正刚度特性的吸振、隔振系统往往难以达到满意效果,这种情况在低频振动控制系统中尤其明显.放大机构与负刚度元件在振动控制领域均表现出良好性能,但是较少有对同时含有放大机构与负刚度装置的动力吸振系统的研究.以Voigt型动力吸振器为基础提出了一种将放大机构应用于含负刚度弹簧元件的动力吸振器模型,对该模型的最优参数进行了研究.首先建立了系统的运动微分方程并得到了其解析解,发现该系统存在两个固定点,利用固定点理论得到了动力吸振器的最优频率比.根据负刚度的特性,在保证系统稳定的前提下得到了最优负刚度比,并推导了系统的近似最优阻尼比.通过数值仿真验证了解析解的正确性.与多种动力吸振器在简谐激励与随机激励下进行了对比,说明了本文模型相比于已有的动力吸振器,能够大幅降低共振振幅、拓宽减振频带并且降低系统的谐振频率,为设计新型动力吸振器模型提供了理论依据.
Mechanical vibration is detrimental in most engineering situations, and it may not only generate noise but also reduce the operational accuracy and working life of the equipment. It is generally difficult for vibration absorption and isolation systems with positive stiffness characteristics to achieve satisfactory performance, which becomes noticeable especially in low-frequency vibration control systems. Amplifying mechanism and negative stiffness element both show good performance in the field of vibration control, but the dynamic vibration absorber with both amplifying mechanism and negative stiffness element is rarely studied. Based on the Voigt type dynamic vibration absorber, a dynamic vibration absorber with negative stiffness element using amplifying mechanism is presented, and the optimal system parameters are studied in detail. Firstly, the differential equation of motion is established and the analytic solution of the system is obtained, and it is found that there are two fixed points independent of damping ratio in the amplitude-frequency curves of the primary system. The optimal frequency ratio of the dynamic vibration absorber is obtained based on the fixed-point theory. According to the characteristics of negative stiffness, the optimal negative stiffness ratio is founded under the premise of ensuring the system stability. A simple method is used to derive the approximate optimal damping ratio of the system. The correctness of the analytical results is verified by the comparison with the results by numerical simulation.Compared with other dynamic vibration absorbers under harmonic and random excitations, it could be found that the model with optimal parameters in this paper can greatly reduce the resonance amplitude, broaden the vibration band, and lower the resonance frequency of the primary system. These results may provide theoretical basis for the optimal design of similar dynamic vibration absorbers.
Mechanical vibration is detrimental in most engineering situations, and it may not only generate noise but also reduce the operational accuracy and working life of the equipment. It is generally difficult for vibration absorption and isolation systems with positive stiffness characteristics to achieve satisfactory performance, which becomes noticeable especially in low-frequency vibration control systems. Amplifying mechanism and negative stiffness element both show good performance in the field of vibration control, but the dynamic vibration absorber with both amplifying mechanism and negative stiffness element is rarely studied. Based on the Voigt type dynamic vibration absorber, a dynamic vibration absorber with negative stiffness element using amplifying mechanism is presented, and the optimal system parameters are studied in detail. Firstly, the differential equation of motion is established and the analytic solution of the system is obtained, and it is found that there are two fixed points independent of damping ratio in the amplitude-frequency curves of the primary system. The optimal frequency ratio of the dynamic vibration absorber is obtained based on the fixed-point theory. According to the characteristics of negative stiffness, the optimal negative stiffness ratio is founded under the premise of ensuring the system stability. A simple method is used to derive the approximate optimal damping ratio of the system. The correctness of the analytical results is verified by the comparison with the results by numerical simulation.Compared with other dynamic vibration absorbers under harmonic and random excitations, it could be found that the model with optimal parameters in this paper can greatly reduce the resonance amplitude, broaden the vibration band, and lower the resonance frequency of the primary system. These results may provide theoretical basis for the optimal design of similar dynamic vibration absorbers.
引文
1 Frahm H.Device for damping vibrations of bodies.U.S.Patent 089958,1909-10-30
2 Ormondroyd J,Den Hartog JP.The theory of the dynamic vibration absorber.Journal of Applied Mechanics,1928,50:9-22
3 Hahnkamm E.The damping of the foundation vibrations at varying excitation frequency.Master of Architecture,1932,4:192-201
4 Brock JE.A note on the damped vibration absorber.Journal of Applied Mechanics,1946,13(4):A284
5 Den Hartog JP.Mechanical Vibrations.New York:McGraw-Hall Book Company,1947:112-132
6 Asami T.Closed-form exact solution to H-infinity optimization of dynamic vibration absorbers:Application to different transfer functions and damping systems.Journal of Vibration and Acoustics,2003,125(3):398-405
7 Nishihara O,Asami T.Close-form solutions to the exact optimizations of dynamic vibration absorber(minimizations of the maximum amplitude magnification factors).Journal of Vibration and Acoustics,2002,124(4):576-582
8 Asami T,Nishihara O,Baz AM.Analytical solutions to H∞and H2optimization of dynamic vibration absorbers attached to damped linear systems.Journal of Vibration and Acoustics,2002,124(2):284-295
9 Ren MZ.A variant design of the dynamic vibration absorber.Journal of Sound and Vibration,2001,245(4):762-770
10 Liu KF,Liu J.The damped dynamic vibration absorbers:Revisited and new result.Journal of Sound and Vibration,2005,284(3):1181-1189
11 Asami T,Nishihara O.Analytical and experimental evaluation of an air damped dynamic vibration absorber:Design optimizations of the three-element type model.Journal of Vibration and Acoustics,1999,121(3):334-342
12 Asami T,Nishihara O.H2 optimization of the three-element type dynamic vibration absorbers.Journal of Vibration and Acoustics,2002,124(4):583-592
13 Zhao YY,Xu J.Effects of delayed feedback control on nonlinear vibration absorber system.Journal of Sound and Vibration,2007,308(1):212-230
14赵艳影,徐鉴.时滞非线性动力吸振器的减振机理.力学学报,2008,40(1):98-106(Zhao Yanying,Xu Jian.Mechanism analysis of delayed nonlinear vibration absorber.Chinese Journal of Theoretical and Applied Mechanics.2008,40(1):98-106(in Chinese))
15赵艳影,徐鉴.利用时滞反馈控制自参数振动系统的振动.力学学报,2011,43(5):894-904(Zhao Yanying,Xu Jian.Using the delayed feedback to control the vibration of the auto-parametric dynamical system.Chinese Journal of Theoretical and Applied Mechanics.2011,43(5):894-904(in Chinese))
16 Shen YJ,Wang L,Yang SP,et al.Nonlinear dynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbers.Journal of Vibration and Control,2013,19(1):143-160
17 Shen YJ,Ahmadian M.Nonlinear dynamical analysis on four semiactive dynamic vibration absorbers with time delay.Shock and Vibration,2013,20(4):649-663
18 Shen YJ,Yang SP,Xing HJ,et al.Design of single degree-offreedom optimally passive vibration isolation system.Journal of Vibration Engineering&Technologies,2015,3(1):25-36
19李帅,周继磊,任传波等.时变参数时滞减振控制研究.力学学报,2018,50(1):99-108(Li Shuai,Zhou Jilei,Ren Chuanbo,et al.The research of time delay vibration control with time-varying parameters.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):99-108(in Chinese))
20 Alabuzhev PM,Rivin EI.Vibration Protection and Measuring Systems with Quasi-zero Stiffness.New York:Hemisphere Publishing Company,Tylor&Francis Group,1989
21彭献,陈树年,宋福磐.负刚度的工作原理及应用初探.湖南大学学报,1992,19(4):89-94(Peng Xian,Chen Shunian,Song Fupan.Research on theory of negative stiffness and its application.Journal of Hunan University(Natural Sciences),1992,19(4):89-94(in Chinese))
22 Trimboli MS,Wimmel R,Breitbach EJ.Quasi-active approach to vibration isolation using magnetic springs//North American Conference on Smart Structures and Materials.International Society for Optics and Photonics,1994:73-83
23彭解华,陈树年.正负刚度并联结构的稳定性及应用研究.振动、测试与诊断,1995,15(2):14-18(Peng Jiehua,Chen Shunian.The stability and application of a structure with positive stiffness element and negative stiffness element.Journal of Vibration,Measurement&Diagnosis,1995,15(2):14-18(in Chinese))
24 Mizuno T.Proposal of a vibration isolation system using zeropower magnetic suspension//Proceedings of the Asia-pacific Vibration Conference,Hangzhou,China,2001,2:423-427
25 Mizuno T.Vibration isolation system using zero-power magnetic suspension//The 15th Triennial World Congress,Barcelona,Spain,2002:955-960
26 Mizuno T,Takemori Y.A transfer-function approach to the analysis and design of zero-power controllers for magnetic suspension systems.Electrical Engineering in Japan,2002,141(2):67-75
27 Mizuno T,Takasaki M,Kishita D,et al.Vibration isolation system combining zero-power magnetic suspension with springs.Control Engineering Practice,2007,15(2):187-196
28 Mizuno T,Toumiya T,Takasaki M.Vibration isolation system using negative stiffness.International Journal Series C,2003,46(3):807-812
29 Park ST,Luu TT.Techniques for optimizing parameters of negative stiffness.Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2007,221(5):505-510
30纪晗,熊世树,袁涌.基于负刚度原理的结构隔振效果分析.华中科技大学学报(自然科学版),2010,38(2):76-79(Ji Han,Xiong Shishu,Yuan Yong.Analyzing vibration isolation effect of structures using negative stiffness principle.Journal of Huazhong University of Science and Technology(Natural Science Edition),2010,38(2):76-79(in Chinese))
31 Lu ZQ,Brennan MJ,Yang TJ,et al.An investigation of a two-stage nonlinear vibration isolation system.Journal of Sound and Vibration,2013,332(6):1456-1464
32 Lu ZQ,Brennan MJ,Yang TJ,et al.Experimental investigation of a two-stage nonlinear vibration isolation system with high-static-lowdynamic stiffness.ASME Journal of Applied Mechanics,2017,84:021001
33 Lu ZQ,Brennan MJ,Chen LQ.On the transmissibilities of a nonlinear vibration isolation system.Journal of Sound and Vibration,2016,375:28-37
34 Acar MA,Yilmaz C.Design of an adaptive-passive dynamic vibration absorber composed of a string-mass system equipped with negative stiffness tension adjusting mechanism.Journal of Sound and Vibration,2013,332(2):231-245
35彭海波,申永军,杨绍普.一种含负刚度元件的新型动力吸振器的参数优化.力学学报,2015,47(2):320-327(Peng Haibo,Shen Yongjun,Yang Shaopu.Parameter optimization of a new type of dynamic vibration absorber with negative stiffness.Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):320-327(in Chinese))
36王孝然,申永军,杨绍普等.含负刚度元件的三要素型动力吸振器的参数优化.振动工程学报,2017,30(2):177-184(Wang Xiaoran,Shen Yongjun,Yang Shaopu,et al.Parameter optimization of three-element type dynamic vibration absorber with negative stiffness.Journal of Vibration Engineering,2017,30(2):177-184(in Chinese)
37李东海,赵寿根,何玉金等.含有时滞控制的准零刚度隔振器动力学分析.振动与冲击,2018,37(13):49-55(Li Donghai,Zhao Shougen,He Yujin,et al.Dynamical analysis of a QZS vibration isolator with time-delay control.Journal of Vibration Engineering,2018,37(13):49-55(in Chinese))
38李强,董光旭,张希农等.新型可调动力吸振器设计及参数优化.航空学报,2018,39(6):128-140(Li Qiang,Dong Guangxu,Zhang Xinong,et al.Design and parameter optimization of a new tunable dynamic vibration absorber.Acta Aeronautica et Astronautica Sinica,2018,39(6):128-140(in Chinese))
39高雪,陈前,刘先斌.一类分段光滑隔振系统的非线性动力学设计方法.力学学报,2016,48(1):192-200(Gao Xue,Chen Qian,Liu Xianbin.Nonlinear dynamics design for piecewise smooth vibration isolation system.Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):192-200(in Chinese))
40张舒,徐鉴.时滞耦合系统非线性动力学的研究进展.力学学报,2017,49(3):565-587(Zhang Shu,Xu Jian.Review on nonlinear dynamics in systems with coulpling delays.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):565-587(in Chinese))
41曹登庆,白坤朝,丁虎等.大型柔性航天器动力学与振动控制研究进展.力学学报,2019,51(1):1-13(Cao Dengqing,Bai Kunchao,Ding Hu,et al.Advances in dynamics and vibration control of large-scale flexible spacecraft.Chinese Journal of Theoretical and Applied Mechanics,2019,51(1):1-13(in Chinese))
42陆泽琦,陈立群.非线性被动隔振的若干进展.力学学报,2017,49(3):550-564(Lu Zeqi,Chen Liqun.Some recent progresses in nonlinear passive isolations of vibrations.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):550-564(in Chinese))
43李春祥,熊学玉.杠杆式调谐质量阻尼器(LT-TMD)新模型策略的动力特性.四川建筑科学研究,2003,29(4):73-75(Li Chunxiang,Xiong Xueyu.Dynamic characteristics of new modeling strategy of lever tuned mass damper(LT-TMD).Building Science Research of Sichuan,2003,29(4):73-75(in Chinese))
44 Zang J,Yuan TC,Lu ZQ,et al.A lever-type nonlinear energy sink.Journal of Sound and Vibration(2018),doi:10.1016/j.jsv.2018.08.058
45张孝良,聂佳梅,袁朝春等.理想天棚阻尼的被动实现方法.北京理工大学学报,2014,34(1):22-26(Zhang Xiaoliang,Nie Jiamei,Yuan Chaochun,et al.A passive realization method of ideal skyhook damping.Transactions of Beijing Institute of Technology,2014,34(1):22-26(in Chinese))
46吴庭,王林翔.复合式低频隔振器的理论建模及其性能分析.振动与冲击,2017,36(19):232-235(Wu Ting,Wang Linxiang.Design and performance analysis of a composite low-frequency vibration isolator.Journal of Vibration and Shock,2017,36(19):232-235(in Chinese))
2 Ormondroyd J,Den Hartog JP.The theory of the dynamic vibration absorber.Journal of Applied Mechanics,1928,50:9-22
3 Hahnkamm E.The damping of the foundation vibrations at varying excitation frequency.Master of Architecture,1932,4:192-201
4 Brock JE.A note on the damped vibration absorber.Journal of Applied Mechanics,1946,13(4):A284
5 Den Hartog JP.Mechanical Vibrations.New York:McGraw-Hall Book Company,1947:112-132
6 Asami T.Closed-form exact solution to H-infinity optimization of dynamic vibration absorbers:Application to different transfer functions and damping systems.Journal of Vibration and Acoustics,2003,125(3):398-405
7 Nishihara O,Asami T.Close-form solutions to the exact optimizations of dynamic vibration absorber(minimizations of the maximum amplitude magnification factors).Journal of Vibration and Acoustics,2002,124(4):576-582
8 Asami T,Nishihara O,Baz AM.Analytical solutions to H∞and H2optimization of dynamic vibration absorbers attached to damped linear systems.Journal of Vibration and Acoustics,2002,124(2):284-295
9 Ren MZ.A variant design of the dynamic vibration absorber.Journal of Sound and Vibration,2001,245(4):762-770
10 Liu KF,Liu J.The damped dynamic vibration absorbers:Revisited and new result.Journal of Sound and Vibration,2005,284(3):1181-1189
11 Asami T,Nishihara O.Analytical and experimental evaluation of an air damped dynamic vibration absorber:Design optimizations of the three-element type model.Journal of Vibration and Acoustics,1999,121(3):334-342
12 Asami T,Nishihara O.H2 optimization of the three-element type dynamic vibration absorbers.Journal of Vibration and Acoustics,2002,124(4):583-592
13 Zhao YY,Xu J.Effects of delayed feedback control on nonlinear vibration absorber system.Journal of Sound and Vibration,2007,308(1):212-230
14赵艳影,徐鉴.时滞非线性动力吸振器的减振机理.力学学报,2008,40(1):98-106(Zhao Yanying,Xu Jian.Mechanism analysis of delayed nonlinear vibration absorber.Chinese Journal of Theoretical and Applied Mechanics.2008,40(1):98-106(in Chinese))
15赵艳影,徐鉴.利用时滞反馈控制自参数振动系统的振动.力学学报,2011,43(5):894-904(Zhao Yanying,Xu Jian.Using the delayed feedback to control the vibration of the auto-parametric dynamical system.Chinese Journal of Theoretical and Applied Mechanics.2011,43(5):894-904(in Chinese))
16 Shen YJ,Wang L,Yang SP,et al.Nonlinear dynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbers.Journal of Vibration and Control,2013,19(1):143-160
17 Shen YJ,Ahmadian M.Nonlinear dynamical analysis on four semiactive dynamic vibration absorbers with time delay.Shock and Vibration,2013,20(4):649-663
18 Shen YJ,Yang SP,Xing HJ,et al.Design of single degree-offreedom optimally passive vibration isolation system.Journal of Vibration Engineering&Technologies,2015,3(1):25-36
19李帅,周继磊,任传波等.时变参数时滞减振控制研究.力学学报,2018,50(1):99-108(Li Shuai,Zhou Jilei,Ren Chuanbo,et al.The research of time delay vibration control with time-varying parameters.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):99-108(in Chinese))
20 Alabuzhev PM,Rivin EI.Vibration Protection and Measuring Systems with Quasi-zero Stiffness.New York:Hemisphere Publishing Company,Tylor&Francis Group,1989
21彭献,陈树年,宋福磐.负刚度的工作原理及应用初探.湖南大学学报,1992,19(4):89-94(Peng Xian,Chen Shunian,Song Fupan.Research on theory of negative stiffness and its application.Journal of Hunan University(Natural Sciences),1992,19(4):89-94(in Chinese))
22 Trimboli MS,Wimmel R,Breitbach EJ.Quasi-active approach to vibration isolation using magnetic springs//North American Conference on Smart Structures and Materials.International Society for Optics and Photonics,1994:73-83
23彭解华,陈树年.正负刚度并联结构的稳定性及应用研究.振动、测试与诊断,1995,15(2):14-18(Peng Jiehua,Chen Shunian.The stability and application of a structure with positive stiffness element and negative stiffness element.Journal of Vibration,Measurement&Diagnosis,1995,15(2):14-18(in Chinese))
24 Mizuno T.Proposal of a vibration isolation system using zeropower magnetic suspension//Proceedings of the Asia-pacific Vibration Conference,Hangzhou,China,2001,2:423-427
25 Mizuno T.Vibration isolation system using zero-power magnetic suspension//The 15th Triennial World Congress,Barcelona,Spain,2002:955-960
26 Mizuno T,Takemori Y.A transfer-function approach to the analysis and design of zero-power controllers for magnetic suspension systems.Electrical Engineering in Japan,2002,141(2):67-75
27 Mizuno T,Takasaki M,Kishita D,et al.Vibration isolation system combining zero-power magnetic suspension with springs.Control Engineering Practice,2007,15(2):187-196
28 Mizuno T,Toumiya T,Takasaki M.Vibration isolation system using negative stiffness.International Journal Series C,2003,46(3):807-812
29 Park ST,Luu TT.Techniques for optimizing parameters of negative stiffness.Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2007,221(5):505-510
30纪晗,熊世树,袁涌.基于负刚度原理的结构隔振效果分析.华中科技大学学报(自然科学版),2010,38(2):76-79(Ji Han,Xiong Shishu,Yuan Yong.Analyzing vibration isolation effect of structures using negative stiffness principle.Journal of Huazhong University of Science and Technology(Natural Science Edition),2010,38(2):76-79(in Chinese))
31 Lu ZQ,Brennan MJ,Yang TJ,et al.An investigation of a two-stage nonlinear vibration isolation system.Journal of Sound and Vibration,2013,332(6):1456-1464
32 Lu ZQ,Brennan MJ,Yang TJ,et al.Experimental investigation of a two-stage nonlinear vibration isolation system with high-static-lowdynamic stiffness.ASME Journal of Applied Mechanics,2017,84:021001
33 Lu ZQ,Brennan MJ,Chen LQ.On the transmissibilities of a nonlinear vibration isolation system.Journal of Sound and Vibration,2016,375:28-37
34 Acar MA,Yilmaz C.Design of an adaptive-passive dynamic vibration absorber composed of a string-mass system equipped with negative stiffness tension adjusting mechanism.Journal of Sound and Vibration,2013,332(2):231-245
35彭海波,申永军,杨绍普.一种含负刚度元件的新型动力吸振器的参数优化.力学学报,2015,47(2):320-327(Peng Haibo,Shen Yongjun,Yang Shaopu.Parameter optimization of a new type of dynamic vibration absorber with negative stiffness.Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):320-327(in Chinese))
36王孝然,申永军,杨绍普等.含负刚度元件的三要素型动力吸振器的参数优化.振动工程学报,2017,30(2):177-184(Wang Xiaoran,Shen Yongjun,Yang Shaopu,et al.Parameter optimization of three-element type dynamic vibration absorber with negative stiffness.Journal of Vibration Engineering,2017,30(2):177-184(in Chinese)
37李东海,赵寿根,何玉金等.含有时滞控制的准零刚度隔振器动力学分析.振动与冲击,2018,37(13):49-55(Li Donghai,Zhao Shougen,He Yujin,et al.Dynamical analysis of a QZS vibration isolator with time-delay control.Journal of Vibration Engineering,2018,37(13):49-55(in Chinese))
38李强,董光旭,张希农等.新型可调动力吸振器设计及参数优化.航空学报,2018,39(6):128-140(Li Qiang,Dong Guangxu,Zhang Xinong,et al.Design and parameter optimization of a new tunable dynamic vibration absorber.Acta Aeronautica et Astronautica Sinica,2018,39(6):128-140(in Chinese))
39高雪,陈前,刘先斌.一类分段光滑隔振系统的非线性动力学设计方法.力学学报,2016,48(1):192-200(Gao Xue,Chen Qian,Liu Xianbin.Nonlinear dynamics design for piecewise smooth vibration isolation system.Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):192-200(in Chinese))
40张舒,徐鉴.时滞耦合系统非线性动力学的研究进展.力学学报,2017,49(3):565-587(Zhang Shu,Xu Jian.Review on nonlinear dynamics in systems with coulpling delays.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):565-587(in Chinese))
41曹登庆,白坤朝,丁虎等.大型柔性航天器动力学与振动控制研究进展.力学学报,2019,51(1):1-13(Cao Dengqing,Bai Kunchao,Ding Hu,et al.Advances in dynamics and vibration control of large-scale flexible spacecraft.Chinese Journal of Theoretical and Applied Mechanics,2019,51(1):1-13(in Chinese))
42陆泽琦,陈立群.非线性被动隔振的若干进展.力学学报,2017,49(3):550-564(Lu Zeqi,Chen Liqun.Some recent progresses in nonlinear passive isolations of vibrations.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):550-564(in Chinese))
43李春祥,熊学玉.杠杆式调谐质量阻尼器(LT-TMD)新模型策略的动力特性.四川建筑科学研究,2003,29(4):73-75(Li Chunxiang,Xiong Xueyu.Dynamic characteristics of new modeling strategy of lever tuned mass damper(LT-TMD).Building Science Research of Sichuan,2003,29(4):73-75(in Chinese))
44 Zang J,Yuan TC,Lu ZQ,et al.A lever-type nonlinear energy sink.Journal of Sound and Vibration(2018),doi:10.1016/j.jsv.2018.08.058
45张孝良,聂佳梅,袁朝春等.理想天棚阻尼的被动实现方法.北京理工大学学报,2014,34(1):22-26(Zhang Xiaoliang,Nie Jiamei,Yuan Chaochun,et al.A passive realization method of ideal skyhook damping.Transactions of Beijing Institute of Technology,2014,34(1):22-26(in Chinese))
46吴庭,王林翔.复合式低频隔振器的理论建模及其性能分析.振动与冲击,2017,36(19):232-235(Wu Ting,Wang Linxiang.Design and performance analysis of a composite low-frequency vibration isolator.Journal of Vibration and Shock,2017,36(19):232-235(in Chinese))
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