THEORY AND EXPERIMENT OF EQUAL-PEAK OPTIMIZATION OF TIME DELAY COUPLED PENDULUM TUNED MASS DAMPER VIBRATING SYSTEM
-
摘要: 摆式调谐质量阻尼器因其便于安装、维修、更换, 且经济实用, 广泛应用于结构减振. 它通过将摆的自振频率调谐到接近主系统的控制频率, 使摆产生与主系统相反的振动, 从而抑制或消除主系统的振动. 本文通过对主系统无阻尼的被动减振系统和主系统有阻尼的时滞反馈主动减振系统进行多目标优化设计, 实现了对主系统幅频响应曲线的等峰控制和共振峰与反共振峰差值的有效控制. 首先, 建立了时滞耦合质量摆动力吸振器减振系统的力学模型和振动微分方程, 通过对主系统无阻尼的被动减振系统进行等峰优化, 获得了减振系统的最优频率比和质量摆的最优阻尼比. 对于主系统存在阻尼的被动减振系统, 在该优化参数下主系统的幅频响应曲线等峰优化失效. 其次, 对于主系统存在阻尼的时滞反馈优化控制系统, 采用CTCR方法得到了反馈增益系数和时滞的稳定区域. 在保证系统稳定的前提下, 通过调节反馈增益系数和时滞量两个控制参数能够实现对主系统幅频响应曲线的等峰控制. 再次, 对共振点处主系统振幅放大因子时滞敏感度和反馈增益系数敏感度进行分析, 表明共振点幅值对反馈增益系数比对时滞更为敏感. 最后, 通过实验分别在频域和时域内对理论结果进行了验证. 研究表明, 通过采用时滞反馈对摆式调谐质量阻尼减振系统进行等峰优化控制, 在较宽的频率区间内抑制了主系统的振幅; 通过控制共振峰和反共振峰的差值, 保证了幅频响应曲线的平坦性.Abstract: Pendulum tuned mass damper is widely used in structural vibration suppression because it is easy to install, maintain, replace economically and practically. The vibration of the primary system could be suppressed by tuning the natural frequency of the pendulum. The vibration of the pendulum is opposite to primary system by tuning the natural frequency of the pendulum to or close to the control frequency of the primary system. The multi-objective optimization designs are analyzed for both passive system when primary system without damping and time delay feedback active system when primary system with damping. It is realized equal-peak control of the amplitude-frequency response curve of the primary system and difference control between the resonance peak and the anti-resonance point. First, the mechanical model and vibration differential equation of the time-delay coupled pendulum tuned mass damper are established. The optimal frequency ratio of system and the optimal damping ratio of the pendulum tuned mass damper are obtained by equal-peak optimization for the passive system when primary system without damping. For the passive system when primary system with damping, equal-peak phenomenon of the amplitude-frequency response curve for the primary system is destroyed under these optimization parameters. Secondly, for the time delay feedback active optimal control system when primary system with damping, the stability region of feedback gain coefficient and time delay are obtained by using the CTCR method. The equal-peak control of the amplitude frequency response curve for primary system could be realized by adjusting the two control parameters of the feedback gain coefficient and time delay under the conditions of system is stable. Thirdly, the time delay sensitivity and feedback gain sensitivity of the primary system amplitude amplification factor at the resonance point are analyzed. It is shown that the resonance point amplitude is more sensitive to the feedback gain coefficient than to the time delay. Finally, the theoretical results are verified by experiments in frequency domain and time domain. The research shows that the amplitude of the primary system is suppressed in a wide frequency range by using the time delay feedback equal-peak optimization. The flatness of the amplitude-frequency response curve is ensured by controlling the difference between the resonance peak and the anti-resonance point.
-
图 1 时滞耦合质量摆减振系统力学模型
Figure 1. Mechanical model of time delay coupled pendulum tuned mass damper system
图 2 主系统阻尼系数对主系统幅频响应曲线的影响
Figure 2. Effects of damping coefficient of primary system on amplitude-frequency response curves of primary system
图 4 不同
$ {\xi _s} $ 下$g$ 和$\tau $ 稳定区域Figure 4. Stable region of
$g$ and$\tau $ under different$ {\xi _s} $ 
图 9
$\left| {{H_{{A}}} - {H_{{C}}}} \right| - g - \tau$ 响应曲线Figure 9. Response curves of
$\left| {{H_{{A}}} - {H_{{C}}}} \right| - g - \tau$ 
图 10
$\left| {{H_{{A}}} - {H_{{C}}}} \right| - g - \tau$ 响应曲线Figure 10. Response curves of
$\left| {{H_{{A}}} - {H_{{C}}}} \right| - g - \tau$ 
图 11
$ \left| {{H_{A} } - {H_{C} }} \right| - g - \tau $ 响应曲线Figure 11. Response curves of
$ \left| {{H_{A} } - {H_{C} }} \right| - g - \tau $ 
图 12
$ \left| {{H_{A} } - {H_{C} }} \right| - g - \tau $ 响应曲线Figure 12. Response curves of
$ \left| {{H_{A} } - {H_{C} }} \right| - g - \tau $ 
图 13 第一共振峰反馈增益系数敏感性响应曲线
${G_{{{A}}g}}$ −$ g $ Figure 13. Response curve
${G_{{{A}}g}}$ −$ g $ of sensitivity respect to gain at first peak point
图 16 第二共振峰时滞敏感性响应曲线
${G_{{{B}}\tau }}$ −$ \tau $ Figure 16. Response curve
${G_{{{B}}\tau }}$ −$ g $ of sensitivity respect to time delay at second peak point
图 14 第二共振峰反馈增益系数的敏感性响应曲线
${G_{{{B}}g}}$ −$ g $ Figure 14. Response curve
${G_{{{B}}g}}$ −$ g $ of sensitivity respect to gain at second peak point
图 15 第一共振峰时滞敏感性响应曲线
${G_{{{A}}\tau }}$ −$ \tau $ Figure 15. Response curve
${G_{{{A}}\tau }}$ −$ g $ of sensitivity respect to time delay at first peak point
图 17 控制参数临界稳定性曲线的CTCR和QPMR法的验证
Figure 17. Verification of critical stability curves of control parameters by CTCR and QPMR methods
图 18 不同主系统阻尼下幅频响应曲线等峰优化结果
Figure 18. Equal peak optimization results of amplitude frequency response curve under different primary system
图 19 不同主系统阻尼下幅频响应曲线等峰优化结果
Figure 19. Equal peak optimization results of amplitude frequency response curve under differ rent primary system
图 22 实验结构照片(1. 非接触激振器, 2. 底座, 3. 主系统质量, 4. 质量摆吸振器, 5. 伺服电机, 6. 控制刚度, 7.8.9. 加速度传感器)
Figure 22. Photo of the experimental structure (1. non-contact vibration exciter, 2. base, 3. primary mass, 4. pendulum tuned mass damper, 5. servo motor, 6. controlled stiffness, 7.8.9. acceleration sensors)
图 24 外激励频率−固有时滞响应图
Figure 24. External excitation frequency−inherent time delay diagram
图 25 实验和理论的频域对比图
Figure 25. Experimental and theoretical comparison in the frequency domain
表 1 实验系统的参数
Table 1. Parameters of the experimental system
m1/kg m2/kg k1/(N·m−1) k2/(N·m−1) $4.50$ $0.45$ $225.13$ $1.16$ c1/(N·m−1·s−1) c2/(N·m−1·s−1) l/m $15.42$ $3.45$ $0.27$ 
下载: 导出CSV
表 2 实验和理论对比表
Table 2. Comparison of theoretical and experimental results
Uncontrolled Passive parameter optimization Time delay feedback control first resonance peak ${H_{{A} } }$/mm ${\varOmega _{{A} } }$/Hz ${H_{{A} } }$/mm ${\varOmega _{{A} } }$/Hz ${H_{{A} } }$/mm ${\varOmega _{{A} } }$/Hz experiment 1.79 7.00 0.51 5.90 0.18 6.50 theory 1.58 8.50 0.51 6.50 0.10 6.50 error 0.21 1.50 0 0.60 0.08 0 second resonance peak ${H_{{B} } }$/mm ${\varOmega _{{B} } }$/Hz ${H_{{B} } }$/mm ${\varOmega _{{B} } }$/Hz ${H_{{B} } }$/mm ${\varOmega _{{B} } }$/Hz experiment — — 0.31 9.00 0.18 9.00 theory — — 0.39 12.00 0.10 10.0 error — — 0.08 3.00 0.08 1.00
下载: 导出CSV
表 3 理论和实验对应的时滞
Table 3. Time delay corresponding to theory and experiment
Frequency/Hz Time delay/ms $ \tau $ $ {\tau _0} $ $ {\tau _{{\text{arti}}}} $ 6.5 270 120 150 7.7 270 110 160 9.0 270 105 165
下载: 导出CSV
-
[1] Frahm H. Device for damping vibrations of bodies. US Patent 0989958. 1909-10-30 [2] Den Hartog JP. Mechanical Vibrations. New York: McGraw-Hall Book Company, 1985: 112-132 [3] Brock JE. A note on the damped vibration absorber. Transaction of the American Society of Mechanical Engineers, 1946, 13(4): A-284 [4] Asami T, Nishihara O. Closed-form exact solution to H∞ optimization of dynamic vibration absorbers (application to different transfer functions and damping systems). Journal of Vibration and Acoustics, 2003, 125(3): 398-405 doi: 10.1115/1.1569514 [5] 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 Yongiun, Yang Shaopu. Parameters 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) doi: 10.6052/0459-1879-14-275 [6] 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) [7] 邢昭阳, 申永军, 邢海军等. 一种含放大机构的负刚度动力吸振器的参数优化. 力学学报, 2019, 51(3): 894-903 (Xing Zhaoyang, Shen Yongjun, Xing Haijun, et al. Parameters optimization of a dynamic vibration absorber with amplifying mechanism and negative stiffness. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 894-903 (in Chinese) [8] 周子博, 申永军, 邢海军, 等. 含惯容和杠杆元件的减振系统参数优化及性能分析. 振动工程学报, 2022, 35(2): 407-416 (Zhou Zibo, Shen Yongjun, Xing Haijun, et al. Parameter optimization and performance analysis of vibration mitiga-tion systems with inertia and lever components. Journal of Vibration Engineering, 2022, 35(2): 407-416 (in Chinese) [9] Nishihara O, Asami T. Closed-form solutions to the exact optimizations of dynamic vibration absorbers (minimizations of the maximum amplitude magnification factors). Journal of Vibration and Acoustics, 2002, 124(4): 576-582 doi: 10.1115/1.1500335 [10] Sun XT, Xu J, Wang F, et al. Design and experiment of nonlinear absorber for equal-peak and de-nonlinearity. Journal of Sound and Vibration, 2019, 449: 274-299 doi: 10.1016/j.jsv.2019.02.033 [11] Gerges RR, Vickery BJ. Optimum design of pendulum-type tuned mass dampers. Struct Des Tall Spec 2005; 14(4): 353-368 [12] Chung LL, Wu LY, Lien HK, et al. Optimal design of friction pendulum tuned mass damper with varying friction coefficient. Structural Control & Health Monitoring, 2013, 20(4): 544-559 [13] Deraemaeker A, Soltani P. Corrigendum to A short note on equal peak design for the pendulum tuned mass dampers. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2019, 235(3): 285-392 [14] Shu Z, Li S, Zhang J, et al. Optimum seismic design of a power plant building with pendulum tuned mass damper system by its heavy suspended buckets. Engineering Structures, 2017, 136: 114-132 [15] 侯洁, 霍林生, 李宏男. 非线性悬吊质量摆对输电塔减振控制的研究, 振动与冲击, 2014, 3(3): 177-181Hou Jie, Huo Linsheng, Li Hongnan, Aseismic control of transmission towers with nonlinear suspended mass pendulums. Journal of Vibration and Shock, 2014, 3(3): 177-181 (in Chinese) [16] 霍林生, 侯洁, 李宏男. 非线性悬吊质量摆减震控制的等效线性化方法研究, 防灾减灾工程学报, 2015, 35(3): 283-289Huo Linsheng, Hou Jie, Li Hongnan, Research on Equivalent Lineariation of Seismic Control with Nonlinear Suspending Mass Pendulum. Journal of Disaster Prevention and Mitigation Engineering, 2015, 35(3): 283-289 (in Chinese) [17] Roffel AJ, Narasimhan S. Extended Kalman filter for modal identification of structures equipped with a pendulum tuned mass damper. Journal of Sound and Vibration, 2014, 333: 6038-6056 doi: 10.1016/j.jsv.2014.06.030 [18] Wang LK, Shi WX, Zhou Y. Study on self-adjustable variable pendulum tuned mass damper. The Structural Design of Tall and Special Buildings, 2018, 28(12): 1561-1-13 [19] Victor JG, Edwin PD, Jose AI, et al. Pendulum tuned mass damper: optimization and performance assessment in structures with elastoplastic behavior. Heliyon, 2021, 7(6): 07221-1-24 [20] Xu K, Hua X, Lacarbonara W, et al. Exploration of the nonlinear effect of pendulum tuned mass dampers on vibration control. Journal of Engineering Mechanics, 2021, 147(8): 4021047-1-19 doi: 10.1061/(ASCE)EM.1943-7889.0001961 [21] Soltani P, Deraemaeker A. Pendulum tuned mass dampers and tuned mass dampers: Analogy and optimum parameters for various combinations of response and excitation parameters. Journal of Vibration and Control, 2022, 28: 15-16 [22] Sun SS, Yang J, Li WH, et al. Development of an MRE adaptive tuned vibration absorber with self-sensing capability. Smart Materials and Structures, 2015, 24(9): 095012-1-10 doi: 10.1088/0964-1726/24/9/095012 [23] Kumbhar SB, Chavan SP, Gawade SS. Adaptive tuned vibration absorber based on magnetorheological elastomer-shape memory alloy composite. Mechanical Systems and Signal Processing, 2018, 100(1): 208-223 [24] Gripp JAB, Goes LCS, Heuss O, et al. An adaptive piezoelectric vibration absorber enhanced by a negative capacitance applied to a shell structure. Smart Material Structures, 2015, 24(12): 125017-1-15 doi: 10.1088/0964-1726/24/12/125017 [25] Olgac N, Holm-Hansen BT. A novel active vibration absorption technique: delayed resonator. Journal of Sound and Vibration, 1994, 176(1): 93-104 doi: 10.1006/jsvi.1994.1360 [26] Sipahi R, Olgac N, Breda D. A stability study on first-order neutral systems with three rationally independent time delays. International Journal of Systems Science, 2010, 41(12): 1445-1455 doi: 10.1080/00207720903353625 [27] Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Transactions on Automatic Control, 2002, 47(5): 793-797 doi: 10.1109/TAC.2002.1000275 [28] Dan P, Vyhlidal T, Michiels W. Optimized design of robust resonator with distributed time-delay. Journal of Sound and Vibration, 2018, 443: 576-590 [29] Wang F, Sun XT, Meng H, et al. Tunable broadband low-frequency band gap of multiple-layer metastructure induced by time-delayed vibration absorbers. Nonlinear Dynamics, 2022, 107(3): 1903-1918 doi: 10.1007/s11071-021-07065-z [30] Wang F, Sun XT, Meng H, et al. Time-delayed feedback control design and its application for vibration absorption. IEEE Transactions on Industrial Electronics, 2020, 68(9): 8593-8602 [31] Meng H, Sun XT, Xu J, et al. The generalization of equal-peak method for delay-coupled nonlinear system. Physica D: Nonlinear Phenomena, 2020, 403: 132340-1-12 doi: 10.1016/j.physd.2020.132340 [32] 代晗, 赵艳影. 负刚度时滞反馈控制动力吸振器的等峰优化. 力学学报, 2021, 53(6): 1720-1732 (Dai Han, Zhao Yanying. Equal-peak optimization of dynamic vibration absorber with negative stiffness and delay feedback control. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1720-1732 (in Chinese) doi: 10.6052/0459-1879-21-074 [33] Hosek M , Elmali H , Olgac N. Centrifugal delayed resonator: theory and experiments//Proceedings of the ASME 1997 Design Engineering Technical Conferences. Volume 1B: 16th Biennial Conference on Mechanical Vibration and Noise. Sacramento, California, USA. September 14-17, 1997. V01 BT08 A006 [34] Hosek M. The centrifugal delayed resonator as a tunable torsional vibration absorber for multi-degree-of-freedom systems. Journal of Vibration and Control, 1999, 5(2): 299-322 doi: 10.1177/107754639900500209 [35] Christie MD, Sun S, Deng L, et al. A variable resonance magnetorheological-fluid-based pendulum tuned mass damper for seismic vibration suppression. Mechanical Systems and Signal Processing, 2019, 116: 530-544 doi: 10.1016/j.ymssp.2018.07.007 [36] Vyhlidal T, Pavel Z. QPmR-Quasi-Polynomial Root-Finder: Algorithm Update and Examples Chapter 22. Springer International Publishing, 2014: 299-312 -
下载:
点击查看大图
计量
- 文章访问数: 215
- HTML全文浏览量: 68
- PDF下载量: 69
- 被引次数: 0
