风激斜拉索多模态自激振动及其最优控制
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摘要
本文研究了斜拉索因雨、雪等改变了截面形状时受到横向风的激励而产生的多模态自激振动及其最优控制。首先通过建立拉索受风力作用的非线性运动微分方程,基于索纵向运动相对较小而导出关于索横向运动的偏微分振动方程,进一步运用Galerkin法将该方程转化为常微分方程组,用以描述索的多模态自激振动；再应用非线性振动的平均法,求解得到该系统的自激振动分析解,根据奇解的稳定性条件,分析了拉索前两阶模态耦合的自激振动以及拉索一三阶模态耦合的自激振动的临界风速和极限环的稳定性及其条件,并通过数值模拟验证,然后通过数值方法计算了拉索一二三阶模态耦合的自激振动的特性,并与拉索前两阶模态耦合的自激振动和一三阶模态耦合的自激振动的情况相比较。接着研究了各参数对拉索前两阶模态耦合的自激振动的影响。
以拉索前两阶模态的耦合的自激振动为对象,研究了斜拉索多模态自激振动的最优控制问题。根据最优控制的动态规划原理确定最优非线性控制力,再应用非线性振动的平均法,求解得到该系统自激振动及其稳定性的分析解,通过分析比较和数值模拟验证控制前后极限环及其稳定性,说明该最优控制能够有效地抑制风激拉索的自激振动。然后比较了非线性最优控制与线性次最优控制控制效果的差异,结果表明：在小初始扰动条件下,非线性最优控制和线性次最优控制均能有效的抑制拉索的振荡,且控制效果基本相同；在大初始扰动条件下,非线性最优控制和线性次最优控制也均能有效的抑制拉索的振荡,但和线性次最优控制相比,非线性最优控制能够更快的减小振幅。接着研究非线性最优控制的实施,建立起拉索横向一点、两点非线性最优控制,结果表明：选取合适的控制参数,拉索横向一点、两点最优控制均能有效抑制拉索的自激振荡,但两点最优控制的效果优于一点最优控制。
考虑拉索的纵向振动对拉索横向多模态振动的影响,建立起拉索面内纵向、横向二维耦合的运动微分方程组,应用非线性振动的平均法,研究了纵向一阶、横向一二阶模态的耦合振动特性,结果表明：稳态响应时,纵向一阶模态振幅与横向二阶模态振幅成正比,相位相差180°,并与第二章简化拉索纵向振动时进行比较,拉索横向一二阶模态耦合振动的极限环的稳定性及其条件具有相同的规律。
The fluid-flow-induced cable vibration is an active research subject in structural engineering and has practical importance for structural improvement and vibration control. This paper focused mainly on the analysis of the self-excited oscillation of an inclined taut cable with multi-vibration modes under wind loading as the cable's cross-section shape changes because of rain or snow. Firstly the nonlinear differential equations of motion were derived for the wind-induced vibration of the cable. Then the partial differential equation for the transverse cable vibration was obtained based on the assumption of longitudinal cable vibration comparatively small. By using the Galerkin approach, this partial differential equation was converted into ordinary differential equations which describe the self-excited oscillation of the cable as multi-modes system. The analytical solutions in self-excited oscillation were obtained further by using the averaging method for nonlinear vibration. According to the stability condition of analytical solutions, the wind-induced limit cycle of self-excited oscillation and its existence conditions and the critical wind velocities for the first two modes and the first, third modes of the cable system in self-oscillation were analytically determined finally and verified by the numerical results. Then the self-oscillation characteristics for the first three modes of the cable system were analyzed by numerical method. The effects of parameters about self-oscillation for the first two modes of the cable were researched at last.
Secondly the cable control for the first two modes in self-excited oscillation was focused on. The performance index for optimal cable control was given to the polynomial control solution. Based on the dynamical programming principle, the Hamilton-Jacobi-Bellman (HJB) equation was established for the cable control with the index. The optimal nonlinear control force was determined from solving the HJB equation. By using the averaging method for nonlinear vibration, the analytical solutions, in particular, limit cycle solutions and stability of the controlled system in self-excited oscillation were obtained further. It was concluded by the analytical and numerical comparison between controlled and uncontrolled self-excited oscillations and stability that the proposed optimal control could effectively suppress the self-excited oscillation of the cable under wind loading. Then the control effectiveness was compared between the nonlinear optimal control and linear sub-optimal control. The results showed that under a small initial disturbance, the nonlinear optimal control and linear sub-optimal control could effectively suppress the cable oscillation and the control effectiveness was almost same; under the large initial disturbance, the nonlinear optimal control and linear sub-optimal control was also able to effectively suppress the cable oscillation but compared with the linear sub-optimal control, nonlinear optimal control could reduce the amplitude faster. Then the implementation of nonlinear optimal control was researched. It was established that one-point and two-point nonlinear optimal control of the cable in transverse direction. The results showed that selecting the appropriate control parameters, one-point and two-points nonlinear optimal control could both effectively suppress the self-excited oscillation of cable but the control effectiveness of two-point nonlinear optimal control was better than that of one-point nonlinear optimal control.
Thirdly it was considered that the effects of the cable's longitudinal vibration on the cable's transverse multi-modal vibration. The differential equations of motion for the two-dimensional longitudinal and transverse coupling of cables were established. Then the self-excited oscillation characteristics for the first mode of cable longitudinal vibration and the first two modes of cable transverse vibration were analyzed by the averaging method for nonlinear vibration. The results showed that the amplitude of the longitudinal first mode was proportional to that of the transverse second mode of the cable and, the phase difference between the longitudinal first mode and the transverse second mode of the cable was 180°. Compared with those in the second chapter under the longitudinal vibration simplified, the stability of limit cycles in self-excited oscillation for the first two modes of cable transverse vibration had the same conclusion.
以拉索前两阶模态的耦合的自激振动为对象,研究了斜拉索多模态自激振动的最优控制问题。根据最优控制的动态规划原理确定最优非线性控制力,再应用非线性振动的平均法,求解得到该系统自激振动及其稳定性的分析解,通过分析比较和数值模拟验证控制前后极限环及其稳定性,说明该最优控制能够有效地抑制风激拉索的自激振动。然后比较了非线性最优控制与线性次最优控制控制效果的差异,结果表明：在小初始扰动条件下,非线性最优控制和线性次最优控制均能有效的抑制拉索的振荡,且控制效果基本相同；在大初始扰动条件下,非线性最优控制和线性次最优控制也均能有效的抑制拉索的振荡,但和线性次最优控制相比,非线性最优控制能够更快的减小振幅。接着研究非线性最优控制的实施,建立起拉索横向一点、两点非线性最优控制,结果表明：选取合适的控制参数,拉索横向一点、两点最优控制均能有效抑制拉索的自激振荡,但两点最优控制的效果优于一点最优控制。
考虑拉索的纵向振动对拉索横向多模态振动的影响,建立起拉索面内纵向、横向二维耦合的运动微分方程组,应用非线性振动的平均法,研究了纵向一阶、横向一二阶模态的耦合振动特性,结果表明：稳态响应时,纵向一阶模态振幅与横向二阶模态振幅成正比,相位相差180°,并与第二章简化拉索纵向振动时进行比较,拉索横向一二阶模态耦合振动的极限环的稳定性及其条件具有相同的规律。
The fluid-flow-induced cable vibration is an active research subject in structural engineering and has practical importance for structural improvement and vibration control. This paper focused mainly on the analysis of the self-excited oscillation of an inclined taut cable with multi-vibration modes under wind loading as the cable's cross-section shape changes because of rain or snow. Firstly the nonlinear differential equations of motion were derived for the wind-induced vibration of the cable. Then the partial differential equation for the transverse cable vibration was obtained based on the assumption of longitudinal cable vibration comparatively small. By using the Galerkin approach, this partial differential equation was converted into ordinary differential equations which describe the self-excited oscillation of the cable as multi-modes system. The analytical solutions in self-excited oscillation were obtained further by using the averaging method for nonlinear vibration. According to the stability condition of analytical solutions, the wind-induced limit cycle of self-excited oscillation and its existence conditions and the critical wind velocities for the first two modes and the first, third modes of the cable system in self-oscillation were analytically determined finally and verified by the numerical results. Then the self-oscillation characteristics for the first three modes of the cable system were analyzed by numerical method. The effects of parameters about self-oscillation for the first two modes of the cable were researched at last.
Secondly the cable control for the first two modes in self-excited oscillation was focused on. The performance index for optimal cable control was given to the polynomial control solution. Based on the dynamical programming principle, the Hamilton-Jacobi-Bellman (HJB) equation was established for the cable control with the index. The optimal nonlinear control force was determined from solving the HJB equation. By using the averaging method for nonlinear vibration, the analytical solutions, in particular, limit cycle solutions and stability of the controlled system in self-excited oscillation were obtained further. It was concluded by the analytical and numerical comparison between controlled and uncontrolled self-excited oscillations and stability that the proposed optimal control could effectively suppress the self-excited oscillation of the cable under wind loading. Then the control effectiveness was compared between the nonlinear optimal control and linear sub-optimal control. The results showed that under a small initial disturbance, the nonlinear optimal control and linear sub-optimal control could effectively suppress the cable oscillation and the control effectiveness was almost same; under the large initial disturbance, the nonlinear optimal control and linear sub-optimal control was also able to effectively suppress the cable oscillation but compared with the linear sub-optimal control, nonlinear optimal control could reduce the amplitude faster. Then the implementation of nonlinear optimal control was researched. It was established that one-point and two-point nonlinear optimal control of the cable in transverse direction. The results showed that selecting the appropriate control parameters, one-point and two-points nonlinear optimal control could both effectively suppress the self-excited oscillation of cable but the control effectiveness of two-point nonlinear optimal control was better than that of one-point nonlinear optimal control.
Thirdly it was considered that the effects of the cable's longitudinal vibration on the cable's transverse multi-modal vibration. The differential equations of motion for the two-dimensional longitudinal and transverse coupling of cables were established. Then the self-excited oscillation characteristics for the first mode of cable longitudinal vibration and the first two modes of cable transverse vibration were analyzed by the averaging method for nonlinear vibration. The results showed that the amplitude of the longitudinal first mode was proportional to that of the transverse second mode of the cable and, the phase difference between the longitudinal first mode and the transverse second mode of the cable was 180°. Compared with those in the second chapter under the longitudinal vibration simplified, the stability of limit cycles in self-excited oscillation for the first two modes of cable transverse vibration had the same conclusion.
引文
[1]严国敏.现代斜拉桥.成都：西南交通大学出版社,1996:1-10.
[2]刘延柱,陈立群.非线性振动.北京：高等教育出版社,2001:153-172.
[3]Fujino Y, Warnitchai P. An experimental and analytical study of autoparametrie resonance on a 3 DOF model of cable-stayed-beam. Nonlinear Dynamics,1993,4:111-138.
[4]Warnitchai P, Fujino Y. A non-linear dynamic model for cables and its application to a cable-structure system. Journal of Sound and Vibration,1995,187(4):695-712.
[5]daCosta AP, Martins JAC, Branco F. Oscillations of bridge stay cables induced by periodic motions of deck and/or towers. Journal of Engineering Mechanics,1996, 122(7):613-622.
[6]Ying ZG, Ni YQ, Ko JM. Parametrically excited instability of a cable under two support motions. International Journal of Structural Stability and Dynamics,2006,6(1):43-58.
[7]陈政清.斜拉索风雨振现场观测与振动控制.建筑科学与工程学报,2005,22(4):6-10.
[8]禹见达,陈政清.基于现场观测的拉索风雨振特性研究.湖南科技大学学报,2006,21(2)： 22-24.
[9]顾明,杜晓庆.不同风向角下斜拉桥拉索模型测压试验研究.振动与冲击,2005,24(6)： 6-8.
[10]Matsumoto M. Response characteristics of rain-wind induced vibration of stay-cable of cable-stayed-bridge. Journal of Wind Engineering and Industrial Aerodynamics,1995, 57:323-333.
[11]Matsumoto M. Vortex-induced cable vibration of cable-stayed. Journal of Wind Engineering and Industrial Aerodynamics,2001,89:633-647.
[12]郭应龙.输电导线的舞动.武汉：武汉水利电力大学出版社,1998:13-20.
[13]Cheng SH, Tanaka H. Cable Aerodynamics:The state-of-the-art. The Second Interna-tional Symposium on Wind and Structures,2002,343-352.
[14]Hikami Y, Shiraishi N. Rain-wind induced vibrations of cables in cable-stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1988,29:409-418.
[15]顾明,杜晓庆,李寿英.斜拉桥拉索风雨激振的实验和理论研究.振动工程学报,2007,20(5)：473-478.
[16]Yamaguchi H. Analytical study on growth mechanism of rain vibration of cable. Journal of Wind Engineering and Industrial Aerodynamics,1990,33:73-80.
[17]Bosdogianni A, Olivari D. Wind-and rain-induced oscillations of cables of stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1996,64:171-185.
[18]Peil U, Nahrath N. Modeling of rain-wind induced vibrations Wind & Structures,2003, 6(1):41-52.
[19]Lemaitre C, Alam MM, Hemon Pascal. Rainwater rivulets on a cable subject to wind. Comptes Rendus Mecanique,2006,334(3):158-163.
[20]Macdonalda JHG, Larose GL. Two-degree-of-freedom inclined cable galloping. Journal of Wind Engineering and Industrial Aerodynamics,2008,96:291-307.
[21]vander Burgh AHP, Hartono. Rain-wind-induced vibrations of a simple oscillator. International Journal of Non-Linear Mechanics,2004,39:93-100.
[22]陈锐林,曾庆元.斜拉桥拉索风雨诱导混沌振动.振动与冲击,2008,27(7):42-45.
[23]李伟义,张琪昌,何学军.斜拉索风雨激振面内运动的非线性分析.天津大学学报,2010,43(2)： 156-160.
[24]Xu YL, Wang LY. Analytical study of wind-rain-induced cable vibration:SDOF model. Journal of Wind Engineering and Industrial Aerodynamics,2003,91:27-40.
[25]顾明,刘慈军,徐幼麟等.带人工雨线的拉索在风雨激励下的响应.应用数学和力学,2002,1(23)：1047-1054.
[26]李寿英,陈政清,顾明.斜拉索风雨激振中拉索和水线之间的耦合运动.振动与冲击,2008,27(10)：1-5.
[27]刘庆宽,乔富贵,张峰.斜拉索表面水线多相复杂摆动对索气动稳定性的影响.工程力学,2008,25(6)：234-240.
[28]李寿英,顾明,陈政清.准运动水线三维连续弹性拉索风雨激振理论模型.工程力学,2007,24(6)：7-14.
[29]李暾,陈政清,李寿英.拉索表面材料对风雨激振的影响研究.第十三届全国结构 风工程学术会议论文集,2007:746-751.
[30]张琪昌,李伟义,王炜.斜拉索风雨振的动力学行为研究.振动与冲击,2010,29(4):173-176.
[31]Flamand O. Rain-wind induced vibration of cables. Journal of Wind Engineering and Industrial Aerodynamics,1995,57:353-362.
[32]陈昭晖.斜拉索参激不稳定性的主动与半主动控制[硕士学位论文].杭州：浙江大学,2006:4-5.
[33]Nakamura A, Kasuga A, Arai H. The effects of mechanical dampers on stay cables with high-damping rubber. Construction and Building Materials,1998,12(3):115-123.
[34]Takano H, Ogssawara M. Vibrational damper for cables of the Tsurumi Tsubasa Bridge, Journal of Wind Engineering and Industrial Aerodynamics,1997,69:807-818.
[35]Jensen CN, Nielsen SRK. Optimal Damping of Stays in Cable-Stayed Bridges for in-Plane Vibrations. Journal of Sound and Vibration 2002,256(3):499-513.
[36]Kobayashi H, Minami Y, Miki M. Prevention of rain-wind induced vibration of inclined cable by surface processing. In Proceeding 9th International Conference on Wind Engineering,1995:753-758.
[37]Matsumoto M, Shirashi N. Aerodynamic behavior of inclined circular cylinders-cable aerodynamics. Journal of Wind Engineering and Industrial Aerodynamics,1990: 3363-72.
[38]Yamaguchi H, Fujino Y. Stayed cable dynamics and its vibration control. Bridge Aerodynamic,1998:235-253.
[39]汪至刚,孙炳楠.斜拉桥参数振动引起的拉索大幅振动.工程力学,2001,18(1):103-109.
[40]Hogsberg J, Krenk S. Adaptive tuning of elasto-plastic damper. International Journal of Non-Linear Mechanics,2007,42:928-940.
[41]Weber F, Hogsberg Jan, Krenk Steen. Optimal Tuning of Amplitude Proportional Coulomb Friction Damper for Maximum Cable Damping. Journal of Structural Engineering-ASCE,2010,136(2):123-129.
[42]Cheng SH, Darivandi N, Ghrib F. The design of an optimal viscous damper for a bridge stay cable using energy-based approach. Journal of Sound and Vibration,2010,329: 4689-4704.
[43]Hwang I, Lee JS, Spencer BF. Isolation System for Vibration Control of Stay Cables. Journal of Structural Engineering-ASCE,2009,135(1):62-65.
[44]Winslow WW. Method and means for translating electrical impulse into mechanical force: US, No 2417850.1947.
[45]路子阳.磁流变阻尼器力学性能及参数辨识研究[硕士学位论文].南京：南京理工大学,2009:1-5.
[46]常银昌.磁流变阻尼器的桥梁振动控制[硕士学位论文].武汉：武汉理工大学,2006：4-6.
[47]左晓宝,李爱群,孙香花.SMA阻尼器-斜拉索系统减振控制试验.南京理工大学学报：自然科学版,2008,32(6)：666-671.
[48]Stanway R, Spreston JL, Stevens NG. Non-linear modeling of an elcctro-rheological vibration damper. Journal of Electrostatics,1987,20(2):167-184.
[49]Gamota DR, Filisko FE. Dynamic mechanical studies of electro-rheological materials: moderate frequencies. Journal of Rheology,1991,35:399-425.
[50]Wen YK. Method of random vibration of hysteretic systems. Journal of Engineering Mechanics-ASCE,1976,102(2):249-263.
[51]Spencer BF, Dyke SJ, Sain Mk, et al. Phenomenological model for magneto-rheological dampers. Journal of Engineering Mechanics-ASCE,1997,123(3):230-238.
[52]Dyke SJ, Spencer BF, Saln MK. et al. Modeling and control of magneto-rhcological dampers for seismic response reduction. Smart Materials and Structures,1996,5: 565-575.
[53]Christenson Richard E, Spencer B F. Experimental Verification of Smart Cable Damp-ing. Journal of Engineering Mechanics-ASCE,2006,132(3):268-277.
[54]Inaudi JA. Modulated homogeneous friction:a semi-active damping strategy. Earth-quake Engineering and Structural Dynamics.1997,26:361-376.
[55]Stammers CW, Sireteanu T. Vibration control of machines by use of semi-active dry friction damping. Journal of Sound and Vibration,1998,209:671-684.
[56]陈勇,孙炳楠,楼文娟.基于时间离散模型的斜拉索半主动振动控制.浙江大学学报：工学版,2003,37(3)：332-336.
[57]陈水生.斜拉索振动的模糊半主动控制研究.计算力学学报,2006,23(6):734-742
[58]Ni YQ, Chen Y, Ko JM, et al. Neuro-control of cable vibration using semi-active magneto-rheological dampers. Engineering Structures,2002,24:295-307.
[59]王修勇,陈政清,倪一清.斜拉索-磁流变阻尼器系统半主动控制的神经网络法.振动与冲击,2003,22(4)：49-53.
[60]Boston C, Weber F, Guzzella L. Optimal semi-active damping of cables:evolutionary algorithms and closed-form solutions. Smart Mater Struct,2009,18:1-9.
[62]邬喆华,楼文娟,陈勇.采用磁流变阻尼器的斜拉索半主动控制.同济大学学报：自然科学版,2006,34(10)：1310-1314.
[63]禹见达,陈政清,王修勇,等.基于遗传算法的磁流变阻尼器模型参数识别.地震工程与工程振动,2009,29(1)：136-138.
[64]Ying ZG, Zhu WQ, Scong TT. A stochastic optimal semi-active control strategy for ER/MR dampers Journal of Sound and Vibration,2003,259(1):45-62.
[65]Le Diouron T, Fujino Y, et al. Control of Wind-Induced Self-Excited Oscillations by Transfer of Internal Energy to Higher Modes of Vibration II:Application to Taut Cables, Journal of Engineering Mechanics-ASCE,2003,129(5):526-536.
[65]王代华,黄尚廉.斜拉桥横向振动的半主动筋腱控制.应用力学学报,2000,17(2):13-17.
[66]陈政清,王修勇,胡建华,等.洞庭湖大桥拉索振动控制工程.第十五届全国桥梁学术会议论文集,2002:476-480.
[67]禹见达,陈政清,王修勇,等.磁流变阻尼器与拉索振动控制研究.功能材料,2006,37(5)：823-826.
[68]Chen JC. Response of large space structures with stiffness control. Journal of Spacecraft and Rockets,1984,21(5):463-467.
[69]Susumpow T, Fujino Y. Active control of multi modal cable vibrations by axial support motion. Smart Material and Structures,1995,121(9):964-970.
[70]Achkire Y, Preumont A. Active Tendon Control of Cable-stayed Bridges. Earthquake Engineering and Structural Dynamics,1996,25:585-597.
[71]王代华.斜拉索横向振动的积分应变反馈控制实验研究.实验力学,1999,14(4):464-474.
[72]陈昭晖,应祖光.斜拉索不稳定振动的主动及半主动控制.噪声与振动控制,2006,(3)： 5-8.
[73]陈昭晖.斜拉索参激不稳定性的主动与半主动控制[硕士学位论文].杭州：浙江大学,2006:42-49.
[74]李寿英,顾明,陈政清.运动水线三维连续弹性拉索风雨激振理论模型.湖南大学学报：自然科学版,2009,36(2)：1-6.
[75]Irvine H M. Cable structures. Cambridge:MIT Press,1981.
[76]应祖光.材料力学中索的张力与变形.力学与实践,2004,26(5):68-70.
[78]金林,应祖光,罗银淼.斜拉索的不稳定振动及控制.噪声与振动控制,2004,(5)：5-7.
[78]Den Hartog JP. Mechanical Vibrations. New York:McGraw-Hill Book Company,1956.
[79]van der Burgh AHP, Hartonoa, Abramian AK. A new model for the study of rain-wind-induced vibrations of a simple oscillator. International Journal of Non-Linear Mechanics,2006,41:345-358.
[80]Nayfeh A H, Mook D T非线性振动.下册.宋家骗,罗惟德,陈守吉,译.北京：高等教育出版社,1990:96-105.
[81]Nayfeh A H, Mook D T.非线性振动.上册.宋家骕,罗惟德,陈守吉,译.北京：高等教育出版社,1990:133-134.
[82]何学军,张琪昌,田瑞兰.索结构风雨振的动力学行为研究.工程力学,2008,25(10):1-5.
[83]Agrawal AK, Yang JN, Wu JC. Application of Optimal Polynomial Controller to A Benchmark Problem. Earthquake Engng.Struct.Dyn.,1998,27:1291-1302.
[84]张明达.最优控制工程.中南工业大学出版社：第一版,1986:258-300.
[85]Yamaguchi H, Dung NN. Active wave control of sagged-cable vibration. Proceedings of 1th International Conference on Motion and Vibration Control,1992,134-139.
[86]Hagedorn P. Active vibration damping in large flexible structures. Proceedings of 17thInternational Congress on Theoretical and Applied Mechanics, Oreooble, France, 1988:83-100.
[87]Tan CA, Ying S. Active wave control of the axially moving string:theory and experiment. Journal of Sound and Vibration,2000,236(5):861-880.
[88]陈予恕.非线性振动.天津：天津科学技术出版社,1983:117-124.
[2]刘延柱,陈立群.非线性振动.北京：高等教育出版社,2001:153-172.
[3]Fujino Y, Warnitchai P. An experimental and analytical study of autoparametrie resonance on a 3 DOF model of cable-stayed-beam. Nonlinear Dynamics,1993,4:111-138.
[4]Warnitchai P, Fujino Y. A non-linear dynamic model for cables and its application to a cable-structure system. Journal of Sound and Vibration,1995,187(4):695-712.
[5]daCosta AP, Martins JAC, Branco F. Oscillations of bridge stay cables induced by periodic motions of deck and/or towers. Journal of Engineering Mechanics,1996, 122(7):613-622.
[6]Ying ZG, Ni YQ, Ko JM. Parametrically excited instability of a cable under two support motions. International Journal of Structural Stability and Dynamics,2006,6(1):43-58.
[7]陈政清.斜拉索风雨振现场观测与振动控制.建筑科学与工程学报,2005,22(4):6-10.
[8]禹见达,陈政清.基于现场观测的拉索风雨振特性研究.湖南科技大学学报,2006,21(2)： 22-24.
[9]顾明,杜晓庆.不同风向角下斜拉桥拉索模型测压试验研究.振动与冲击,2005,24(6)： 6-8.
[10]Matsumoto M. Response characteristics of rain-wind induced vibration of stay-cable of cable-stayed-bridge. Journal of Wind Engineering and Industrial Aerodynamics,1995, 57:323-333.
[11]Matsumoto M. Vortex-induced cable vibration of cable-stayed. Journal of Wind Engineering and Industrial Aerodynamics,2001,89:633-647.
[12]郭应龙.输电导线的舞动.武汉：武汉水利电力大学出版社,1998:13-20.
[13]Cheng SH, Tanaka H. Cable Aerodynamics:The state-of-the-art. The Second Interna-tional Symposium on Wind and Structures,2002,343-352.
[14]Hikami Y, Shiraishi N. Rain-wind induced vibrations of cables in cable-stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1988,29:409-418.
[15]顾明,杜晓庆,李寿英.斜拉桥拉索风雨激振的实验和理论研究.振动工程学报,2007,20(5)：473-478.
[16]Yamaguchi H. Analytical study on growth mechanism of rain vibration of cable. Journal of Wind Engineering and Industrial Aerodynamics,1990,33:73-80.
[17]Bosdogianni A, Olivari D. Wind-and rain-induced oscillations of cables of stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1996,64:171-185.
[18]Peil U, Nahrath N. Modeling of rain-wind induced vibrations Wind & Structures,2003, 6(1):41-52.
[19]Lemaitre C, Alam MM, Hemon Pascal. Rainwater rivulets on a cable subject to wind. Comptes Rendus Mecanique,2006,334(3):158-163.
[20]Macdonalda JHG, Larose GL. Two-degree-of-freedom inclined cable galloping. Journal of Wind Engineering and Industrial Aerodynamics,2008,96:291-307.
[21]vander Burgh AHP, Hartono. Rain-wind-induced vibrations of a simple oscillator. International Journal of Non-Linear Mechanics,2004,39:93-100.
[22]陈锐林,曾庆元.斜拉桥拉索风雨诱导混沌振动.振动与冲击,2008,27(7):42-45.
[23]李伟义,张琪昌,何学军.斜拉索风雨激振面内运动的非线性分析.天津大学学报,2010,43(2)： 156-160.
[24]Xu YL, Wang LY. Analytical study of wind-rain-induced cable vibration:SDOF model. Journal of Wind Engineering and Industrial Aerodynamics,2003,91:27-40.
[25]顾明,刘慈军,徐幼麟等.带人工雨线的拉索在风雨激励下的响应.应用数学和力学,2002,1(23)：1047-1054.
[26]李寿英,陈政清,顾明.斜拉索风雨激振中拉索和水线之间的耦合运动.振动与冲击,2008,27(10)：1-5.
[27]刘庆宽,乔富贵,张峰.斜拉索表面水线多相复杂摆动对索气动稳定性的影响.工程力学,2008,25(6)：234-240.
[28]李寿英,顾明,陈政清.准运动水线三维连续弹性拉索风雨激振理论模型.工程力学,2007,24(6)：7-14.
[29]李暾,陈政清,李寿英.拉索表面材料对风雨激振的影响研究.第十三届全国结构 风工程学术会议论文集,2007:746-751.
[30]张琪昌,李伟义,王炜.斜拉索风雨振的动力学行为研究.振动与冲击,2010,29(4):173-176.
[31]Flamand O. Rain-wind induced vibration of cables. Journal of Wind Engineering and Industrial Aerodynamics,1995,57:353-362.
[32]陈昭晖.斜拉索参激不稳定性的主动与半主动控制[硕士学位论文].杭州：浙江大学,2006:4-5.
[33]Nakamura A, Kasuga A, Arai H. The effects of mechanical dampers on stay cables with high-damping rubber. Construction and Building Materials,1998,12(3):115-123.
[34]Takano H, Ogssawara M. Vibrational damper for cables of the Tsurumi Tsubasa Bridge, Journal of Wind Engineering and Industrial Aerodynamics,1997,69:807-818.
[35]Jensen CN, Nielsen SRK. Optimal Damping of Stays in Cable-Stayed Bridges for in-Plane Vibrations. Journal of Sound and Vibration 2002,256(3):499-513.
[36]Kobayashi H, Minami Y, Miki M. Prevention of rain-wind induced vibration of inclined cable by surface processing. In Proceeding 9th International Conference on Wind Engineering,1995:753-758.
[37]Matsumoto M, Shirashi N. Aerodynamic behavior of inclined circular cylinders-cable aerodynamics. Journal of Wind Engineering and Industrial Aerodynamics,1990: 3363-72.
[38]Yamaguchi H, Fujino Y. Stayed cable dynamics and its vibration control. Bridge Aerodynamic,1998:235-253.
[39]汪至刚,孙炳楠.斜拉桥参数振动引起的拉索大幅振动.工程力学,2001,18(1):103-109.
[40]Hogsberg J, Krenk S. Adaptive tuning of elasto-plastic damper. International Journal of Non-Linear Mechanics,2007,42:928-940.
[41]Weber F, Hogsberg Jan, Krenk Steen. Optimal Tuning of Amplitude Proportional Coulomb Friction Damper for Maximum Cable Damping. Journal of Structural Engineering-ASCE,2010,136(2):123-129.
[42]Cheng SH, Darivandi N, Ghrib F. The design of an optimal viscous damper for a bridge stay cable using energy-based approach. Journal of Sound and Vibration,2010,329: 4689-4704.
[43]Hwang I, Lee JS, Spencer BF. Isolation System for Vibration Control of Stay Cables. Journal of Structural Engineering-ASCE,2009,135(1):62-65.
[44]Winslow WW. Method and means for translating electrical impulse into mechanical force: US, No 2417850.1947.
[45]路子阳.磁流变阻尼器力学性能及参数辨识研究[硕士学位论文].南京：南京理工大学,2009:1-5.
[46]常银昌.磁流变阻尼器的桥梁振动控制[硕士学位论文].武汉：武汉理工大学,2006：4-6.
[47]左晓宝,李爱群,孙香花.SMA阻尼器-斜拉索系统减振控制试验.南京理工大学学报：自然科学版,2008,32(6)：666-671.
[48]Stanway R, Spreston JL, Stevens NG. Non-linear modeling of an elcctro-rheological vibration damper. Journal of Electrostatics,1987,20(2):167-184.
[49]Gamota DR, Filisko FE. Dynamic mechanical studies of electro-rheological materials: moderate frequencies. Journal of Rheology,1991,35:399-425.
[50]Wen YK. Method of random vibration of hysteretic systems. Journal of Engineering Mechanics-ASCE,1976,102(2):249-263.
[51]Spencer BF, Dyke SJ, Sain Mk, et al. Phenomenological model for magneto-rheological dampers. Journal of Engineering Mechanics-ASCE,1997,123(3):230-238.
[52]Dyke SJ, Spencer BF, Saln MK. et al. Modeling and control of magneto-rhcological dampers for seismic response reduction. Smart Materials and Structures,1996,5: 565-575.
[53]Christenson Richard E, Spencer B F. Experimental Verification of Smart Cable Damp-ing. Journal of Engineering Mechanics-ASCE,2006,132(3):268-277.
[54]Inaudi JA. Modulated homogeneous friction:a semi-active damping strategy. Earth-quake Engineering and Structural Dynamics.1997,26:361-376.
[55]Stammers CW, Sireteanu T. Vibration control of machines by use of semi-active dry friction damping. Journal of Sound and Vibration,1998,209:671-684.
[56]陈勇,孙炳楠,楼文娟.基于时间离散模型的斜拉索半主动振动控制.浙江大学学报：工学版,2003,37(3)：332-336.
[57]陈水生.斜拉索振动的模糊半主动控制研究.计算力学学报,2006,23(6):734-742
[58]Ni YQ, Chen Y, Ko JM, et al. Neuro-control of cable vibration using semi-active magneto-rheological dampers. Engineering Structures,2002,24:295-307.
[59]王修勇,陈政清,倪一清.斜拉索-磁流变阻尼器系统半主动控制的神经网络法.振动与冲击,2003,22(4)：49-53.
[60]Boston C, Weber F, Guzzella L. Optimal semi-active damping of cables:evolutionary algorithms and closed-form solutions. Smart Mater Struct,2009,18:1-9.
[62]邬喆华,楼文娟,陈勇.采用磁流变阻尼器的斜拉索半主动控制.同济大学学报：自然科学版,2006,34(10)：1310-1314.
[63]禹见达,陈政清,王修勇,等.基于遗传算法的磁流变阻尼器模型参数识别.地震工程与工程振动,2009,29(1)：136-138.
[64]Ying ZG, Zhu WQ, Scong TT. A stochastic optimal semi-active control strategy for ER/MR dampers Journal of Sound and Vibration,2003,259(1):45-62.
[65]Le Diouron T, Fujino Y, et al. Control of Wind-Induced Self-Excited Oscillations by Transfer of Internal Energy to Higher Modes of Vibration II:Application to Taut Cables, Journal of Engineering Mechanics-ASCE,2003,129(5):526-536.
[65]王代华,黄尚廉.斜拉桥横向振动的半主动筋腱控制.应用力学学报,2000,17(2):13-17.
[66]陈政清,王修勇,胡建华,等.洞庭湖大桥拉索振动控制工程.第十五届全国桥梁学术会议论文集,2002:476-480.
[67]禹见达,陈政清,王修勇,等.磁流变阻尼器与拉索振动控制研究.功能材料,2006,37(5)：823-826.
[68]Chen JC. Response of large space structures with stiffness control. Journal of Spacecraft and Rockets,1984,21(5):463-467.
[69]Susumpow T, Fujino Y. Active control of multi modal cable vibrations by axial support motion. Smart Material and Structures,1995,121(9):964-970.
[70]Achkire Y, Preumont A. Active Tendon Control of Cable-stayed Bridges. Earthquake Engineering and Structural Dynamics,1996,25:585-597.
[71]王代华.斜拉索横向振动的积分应变反馈控制实验研究.实验力学,1999,14(4):464-474.
[72]陈昭晖,应祖光.斜拉索不稳定振动的主动及半主动控制.噪声与振动控制,2006,(3)： 5-8.
[73]陈昭晖.斜拉索参激不稳定性的主动与半主动控制[硕士学位论文].杭州：浙江大学,2006:42-49.
[74]李寿英,顾明,陈政清.运动水线三维连续弹性拉索风雨激振理论模型.湖南大学学报：自然科学版,2009,36(2)：1-6.
[75]Irvine H M. Cable structures. Cambridge:MIT Press,1981.
[76]应祖光.材料力学中索的张力与变形.力学与实践,2004,26(5):68-70.
[78]金林,应祖光,罗银淼.斜拉索的不稳定振动及控制.噪声与振动控制,2004,(5)：5-7.
[78]Den Hartog JP. Mechanical Vibrations. New York:McGraw-Hill Book Company,1956.
[79]van der Burgh AHP, Hartonoa, Abramian AK. A new model for the study of rain-wind-induced vibrations of a simple oscillator. International Journal of Non-Linear Mechanics,2006,41:345-358.
[80]Nayfeh A H, Mook D T非线性振动.下册.宋家骗,罗惟德,陈守吉,译.北京：高等教育出版社,1990:96-105.
[81]Nayfeh A H, Mook D T.非线性振动.上册.宋家骕,罗惟德,陈守吉,译.北京：高等教育出版社,1990:133-134.
[82]何学军,张琪昌,田瑞兰.索结构风雨振的动力学行为研究.工程力学,2008,25(10):1-5.
[83]Agrawal AK, Yang JN, Wu JC. Application of Optimal Polynomial Controller to A Benchmark Problem. Earthquake Engng.Struct.Dyn.,1998,27:1291-1302.
[84]张明达.最优控制工程.中南工业大学出版社：第一版,1986:258-300.
[85]Yamaguchi H, Dung NN. Active wave control of sagged-cable vibration. Proceedings of 1th International Conference on Motion and Vibration Control,1992,134-139.
[86]Hagedorn P. Active vibration damping in large flexible structures. Proceedings of 17thInternational Congress on Theoretical and Applied Mechanics, Oreooble, France, 1988:83-100.
[87]Tan CA, Ying S. Active wave control of the axially moving string:theory and experiment. Journal of Sound and Vibration,2000,236(5):861-880.
[88]陈予恕.非线性振动.天津：天津科学技术出版社,1983:117-124.