大跨度桥梁斜拉索的参数振动及索力识别研究
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摘要
随着斜拉桥跨度的不断增大,斜拉索变得越来越长(苏通大桥最长拉索已达到580m),因此为保证整个桥梁结构的安全性、耐久性以及延长桥梁结构的使用寿命,拉索的振动问题越来越引起各国学者的高度重视。本文针对可能引起拉索大幅振动的参数振动问题,以理论分析、模型试验和实际工程应用相结合的方法,深入系统的研究了大跨度桥梁斜拉索的参数振动及索力测识别问题。本文的研究内容具体包括以下几个部分
首先在不考虑拉索倾角的情况下,分别建立了水平拉索的索-桥非耦合以及索-桥耦合两种参数振动模型。采用Galerkin法对两种参数振动模型的微分方程进行解耦并考虑前两阶模态组合的影响,通过多尺度法对振动方程近似求解分析,讨论模型的内共振响应特性,并利用数值方法模拟两种模型下,拉索在各共振频点处发生内共振的位移响应特性及索力时程变化规律。研究表明:拉索的第一阶模态在内共振频点2:1和1:1处发生大幅“拍”振,而第二阶模态则仅在2:1处发生大幅“拍”振,理想激励下的索-桥非耦合模型,由于端部有外界能量的不断输入,使其较索-桥耦合模型的位移响应较大,且衰减较慢。
在考虑拉索倾角影响的情况下,建立了斜拉索在理想激励下的索-桥非耦合模型,以及在非理想激励下的索-桥耦合和索-桥-塔耦合的参数振动模型,然后对三种模型振动微分方程做无量纲化推导,并利用多尺度法进行近似求解分析,最后结合数值模拟方法对耦合方程组进行数值求解。讨论了拉索发生参数振动的起振时间,对比不同长度拉索发生参数振动的频比范围,并进行参数分析。重点针对索-桥-塔耦合模型进行讨论分析,在三者共同参与耦合振动的情况下,当桥面与拉索的频率满足参数振动频比关系2:1时,即系统发生1:2:1或2:2:1内共振时,振动在持续一段时间后将发散;而当系统发生1:1:1或2:1:1内共振时,拉索、桥塔和桥面位移同时出现耦合“拍”振的情况,且桥面和桥塔的振动位移与远大于初始位移条件,系统各部分都被完全激励起来。
除了将桥面激励简化为理想激励和弹簧-质量块耦合系统进行参数振动研究外,建立斜拉桥索-梁组合结构的非线性振动模型,进一步研究索-梁耦合模型下斜拉索的参数振动问题。同样利用多尺度法对无量纲形式的耦合振动微分方程组进行近似求解分析,并探讨该模型的内共振响应特性,最后结合数值模拟方法,讨论索-梁耦合模型在各内共振频比点下的动力响应规律。研究表明:在梁初始位移条件较大的情况下,当满足ωb = 2和ωb = 1两种频比关系时,拉索都发生大幅“拍”振,但在梁初始位移较小的情况下,当满足ωb =2的频比关系时,小幅激励仍然引起拉索的大幅“拍”,由此说明即便是桥面梁的微幅振动,也将引起斜拉索的大幅振动。
在对考虑索-梁组合结构的参数振动模型研究的基础上,进行支座激励下斜拉索的参数问题研究。支座的位移激励方向与斜拉索轴向成一定的夹角,有别于理想轴向激励和将桥面简化为弹簧-质量块的参数振动问题。建立了斜拉索在端部支座激励下的非线性振动模型,利用多尺度法对无量纲形式的振动微分方程进行近似求解,通过讨论可知,支座激励频率除了单独或同时与斜拉索频率满足内共振频比关系时将引起拉索的大幅振动外,当支座激励频率的组合满足|ω|-_B±ω|-_A|= 0、|ω|-_B±ω|-_A|= 1和|ω|-_B±ω|-_A|= 2关系时,也将引起拉索的大幅振动,尤其是满足参数共振时的共振响应更大。
为了进一步认识支座激励下斜拉索的参数振动问题,进行了室内斜拉索模型在支座激励下的参数振动试验研究。利用作动器模拟桥面,给斜拉索端部施加谐波激励,对内共振的频比关系2:1、1:1及1:2进行不同激励幅值下的多工况试验分析。由分析结果可知,由于参数振动频比关系2:1引起的拉索振动幅值较大,而拉索在振动过程中受端部刚性连接件及面外振动的影响较大,故试验分析与数值模拟结果的幅值一致,而试验索“拍”振时间间隔较大,且衰减较快,但1:1及1:2两种频比关系在小激励幅值下,试验分析和数值模拟结果吻合较好。综合试验与数值对比结果可知,该试验很好的验证了斜拉索在支座激励下发生大幅参数振动现象的这一理论。
最后针对现场索力识别的精度问题,以实际工程的现场试验为研究背景,主要针对索力识别方法、索力识别精度等问题进行研究,并结合得到的部分研究成果,开发高精度的索力测试仪器。
Stay cables become longer and longer as the span of cable stayed bridge gets larger and larger (The longest cable of the Suzou-Nantong bridge is 580m). To guarantee safety, durability of bridges and extend useful time of bridge, vibration of cables is paid more attention to by researchers. According to parametric vibration which probably causes large amplitude vibration, parametric vibration and cable force identification of large span cable stayed bridge are studied systematically by theoretical analysis, experiment and project application. The research contents are as follows.
Firstly cable - bridge coupled and uncoupled parametric vibration models of lateral cable are built without regard to cable angle. Using Galerkin method, the differential equations of two kinds of parametric vibration models are decoupled considering the first two orders of modal combinations. Then the vibration equation is solved approximately and internal resonant response of the model is discussed via multi-scale method, and The rule of internal resonant response and cable force history is studied by numerical simulation. The results have shown that substantial 'beat' vibration of cable first-order mode occurs at internal resonant point 2:1and 1:1 and substantial 'beat' vibration of cable second-order mode occurs at internal resonant point 2:1. Because of the external constant input of energy at ends, displacement response of cable - bridge non-coupled model under ideal excitation is larger than that of cable - bridge coupled model, and attenuates more slowly.
Cable-bridge uncoupled parametric vibration models of stayed cable under ideal excitation, as well as cable-bridge and cable-bridge-tower coupled parametric vibration models of stayed cable under non-ideal excitation are built in view of cable angle. Vibration differential equations of the three models are derived by dimensionless method, and the vibration equation is solved approximately by multi-scale method. At last, coupled equations are solved numerically by numerical simulation method. Start-up time of parametric vibration is discussed. The ranges of parametric vibration frequency ratio of different length cable are compared and parametric analysis is processed. Under the state of cable-bridge-tower coupled vibration, when the ratio of bridge frequency and cable frequency is 2:1, namely 1:2:1 or 2:2:1 internal resonant occur, vibration diverges after period of time. But while 1:1:1 or 2:1:1 internal resonant occurs, coupled beat vibration of displacement of cable, tower and bridge, as well as vibration displacement of bridge and tower is much larger than initial displacement. Each part of system is excited completely.
Besides parametric vibration is studied by simplifying bridge excitation as ideal excitation and spring-mass coupled system, nonlinear vibration model of stayed cable-beam composite structure is set up. Parametric vibration of stayed cable of cable-beam coupled model is studied. And coupled vibration differential equations in dimensionless style are solved approximately by multi-scale method. Afterwards property of internal resonant response of the model is discussed. Finally the rule of dynamic response of cable-beam coupled model at different internal resonant points is discussed by numerical simulation. The achievements have shown that under the state of large initial displacement of beam, whenωb equals 2 or 1, substantial 'beat' vibration of cable occurs. But under the state of small initial displacement of beam, whenωb is 2, substantial 'beat' vibration of cable occurs due to small excitation. It is concluded that substantial vibration of cable can be caused by small vibration of beam of bridge deck.
Parametric vibration of stayed cable under bearing excitation is studied, based on the research of parametric vibration model considering cable-beam composite structure. There is an angle between the direction of bearing excitation and axial direction of stayed cable, which is different from the method which simplifies bridge deck as spring-mass under ideal axial excitation. Moreover nonlinear vibration model of stayed cable under bearing excitation is built, and vibration differential equations in dimensionless style are solved approximately by means of multi-scale method. It is discussed that while ratio of excitation frequency of bearing and stayed cable frequency satisfies internal resonant frequency ratio, substantial vibration of cable occurs. And when the combination of excitation frequency of bearing satisfy |ω|-_B±ω|-_A|= 0, |ω|-_B±ω|-_A|= 1 or |ω|-_B±ω|-_A|= 2, substantial vibration of cable occurs similarly. In particular, while meeting the parametric resonance, resonance response gets bigger.
To make clear parametric vibration of stayed cable under bearing excitation, the indoor test is processed. According to actuator imitating bridge deck and exciting harmonic wave on the tip of stayed cable, multi-condition tests at the varying internal resonance frequency ratios 2:1, 1:1, 1:2 under different amplitude are carried out. It can be seen that vibration amplitude is larger as the frequency ratio is 2:1, and cable vibration is influenced greatly by rigid link component at the end and out-of-plan vibration. Thus the amplitude of experiment is identical with that of numerical simulation, and time interval of“beat’vibration of experimental cable is larger and attenuates faster. When the frequency ratios are 1:1 and 1:2, the results of experiment and numerical simulation are in good agreement. It can be shown that parametric vibration of stayed cable appearing under bearing excitation is verified by this experiment according to the comparison of the outcomes of experiment and numerical simulation.
At last, in view of the precision of cable force identification on the spot, the method and the precision of cable force identification are investigated systematically. Afterwards the high-precision cable force tester is developed via the partial above-mentioned achievements.
首先在不考虑拉索倾角的情况下,分别建立了水平拉索的索-桥非耦合以及索-桥耦合两种参数振动模型。采用Galerkin法对两种参数振动模型的微分方程进行解耦并考虑前两阶模态组合的影响,通过多尺度法对振动方程近似求解分析,讨论模型的内共振响应特性,并利用数值方法模拟两种模型下,拉索在各共振频点处发生内共振的位移响应特性及索力时程变化规律。研究表明:拉索的第一阶模态在内共振频点2:1和1:1处发生大幅“拍”振,而第二阶模态则仅在2:1处发生大幅“拍”振,理想激励下的索-桥非耦合模型,由于端部有外界能量的不断输入,使其较索-桥耦合模型的位移响应较大,且衰减较慢。
在考虑拉索倾角影响的情况下,建立了斜拉索在理想激励下的索-桥非耦合模型,以及在非理想激励下的索-桥耦合和索-桥-塔耦合的参数振动模型,然后对三种模型振动微分方程做无量纲化推导,并利用多尺度法进行近似求解分析,最后结合数值模拟方法对耦合方程组进行数值求解。讨论了拉索发生参数振动的起振时间,对比不同长度拉索发生参数振动的频比范围,并进行参数分析。重点针对索-桥-塔耦合模型进行讨论分析,在三者共同参与耦合振动的情况下,当桥面与拉索的频率满足参数振动频比关系2:1时,即系统发生1:2:1或2:2:1内共振时,振动在持续一段时间后将发散;而当系统发生1:1:1或2:1:1内共振时,拉索、桥塔和桥面位移同时出现耦合“拍”振的情况,且桥面和桥塔的振动位移与远大于初始位移条件,系统各部分都被完全激励起来。
除了将桥面激励简化为理想激励和弹簧-质量块耦合系统进行参数振动研究外,建立斜拉桥索-梁组合结构的非线性振动模型,进一步研究索-梁耦合模型下斜拉索的参数振动问题。同样利用多尺度法对无量纲形式的耦合振动微分方程组进行近似求解分析,并探讨该模型的内共振响应特性,最后结合数值模拟方法,讨论索-梁耦合模型在各内共振频比点下的动力响应规律。研究表明:在梁初始位移条件较大的情况下,当满足ωb = 2和ωb = 1两种频比关系时,拉索都发生大幅“拍”振,但在梁初始位移较小的情况下,当满足ωb =2的频比关系时,小幅激励仍然引起拉索的大幅“拍”,由此说明即便是桥面梁的微幅振动,也将引起斜拉索的大幅振动。
在对考虑索-梁组合结构的参数振动模型研究的基础上,进行支座激励下斜拉索的参数问题研究。支座的位移激励方向与斜拉索轴向成一定的夹角,有别于理想轴向激励和将桥面简化为弹簧-质量块的参数振动问题。建立了斜拉索在端部支座激励下的非线性振动模型,利用多尺度法对无量纲形式的振动微分方程进行近似求解,通过讨论可知,支座激励频率除了单独或同时与斜拉索频率满足内共振频比关系时将引起拉索的大幅振动外,当支座激励频率的组合满足|ω|-_B±ω|-_A|= 0、|ω|-_B±ω|-_A|= 1和|ω|-_B±ω|-_A|= 2关系时,也将引起拉索的大幅振动,尤其是满足参数共振时的共振响应更大。
为了进一步认识支座激励下斜拉索的参数振动问题,进行了室内斜拉索模型在支座激励下的参数振动试验研究。利用作动器模拟桥面,给斜拉索端部施加谐波激励,对内共振的频比关系2:1、1:1及1:2进行不同激励幅值下的多工况试验分析。由分析结果可知,由于参数振动频比关系2:1引起的拉索振动幅值较大,而拉索在振动过程中受端部刚性连接件及面外振动的影响较大,故试验分析与数值模拟结果的幅值一致,而试验索“拍”振时间间隔较大,且衰减较快,但1:1及1:2两种频比关系在小激励幅值下,试验分析和数值模拟结果吻合较好。综合试验与数值对比结果可知,该试验很好的验证了斜拉索在支座激励下发生大幅参数振动现象的这一理论。
最后针对现场索力识别的精度问题,以实际工程的现场试验为研究背景,主要针对索力识别方法、索力识别精度等问题进行研究,并结合得到的部分研究成果,开发高精度的索力测试仪器。
Stay cables become longer and longer as the span of cable stayed bridge gets larger and larger (The longest cable of the Suzou-Nantong bridge is 580m). To guarantee safety, durability of bridges and extend useful time of bridge, vibration of cables is paid more attention to by researchers. According to parametric vibration which probably causes large amplitude vibration, parametric vibration and cable force identification of large span cable stayed bridge are studied systematically by theoretical analysis, experiment and project application. The research contents are as follows.
Firstly cable - bridge coupled and uncoupled parametric vibration models of lateral cable are built without regard to cable angle. Using Galerkin method, the differential equations of two kinds of parametric vibration models are decoupled considering the first two orders of modal combinations. Then the vibration equation is solved approximately and internal resonant response of the model is discussed via multi-scale method, and The rule of internal resonant response and cable force history is studied by numerical simulation. The results have shown that substantial 'beat' vibration of cable first-order mode occurs at internal resonant point 2:1and 1:1 and substantial 'beat' vibration of cable second-order mode occurs at internal resonant point 2:1. Because of the external constant input of energy at ends, displacement response of cable - bridge non-coupled model under ideal excitation is larger than that of cable - bridge coupled model, and attenuates more slowly.
Cable-bridge uncoupled parametric vibration models of stayed cable under ideal excitation, as well as cable-bridge and cable-bridge-tower coupled parametric vibration models of stayed cable under non-ideal excitation are built in view of cable angle. Vibration differential equations of the three models are derived by dimensionless method, and the vibration equation is solved approximately by multi-scale method. At last, coupled equations are solved numerically by numerical simulation method. Start-up time of parametric vibration is discussed. The ranges of parametric vibration frequency ratio of different length cable are compared and parametric analysis is processed. Under the state of cable-bridge-tower coupled vibration, when the ratio of bridge frequency and cable frequency is 2:1, namely 1:2:1 or 2:2:1 internal resonant occur, vibration diverges after period of time. But while 1:1:1 or 2:1:1 internal resonant occurs, coupled beat vibration of displacement of cable, tower and bridge, as well as vibration displacement of bridge and tower is much larger than initial displacement. Each part of system is excited completely.
Besides parametric vibration is studied by simplifying bridge excitation as ideal excitation and spring-mass coupled system, nonlinear vibration model of stayed cable-beam composite structure is set up. Parametric vibration of stayed cable of cable-beam coupled model is studied. And coupled vibration differential equations in dimensionless style are solved approximately by multi-scale method. Afterwards property of internal resonant response of the model is discussed. Finally the rule of dynamic response of cable-beam coupled model at different internal resonant points is discussed by numerical simulation. The achievements have shown that under the state of large initial displacement of beam, whenωb equals 2 or 1, substantial 'beat' vibration of cable occurs. But under the state of small initial displacement of beam, whenωb is 2, substantial 'beat' vibration of cable occurs due to small excitation. It is concluded that substantial vibration of cable can be caused by small vibration of beam of bridge deck.
Parametric vibration of stayed cable under bearing excitation is studied, based on the research of parametric vibration model considering cable-beam composite structure. There is an angle between the direction of bearing excitation and axial direction of stayed cable, which is different from the method which simplifies bridge deck as spring-mass under ideal axial excitation. Moreover nonlinear vibration model of stayed cable under bearing excitation is built, and vibration differential equations in dimensionless style are solved approximately by means of multi-scale method. It is discussed that while ratio of excitation frequency of bearing and stayed cable frequency satisfies internal resonant frequency ratio, substantial vibration of cable occurs. And when the combination of excitation frequency of bearing satisfy |ω|-_B±ω|-_A|= 0, |ω|-_B±ω|-_A|= 1 or |ω|-_B±ω|-_A|= 2, substantial vibration of cable occurs similarly. In particular, while meeting the parametric resonance, resonance response gets bigger.
To make clear parametric vibration of stayed cable under bearing excitation, the indoor test is processed. According to actuator imitating bridge deck and exciting harmonic wave on the tip of stayed cable, multi-condition tests at the varying internal resonance frequency ratios 2:1, 1:1, 1:2 under different amplitude are carried out. It can be seen that vibration amplitude is larger as the frequency ratio is 2:1, and cable vibration is influenced greatly by rigid link component at the end and out-of-plan vibration. Thus the amplitude of experiment is identical with that of numerical simulation, and time interval of“beat’vibration of experimental cable is larger and attenuates faster. When the frequency ratios are 1:1 and 1:2, the results of experiment and numerical simulation are in good agreement. It can be shown that parametric vibration of stayed cable appearing under bearing excitation is verified by this experiment according to the comparison of the outcomes of experiment and numerical simulation.
At last, in view of the precision of cable force identification on the spot, the method and the precision of cable force identification are investigated systematically. Afterwards the high-precision cable force tester is developed via the partial above-mentioned achievements.
引文
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33 G.. Tagata. Harmonically Forced, Finite Amplitude Vibration of A String. Journal of Sound and Vibration. 1977, 51(4):483-492.
34 I. Kovacs. Zur Frage der Seilschwingungen und der Seildampfung. Die Bautechnik. 1982.
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37 Michel Virlogeux. Cable Vibration in Cable-stayed Bridges. Bridge Dynamics. 1998, 127(3): 213-233.
38 Q-L. Zhang, U. Peil. Dynamic Behaviors of Cables in Parametrically Unstable Zones. Computers and Structures. 1999, 73(5):437-443.
39 S. R. K. Nielsen, P. H. Kirkegaard. Super and Combinatorial Harmonic Response of Flexible Elastic with Small Sag. Journal of Sound and Vibration. 2002, 251(1):79~102.
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24 M. Marsumoto, M.Yamagishi, J. Aoki, N. Shiraishi. Various Mechanism of Induced Cable Aerodynamics, 9ICWE, New delhi India, 1995.
25 Verwiebe, C. Ruschweyh, H. Recent. Research Results Concerning the Exciting Mechanisms of Rain-Wind Induced Vibration. Journal of Wind Engineering andIndustrial Aerodynamics. 1998, (74):1005~1013.
26 A. Bosdogianni, D. Olivari. Wind-and-Rain-induced Oscillations of Cables of Stayed Bridge. Journal of Wind Engineering and Industrial Aerodynamics. 1996, (64):171~185.
27 L. Y. Wang, Y. L. Xu. Analytical study of Wind-Rain-induced Cable Vibration: 2DOF Model. Wind and Structures. 2003, 6(4):291~306.
28李国豪.桥梁结构稳定与振动.中国铁道出版社, 1996.
29李寿英.斜拉桥拉索风雨激振机理及其控制理论研究.同济大学博士学位论文, 2005.
30彭天波.斜拉桥拉索风雨激振的机理研究.同济大学硕士学位论文, 2000.
31 A. H.奈克, D. T.穆克.非线性振动.宋家肃,罗惟德,陈守吉译.高等教育出版社,1996.
32 T. Iwatsubo, M. Saigo. Parametric Instability of Clamped-clamped and Clamped-simply Supported Columns Under Periodic Axial Load. Journal of Sound and Vibration. 1973, 30(1):65-77.
33 G.. Tagata. Harmonically Forced, Finite Amplitude Vibration of A String. Journal of Sound and Vibration. 1977, 51(4):483-492.
34 I. Kovacs. Zur Frage der Seilschwingungen und der Seildampfung. Die Bautechnik. 1982.
35 K. Takahashi. Dynamic Stability of Cables Subjected to an Axial Periodic Load. Journal of Sound and Vibration. 1991, 144(2):323-330.
36 A. Pinto Da Costa, J. A. C. Martins, et.al. Oscillations of Bridge Stsy Cables Induced by Periodic Motions of Deck and/or Towers. Journal of Engineering Mechanics. Journal of Engineering Mechanics. 1996, 122(7):613~622.
37 Michel Virlogeux. Cable Vibration in Cable-stayed Bridges. Bridge Dynamics. 1998, 127(3): 213-233.
38 Q-L. Zhang, U. Peil. Dynamic Behaviors of Cables in Parametrically Unstable Zones. Computers and Structures. 1999, 73(5):437-443.
39 S. R. K. Nielsen, P. H. Kirkegaard. Super and Combinatorial Harmonic Response of Flexible Elastic with Small Sag. Journal of Sound and Vibration. 2002, 251(1):79~102.
40 P. Warnitchai, Y. Fujino, T. Susumpow. A Non-linear Dynamic Model for Cable and its Application to a Cable-structure System. Journal of Sound and Vibration. 1995, 187(4):695~712.
41 Fuheng Yang, Ghislain A. Fonder. Dynamic Response of Cable-stayed Bridges under Moving Load. Journal of Engineering Mechanics. 1995, 124(7):741~747.
42 S. H. Cheng, David T. Lau. Modeling of Cable Vibration Effects of Cable-stayed Bridges. Earthquake Engineering and Engineering Vibration. 2002, 1(1):74~85.
43 F. Benedettini, G. Rega, R. Alaggio. Non-linear oscillations of a Four-degree-of Freedom Model of a Suspended Cable under Multiple Internal Resonance Conditions. Journal of Sound and Vibration. 1995, 182(5):775~798.
44 M. El-Attar, A. Ghobarah, T. S. Aziz. Non-linear Cable Response to Multiple Excitations. Engineering Structures. 2002, (22):1301~1312.
45 Jonghwa Kim, Sung Pil Chang. Dynamic Stiffness Matrix of an Inclined Cable. Engineering Structures. 2001, (23):1614~1621.
46 M. Pakdemirli. Vibrations of Continuous Systems with a General Operator Suitable for Perturbative Calculations. Journal of Sound and Vibration. 2001, 246(5):841~851.
47亢战,钟万勰.斜拉桥参数共振问题的数值研究.土木工程学报.1998,
31(4): 14-22.
48陈文礼,李惠.黏滞阻尼器对拉索参数振动的控制分析.地震工程与工程振动. 2006, 27(2):137~144.
49 H. Li, W. L. Chen, J. P. Ou. Semiactive Variable Stiffness Control for Parametric Vibration of Cables. Earthquake Engineering and Engineering Vibration. 2006, 5(2):215~222.
50汪至刚.大跨度斜拉桥拉索的参数振动与控制.浙江大学博士学位论文. 2000.
51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.
52魏建东,杨佑发,车惠民.斜拉桥拉索的参数振动有限元分析.工程力学. 2000, 17(6):130~134.
53李忠献,陈海泉,李延涛.斜拉桥参数振动有限元分析与半主动控制.工程力学. 2004, 21(1):131~135.
54陈水生,孙炳楠,胡隽.斜拉索轴向激励引起的面内参数振动分析.振动工程学报. 2002, 15(2):144-150.
55陈水生,孙炳楠.斜拉桥索-桥耦合非线性参数振动数值研究.土木工程学报. 2003, 36(4):70-75.
56陈水生.大跨度斜拉桥拉索的振动及被动、半主动控制研究.浙江大学博士学位论文. 2002.
57任淑琰.斜拉桥拉索参数振动研究.同济大学博士学位论文. 2007.
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