斜拉索—辅助索系统动力特性和减振研究
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摘要
随着斜拉桥跨度的增加，斜拉索的长度也不断增大，其振动问题更加突出。仅采用气动和索端阻尼器措施已难以满足斜拉索的减振要求，辅助索减振措施将成为大跨度斜拉桥超长斜拉索减振措施的重要选择。
本文针对斜拉索辅助索减振措施，研究了单索和索网结构动力特性分析理论和方法，着重讨论了索网的阻尼特性和影响因素；进行了索网结构的模型验证试验；以实际大跨度斜拉桥工程为背景，通过算例讨论了采取辅助索措施的必要性，进行了辅助索设计参数分析。研究成果为大跨度斜拉桥拉索的辅助索减振措施设计提供了理论基础和方法。
本文的主要研究包括：
(1)首先采用三单元Maxwell阻尼器模型模拟考虑阻尼器内刚度和支架柔度影响的拉索阻尼器，运用子结构方法建立了单根斜拉索-三单元Maxwell阻尼器系统的运动方程，推导了系统的复频率方程。
实际的阻尼器都在在一定的内刚度且阻尼器支架都具有一定的柔度，为了定量研究这些因素的影响，在假定阻尼器安装位置接近锚固端(l／L(?)1)的前提下得到了拉索—三单元Maxwell阻尼器系统复特征频率方程的近似解析式，并给出了拉索最大模态阻尼和相应的阻尼器最优阻尼系数。研究表明，拉索的最大模态阻尼比随阻尼器内刚度和支架柔度的增大而减小；阻尼器内刚度和支架柔度对系统的最大模态阻尼比的影响是相互耦合的，而阻尼器内刚度和支架柔度对阻尼器最优阻尼系数的影响是可以线性叠加的。
本文建议在辅助索位置安装阻尼器以增加索网结构的阻尼减振效果。针对这一问题，以与阻尼器并联的弹簧模拟辅助索的刚度，与阻尼器串联的弹簧模拟相邻斜拉索的支撑柔度，其模型同样可化简为三单元Maxwell阻尼器模型。由于安装在辅助索位置的阻尼器不再满足l／L(?)1的条件，所以推导了复特征频率方程的迭代函数形式。根据复特征频率方程的迭代函数形式，运用数值方法分析了辅助索位置的集中阻尼对索网结构模态阻尼特性的影响。研究表明，索网的模态阻尼与辅助索集中阻尼的位置有很大关系；集中阻尼的位置越接近振型波峰时，索网的模态阻尼越大；集中阻尼的位置接近振型节点时，索网的模态阻尼比迅速减小；索网的支撑柔度和辅助索的刚度都降低了索网的模态阻尼。
(2)以Kelvin阻尼器模型模拟具有并联刚度的阻尼器，运用传递函数法推导了拉索-任意位置、任意数目Kelvin阻尼器系统的复频率方程。
运用该理论分析了斜拉索套管口橡胶垫圈对斜拉索阻尼特性的影响，研究表明，梁端橡胶圈降低了索端阻尼器的减振效果；而塔端橡胶圈对斜拉索阻尼特性的影响可以忽略不计。
该理论的另一应用是分析梁、塔两端同时安装阻尼器时对斜拉索阻尼特性的影响，研究表明，梁、塔两端同时安装阻尼器时，系统的最大模态阻尼比约等于分别安装梁端阻尼器和塔端阻尼器时拉索所能获得的最大模态阻尼比之和。
最后运用该理论分析了辅助索刚度对索端阻尼器效果的影响，研究表明，对于位于主索等分点位置的辅助索，其刚度对系统的最大模态阻尼和阻尼器最优阻尼系数均有一定的影响，增加辅助索的刚度可以提高系统的最大模态阻尼，阻尼器的最优阻尼系数也相应增大。
(3)在确定索网结构的几何非线性静力平衡位置的基础上，运用有限单元法分析了索网结构的模态频率和振型，并结合能量法对索网结构的分布阻尼特性进行了分析。其中主要考虑了辅助索的道数、张拉方式、初张力和刚度等的影响，并根据辅助索的刚度对索网结构模态振型和频率的影响定义了刚性辅助索和柔性辅助索。
研究表明，增加辅助索道数，不仅可以提高索网结构的面内频率，也可以提高主索和辅助索的分布阻尼对索网结构模态阻尼的贡献。辅助索的张拉方式和张拉力对索网结构模态频率和分布阻尼贡献均有一定影响。刚性辅助索对提高索网结构频率的效果较好，而柔性辅助索对提高索网结构分布阻尼贡献的效果较好。
(4)为了验证索网结构模态频率的理论分析方法并定量研究索网结构的分布阻尼特性，采用三根斜拉索和一根辅助索组成的索网模型进行了试验研究。采用钢丝绳和橡胶绳两种材料分别模拟刚性辅助索和柔性辅助索，研究了辅助索的张拉方式、张拉力、刚度及分布阻尼对索网结构面内面外减振性能的影响，并基于理论分析对试验结果进行了解释。结果表明：索网结构模态频率的试验结果与理论分析基本一致，索网结构模态阻尼的试验结果与理论分析趋势相同。
(5)最后运用本文提出的理论分析了一个实际大跨度斜拉桥的算例，进行了辅助索设计参数分析。
The span of cable-stayed bridges has been greatly extended since the beginning of the 1990s, accordingly the length of stay cable also becomes longer. To mitigate the vibration of very long stay cables induced by wind/rain, the countermeasures, such as installation of dampers near to anchorage of cables or aerodynamic means, may hardly meet the requirements. Therefore, cable cross ties connecting stay cables together would be a significant choice for mitigating the vibration of very long stay cables.
To investigate the characteristics of the cable system with cables and cross ties, theoretical analysis methods about a single cable or a cable system were derived with emphasis on the damping performance and its influencing factors. The free vibration test using a scaled model of cable system was conducted to verify the theoretical results. Numerical simulation was carried out for a large span cable-stayed bridge to discuss the necessity of using cross ties for cable vibration mitigation and to optimize the design parameters of cross ties. The research findings will provide a theoretical basis for the design of vibration mitigation using cross ties.
The main contents of this paper are as follows:
(1) A three-element Maxwell damper model was adopted to simulate the inherent stiffness of a damper installed at a cable and the flexibility of its support, the equation of motion of a single cable with the Maxwell model was formulated, then the complex eigen-equation of the system was obtained.
Usually a real damper has inherent stiffness and its support has flexibility. To investigate the influences of these factors, an approximate solution of the complex eigen-equation of cable system was obtained with the assumption that the location of damper is near to the anchorage of cable. Subsequently the maximum damping of cable and the optimal damping coefficient of damper can be calculated. The theoretical analyses show that the inherent stiffness of damper and the flexibility of damper support both reduce the effectiveness of damper; the optimum damping coefficient of damper increases when the inherent stiffness of the damper increases, but decreases when the flexibility of support increases.
The author suggested that dampers should be installed at the connections of cross ties to stay cables to increase the damping of cable system. To analyze such problem, the stiffness of cross tie could be simulated by a spring parallely connected to the damper element and the flexibility of the adjacent cables could be simulated by a spring serially connected to the damper. Thus the model is also a three-element Maxwell model. Because the locations of dampers are not close to the anchorage of cable, the approximate solution is not applicable anymore. The iteration scheme of the complex eigen-equation was therefore deduced. Based on the iteration scheme, the numerical analyses were conducted to investigate the effect of cross tie damper on the modal damping of cable system. The analyses show that the modal damping of cable system is closely related to the locations of cross tie damper. When the damper is closer to the wave crest of modal shape, the damping ratio of the cable system is larger; and when the damper gets closer to the node of the modal shape, the damping
ratio of the cable system decreases rapidly. The flexibility of the adjacent cables and the stiffness of the cross ties both reduce the modal damping of the cable system.
(2) A damper with parallel stiffness was simulated by a Kelvin damper model. A cable with arbitrary numbers of Kelvin model at arbitrary locations was analyzed using a transfer matrix method, and the complex eigen-equation of the system was obtained.
The effect of rubber rings installed near the anchorages of cable on the damping characteristics of stay cable was investigated. The results show that the rubber ring installed at the girder anchorage of cable decrease the effectiveness of the girder-side damper while the rubber ring at the tower anchorage have little influence.
Anther application of this theory is to investigate the damping of cable with two dampers installed near to the girder and tower anchorage of cable respectively. The results show that the maximum damping of the cable approximately equals to the summation of the cable damping with a single damper installed at girder or tower anchorage respectively.
The theory is also used to analyze the effect of the stiffness of cross tie on the effectiveness of damper located near the cable anchorage. It is shown that for cross ties equally spaced along the longest cable, increasing the stiffness the cross ties could increase the effectiveness of damper and accordingly the optimum damping coefficient of damper should be increased.
(3) On the basis of static equilibrium configuration of cable, Finite Element Method was applied to analyze the modal frequency and shape. The characteristics of distributed damping were then evaluated by using energy method. The factors mainly considered are the number of cross ties, the tensioning method, the initial tension force, and the stiffness of cross ties. Stiff type cross tie and flexible type cross tie were also defined.
The analyses show that when the number of cross ties increases, the in-plane frequencies of the cable system increases and the damping of cable system is also increased. The tensioning method and the initial tension force both have some effects on the modal frequencies and damping. The stiff type cross ties may be mainly used to increase the frequencies of cable system and the flexible type cross ties to increase the damping of cable system.
(4) To verify the theoretical results and to investigate quantitatively the characteristics of distributed structural damping, free vibration test was conducted by using a cable system model which consists of three stayed cables and one cross tie. Steel wire ropes and rubber ropes were adopted to model the stiff type and the flexible type cross ties respectively. The effects of tension method, initial tension and stiffness of cross ties on in-plane and out-of-plane vibration mitigation of the cable system were investigated. The test results were explained based on the theoretical studies. The test results of frequencies of cable system are approximately the same as the theoretical ones and the test results on damping characteristics have the same tendency as the theoretical studies.
(5) Finally, a large span cable-stayed bridge was used as a sample to illustrate the necessity of using cross ties for mitigating the vibrations of stay cables, and the optimum parameters for designing cross ties were analyzed.
本文针对斜拉索辅助索减振措施，研究了单索和索网结构动力特性分析理论和方法，着重讨论了索网的阻尼特性和影响因素；进行了索网结构的模型验证试验；以实际大跨度斜拉桥工程为背景，通过算例讨论了采取辅助索措施的必要性，进行了辅助索设计参数分析。研究成果为大跨度斜拉桥拉索的辅助索减振措施设计提供了理论基础和方法。
本文的主要研究包括：
(1)首先采用三单元Maxwell阻尼器模型模拟考虑阻尼器内刚度和支架柔度影响的拉索阻尼器，运用子结构方法建立了单根斜拉索-三单元Maxwell阻尼器系统的运动方程，推导了系统的复频率方程。
实际的阻尼器都在在一定的内刚度且阻尼器支架都具有一定的柔度，为了定量研究这些因素的影响，在假定阻尼器安装位置接近锚固端(l／L(?)1)的前提下得到了拉索—三单元Maxwell阻尼器系统复特征频率方程的近似解析式，并给出了拉索最大模态阻尼和相应的阻尼器最优阻尼系数。研究表明，拉索的最大模态阻尼比随阻尼器内刚度和支架柔度的增大而减小；阻尼器内刚度和支架柔度对系统的最大模态阻尼比的影响是相互耦合的，而阻尼器内刚度和支架柔度对阻尼器最优阻尼系数的影响是可以线性叠加的。
本文建议在辅助索位置安装阻尼器以增加索网结构的阻尼减振效果。针对这一问题，以与阻尼器并联的弹簧模拟辅助索的刚度，与阻尼器串联的弹簧模拟相邻斜拉索的支撑柔度，其模型同样可化简为三单元Maxwell阻尼器模型。由于安装在辅助索位置的阻尼器不再满足l／L(?)1的条件，所以推导了复特征频率方程的迭代函数形式。根据复特征频率方程的迭代函数形式，运用数值方法分析了辅助索位置的集中阻尼对索网结构模态阻尼特性的影响。研究表明，索网的模态阻尼与辅助索集中阻尼的位置有很大关系；集中阻尼的位置越接近振型波峰时，索网的模态阻尼越大；集中阻尼的位置接近振型节点时，索网的模态阻尼比迅速减小；索网的支撑柔度和辅助索的刚度都降低了索网的模态阻尼。
(2)以Kelvin阻尼器模型模拟具有并联刚度的阻尼器，运用传递函数法推导了拉索-任意位置、任意数目Kelvin阻尼器系统的复频率方程。
运用该理论分析了斜拉索套管口橡胶垫圈对斜拉索阻尼特性的影响，研究表明，梁端橡胶圈降低了索端阻尼器的减振效果；而塔端橡胶圈对斜拉索阻尼特性的影响可以忽略不计。
该理论的另一应用是分析梁、塔两端同时安装阻尼器时对斜拉索阻尼特性的影响，研究表明，梁、塔两端同时安装阻尼器时，系统的最大模态阻尼比约等于分别安装梁端阻尼器和塔端阻尼器时拉索所能获得的最大模态阻尼比之和。
最后运用该理论分析了辅助索刚度对索端阻尼器效果的影响，研究表明，对于位于主索等分点位置的辅助索，其刚度对系统的最大模态阻尼和阻尼器最优阻尼系数均有一定的影响，增加辅助索的刚度可以提高系统的最大模态阻尼，阻尼器的最优阻尼系数也相应增大。
(3)在确定索网结构的几何非线性静力平衡位置的基础上，运用有限单元法分析了索网结构的模态频率和振型，并结合能量法对索网结构的分布阻尼特性进行了分析。其中主要考虑了辅助索的道数、张拉方式、初张力和刚度等的影响，并根据辅助索的刚度对索网结构模态振型和频率的影响定义了刚性辅助索和柔性辅助索。
研究表明，增加辅助索道数，不仅可以提高索网结构的面内频率，也可以提高主索和辅助索的分布阻尼对索网结构模态阻尼的贡献。辅助索的张拉方式和张拉力对索网结构模态频率和分布阻尼贡献均有一定影响。刚性辅助索对提高索网结构频率的效果较好，而柔性辅助索对提高索网结构分布阻尼贡献的效果较好。
(4)为了验证索网结构模态频率的理论分析方法并定量研究索网结构的分布阻尼特性，采用三根斜拉索和一根辅助索组成的索网模型进行了试验研究。采用钢丝绳和橡胶绳两种材料分别模拟刚性辅助索和柔性辅助索，研究了辅助索的张拉方式、张拉力、刚度及分布阻尼对索网结构面内面外减振性能的影响，并基于理论分析对试验结果进行了解释。结果表明：索网结构模态频率的试验结果与理论分析基本一致，索网结构模态阻尼的试验结果与理论分析趋势相同。
(5)最后运用本文提出的理论分析了一个实际大跨度斜拉桥的算例，进行了辅助索设计参数分析。
The span of cable-stayed bridges has been greatly extended since the beginning of the 1990s, accordingly the length of stay cable also becomes longer. To mitigate the vibration of very long stay cables induced by wind/rain, the countermeasures, such as installation of dampers near to anchorage of cables or aerodynamic means, may hardly meet the requirements. Therefore, cable cross ties connecting stay cables together would be a significant choice for mitigating the vibration of very long stay cables.
To investigate the characteristics of the cable system with cables and cross ties, theoretical analysis methods about a single cable or a cable system were derived with emphasis on the damping performance and its influencing factors. The free vibration test using a scaled model of cable system was conducted to verify the theoretical results. Numerical simulation was carried out for a large span cable-stayed bridge to discuss the necessity of using cross ties for cable vibration mitigation and to optimize the design parameters of cross ties. The research findings will provide a theoretical basis for the design of vibration mitigation using cross ties.
The main contents of this paper are as follows:
(1) A three-element Maxwell damper model was adopted to simulate the inherent stiffness of a damper installed at a cable and the flexibility of its support, the equation of motion of a single cable with the Maxwell model was formulated, then the complex eigen-equation of the system was obtained.
Usually a real damper has inherent stiffness and its support has flexibility. To investigate the influences of these factors, an approximate solution of the complex eigen-equation of cable system was obtained with the assumption that the location of damper is near to the anchorage of cable. Subsequently the maximum damping of cable and the optimal damping coefficient of damper can be calculated. The theoretical analyses show that the inherent stiffness of damper and the flexibility of damper support both reduce the effectiveness of damper; the optimum damping coefficient of damper increases when the inherent stiffness of the damper increases, but decreases when the flexibility of support increases.
The author suggested that dampers should be installed at the connections of cross ties to stay cables to increase the damping of cable system. To analyze such problem, the stiffness of cross tie could be simulated by a spring parallely connected to the damper element and the flexibility of the adjacent cables could be simulated by a spring serially connected to the damper. Thus the model is also a three-element Maxwell model. Because the locations of dampers are not close to the anchorage of cable, the approximate solution is not applicable anymore. The iteration scheme of the complex eigen-equation was therefore deduced. Based on the iteration scheme, the numerical analyses were conducted to investigate the effect of cross tie damper on the modal damping of cable system. The analyses show that the modal damping of cable system is closely related to the locations of cross tie damper. When the damper is closer to the wave crest of modal shape, the damping ratio of the cable system is larger; and when the damper gets closer to the node of the modal shape, the damping
ratio of the cable system decreases rapidly. The flexibility of the adjacent cables and the stiffness of the cross ties both reduce the modal damping of the cable system.
(2) A damper with parallel stiffness was simulated by a Kelvin damper model. A cable with arbitrary numbers of Kelvin model at arbitrary locations was analyzed using a transfer matrix method, and the complex eigen-equation of the system was obtained.
The effect of rubber rings installed near the anchorages of cable on the damping characteristics of stay cable was investigated. The results show that the rubber ring installed at the girder anchorage of cable decrease the effectiveness of the girder-side damper while the rubber ring at the tower anchorage have little influence.
Anther application of this theory is to investigate the damping of cable with two dampers installed near to the girder and tower anchorage of cable respectively. The results show that the maximum damping of the cable approximately equals to the summation of the cable damping with a single damper installed at girder or tower anchorage respectively.
The theory is also used to analyze the effect of the stiffness of cross tie on the effectiveness of damper located near the cable anchorage. It is shown that for cross ties equally spaced along the longest cable, increasing the stiffness the cross ties could increase the effectiveness of damper and accordingly the optimum damping coefficient of damper should be increased.
(3) On the basis of static equilibrium configuration of cable, Finite Element Method was applied to analyze the modal frequency and shape. The characteristics of distributed damping were then evaluated by using energy method. The factors mainly considered are the number of cross ties, the tensioning method, the initial tension force, and the stiffness of cross ties. Stiff type cross tie and flexible type cross tie were also defined.
The analyses show that when the number of cross ties increases, the in-plane frequencies of the cable system increases and the damping of cable system is also increased. The tensioning method and the initial tension force both have some effects on the modal frequencies and damping. The stiff type cross ties may be mainly used to increase the frequencies of cable system and the flexible type cross ties to increase the damping of cable system.
(4) To verify the theoretical results and to investigate quantitatively the characteristics of distributed structural damping, free vibration test was conducted by using a cable system model which consists of three stayed cables and one cross tie. Steel wire ropes and rubber ropes were adopted to model the stiff type and the flexible type cross ties respectively. The effects of tension method, initial tension and stiffness of cross ties on in-plane and out-of-plane vibration mitigation of the cable system were investigated. The test results were explained based on the theoretical studies. The test results of frequencies of cable system are approximately the same as the theoretical ones and the test results on damping characteristics have the same tendency as the theoretical studies.
(5) Finally, a large span cable-stayed bridge was used as a sample to illustrate the necessity of using cross ties for mitigating the vibrations of stay cables, and the optimum parameters for designing cross ties were analyzed.
引文
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