Dynamic Characteristics And Vibration Mitigation Of Stay Cables Using Cross Ties

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.