大跨度悬索桥悬吊体系参数振动研究
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摘要
悬索桥是一种古老的桥梁型式。悬索桥由于其力线明确，造型美观和跨越能力大等优点为越来越多的跨海连岛，跨谷工程所采用。由于跨度不断增大，其主要受力索构件（包括主缆，吊索）的长度也越长，由此导致的大柔度索结构振动问题越来越突出。为了保证桥梁结构的安全性，耐久性，甚至使用舒适度，索桥的振动问题引起了各国学者的的高度重视。本文针对大跨度悬索桥悬吊体系各主要构件及其构成的系统有可能发生大幅度参数振动的问题进行较系统的研究。主要研究工作如下：
（1）利用精确悬链单元迭代计算方法（Parabola to Catenary Iteration method，后简称PTCI法）对悬索进行找形分析并结合有限元及实测结果进行验证。从索单元静力平衡关系出发，在分析其解析解的过程中，根据索单元的竖向分布荷载按照沿弦长和弧长分布分为两类：一种对应的索单元的形状曲线为抛物线，另一种为悬链线；并以抛物线结果作为初始值对非线性解进行迭代，索单元最终收敛为悬链线。基于PTCI找形算法提出了无应力索长的计算方法和流程。通过比较该方法和有限元软件计算的找形和无应力索长结果，并与矮寨特大悬索桥实测值进行对比，验证该计算理论的优点及工程适用性。
（2）提出了基于PTCI理论的悬索自振特性值计算方法。在得到悬索精确线形的基础上,同样以悬索微元为基本研究对象，从静力学平衡关系为出发，推导出索单元的自激振动特征方程。然后用非线性单元（悬链线单元）对悬索微元进行模拟，得到相应单元的刚度矩阵及质量矩阵，最后集成总刚度矩阵和总质量矩阵代入悬索系统的弹性动力学方程，可以计算得悬索的面内各阶自振频率和相应振型。本章依托矮寨悬索桥的工程实例将其计算结果与有限元软件计算结果和实测值进行对比，验证本计算方法的准确性。
（3）利用大型有限元软件建立悬索桥动力分析的基础模型，分析了设计成桥状态的全桥动力特性以及在动力特性分析模型中主缆及吊索的局部动力特性。最后分析了矮寨悬索桥斜拉中央扣，端部弹性索及桁梁高度等参数对全桥动力特性的影响。
（4）针对大跨度悬索桥施工阶段中的空缆状态，分析了非水平大垂度悬索在端部激励下发生参数振动的响应频比关系及响应特性。推导了主缆参数运动方程,并考虑了由于主缆的大垂度、大位移而引起的几何非线性因素的影响,并利用Galerkin方法对方程进行离散，再运用多尺度摄动方法，对主缆可能发生的参数共振进行了解析分析和数值分析，确定了主缆发生参数共振的条件及响应特性。考虑了大垂度悬索受低高程端桥塔和高高程端桥塔激励这两种情况，分别进行主缆的参数振动分析。分析的主要内容包括产生参数振动时主缆振幅与频率比之间的关系、主缆振幅与激励幅值之间的关系、振动过程中主缆张力的变化时程及悬索中点位移变化时程等。对主缆参数振动影响因素进行了分析：考虑激励幅值、初张力、主缆的长度、垂度及主缆的单位长度质量这五个影响因素。每个影响因素分为其值增加30%和降低30%两种工况进行参数振动分析。研究发生参数振动时，主缆各个参数对振幅与频率比之间的关系、振幅与激励幅值之间的关系、振动过程中主缆张力的变化时程及中点位移变化时程等的影响程度。
（5）针对大跨度悬索桥上长吊索的不稳定特性，建立了简易的吊索力学模型并根据牛顿定律推导了吊索中点位移的参数振动微分方程，根据Galerkin方法对方程进行离散分离变量，利用多尺度解析法并结合工程实例参数分析了吊索单模态及双模态情况下在理想端部激励下竖直吊索发生参数振动的频比条件及响应特性。并分析了激励幅值，吊索长度及初始扰动等因素对吊索发生参数振动响应的影响。
（6）针对大跨度悬索桥悬吊体系的整体柔性特性，在研究理想激励的前提下进一步考虑了竖直吊索与柔性主梁的耦合参数振动，建立了二自由度力学简化模型，运用多尺度法求解相应微分方程得出解析解，并结合矮寨特大悬索桥工程实例，利用Runge-Kutta法对振动方程进行数值求解得出二自由度系统发生耦合参数振动的发生条件及响应特性。
（7）在前面考虑二自由度系统的基础上，进一步考虑柔性主缆的影响，建立三自由度（主缆-吊索-主梁）耦合参数振动力学模型，根据牛顿定律推导振动微分方程，运用多尺度法对其进行解析求解，得出了三自由度组合频比参数共振的相关结果，并结合实际工程参数用四阶Runge-Kutta法进行数值求解进一步得到三自由度耦合参数振动的数值结果对解析解进行补充及验证。
Suspension bridge is an ancient type of bridge. Because of its many advantages such asstraightforward force line, slinky modeling and its large span ability, it has been used in theproject that across the sea and link the islands or across the valley more and more frequently。Because the span is increasing very quickly, the main load bearing member (including themain cable and the sling)will become more and more longer, as a result, the vibration problemof cable structures of great flexibility will be more and more prominent. In order to guaranteethe safety, durability, and using comfort level, the vibration problems caused the attention ofmany scholars. Based on a long-span suspension bridge suspension system's maincomponents or its composition of the system has a possibility of engendering parametricvibration with great amplitude, this problem are studied systematically in the paper. Mainresearch work is as follows:
(1) The Parabola to Catenary Iteration method(PTCI method for short)was carried outon the form-finding of suspension cable, and it is validated combined with the finite elementand the measured result. Analysis is starting from the cable unit static equilibrium relationship,in the process of gain the analytical solutions, according to the vertical distribution load lineon the cable element is a arc line or a chord line, it can be divided into two categories: onecorresponding cable form curve is parabola, another is catenary; then using the parabolaresults as the initial value for nonlinear iteration solution, eventually the cable element lineconverge for catenary. On the bases of the accurate element method, the calculation methodand process has been presented for unstressed cable length. Finally, comparing the results ofform-finding and unstressed cable length which are obtained from finite element software andthe measured values and the accurate method, we can get that this calculation theory needssmaller density for elements dividing, and get more quick iteration convergence rate, and canguarantee the necessary calculation accuracy.
(2) This paper puts forward a calculation method of the natural characteristic values forthe suspension cable based on the PTCI method. After getting the accurate form of thesuspension cable, same with the micro-unit of suspension cable as the basic study object, starting from the static equilibrium relation, self-excited vibration of the cable element isdeduced. So we can get the right stiffness matrix and mass matrix of every suspension cablemicro-unit simulated with nonlinear catenary units, the following step is cable unit matrixintegrating into global stiffness and mass matrix of the whole cable, which will be substitutedinto the elastic dynamics equation of suspension cable system. So we can get the in-planenatural frequency and the corresponding vibration mode of every orders of the cable. Thenverifying the accuracy of the calculation method based on the comparison between the threeresults: one is the infinite software result, another is the measured value, the third result isfrom the right calculation method.
(3) In the paper, a dynamic analysis finite element model of the AiZhai suspensionbridge was created by using a kind of finite element software. And the dynamiccharacteristics of the whole bridge on designed completed state had been analyzed, as well asthe local dynamic characteristics of the main cable and the sling in the same finite elementmodel. Finally, the effect of the suspension central buckle, end elastic cable and girder beamheight produced on the dynamic characteristics of the whole bridge.
(4) In view of the empty loading state in the bridge construction stage of suspensionbridge, this paper analyzes the frequency ratio relations and response characteristics ofparametric vibration of the large span unhorizontal cable with large sag under the endexcitation. The parametric vibration dynamic equation of the main cable was deduced, inwhich the geometric nonlinear factors caused by the large sag and the large displacement hasbeen taken into account; then discrete the right equation by using Galerkin method; finallyanalytical analyzes and numerical analyzes the parametric vibration of the main cable byusing the Multi-scale perturbation method, then we can get the occurring conditions and theresponse characteristics of parametric vibration of the main cable.
Two cases have been considering about the parametric vibration of the main cable,which respectively are it was excited by the low attitude tower and the high attitude tower.The main analysis include the relationship between the mid-span amplitude of the main cableand frequency, and the relationship between the vibration amplitude of the main cable and the excitation amplitude, and the time-history change of the main cable tension, and thetime-history change of the suspension cable midpoint displacement when parametric vibrationoccurred.
Influence factors of the main cable parametric vibration have been analyzed, whichinclude the exciting amplitude, the initial tension of the cable, the cable length, the cable sag,and the quality of per unit cable length. Each influence factor are analyzed in two conditionsinclude its value increased by30%and reduced by30%. Analyze the influence degree of eachparameter on the relationship between the amplitude and the frequency ratio, the relationshipbetween the amplitude and the excite amplitude, the time-history change of cable tension, thetime-history change of the midpoint displacement when the parametric vibration occurred.
(4)Aiming at the unstable characteristics of the long sling of the large span suspensionbridge, Simple sling mechanics model is established and deduced the parametric vibrationdifferential equation of the sling midpoint according to Newton's law; then the variable wasdiscrete based on Galerkin method. Analyzed the single-mode and double-mode parametricvibration of the vertical sling under the ideal end excite using Multi-scale method andcombined with engineering instance parameters and get the right frequency ratio and responsecharacteristics. Finally the paper analyzed the difference to the parametric vibration responseof sling, which have made by some factors such as excite amplitude, cable length and theinitial disturbance.
(5)In consideration of the flexible features of the whole suspended system of the largespan suspension bridge, on the premise of studying the parametric vibration under ideal exciteof the sling, this part analyzed the parametric vibration of vertical sling coupling with theflexible girder. Two degrees of freedom simplified mechanics model had been established.Then get the analytical resolution using the Multi-scale method, and combined with theAizhai suspension bridge engineering example, got the numerical occurring condition andresponse characteristics resolution of the two degree of freedom coupling system occurringparametric vibration by using Runge-Kutta method.
On the basis of studying the two degrees of freedom system, this part further considering the influence of the flexible main cable, established three degrees of freedom(maincable-sling-girder)coupling parametric vibration mechanical model, and vibration differentialequation was derived according to Newton's law, then carrying on the analytical resolution byusing Multi-scale method, following we can get the result about the combination resonancefrequency ratio of the parametric vibration of three degrees of freedom system. Finally thenumerical solution of three degrees of freedom coupling parametric vibration system wasavailable by using four orders Runge-Kutta method combined with the actual projectparameters, which are able to complement and verify the analytical solution.
（1）利用精确悬链单元迭代计算方法（Parabola to Catenary Iteration method，后简称PTCI法）对悬索进行找形分析并结合有限元及实测结果进行验证。从索单元静力平衡关系出发，在分析其解析解的过程中，根据索单元的竖向分布荷载按照沿弦长和弧长分布分为两类：一种对应的索单元的形状曲线为抛物线，另一种为悬链线；并以抛物线结果作为初始值对非线性解进行迭代，索单元最终收敛为悬链线。基于PTCI找形算法提出了无应力索长的计算方法和流程。通过比较该方法和有限元软件计算的找形和无应力索长结果，并与矮寨特大悬索桥实测值进行对比，验证该计算理论的优点及工程适用性。
（2）提出了基于PTCI理论的悬索自振特性值计算方法。在得到悬索精确线形的基础上,同样以悬索微元为基本研究对象，从静力学平衡关系为出发，推导出索单元的自激振动特征方程。然后用非线性单元（悬链线单元）对悬索微元进行模拟，得到相应单元的刚度矩阵及质量矩阵，最后集成总刚度矩阵和总质量矩阵代入悬索系统的弹性动力学方程，可以计算得悬索的面内各阶自振频率和相应振型。本章依托矮寨悬索桥的工程实例将其计算结果与有限元软件计算结果和实测值进行对比，验证本计算方法的准确性。
（3）利用大型有限元软件建立悬索桥动力分析的基础模型，分析了设计成桥状态的全桥动力特性以及在动力特性分析模型中主缆及吊索的局部动力特性。最后分析了矮寨悬索桥斜拉中央扣，端部弹性索及桁梁高度等参数对全桥动力特性的影响。
（4）针对大跨度悬索桥施工阶段中的空缆状态，分析了非水平大垂度悬索在端部激励下发生参数振动的响应频比关系及响应特性。推导了主缆参数运动方程,并考虑了由于主缆的大垂度、大位移而引起的几何非线性因素的影响,并利用Galerkin方法对方程进行离散，再运用多尺度摄动方法，对主缆可能发生的参数共振进行了解析分析和数值分析，确定了主缆发生参数共振的条件及响应特性。考虑了大垂度悬索受低高程端桥塔和高高程端桥塔激励这两种情况，分别进行主缆的参数振动分析。分析的主要内容包括产生参数振动时主缆振幅与频率比之间的关系、主缆振幅与激励幅值之间的关系、振动过程中主缆张力的变化时程及悬索中点位移变化时程等。对主缆参数振动影响因素进行了分析：考虑激励幅值、初张力、主缆的长度、垂度及主缆的单位长度质量这五个影响因素。每个影响因素分为其值增加30%和降低30%两种工况进行参数振动分析。研究发生参数振动时，主缆各个参数对振幅与频率比之间的关系、振幅与激励幅值之间的关系、振动过程中主缆张力的变化时程及中点位移变化时程等的影响程度。
（5）针对大跨度悬索桥上长吊索的不稳定特性，建立了简易的吊索力学模型并根据牛顿定律推导了吊索中点位移的参数振动微分方程，根据Galerkin方法对方程进行离散分离变量，利用多尺度解析法并结合工程实例参数分析了吊索单模态及双模态情况下在理想端部激励下竖直吊索发生参数振动的频比条件及响应特性。并分析了激励幅值，吊索长度及初始扰动等因素对吊索发生参数振动响应的影响。
（6）针对大跨度悬索桥悬吊体系的整体柔性特性，在研究理想激励的前提下进一步考虑了竖直吊索与柔性主梁的耦合参数振动，建立了二自由度力学简化模型，运用多尺度法求解相应微分方程得出解析解，并结合矮寨特大悬索桥工程实例，利用Runge-Kutta法对振动方程进行数值求解得出二自由度系统发生耦合参数振动的发生条件及响应特性。
（7）在前面考虑二自由度系统的基础上，进一步考虑柔性主缆的影响，建立三自由度（主缆-吊索-主梁）耦合参数振动力学模型，根据牛顿定律推导振动微分方程，运用多尺度法对其进行解析求解，得出了三自由度组合频比参数共振的相关结果，并结合实际工程参数用四阶Runge-Kutta法进行数值求解进一步得到三自由度耦合参数振动的数值结果对解析解进行补充及验证。
Suspension bridge is an ancient type of bridge. Because of its many advantages such asstraightforward force line, slinky modeling and its large span ability, it has been used in theproject that across the sea and link the islands or across the valley more and more frequently。Because the span is increasing very quickly, the main load bearing member (including themain cable and the sling)will become more and more longer, as a result, the vibration problemof cable structures of great flexibility will be more and more prominent. In order to guaranteethe safety, durability, and using comfort level, the vibration problems caused the attention ofmany scholars. Based on a long-span suspension bridge suspension system's maincomponents or its composition of the system has a possibility of engendering parametricvibration with great amplitude, this problem are studied systematically in the paper. Mainresearch work is as follows:
(1) The Parabola to Catenary Iteration method(PTCI method for short)was carried outon the form-finding of suspension cable, and it is validated combined with the finite elementand the measured result. Analysis is starting from the cable unit static equilibrium relationship,in the process of gain the analytical solutions, according to the vertical distribution load lineon the cable element is a arc line or a chord line, it can be divided into two categories: onecorresponding cable form curve is parabola, another is catenary; then using the parabolaresults as the initial value for nonlinear iteration solution, eventually the cable element lineconverge for catenary. On the bases of the accurate element method, the calculation methodand process has been presented for unstressed cable length. Finally, comparing the results ofform-finding and unstressed cable length which are obtained from finite element software andthe measured values and the accurate method, we can get that this calculation theory needssmaller density for elements dividing, and get more quick iteration convergence rate, and canguarantee the necessary calculation accuracy.
(2) This paper puts forward a calculation method of the natural characteristic values forthe suspension cable based on the PTCI method. After getting the accurate form of thesuspension cable, same with the micro-unit of suspension cable as the basic study object, starting from the static equilibrium relation, self-excited vibration of the cable element isdeduced. So we can get the right stiffness matrix and mass matrix of every suspension cablemicro-unit simulated with nonlinear catenary units, the following step is cable unit matrixintegrating into global stiffness and mass matrix of the whole cable, which will be substitutedinto the elastic dynamics equation of suspension cable system. So we can get the in-planenatural frequency and the corresponding vibration mode of every orders of the cable. Thenverifying the accuracy of the calculation method based on the comparison between the threeresults: one is the infinite software result, another is the measured value, the third result isfrom the right calculation method.
(3) In the paper, a dynamic analysis finite element model of the AiZhai suspensionbridge was created by using a kind of finite element software. And the dynamiccharacteristics of the whole bridge on designed completed state had been analyzed, as well asthe local dynamic characteristics of the main cable and the sling in the same finite elementmodel. Finally, the effect of the suspension central buckle, end elastic cable and girder beamheight produced on the dynamic characteristics of the whole bridge.
(4) In view of the empty loading state in the bridge construction stage of suspensionbridge, this paper analyzes the frequency ratio relations and response characteristics ofparametric vibration of the large span unhorizontal cable with large sag under the endexcitation. The parametric vibration dynamic equation of the main cable was deduced, inwhich the geometric nonlinear factors caused by the large sag and the large displacement hasbeen taken into account; then discrete the right equation by using Galerkin method; finallyanalytical analyzes and numerical analyzes the parametric vibration of the main cable byusing the Multi-scale perturbation method, then we can get the occurring conditions and theresponse characteristics of parametric vibration of the main cable.
Two cases have been considering about the parametric vibration of the main cable,which respectively are it was excited by the low attitude tower and the high attitude tower.The main analysis include the relationship between the mid-span amplitude of the main cableand frequency, and the relationship between the vibration amplitude of the main cable and the excitation amplitude, and the time-history change of the main cable tension, and thetime-history change of the suspension cable midpoint displacement when parametric vibrationoccurred.
Influence factors of the main cable parametric vibration have been analyzed, whichinclude the exciting amplitude, the initial tension of the cable, the cable length, the cable sag,and the quality of per unit cable length. Each influence factor are analyzed in two conditionsinclude its value increased by30%and reduced by30%. Analyze the influence degree of eachparameter on the relationship between the amplitude and the frequency ratio, the relationshipbetween the amplitude and the excite amplitude, the time-history change of cable tension, thetime-history change of the midpoint displacement when the parametric vibration occurred.
(4)Aiming at the unstable characteristics of the long sling of the large span suspensionbridge, Simple sling mechanics model is established and deduced the parametric vibrationdifferential equation of the sling midpoint according to Newton's law; then the variable wasdiscrete based on Galerkin method. Analyzed the single-mode and double-mode parametricvibration of the vertical sling under the ideal end excite using Multi-scale method andcombined with engineering instance parameters and get the right frequency ratio and responsecharacteristics. Finally the paper analyzed the difference to the parametric vibration responseof sling, which have made by some factors such as excite amplitude, cable length and theinitial disturbance.
(5)In consideration of the flexible features of the whole suspended system of the largespan suspension bridge, on the premise of studying the parametric vibration under ideal exciteof the sling, this part analyzed the parametric vibration of vertical sling coupling with theflexible girder. Two degrees of freedom simplified mechanics model had been established.Then get the analytical resolution using the Multi-scale method, and combined with theAizhai suspension bridge engineering example, got the numerical occurring condition andresponse characteristics resolution of the two degree of freedom coupling system occurringparametric vibration by using Runge-Kutta method.
On the basis of studying the two degrees of freedom system, this part further considering the influence of the flexible main cable, established three degrees of freedom(maincable-sling-girder)coupling parametric vibration mechanical model, and vibration differentialequation was derived according to Newton's law, then carrying on the analytical resolution byusing Multi-scale method, following we can get the result about the combination resonancefrequency ratio of the parametric vibration of three degrees of freedom system. Finally thenumerical solution of three degrees of freedom coupling parametric vibration system wasavailable by using four orders Runge-Kutta method combined with the actual projectparameters, which are able to complement and verify the analytical solution.
引文
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