Elimination of resonant vibration in cable stayed bridges and footbridges by active steering of tension in cables

The paper presented proposes a new approach to the elimination of vibration in cable stayed bridges and footbridges which may be applied to dynamic analysis of modern cable stayed bridges with typical structure systems. The reduction of the vibration is to be achieved by way of active steering of the tension in cables. The problem of eigenproblem sensitivity analysis with respect to change of the cable tension in cable stayed footbridges and bridges is considered. In the future, the information gained from the sensitivity analysis may be used to calculate how to steer the cable tension in a way that will ensure the most efficient resonant vibration elimination. Particular notice has been paid to the vibration of cables. The matrix equation of motion of a three–dimensional computer model of the structure has been created with the help of a non–standard use of standard computer software leaning on Finite Element Method (FEM). The system Mathematica and the authors' own program have also been used to analyze the eigenproblem sensitivity of bridges. An example has been provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and dynamic behavior of a selected cable-stayed footbridge with respect to change of the cable tension

The paper presents the theoretical analysis of the possibilities of using the authors' own method of actively reducing resonant vibration in cable-stayed footbridges. The method of actively reducing vibration was earlier tested on a laboratory model of a steel footbridge, [1]. The method relies on changing the static tension in chosen cables. The term “active” refers to the fact that the tension changes would only be introduced when resonant vibration of the structure occurred. A change in the cable tensions should cause a change of the natural frequencies for which resonant vibration amplification appeared earlier. Eigenproblem sensitivity analysis, formulated for the structure according to the second order theory, is the basis for choosing the cables in which the tension should change. Apart from analyzing the effectiveness of the active method for reducing resonant vibration, the influence of cable tension change on other values was also studied. These values included the changes in vibration amplitudes, as well as changes of cross-section forces, tensions, and static bend of the structure elements that are most sensitive to tension change. It was also established whether the tension changes in chosen cables, which were large enough to cause a practically significant reduction of resonant vibration, cause exceeding the levels of Serviceability Limit State (SLS) and Ultimate Limit State (ULS) for the structure as a whole or for some of its elements. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrations of spatially suspended flexible pipe-beam with moving fluid

The nonlinear dynamic model of flexible pipe–beam suspended by spatial system of cables is proposed for vibration analysis of pipeline suspension bridges. The model, based on substructure technique, is considered as an assemblage of the following substructures: cables, hangers and pipe–beam. Equation of motion of pipe–beam is derived by Galerkin's FEM with the original finite element formulated in order to include moving mass of transported fluid. To describe cable vibrations, general continuum approach proposed in Ref.[1] is adopted with application into 3D model. Cable model takes into account initial sag, pre–tension force, large displacements and hangers' point reactions. Equation governing the motion of pipe–beam with cables and hangers is obtained regarding equilibrium conditions of interaction forces and compatibility of displacements at connection points between substructures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instability of Oscillations in Cable-Stayed Bridges

In this paper the stability of two basic types of cable stayed bridges, suspended by one or two rows of cables, is studied. Two linearized models of the center span describing the vertical and torsional oscillations are investigated. After the analysis of these models, a stability criterion is formulated. The criterion expresses a relation between the eigenvalues of the vertical and torsional oscillations of the center span. The continuous dependence of the eigenvalues on some data is studied and a stability problem for the center span is formulated. The existence of a solution to the stability problem is proved. Some other qualitative results concerning the stability/instability of oscillations are studied as well. This work was supported by Grant ASCR K1019101.     

Application of a revised cable theory to Rain-Wind-induced vibrations

Simultaneous occurrence of rain and wind can lead to vibrations with large amplitudes of inclined cables and hangers of arch bridges. These motion-induced vibrations are usually described with models based on the strip theory, or on the cable theory which is the best developed model [1]. The own solution concept derives complete nonlinear cable equations being mechanical equal to those known in literature but with a significant more simple structure by using longitudinal Green's strains and non-physical cable forces. The nonlinear effects of wind and rain forces are included. As an example of application a cable of the Erasmus bridge in Rotterdam (NL) will be analyzed. The own results are compared with the vibration amplitudes in the literature and with observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An equivalent moving force model for consideration of human-structure interaction

To predict the vibration response of footbridges, many codes of practice use a deterministic moving force (MF) model. This approach may not be well suited for the design of slender, lightweight, low-damping, and low-frequency footbridges because it ignores the pedestrian interaction with the vibrating footbridge. On the other hand, a spring-mass-damper (SMD) model is able to incorporate human mass, stiffness, and damping into the vibration response prediction. However, the SMD model is computationally demanding and not commonly available in engineering practice. To address this shortfall, a framework is proposed to derive a computationally-efficient equivalent MF-structure system to the reference SMD-structure system such that both systems give a similar vibration response metric. Analytical and numerical approaches to the equivalent MF (EMF) system are described in detail and applied to bridges with approximately simply-supported mode shapes. A sensitivity analysis is carried out to show the effects of different pedestrian parameters on the equivalent damping of the EMF system. The effects of pedestrian damping, frequency, and weight are found to be pronounced, while those of dynamic load factors and pedestrian step length are insignificant. Finally, empirical expressions are proposed in a probabilistic framework to determine the equivalent damping for simply-supported low-frequency footbridges as a function of bridge frequency. This work should find use in the serviceability assessment of low-frequency footbridges in engineering practice.     

时滞影响下受控斜拉索的参数振动稳定性

研究了轴向激励作用下受控斜拉索系统主参数共振的时滞效应,考虑了拉索垂度和几何非线性的影响,基于Hamilton变分原理建立了受控斜拉索系统轴向激励下的非线性参数振动方程,利用Galerkin方法得到时滞动力系统,运用多尺度法对受控系统的主参数共振进行了分析,得到了不同时滞值、控制增益时参数振动稳定域和受控拉索的时程曲线.研究表明,时滞影响下斜拉索振动控制系统的效果变差,参数共振的稳定域发生偏移,对受控斜拉索系统的控制效果随着时滞的增大而变差,从而对控制系统的参数设计起到指导作用.     

Nonlinear models of suspension bridges

Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.     

Mathematical Modelling of Cable Stayed Bridges: Existence, Uniqueness, Continuous Dependence on Data, Homogenization of Cable Systems

A model of a cable stayed bridge is proposed. This model describes the behaviour of the center span, the part between pylons, hung on one row of cable stays. The existence, the uniqueness of a solution of a time independent problem and the continuous dependence on data are proved. The existence and the uniqueness of a solution of a linearized dynamic problem are proved. A homogenizing procedure making it possible to replace cables by a continuous system is proposed. A nonlinear dynamic problem connected with the homogenizing procedure is proposed and the existence and uniqueness of a solution are proved.     

Prestress Losses in the Stabilizing Cables of a Composite Saddle-Shaped Cable Roof

The paper deals with a square composite saddle-shaped cable roof 30 × 30 m in the plan, which is formed by two orthogonal cable groups joined with a compliant support contour. The kinematic invariability of the roof is achieved by prestressing the cable net. From the viewpoint of material consumption, the cable roof has rational geometrical characteristics. A hybrid composite cable on the basis of CFRP, GFRP, and steel is considered as a material for contour cables. The load-carrying and the stabilizing cables are made of steel. Prestress losses in the stabilizing cables are determined for three variants of prestressing. The possibility of reducing the consumption of cable materials by prestressing each cable of the net by an individual force is revealed.     

Modeling problems of steeply inclined cableway subjected to moving load

Problems of structural modeling of multi-span bicable ropeway systems due to significant inclination of a track are discussed in this paper with reference to the vibration analysis. The analytical continuous model of a cableway is formulated in a form which enables to describe in-plane vibrations of steeply inclined carrying cable subjected to traveling passenger carriers modeled by moving pendulums. The nonlinear interaction between carrying cable and moving carriers is taken into account in the moving load definition. Effects of a steep track slope are discussed on the basis of equations of motion of the system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

带人工雨线的拉索在风激励下的响应

通过在风洞中对一具有可调动力特性、雨线位置和风向角的带人工雨线的拉索模型进行了试验,研究了拉索的风雨激振特性.试验结果和其他研究人员的结果进行了仔细对比,得到了一些新的结论,澄清了过去的一些模糊认识.结果表明,带人工雨线的水平索在风向为零时的响应可以用Den Hartog驰振机制来解释,而风向不为零时,拉索表现为限速振动或限速和驰振的混合型振动.     

Dynamic analysis of cable-driven parallel manipulators with time-varying cable lengths

In conventional researches, cables of cable-driven parallel manipulators are treated as simple linear elements that can only work in tension. This results in the fact that the effect of cable dynamics on the positioning precision of the end-effector is not adequately taken into account. To overcome this shortcoming, a dynamic model for cable-driven parallel manipulators with cables of slowly time-varying length is presented in this paper. The partial differential equation characterizing the dynamics of a cable with varying-length is deduced, and converted into ordinary differential equations through spatial discretization by finite difference approximation. Then, the dynamic model for cable-driven parallel manipulators is achieved considering the relationship between the motion of the end-effector and the cable end force, in which the degrees of freedom of cables and the end-effector are all involved. Two numerical examples are demonstrated to validate the dynamic model, and also show that it is necessary to take into consideration the cable dynamics for manipulators of long-span cables.     

Dynamic modelling of a multi-cable driven parallel platform with guiding devices

ABSTRACT

In this paper, a mathematical model is presented to numerically simulate the dynamical responses in a multi-cable suspension platform taking into account the slack cables and guiding devices. The state change of the cable (slack versus tensioned) is considered and is described mathematically by a complementary condition equation, and the interactions between the guiding wheels and the shaft wall are described by the Heaviside step function. The Lagrange’s equation with constraints is used to derive the dynamic equations of the system, and a non-smooth generalized-α algorithm for non-smooth phenomena of multibody dynamics is applied to numerically solve the equations. The simulation results have shown the dynamic responses of the platform and the cable tension characters when different cables are excited by different longitudinal excitations. Moreover, the results have illustrated how the cable tension differences may affect the pressure on the shaft wall applied by the guiding devices.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

A probabilistic description of footbridges vibration serviceability

Vibration serviceability of pedestrian bridges is considered. Imperfections in the human walking force are described as changes in walking factors. Probability distributions of walking frequencies, step length, and amplitude of walking force are taken into account. The imperfections can influence the level of footbridge vibration response. The Monte Carlo method is applied. The generated realizations of the loads formulate a set of initial data for numerical calculations. The probability distribution of maximal vertical acceleration is obtained. The results are presented as a probability that the required level of vibration will not be exceeded. A reliability describing the footbridges serviceability is also estimated. The probabilistic based approach can be used for future optimalizations of footbridge design codes due to the dynamic force loads. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

悬索桥竖向自由振动的传递矩阵解

该文将传递矩阵方法引入悬索桥竖向自由振动求解.首先,提出了悬索桥竖向自由振动微分方程的求解方法,由此求解方法可直接得到结构自由振动解析式.在此基础上,得出了悬索桥竖向自由振动分析的传递矩阵.应用此方法,该文研究了等截面悬索桥及变截面悬索桥竖向自由振动问题.研究结果表明,传递矩阵法能高效地进行悬索桥竖向自由振动分析,并对变截面问题的处理有独特的优势.研究工作对悬索桥在方案设计阶段的动力学分析有一定的参考价值和借鉴意义.     

斜拉桥拉索-阻尼器系统非线性瞬态响应分析

考虑索的抗弯刚度、垂度及几何非线性的影响,得出了索-阻尼器系统的空间非线性振动偏微分方程,用中心差分法将偏微分方程在空间内离散,导出了系统的非线性振动常微分方程组．结合Newmark法及虚拟力法提出了一种用于求解非线性振动瞬态响应的杂交分析算法．并以典型的斜拉桥拉索为研究对象,给出了数值算例,并与Runge-Kutta直接积分法进行了比较,说明了杂交算法的准确性及有效性．为大跨斜拉桥拉索的振动控制研究提供了一种简便、有效、快速的时程分析方法．     

The Index of PDAEs Applied to the Modelling of a Flexible Mechanical System

The numerical solution of systems of partial differential and algebraic equations (PDAEs) is strictly related to a property of the system, the index, whose definition and role are discussed in this paper. The notion of algebraic index is reviewed and compared to the more general notion of perturbation index. Extensions to nonlinear PDAEs are also proposed. Reference is then made to the case of a flexible mechanical system (an inextensible cable), whose model is formulated in three different, yet dynamically equivalent, ways, with different properties with respect to the feasibility of an accurate numerical integration. The methodology used in this analysis is finally formalized in an algorithm for index reduction.     

Free vibrations and stability of hydraulic cylinder fixed elastically on both ends

The boundary value problem of the stability and free vibration of a hydraulic cylinder has been formulated and solved in this paper. The considered hydraulic cylinder has been elastically fixed at both ends and loaded by Euler force. An elastical fixation has been modelled by rotational springs. The mentioned above hydraulic cylinder consists of a piston rod and cylinder replaced by rods. There are conditions of continuity between the rods. The boundary value problem has been formulated on the basis of minimum potential energy (static problem) and on the basis of Hamilton's principle (free vibration problem). In this paper example results of numerical calculations of the stability and free vibration have been presented. Experimental research has been performed in order to verify the correctness of the assumed mathematical model. Professional measuring apparatus and special stand for research into the slender systems have been used in experiment. Natural frequencies have been measured in dependence on the values of an external load. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)