Non-linear vibrations of cables considering loosening

A cable cannot resist the axial compressive force that may be induced during large amplitude vibrations. In this paper, the effect of cable loosening on non-linear vibrations of flat-sag cables is discussed by using the finite difference method that can express cable loosening. In the present method, flexural rigidity and damping of the cable are considered in the equations of motion of a cable in order to handle the numerical instability. The effect of cable loosening is evaluated explicitly in the present paper. Furthermore, non-linear vibration properties are evaluated for various parameters under periodic and step vertical loading. The effect of cable loosening on response under vertical periodic time-varying load is small and it is possible for the sag-to-span ratio to roughly equal the ratio for modal transition. The loosening under the vertical step loading in the direction opposite to the gravity appears at almost the same sag-to-span ratio.     

Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge

The in-plane vibration of a complex cable-stayed bridge that consists of a simply-supported four-cable-stayed deck beam and two rigid towers is studied. The nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of the cables and transverse vibrations of segments of the deck beam, respectively, are derived, along with their boundary and matching conditions. The undamped natural frequencies and mode shapes of the linearized model of the cable-stayed bridge are determined, and orthogonality relations of the mode shapes are established. Numerical analysis of the natural frequencies and mode shapes of the cable-stayed bridge is conducted for various symmetrical and non-symmetrical bridge cases with regards to the sizes of the components of the bridge and the initial sags of the cables. The results show that there are very close natural frequencies when the bridge model is symmetrical and/or partially symmetrical, and the mode shapes tend to be more localized when the bridge model is less symmetrical. The relationships between the natural frequencies and mode shapes of the cable-stayed bridge and those of a single fixed–fixed cable and the single simply-supported deck beam are analyzed. The results, which are validated by commercial finite element software, demonstrate some complex classical resonance behavior of the cable-stayed bridge.     

Response characteristics of local vibrations in stay cables on an existing cable-stayed bridge

This paper examines local parametric vibrations in the stay cables of a cable-stayed bridge. The natural frequencies of the global modes are obtained by using a three-dimensional FE model. The global motions generated by (1) sinusoidal excitations using exciter, (2) a traffic loading, and (3) an earthquake are analyzed by using the modal analysis method or the direct integration method. The local vibration of stay cable is calculated by using a model in which inclined cable is subjected to time-varying displacement at one support during global motions. This paper describes the properties of the local vibrations in stay cables under these dynamic loadings by using an existing cable-stayed bridge.     

Parametric-resonance-induced cable vibrations in network cable-stayed bridges. A continuum approach

There is a wealth of evidence to suggest that the bearing cables of cable-stayed bridges may experience large-amplitude oscillations, attributed in general to parametric resonance with the girder vibrations. A common coutermeasure consists of connecting the principal stays together with secondary cables to form a network and, here, optimal cable arrangments will be discussed when such a network is uniform and triangular meshed. The present approach is qualitative, and basically consists of homogenizing the cable net to an orthotropic elastic membrane, and then considering an auxiliary structure where the bridge girder, instead of being supported by the cable network, is supported by wedge-shaped membranes. The elastic solution under uniformly distributed loads, found using Lekhnitskii's approach, is the starting point for the discussion of the system in dynamic equilibrium. Having established a correspondence between the cable-net size and shape and the elastic moduli of the homogenized membrane, simple formulas are obtained to describe the global bridge vibration, as well as the local oscillations of the cables. It is then possible to estimate the girder and cable-net characteristic frequencies, to evaluate those conditions possibly leading to parametric resonance and, with respect to these variables, to determine optimal cable arrangements. This method is finally applied to the paradigmatic example of the Normandy Bridge.     

DYNAMIC MODELLING AND VIBRATION ANALYSIS OF A FLEXIBLE CABLE-STAYED BEAM STRUCTURE

In this paper, the non-linear vibration of a cable-stayed beam with time-varying length and tension in the cable is investigated. A set of non-linear, time-varying differential equations describing this coupling system is derived by Hamilton's principle and the finite element method. According to the results of numerical simulation, the tension of the cable is related to the cable length, which in turn is a function of the longitudinal and transverse displacements of the cable. Furthermore, it is shown that the tension and length of the cable can be considerably different by using linear and non-linear models.     

拉索锚固系统隐蔽部分安防预警检测装置的设计

斜拉桥锚固区拉索索体隐蔽在预埋管内,肉眼无法检测其损伤情况。搭建了一套基于高分辨率的视频采集检测技术装置,实现了隐蔽段拉索索体的远距离观测和狭小空间内的外表损伤检测。系统主要由光学支架与精密机械支撑装置、光学成像系统、照明光源、驱动电机控制、IP摄像机视频采集系统等部分组成。设计了一套3个不同焦距（长焦、中焦和短焦）的望远物镜,通过转换器实现在不同监控距离上的适配选用,物镜系统与照明系统远分离,使成像光路有效地避开隐蔽部分的恶劣环境。实践证明,该装置能有效采集到清晰的索体外表面监控视频图像,可用于预应力构筑物安防预警等方面。     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.     

DYNAMIC ANALYSIS OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT FLEXURAL RIGIDITY

For the purpose of developing a vibration-based tension force evaluation procedure for bridge cables using measured multimode frequencies, an investigation on accurate finite element modelling of large-diameter sagged cables taking into account flexural rigidity and sag extensibility is carried out in this paper. A three-node curved isoparametric finite element is formulated for dynamic analysis of bridge stay cables by regarding the cable as a combination of an “ideal cable element” and a fictitious curved beam element in the variational sense. With the developed finite element formulation, parametric studies are conducted to evaluate the relationship between the modal properties and cable parameters lying in a wide range covering most of the cables in existing cable-supported bridges, and the effect of cable bending stiffness and sag on the natural frequencies. A case study is eventually provided to compare the measured natural frequencies of main cables of the Tsing Ma Bridge and the computed frequencies with and without considering cable bending stiffness. The results show that ignoring bending stiffness gives rise to unacceptable errors in predicting higher order natural frequencies of the cables, and the proposed finite element formulation provides an accurate baseline model for cable tension identification from measured multimode frequencies.     

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.     

Bifurcation analysis of a parametrically excited inclined cable close to two-to-one internal resonance

This paper presents a study of how different vibration modes contribute to the dynamics of an inclined cable that is parametrically excited close to a 2:1 internal resonance. The behaviour of inclined cables is important for design and analysis of cable-stayed bridges. In this work the cable vibrations are modelled by a four-mode model. This type of model has been used previously to study the onset of cable sway motion caused by internal resonances which occur due to the nonlinear modal coupling terms. A bifurcation study is carried out with numerical continuation techniques applied to the scaled and averaged modal equations. As part of this analysis, the amplitudes of the cable vibration response to support inputs is computed. These theoretical results are compared with experimental measurements taken from a 5.4 m long inclined cable with a vertical support input at the lower end. In general this comparison shows a very high level of agreement.     

Applications of optical fibre Bragg gratings sensing technology-based smart stay cables

Stay cable is one of the most critical structural components of a bridge. However, it readily suffers from fatigue damage, corrosion damage, and their coupled effects. Thus, health monitoring of stay cables is important for ensuring the integrity and safety of a bridge. A smart stay cable assembled with optical fibre Bragg grating (OFBG) strain and temperature sensors was proposed in this study. To protect the OFBG sensors against breakage in application, the OFBG sensors were first incorporated into a glass-fibre-reinforced polymer (GFRP) bar (GFRP-OFBG bar) when the bar was fabricated. To fabricate cables assembled with OFBG sensors, several GFRP-OFBG bars were inserted into the hollows of steel wires and fixed with the steel wires together at the anchorages of the cable. Therefore, the GFRP-OFBG bars can consistently deform with the steel wires in a cable and the smart stay cable can sense its own strain and temperature through OFBG sensors. The fabrication procedure of the smart stay cable was developed and the self-sensing property of the smart stay cable was calibrated. Finally, the application of the smart stay cables on the Tianjing Yonghe Bridge was demonstrated. The fatigue accumulative damage of the smart stay cables was evaluated based on field monitoring strain.     

OPTIMAL DAMPING OF STAYS IN CABLE-STAYED BRIDGES FOR IN-PLANE VIBRATIONS

Significant vibrations have been reported in stays of recently constructed cable stayed bridges. The vibrations appear as in-plane vibrations that may be caused by rain-wind- induced aeroelastic interaction or by resonance excitation of the cables from the motion of the pylons. The stays of modern cable-stayed bridges are often designed as twin cables with a spacing of, say 1m. In such cases, it is suggested in the paper to suppress the mentioned in-plane types of vibrations by means of a tuned mass-damper (TMD) placed between the twin cables at their midpoints. The TMD divides the stay into four half-cables, and resonance may occur in each of the half-cables as well as in the entire stay. The optimal tuning of the TMD is investigated based on a mathematical model, where the motion of the support points on the pylons is considered to be the main cause of excitation. The indicated motion is modelled as a band-limited Gaussian white noise process. Three load scenarios are considered: narrow-banded excitations, with the central frequency of the autospectrum close to the lowest eigenfrequency of each of the two cables constituting the stay, and a broadbanded excitation which encompasses both of the mentioned frequencies. The spring and the damper constants of the TMD are optimized so that the variances of the displacement of the adjacent four half-cables, the support point of the TMD and the secondary mass are minimized. At optimal design, it is shown that the variances reduce below 14% of those of the unprotected stay.     

A stochastic approach to cable dynamics with moving rivulets

This article constructs a stochastic model for the response of stay cables of cable-stayed bridges to the combined effect of wind and rain. It describes a spring-mounted section model of a stay cable in a steady wind where aerodynamic forces are modified by the dynamics of a mobile liquid rivulet. The motion of the rivulet is described by a simple stochastic process that, together with aerodynamic forces, models the complex fluid-structure interaction. Based on measured data for drag and lift coefficients and a static rivulet location, an analysis of the model suggests a new stochastic excitation mechanism for the rain-wind induced vibrations of stay cables.     

In-plane free vibration analysis of cable-arch structure

Cable-stayed arch bridge is a new type of composite bridge, which utilizes the mechanical characters of cable and arch. Based on the supporting members of cable-stayed arch bridge and of erection of arch bridge using of the cantilever construction method with tiebacks, we propose a novel mechanical model of cable-arch structure. In this model, the equations governing vibrations of the cable-arch are derived according to Hamilton's principle for dynamic problems in elastic body under equilibrium state. Then, the program of solving the dynamic governing equations is ultimately established by the transfer matrix method for free vibration of uniform and variable cross-section, and the internal characteristics of the cable-arch are investigated. After analyzing step by step, the research results approve that the program is accurate; meanwhile, the mechanical model and method are both valuable and significant not only in theoretical research and calculation but also in design of engineering.     

Modeling “unilateral” response in the cross-ties of a cable network: Deterministic vibration

Cross-ties are employed as passive devices for the mitigation of stay-cable vibrations, exhibited on cable-stayed bridges under wind and wind-rain excitation. Large-amplitude oscillation can result in damage to the cables or perceived discomfort to bridge users. The “cable-cross-ties system” derived by connecting two or more stays by transverse cross-ties is often referred to as an “in-plane cable network”. Linear modeling of network dynamics has been available for some time. This framework, however, cannot be used to detect incipient failure in the restrainers due to slackening or snapping. A new model is proposed in this paper to analyze the effects of a complete loss in the pre-tensioning force imparted to the cross-ties, which leads to the “unilateral” free-vibration response of the network (i.e., a cross-tie with linear-elastic internal force in tension and partially inactive in compression).     

Transient vibration phenomena in deep mine hoisting cables. Part 1: Mathematical model

The classical moving co-ordinate frame approach and Hamilton's principle are employed to derive a distributed-parameter mathematical model to investigate the dynamic behaviour of deep mine hoisting cables. This model describes the coupled lateral-longitudinal dynamic response of the cables in terms of non-linear partial differential equations that accommodate the non-stationary nature of the system. Subsequently, the Rayleigh-Ritz procedure is applied to formulate a discrete mathematical model. Consequently, a system of non-linear non-stationary coupled second order ordinary differential equations arises to govern the temporal behaviour of the cable system. This discrete model with quadratic and cubic non-linear terms describes the modal interactions between lateral oscillations of the catenary cable and longitudinal oscillations of the vertical rope. It is shown that the response of the catenary-vertical rope system may feature a number of resonance phenomena, including external, parametric and autoparametric resonances. The parameters of a typical deep mine winder are used to identify the depth locations of the resonance regions during the ascending cycles with various winding velocities.     

桥梁拉索损伤声发射监测研究进展

大量索承体系桥梁即将达到20–30年的拉索寿命期,开展有效的拉索损伤监测方法研究有利于保障大桥结构安全。本文简要论述了现有桥梁拉索损伤检测及监测方法的适用性,重点综述了近40年来声发射技术应用于桥梁拉索监测的研究进展,以及在腐蚀、疲劳、断丝等损伤监测方面所取得的研究成果。结合实测数据在采集、数据处理过程中的难点及参数分析盲点,探讨了现有研究成果应用于特大桥梁拉索损伤监测仍需解决的问题,并针对性地提出了研究思路。     

Eigenproperties of suspension bridges with damage

The vertical vibration of suspension bridges with a damage in the main cables is studied using a continuum formulation. Starting from a model for damaged suspended cables recently proposed in the literature, an improved expression for the dynamic increment of cable tension is derived. The nonlinear equation of motion of the damaged bridge is obtained by extending this model to include the stiffening girder. The linear undamped modal eigenproperties are then extracted, in closed-form, from the linearized equation of motion, thus generalizing to the presence of an arbitrary damage the expressions known from the literature for undamaged suspension bridges. The linear dynamics of the damaged bridge reveals to be completely described by means of the same two non-dimensional parameters that govern the linear dynamics of undamaged bridges and which account for the mechanical characteristics of both the main cable and the girder, with the addition of three non-dimensional parameters characterizing damage intensity, position and extent. After presenting the mathematical formulation, a parametric analysis is conducted with the purpose of investigating the sensitivity of natural frequencies and mode shapes to damage, which, in fact, is a crucial point concerning damage detection applications using inverse methods. All through the paper, systematic comparisons with finite element simulations are presented for the purpose of model validation.     

Snap loads and torsional oscillations of the original Tacoma Narrows Bridge

In 1940, the original Tacoma Narrows Bridge was completed on June 10 and opened to traffic on July 1. On November 7, the deck collapsed. Before that day, significant vertical oscillations had occurred, but no torsion. The bridge as built was stable with respect to torsional motion under the winds of November 7 and previous winds with higher speeds. However, snap loads in the diagonal ties attached to the north midspan cable band helped to loosen the band, and the frictional resistance between the band and the north suspension cable passing through it was overcome. The cable began to slip through the band. For this new structural system, with longitudinal motion of the north cable, the wind speed was higher than the critical speed for torsional flutter, and torsional motion was initiated. Approximately 700 cycles of torsional oscillations occurred during the hour prior to the collapse. In the present study, the snap loads on the cable band are discussed first. Then a continuum model of the central span (deck, cables, and hangers) is formulated. The longitudinal motions of the cables are included, so that the slippage can be incorporated. Known information from the observed steady-state torsional motion is utilized with assumed forms of the vertical cable displacements, and the governing equations provide the horizontal cable displacements, the dynamic tensions in the cables, the vertical and torsional motions of the deck, and the resultant lift force and pitching moment (including damping) acting on the deck during its final hour.