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.     

Multi-Zone Ice Accretion and Roughness Models for Aircraft Icing Numerical Simulation

A mathematical multi-zone ice accretion model used in the numerical simulation of icing on airfoil surface based on three water states, namely, continuous film, rivulets and beads is studied in this paper. An improved multi-zone roughness model is proposed. According to the flow state of liquid water and film flow, rivulets flow governing equations are established to calculate film mass distribution, film velocity, rivulet wetness factor and rivulet mass distribution. Force equilibrium equations of droplet are used to establish the critical conditions of water film broken into rivulets and rivulets broken into beads. The temperature conduction inside the water layer and ice layer is considered. Using the proposed model ice accretion on a NACA0012 airfoil profile with a $4^◦$ angle of attack under different icing conditions is simulated. Different ice shapes like glaze ice, mixed ice and rime ice are obtained, and the results agree well with icing wind tunnel experiment data. It can be seen that, water films are formed on the surface, and heights of the films vary with icing time and locations. This results in spatially-temporally varying surface roughness and heat transfer process, ultimately affects the ice prediction. Model simulations indicate that the process of water film formation and evolution cannot be ignored, especially under glaze ice condition.     

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.     

Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables

A study is carried out on nonlinear multimodal galloping of suspended cables. A consistent model of a curved cable-beam, geometrically nonlinear and able to torque, recently formulated by the authors, is used. The model accounts for quasi-steady aerodynamic forces, including the effect of static swing of the cable and dynamic twist of the cross-section. Complementary solution methods are employed, namely, finite-difference and Galerkin spatial discretization, followed by numerical time-integration, or Galerkin spatial discretization in conjunction with Multiple Scale perturbation analysis. The different techniques are applied to a cable close to the first cross-over point, at which a number of internal resonances exist. Branches of periodic solutions and their stability are evaluated as functions of wind velocity. The existence of branches of quasi-periodic solutions, originating from narrow unstable intervals and propagating elsewhere, is also proved. Qualitative and quantitative results furnished by the different investigation tools are compared among them, and the importance of the various components of motion, accounted or neglected in the reduced models, is discussed.     

The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge

In conventional non-linear seismic analyses of cable-stayed bridges, the non-linear characteristics of the girders, stay cables and towers are considered. The non-linearity caused by cable loosening should also be considered because a large axial force fluctuation is generated in the cables of a prestressed concrete (PC) cable-stayed bridge that is subjected to strong seismic motion. In this paper, the possibility of the cable loosening in a PC cable-stayed bridge is discussed by using a cable model that can express the cable loosening. Furthermore, the effect of the cable loosening on the responses of the cables, girder and towers is evaluated using the mean value for three seismic waves. Numerical analytic results imply that the cable loosening appears in the bottom cables of the multi-cable system and the dynamic response of the bridge is slightly increased.     

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.     

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.     

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.     

A parametric multi-body section model for modal interactions of cable-supported bridges

A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.     

Rivulet flow on the outer surface of an inclined cylinder

A study of the three-dimensional flow of a liquid film (rivulet) over the external part of an inclined cylinder was conducted for liquids with various physical properties. Patterns of the flow regimes were constructed. Good agreement is observed between the experimental data on the thickness and wall friction with the calculation with an asymptotic model in the case of a waveless rivulet. A comparison of the evolution of natural waves on rivulets with the theory of waves of maximal growth shows good agreement for small Re numbers. During the experiments, the wave characteristics of excited waves on a rivulet were investigated. The thickness, amplitude, frequency, and phase velocity of the waves over a wide range of variable parameters are given. Phase velocity integrated functions of the amplitude are constructed for various liquids. The friction on the cylinder wall is measured in the presence of natural and excited waves. The effects of wave regimes on the average values and RMS (root-mean-square) friction pulsations are studied.     

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.     

考虑风剪切的漂浮式风力机气动特性研究

波浪与风载的共同作用下,海上风电机组平台存在多自由度运动,同时伴随着海上复杂的风况,其气动特性变化较为复杂。以NREL5MW风电机组为研究对象,在风剪切来流下,依据波浪和风载的作用规律,研究平台纵摇和纵荡运动对机组气动特性和绕流场细节的影响。结果表明:平台运动会造成风电机组气动性能的周期性波动,而风剪切作用使得风电机组平均发电量降低并加剧功率和推力的波动。风剪切会增大纵摇运动过程中展向截面的法向载荷幅值且波动加剧,但会降低与功率输出有关的切向受力;风剪切作用推迟了纵荡运动过程中展向截面法向和切向载荷峰值出现时刻,加剧载荷的波动。外叶展截面的法向和切向力系数曲线出现平台,加剧了叶片疲劳载荷,减小了功率输出。     

Experimental study of the nonlinear dynamic characteristics of suspended taut steel cables using a 3-D motion analysis system

This paper presents an experimental study of the nonlinear dynamic characteristics of taut steel cables using a 3-D motion analysis system. In the experiment, the taut cables have one end fixed and the other end subject to harmonic vertical excitation. The 3-D motion analysis system can simultaneously record (with high resolution) the instant 3-D coordinates of the multiple markers fixed on a vibrating cable; this distinguishes it from other experimental systems used in vibration studies, in which the vibration of only one single point can be recorded during each individual testing. With the 3-D motion analysis system, this experimental study presents a distinctive interpretation of the dynamic characteristics of taut cables in spatial domain (based on the mode-shape information of the entire cable), in addition to one in time domain (based on real-time traces of one single point). This paper introduces the 3-D motion analysis system and experimental setup, discusses practical experimental procedures, and presents a detailed analysis of three sets of experimental vibration data of three taut steel cables with different small sags. The frequency response curves were obtained for three cables. For one of the three taut cables, more informative vibration data were recorded; this cable was studied in greater detail via modal analysis using a modal decomposition technique and nonlinear time-series analysis.     

Lightning-induced currents in buried coaxial cables: A frequency-domain approach and its validation using rocket-triggered lightning

The aim of this paper is to present an experimental validation of a frequency-domain approach to the solution of the lightning electromagnetic field-to-buried cable coupling equations. The coupling to the inner conductor is evaluated using the concept of cable transfer impedance. The theoretical model and relevant computer code are tested using experimental data on lightning-induced currents in buried cables carried out in 2002 and 2003 at the International Center for Lightning Research & Testing (ICLRT) at Camp Blanding, Florida where currents induced by triggered lightning events were measured at the ends of a buried coaxial cable, both in the shield and in the inner cable conductors. Reasonably good agreement has been found between numerical simulations and recorded waveforms. In particular, the early-time response of the cable and the peak value of the induced currents were generally well reproduced by the simulations.     

The effect of wave characteristics on rivulet formation in heated liquid films

Formation of rivulets on the surface of non-isothermal water film falling vertically over the heaters with different sizes and boundary conditions was studied experimentally. The distances between rivulets were measured depending on Reynolds number, heat flux density and film path (a distance between the lower edge of a film-former nozzle and the measurement point of film flow characteristics). The breakdown of solitary waves at liquid film heating was revealed. Four zones of film path influence on rivulet formation were distinguished.     

Investigation of a hybrid approach combining experimental tests and numerical simulations to study vortex-induced vibration in a circular cylinder

In this study, a hybrid approach based on computational fluid dynamics (CFD) was used to investigate the aerodynamic forces associated with vortex-induced vibration (VIV) in a circular cylinder. The circular cylinder and the flow field were considered as two substructures of a system. Circular cylinder motion was produced in a wind tunnel test of the VIV prior to the numerical simulation; this motion was used as a known cylinder boundary condition and applied to the flow field. The flow field with the known moving boundary condition was then numerically simulated by the ANSYS CFX code. The transient aerodynamic coefficients of the circular cylinder with predetermined motion were obtained from the numerical simulation. To verify the feasibility and accuracy of the proposed hybrid approach and to calculate cylinder vibrations, the transient aerodynamic coefficients were applied to a single degree of freedom (SDOF) model of the circular cylinder. The oscillation responses of the circular cylinder from the calculated (SDOF model) and experimental results were compared, and the results indicate that the hybrid approach accurately simulated the transient aerodynamic coefficients of the circular cylinder. For further comparison, a nonlinear aerodynamic coefficient model based on a nonlinear least square technique was applied to the SDOF model. The nonlinear aerodynamic model can predict well the amplitude and lock-in region of the VIV of the circular cylinder model.     

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.     

Wall effect on fluid–structure interactions of a tethered bluff body

Wind tunnel experiments have shown an unexplained amplification of the free motion of a tethered bluff body in a small wind tunnel relative to that in a large wind tunnel. The influence of wall proximity on fluid–structure interaction is explored using a compound pendulum motion in the plane orthogonal to a steady freestream with a doublet model for aerodynamic forces. Wall proximity amplifies a purely symmetric single degree of freedom oscillation with the addition of an out-of-phase force. The success of this simple level of simulation enables progress to develop metrics for unsteady wall interference in dynamic testing of tethered bluff bodies.     

Determination of the axial force on stay cables accounting for their bending stiffness and rotational end restraints by free vibration tests

Determination of the axial force in terms of its natural frequencies may be significantly influenced by the bending stiffness of the cable and the rotational elastic restraints at the ends, depending on the geometrical and mechanical parameters of the cable and its supports and restraints, particularly in cement-grouted parallel-bundle wire cables. The paper presents an explicit analytical expression for the natural frequencies taking into account both the bending stiffness of the cable and the rotational restraint at the ends that may be used to determine the axial force. While the bending stiffness of the cable and the axial force are selected as variables to attain an optimal match between analytical and experimental data, the rotational stiffness at the ends is treated as a known parameter in that process. The degree of rotational restraint at the ends cannot be accurately inferred from the sequence of the experimentally determined natural frequencies, since this parameter does not appreciably affect the progression of their values. Techniques are discussed that allow approximate determination of the rotational stiffness at the ends for the most common arrangements of anchors and cables with, and without, intermediate supports provided by deviators located near the ends. The axial force and the bending stiffness of the cable are both simultaneously adjusted by matching the natural frequencies of the analytical model with the experimental values. The proposed approach leads to a reduction of the error in the estimation of the axial force for short cables with relatively high bending stiffness such as those typical of cement-grouted parallel-bundle wire cables often used as cable stays for bridges until the early 1990s.     

Transient vibration phenomena in deep mine hoisting cables. Part 2: Numerical simulation of the dynamic response

A simulation model is presented which investigates the dynamic response of a deep mine hoisting cable system during a winding cycle. The response, namely the lateral motions of the catenary cable and the longitudinal motion of the vertical rope with conveyance is observed on the fast time scale, and the slow time scale is introduced to monitor the variation of slowly varying parameters of the system. The cable equivalent proportional damping parameters, and periodic excitation functions resulting from the cross-over cable motion on the winder drum are identified. Subsequently, the model is solved numerically using parameters of a double-drum multi-rope system. Since the system eigenvalues are widely spread and the problem is of stiff nature, the numerical simulation is conducted using a stiff solver. The results of the simulation demonstrate various transient non-linear resonance phenomena arising in the system during the wind. The nominal ascending cycle simulation results reveal adverse dynamic behaviour of the catenary largely due to the autoparametric interactions between the in- and out-of-plane modes. Principal parametric resonances of the lateral modes also occur, and conditions for autoparametric interactions between the lateral and longitudinal modes arise. Additionally, a transition through a number of primary longitudinal resonances takes place during the wind. The adverse dynamic motions in the system promote large oscillations in the cable tension which must be considered significant with respect to fatigue of the cable. It is noted that a small change in the winding velocity may cause large changes in the dynamic response due to the resonance region shifts. Consequently, the resonance modal interactions can be avoided, to a large extent, if the winding velocity is increased to an appropriate level.