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.     

In-plane antisymmetric response of cables through bifurcation under symmetric sinusoidally time-varying load

Analysis and results for in-plane non-linear antisymmetric responses of a cable, supported at the same level, through bifurcation under in-plane symmetric sinusoidally time-varying load are presented. The non-linear equation of the in-plane motion of the cable is solved by a Galerkin method and the harmonic balance method. From the computed results the frequency range, where the antisymmetric response occurs, varies with the sag-to-span ratio of the cable and is broad in the particular sag-to-span ratios. The second unstable region is important compared with the principal unstable region. Strong coupling between symmetric and antisymmetric modes is observed in the unstable regions for the particular sag-to-span ratios.     

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.     

Parametric resonance of plates in a sheet metal coating process

The vibration of sheet metal during zinc coating processes can lead to non-uniform coating thickness and overall poor product quality. This vibration develops from two principal mechanisms, namely, the run-out associated with the supporting rollers and/or bearings, and the time-varying tension associated with the manufacturing process. This study focuses on the second of these mechanisms (time-varying tension) that becomes significant under conditions leading to parametric resonance. The parametric resonance of the sheet metal is captured in a proposed model of a plate subject to time-varying and non-uniform edge tension. The model accounts for these effects as well as the non-linear stretching of the plate mid-plane as a result of transverse plate vibration. The linear vibration characteristics of the plate are studied first and are then used in deriving a single mode approximation of the non-linear, parametrically excited plate model. A perturbation solution of this model reveals the major parameters that influence parametric resonance in this application. Theoretical results for plate vibration are compared to experimental measurements of sheet metal vibration in a production facility. This comparison demonstrates that the model accurately captures the physical mechanisms responsible for sheet metal vibration and therefore, the physical parameters (such as damping) have the greatest impact on this vibration.     

Optical fiber cables using V-grooved cylindrical units: High performance cables

Some practical and theoretical aspects of a V-grooved cylindrical cable (high performance optical fiber cable) are now being developed in several countries. Design of cables is important, because bendings may cause distortions. In order to limit micro-bending losses in an optical fiber cable, it is possible to modify parameters relating to the fiber or the cable. Some of the possibilities are: to decrease the core diameter in order to decrease the attenuation coefficient; to increase the diameter of the cladding so as to increase the tension above which micro-bendings occur; to increase the continuous bending radius due to cabling; to decrease the effects of roughness (μ); to integrate the effects of roughness as much as possible by decreasing the Young modulus of the coating and of the cable material and by increasing the cladding thickness; and to suppress the tension T of the fiber in the cable. The cabling element is made of a V-grooved cylindrical core, in which the fiber with an outside diameter d_e (outside diameter or primary coating) are laid without tension, with a slight excess length. The slots with a depth h are helical or alternated helical, providing the possibility of additional excess length, and the fiber can be “cabled” directly without stresses. A central strength member with a diameter Dp reinforces the cylindrical rod, ensuring the mechanical and thermal qualities.     

Optical fiber cables using V-grooved cylindrical units: High performance cables

Some practical and theoretical aspects of a V-grooved cylindrical cable (high performance optical fiber cable) are now being developed in several countries. Design of cables is important, because bendings may cause distortions. In order to limit micro-bending losses in an optical fiber cable, it is possible to modify parameters relating to the fiber or the cable. Some of the possibilities are: to decrease the core diameter in order to decrease the attenuation coefficient; to increase the diameter of the cladding so as to increase the tension above which micro-bendings occur; to increase the continuous bending radius due to cabling; to decrease the effects of roughness (μ); to integrate the effects of roughness as much as possible by decreasing the Young modulus of the coating and of the cable material and by increasing the cladding thickness; and to suppress the tension T of the fiber in the cable. The cabling element is made of a V-grooved cylindrical core, in which the fiber with an outside diameter d_e (outside diameter or primary coating) are laid without tension, with a slight excess length. The slots with a depth h are helical or alternated helical, providing the possibility of additional excess length, and the fiber can be “cabled” directly without stresses. A central strength member with a diameter Dp reinforces the cylindrical rod, ensuring the mechanical and thermal qualities.     

辐照和电流注入下电缆耦合响应的计算

 采用传输线模型，利用时域有限差分方法计算了辐照和电流注入两种试验环境中电缆屏蔽层电流对芯线的耦合响应，并对响应规律进行了研究。计算结果表明：电流注入时近端负载电压峰值最小，辐照时次之，电流注入时远端最大；负载电压峰值、负载能量与屏蔽层电流源幅度等比例变化；相比较前沿的变化而言，改变屏蔽层电流源前沿对负载电压峰值和负载能量的影响不大；屏蔽层电流源半高宽较小时，负载电压峰值、负载能量与半高宽是非线性关系，屏蔽层电流源半高宽较大时，负载电压峰值、负载能量与半高宽成线性关系；电缆较短时，改变电缆长度对负载电压峰值有影响，而电缆较长时，只会影响电缆负载能量。     

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.     

NON-LINEAR DYNAMIC ANALYSIS OF THE TWO-DIMENSIONAL SIMPLIFIED MODEL OF AN ELASTIC CABLE

The non-linear behavior of an elastic cable subjected to a harmonic excitation is investigated in this paper. Using Garlerkin's method and method of multiple scales, the discrete dynamical equations and a set of first order non-linear differential equations are obtained. A numerical simulation is used to obtain the steady state response and stable solutions. Finally the coupled dynamic features between the out-planar pendulation and the in-planar vibration of an elastic cable are analyzed.     

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.     

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.     

The evolution of circular loops of a cosmic string with periodic tension

In this Letter the equation of circular loops of cosmic string with periodic tension is investigated in the Minkowski spacetime and Robertson–Walker universe respectively. We find that the cosmic string loops possessing this kind of time-varying tension will evolve to oscillate instead of collapsing to form a black hole if their initial radii are not small enough.     

A parametrically driven harmonic analysis of a non-linear stretched string with time-varying length

This paper is concerned with the first, second and third harmonic lateral vibrations of a string having a forced, periodic, small length variation. Account is taken of the non-linear term arising from the correction to the tension of the string due to local elongation. It is found that increasing the amplitude of the length variation of the string results in a narrowing of the frequency response range of each of the harmonic oscillation components. Both numerical and analytical calculations of the harmonic frequency responses have been carried out The relationships for initial values between the original differential equation and the approximate differential equations for the harmonic responses are discussed in detail.     

Design and analysis of a scaled model of a high-rise, high-speed elevator

A novel scaled model is developed to simulate the linear lateral dynamics of a hoist cable with variable length in a high-rise, high-speed elevator. The dimensionless groups used to formulate the scaling laws are derived through dimensional analysis. The model parameters are selected based on the scaling laws and are subject to the material, size, and hardware constraints. It is demonstrated that while it is impossible to obtain a fully scaled model unless the model is extremely tall, a reasonably sized model can be designed and the scaling laws that are not satisfied can be rendered to have a minimal effect on the scaling between the model and prototype. In conjunction with the model design, an analysis of model tension in a closed band loop is developed. A new movement profile that ensures a continuous jerk function during the entire period of motion is derived. The dynamic response of the prototype cable and that of the model band under consideration are compared numerically. Practical considerations that occur in the design of the model are addressed. The methodology can be used to investigate the vibration of a very long cable in other applications.     

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.     

On variable length induced vibrations of a vertical string

The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin?s truncation method cannot be applied in all cases to obtain approximations valid on long timescales.     

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.     

THE DYNAMICS OF A VIBROMACHINE WITH PARAMETRIC EXCITATION

The dynamics of a vibration machine with piecewise linear elastic ties under parametric harmonic excitation is investigated. Different designs of elastic elements with periodically time-varying elasticity are described. Specific non-linear features of parametric oscillations in the system under study are revealed (the invariance of parametric vibration regime to possible disturbance of phase co-ordinates, conditions of limitedness of amplitude of parametric vibrations, spectral features of non-linear parametric regimes, etc.). By the utilization of these non-linear effects, a procedure for the design of the main parameters of a parametric vibromachine is proposed.     

基于近场辐射声压信号的索力识别方法*

为准确、快速获取拉索结构索力,该文提出利用结构近场固定位置辐射声压信号对拉索进行索力测试的方法。基于动力学理论推导出结构振动加速度响应与其近场辐射声压响应的线性比例关系,并根据希尔伯特-黄变换推导了声压响应信号与结构频率的关系。对单根拉索依次施加3种不同工况的初始张拉力,并对结构中心表面处施加一瞬时脉冲力,使拉索受迫振动发声,通过对结构近场范围内拉索振动辐射声压进行测量,进而获得不同工况下不同位置声压响应信号,并探讨了在有高斯噪声干扰情况下通过希尔伯特-黄变换方法获取拉索固有频率的可靠性。数值模拟分析验证了该方法能较为准确地得到拉索固有频率并利用索力计算实用公式有效地计算出索力值,为实际工程中拉索索力测试提供了一种新的简单有效的方法。     

On the oscillatory motions of translating elastic cables

The equations of in-plane motion for an elastic catenary translating uniformly between its end supports are derived in what is, essentially, an Eulerian frame of reference. These equations are solved analytically, albeit approximately, for the case where the catenary is shallow and the tension dominated by the cable section modulus. Computed solutions for the natural frequencies and modes of a catenary of sag-span ratio 1:20 are presented and the modal characteristics are shown to be of an unusual form involving phase disparities from point to point on the cable as the cable oscillates at a natural frequency.