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.     

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.     

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.     

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.     

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.     

Effects of cables damage on vertical and torsional eigenproperties of suspension bridges

This paper presents a continuum model for the nonlinear coupled vertical and torsional vibrations of suspension bridges with arbitrary damage in one main cable and, after pursuing a suitable linearization of the equations of motion, an investigation of damage effects on modal parameters. Damage is modeled as a diffused loss of cross-section representing the typical effect of fretting fatigue and it is introduced in the formulation by enforcing relevant literature results providing analytical solution for the static response of damaged suspended cables. The coupled nonlinear equations of motion of the damaged bridge, including the effects of shear deformation, rotary inertia and warping of the cross-section of the girder, are derived by application of Hamilton?s principle. In this way, the equations of motion available in the literature for undamaged suspension bridges are generalized to the presence of arbitrary damage in one main cable and the resulting eigenfrequencies and eigenfunctions are derived in an analytical fashion. An extensive parametric investigation is finally presented to discuss damage effects on eigenfunctions and eigenfrequencies under variation of practically meaningful parameters.     

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.     

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.     

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.     

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.     

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.     

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.     

Nonlinear longitudinal oscillations of relativistic plasma

The investigation has been made of nonlinear forced longitudinal oscillations of a relativistic plasma. It was demonstrated that: a) the nonlinear terms (v▽)p arising in the equations of motion do not restrict the electron oscillation amplitude increase in the region of plasma resonance; b) the nonlinear terms due to the relativistic nature of oscillations, firstly, limit the oscillation amplitude in the resonance region and, secondly, lead to the development of a parametric instability. The harmonic, sub- and ultraharmonic oscillations turned out to be unstable. The conditions for the instability as well as expressions for the growth rates of unstable oscillations were obtained.     

弹性微管内气泡的非线性受迫振动

将弹性管壁视为膜弹性结构, 探索在外部声场作用下弹性微管内液柱-气泡-管壁构成耦合振动系统的非线性特征. 利用逐级近似法对系统非线性共振频率、基频和三倍频振动幅值响应、 分频激励共振机理等进行了理论分析. 基频和三倍频振动的幅-频响应数值结果表明: 气泡的轴向共振和管壁共振不能同时出现; 两垂直方向的振动均表现出幅值响应多值性, 进而可能引起系统的不稳定声响应; 三倍频振动在低频区的声响应强于高频区. 关键词： 弹性微管 受迫振动 非线性振动 气泡声响应     

Parametric resonance in neutrino oscillations in matter

Neutrino oscillations in matter can exhibit a specific resonance enhancement — parametric resonance, which is different from the MSW resonance. Oscillations of atmospheric and solar neutrinos inside the earth can undergo parametric enhancement when neutrino trajectories cross the core of the earth. In this paper we review the parametric resonance of neutrino oscillations in matter. In particular, physical interpretation of the effect and the prospects of its experimental observation in oscillations of solar and atmospheric neutrinos in the earth are discussed. On leave from National Research Centre, Kurchatov Institute, Moscow 123182, Russia     

Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

This paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling.     

Dynamics of a parametric exciton-polariton oscillator in a microcavity

We study the dynamics of parametric oscillations of polaritons in a microcavity that consists of a periodic conversion of a pair of pump polaritons into polaritons of signal and idle modes and vice versa. The period and amplitude of oscillations considerably depend on the initial polariton density, the initial phase difference, and the resonance detuning. We show that there is a possibility of phase controlling the polariton dynamics in the microcavity.     

磁声参量振荡的理论

在本文中,我们从磁-弹性耦合的宏观表达式,通过经典场论的方法,求得弹性振动和磁振璗的耦合方程,用来分析了伴随波长约等于铁氧体样品的线度的声振动而存在的磁振璗(磁声模)。文中指出,Spencer和LeCraw所发现的磁声效应是磁声模和静磁模在注入场的激发下产生的参量振璗现象(也可以说是热声子的电磁讯号的放大)。我们引用Berk等人在讨论一种半静磁操作放大器的文章中给出的公式,算出Spencer-LeCraw实验所需要的功率,其结果与记录的数据相接近。我们提出了使任一静磁模配合磁声模产生振璗的调谐条件以及降低激发功率和观测几十到几百兆赫的声频的办法。通过磁声模和静磁模的交变场向量的空间对称性的分析,我们推导出磁声参量振璗的选择定则:对于球体三个主要弹性振动模(旋转模、向径模和椭球模),(1)静磁模(n,m,r)的Walker指标n是偶数者不产生磁声效应;(2)指标m是奇数者不与旋转模产生磁声效应,m是偶数者不与向径模或椭球模产生磁声效应。我们也举出第一类本征振动中有只可能和n是偶数、m是奇数的静磁模产生参量振璗的例子。Spencer-LeCraw局限于使静磁场调谐在(110)模上,所观察到的现象仅仅是本文所给出理论预见的一个特殊情况。他们发现了椭球模和向径模的频率显著地出现,但并无旋转模的频率,这是上述的选择定则的具体验证。最后,我们指出,热声子的参量放大可形成铁氧体微波放大器的噪声的来源。     

Mechanism of resonance and cancellation for train-induced vibrations on bridges with elastic bearings

In this paper, the mechanism involved in the phenomena of resonance and cancellation in the train-induced vibrations of railway bridges with elastic bearings is explained using an analytical approach. The train is modelled as a sequence of moving loads of constant intervals. The vibration shape of the elastically supported beam is approximated by the combination of a flexural sine mode and a rigid body mode. The present results indicate that under certain conditions, resonances of much higher peaks can be excited on elastically supported beams by moving trains at much lower speeds than those on simply supported beams. The cancellation is a phenomenon more decisive than that of resonance, in that it can suppress the latter even when the condition of resonance is met. Moreover, the speed for cancellation to occur is generally independent of the support stiffness. To verify the analytical results presented herein, a field test was conducted on two adjacent elastically supported bridges in existing railway lines. In the design of railway bridges, it is important that the phenomenon of resonance not be overlooked, as it is harmful not only to the riding comfort of passengers, but to the maintenance of railway tracks.     

Dynamic gas anisotropy under optomechanical parametric resonance

An approximate kinetic equation describing a gas of two-or three-level atoms in an external laser field is found under conditions of optomechanical parametric resonance, which causes atomic oscillations along the field wave vector. The approximation is applicable to a gas of cylindrical particles whose axes retain their spatial orientation, making the gas anisotropic. The closed set of equations for macroparameters allows for a numerical solution. Expressions for direction-dependent transfer coefficients such as diffusion, viscosity, and thermal conductivity are derived phenomenologically. A two-temperature medium is shown to arise if a fluid consists of several gases with the optomechanical parametric resonance conditions satisfied for one of them.