Transient vibrations of a tau inclined cable with a riding accelerating mass

The objective of this paper is a theoretical and numerical study of the dynamics of a taut inclined cable with a riding accolerating mass, which is suspended from two points of different elevation. The moving mass is a trolley that is accelerated by a solid fuel rocket down the inclined cable and is aerodynamically brought to a halt. The thrust of the rocket is tangential to the deformed configuration of the cable.Methods of analysis consist of the dynamics of small deformations superimposed on the static catenary state. The problem is nonlinear due to presence of friction and the convective acceleration interaction of the moving mass and the cable. Galerkin's procedure for removal of spatial dependence and numerical integration are used to obtain convergent solutions.Deceased     

Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.     

风作用下覆冰悬索的非平面非线性动力学

主要研究侧向风载荷作用下小垂度覆冰悬索的非线性非平面运动的复杂动力学.根据分析力学、弹性力学和空气动力学理论,建立覆冰悬索3个自由度非线性振动的偏微分运动方程,并对其进行无量纲化,运用Galerkin方法对偏微分运动方程进行离散得到3个自由度的常微分方程,再利用多尺度法得到面内主共振2:1内共振的平均方程.利用数值方法研究悬索的非线性运动,结果表明系统呈现周期、多倍周期、概周期和混沌运动的规律.     

Steady state responses of poroelastic half-space soil medium to a moving rectangular load

The dynamic steady state responses of a poroelastic half-space soil medium subjected to a moving rectangular load are investigated analytically/numerically. The full dynamic poroelastic theory of Biot is employed, under the assumption of an incompressible solid grain and neglecting the apparent mass density. Using the Fourier transform, the governing equations of motion are then reduced to a system of four coupled ordinary differential equations which are solved semi-analytically. Soil vertical displacements, accelerations and pore water pressures induced by moving load are calculated. Computed result shows that load velocity and intrinsic permeability of the soil medium shows an apparent effect on its dynamic responses and pore water pressures.     

行进绳索在横向流体激励下的运动

给出横向流体对行进绳索的作用力描述,建立了绳索的动力学方程。由于该方程具有零刚度特征,引入Pilipchuk变换,以自平衡状态起度量的径向振动和回转运动来描述绳的运动,获得非零刚度系统。然后,用两变量参数摄动法求得绳索关于扩张振动和回转运动的约化型,使得绳索运动可近似由一个二维动力系统来描述。最后,用数值方法讨论了行进速度和重力对绳索运动形态的影响。     

Modelling and transient planar dynamics of suspended cables with moving mass

In this study, the 3D nonlinear equations of motion of the suspended cable with moving mass are obtained via the Hamilton principle, and its transient linear planar dynamics is investigated. Considering the quasi-static assumption, the condensed planar model accounting for the effect of the moving mass is derived, and it is then discretized by choosing the static deflection and sine series as shape functions. It is shown that this expansion shows good convergence features. The Newmark method is used to investigate the transient response. The effects of the inertia force, mass, sag and velocity of the moving mass on the transient dynamics of the suspended cable are systematically investigated. Finally, the horizontal tension of the suspended cable and the case of sequentially moving masses are examined.     

Coupled global dynamics of an axially moving viscoelastic beam

The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincaré maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.     

悬索桥负载移梁过程静力学分析与试验研究

针对轨索移梁新工艺在悬索桥建设中的关键技术问题进行了理论分析和整体模型试验研究。根据柔索结构基本假定,建立了轨索移梁系统负载状态下的整体力学分析模型,并推导了主缆、吊索和轨索等各构件受力状态的计算方程。理论模型中,移梁小车所在轨索节段采用4个直线索单元模拟,其他位置的轨索节段采用1个直线索单元模拟。以矮寨悬索桥为工程背景,设计制作了轨索移梁体系的整体缩尺模型用以模拟移梁过程,测试了第一段梁从入轨到安装多个子工况的体系响应。理论计算结果与模型试验结果吻合良好,表明本文的计算方法正确有效,能解决轨索移梁工艺负载移梁过程中体系整体受力分析和轨索局部变形分析求解问题。该方法可简化计算过程,计算精度和结果能够应用于工程计算分析和求解,是一种适合于轨索移梁工艺负载状态下体系的分析方法。     

Finding vibrations of inclined cable structures by approximately solving governing equations for exterior matrix

In this paper, how to compute the eigenfrequencies of the structures composed of a series of inclined cables is shown. The physics of an inclined cable can be complicated, so solving the differential equations even approximately is difficult. However, rather than solving the system of 4 first-order equations governing the dynamics of each cable, the governing equations are instead converted to a set of equations that the exterior matrix satisfies. Therefore, the exterior matrix method (EMM) is used without solving the original governing equations. Even though this produces a system of 6 first-order equations, the simple asymptotic techniques to find the first three terms of the perturbative solution can be used. The solutions can then be assembled to produce a 6 × 6 exterior matrix for a cable section. The matrices for each cable in the structure are multiplied together, along with the exterior matrices for each joint. The roots of the product give us the eigenfrequencies of the system.     

Three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot with non-negligible cable mass and elasticity

This study investigates the three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot by considering cable mass, elasticity, and mass of end-effector. According to these models, optimization of cable tensions and cable lengths using fminimax solver is performed to determine the static stiffness. Static and dynamic stiffness of the cables are obtained with simulations. Results show that analyses in three-dimension are very important to measure the actual performance of the cable driven parallel robot. This demonstrates potential for general applicability and motion of the cable driven parallel robot.     

温度变化对端部激励斜拉索共振响应影响

基于增量热场理论,利用Hamilton变分原理,通过引入与张拉力和垂度相关的无量纲参数,建立了考虑温度变化影响下斜拉索非线性动力学模型,并推导其面内/外非线性运动微分方程。考虑斜拉索受端部激励,利用Galerkin法得到离散后的无穷维常微分方程组。面内和面外运动各取前两阶模态,向前和向后扫频,利用龙格-库塔法数值积分求解常微分方程组,得到共振区域的幅频响应曲线。算例分析表明,温度变化和斜拉索固有频率呈反比例关系;温度变化会导致斜拉索共振特性发生定性和定量的改变,如共振区间发生漂移、跳跃点位置发生移动、共振响应幅值发生改变;端部位移激励下,温度变化有可能导致斜拉索更多模态受到激发,从而影响各阶模态的能量以及模态间的能量传递。     

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先基于哈密顿变分原理，引入与损伤程度、范围和位置相关的三个无量纲参数，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．     

斜拉桥索力优化及其工程应用

本文首先介绍斜拉桥索力优化的概念，然后通过广义影响矩阵，介绍了一种实用的斜拉桥索力优化的影响矩阵法，这种方法既可用于确定成桥合理状态的索力，也可用于施工阶段的索力优化和成桥后的索力调整，实现程序化计算十分方程，最后给出其工程应用。     

一种连续索滑移的处理方法

目的是建立一种快速简便的拉索滑移问题的有限元计算方法。采用基于弹性悬链线精确解的两节点悬链线单元模拟索结构中的拉索，利用求解非线性方程组的延拓法，再配以常用的牛顿法，建立了快速的给定索原长情况下的拉索刚度计算方法。此法对于一般的给定初值，能很快逼近真实解，从而扩大了解的收敛范围，更重要的是与牛顿法相比更加高效。进而采用二分法建立了索结构中拉索滑移问题处理的算法，编制了相应的程序段，并通过算例证明了其正确性。本文的拉索滑移处理方法可用于索结构的设计与施工。     

Parametric excitation of beams moving over supports

Studied in this work are the formulation of equations of motion and the response to parametric excitation of a uniform cantilever beam moving longitudinally over a single bilateral support. The equations of motion are generated by using assumed modes to discretize the beam, by regarding the support as a kinematic constraint, and by employing an alternate form of Kane's method that is particularly well suited to systems subject to constraints. Instability information is developed using the results of perturbation analysis for harmonic longitudinal motions of small amplitude and with Floquet theory for general periodic motions of any amplitude. Results demonstrate that definitive instability information can be obtained for a beam moving longitudinally over supports based on the frequencies of free transverse vibration of a beam that is longitudinally fixed.     

Global dynamics of the cable under combined parametrical and external excitations

The chaotic dynamics and global bifurcations of the suspended elastic cable under combined parametric and external excitations are investigated. The non-linear equations of motion of the elastic cable to small vibration of one support are derived. The averaged equations are obtained by using the method of multiple scales. Based on the averaged equations, the theory of normal form and Maple program are used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. On the basis of the normal form, global bifurcation analysis of the parametrically and externally excited suspended elastic cable is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of the elastic cable is also found by numerical simulation.     

Steady-state analysis of multibody systems with reference to vehicle dynamics

This paper presents a systematic methodology and formulation for determining the steady-state response of multibody systems. The equations of motion for a general multibody system are described in terms of a set of relative joint accelerations. Then, the differential equations of motion are converted to a set of algebraic equations for the steady-state response. These equations are derived based upon a set of conditions that must exist for the steady state. The application of this formulation in determining the steady-state response of a vehicle moving in a circular path is shown. The multibody model of the vehicle for two- or four-wheel steering is presented. The results of the steady-state simulation are compared with those obtained from a transient dynamic analysis.     

Large Amplitude Three-Dimensional Free Vibrations of Inclined Sagged Elastic Cables

The nonlinear characteristics in the large amplitude three-dimensionalfree vibrations of inclined sagged elastic cables are investigated. Amodel formulation which is not limited to cables having smallsag-to-span ratios and takes into account the axial deformation effectis considered. Based on a multi-degree-of-freedom cable model, a finitedifference discretization is employed within a numerical solution of thegoverning equations of three-dimensional coupled motion. Variousnumerical examples of arbitrarily inclined sagged cables with initialout-of-plane or in-plane motions are carried out for the case of aspecified end tension. The major findings consist of highlighting theextent of two-and three-dimensional nonlinear couplings, the occurrenceof nonlinear dynamic tensions, and the meaningfulness of modaltransition phenomena ensuing from the activation of various internalresonance conditions. The influence of cable inclination on thenonlinear dynamic behavior is also evaluated. Comprehensive discussionand comparison of large amplitude free vibrations of horizontal andinclined sagged cables are presented.