Static and dynamic response of elastic suspended cables with damage

In cable-stayed structures cables are subjected to potential damage, mainly due to fatigue and galvanic corrosion. The paper presents an analysis of damage effects on the statics and dynamics of suspended cables. An elastic continuous monodimensional model for damaged cables, including geometric nonlinearities, is formulated for the purpose. The damage is described as a diffused reduction of the cable axial stiffness, and defined through its intensity, extent and position. Exact solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the tension loss and sag augmentation resulting from damage are investigated under variation of practically significant parameters. The system spectral properties characterizing the free undamped dynamics are obtained in a closed-form fashion for shallow cables within the low frequency range. The sensitivity of the frequencies to the intensity and extent of damage is discussed, outlining two damage effects, which alternatively stiffen or soften the cable modes, whose respective static and geometric origin is recognized. Finally, the symmetry-breaking induced by damage on the static profile is verified to destroy the crossing phenomenon (crossover) characterizing the frequency loci of undamaged cables, which degenerates into a narrow frequency veering phenomenon.     

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

An elastic section model is proposed to analyze some characteristic issues of the cable-supported bridge dynamics through an equivalent planar multi-body system. The quadratic non-linearities of the four-degree-of-freedom model essentially describe the geometric coupling which may strongly characterize the dynamic interactions of the bridge deck and a pair of identical suspension cables (hangers or stays). The linear modal solution shows that the flexural and torsional modes of the deck (global modes) typically co-exist with symmetric or anti-symmetric modes of the cables (local modes). The combinations of parameters which realize remarkable 2:1:1 internal resonance conditions among one of the global modes (with higher natural frequency) and two local modes (with lower and close natural frequencies) are obtained by virtue of a multiparameter perturbation method. The non-linear response of the resonant systems shows that the global deck motion – directly forced at primary resonance by an external harmonic load – can parametrically excite the local cable motion, when the deck vibration amplitude overcomes the critical value at which a period-doubling bifurcation occurs. The relevant effects of both viscous damping and internal detuning on the instability boundaries are parametrically investigated. All the internal resonance conditions as well as the critical vibration amplitudes are expressed as an explicit, though asymptotically approximate, function of the structural parameters.     

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。     

Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.A portion of this work was presented at the 1992 ASME Winter Annual Meeting, Anaheim, CA.     

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

系留无人机系统振动特性分析

为了对系留无人机系统的振动特性进行分析,本文将无人机的运动作为系留缆绳的边界条件,得到了系留无人机系统的面内运动方程。通过平衡分析得到了系留缆绳在风场中的平衡张力和平衡曲率的近似表达式,然后对系留无人机系统的运动方程进行线性化处理,最终求得了系留缆绳法向的频率方程和振型。在此基础上,数值分析了无人机驱动力和无人机倾角对系留缆绳振动固有特性的影响。研究表明:随着无人机驱动力的变化,系留缆绳法向运动的不同阶固有频率之间会出现频率转向现象;随着无人机倾角的增加,系留缆绳的频率转向现象易于发生,且频率转向会影响系留缆绳的振型。研究结果可为系留无人机系统的设计提供理论参考。     

考虑温度效应的索梁面内动力学建模及特性分析

根据增量热场理论,温度变化影响下索梁结构会形成新的热应力平衡状态．因此基于已有的索梁结构非线性动力学模型,结合与斜拉索张拉力和垂度相关的无量纲参数,重新建立考虑温度变化影响下索梁结构面内振动的动力学模型,并推导其面内非线性运动方程．接着开展特征值分析,得到包含温度效应的索梁结构面内振动频率的超越方程及模态振型函数．通过算例研究温度变化对不同刚度比的索梁结构影响,得到其前四阶面内振动的模态频率与温度变化的关系曲线．研究结果表明：面内模态频率受温度变化影响明显,其影响程度与刚度比大小和模态的阶数密切相关,温度变化对低阶模态频率的影响比对高阶模态频率影响更为复杂;升温和降温对索梁结构面内振动特性的影响不对称;此外温度变化会导致频率偏转点的位置发生漂移．     

Non-linear stochastic response of a shallow cable

The paper considers the stochastic response of geometrical non-linear shallow cables. Large rain-wind induced cable oscillations with non-linear interactions have been observed in many large cable stayed bridges during the last decades. The response of the cable is investigated for a reduced two-degrees-of-freedom system with one modal coordinate for the in-plane displacement and one for the out-of-plane displacement. At first harmonic varying chord elongation at excitation frequencies close to the corresponding eigenfrequencies of the cable is considered in order to identify stable modes of vibration. Depending on the initial conditions the system may enter one of two states of vibration in the static equilibrium plane with the out-of-plane displacement equal to zero, or a whirling state with the out-of-plane displacement different from zero. Possible solutions are found both analytically and numerically. Next, the chord elongation is modelled as a narrow-banded Gaussian stochastic process, and it is shown that all the indicated harmonic solutions now become instable with probability one. Instead, the cable jumps randomly back and forth between the two in-plane and the whirling mode of vibration. A theory for determining the probability of occupying either of these modes at a certain time is derived based on a homogeneous, continuous time three states Markov chain model. It is shown that the transitional probability rates can be determined by first-passage crossing rates of the envelope process of the chord wise component of the support point motion relative to a safe domain determined from the harmonic analysis of the problem.     

斜拉索模态试验参数研究

试验模态分析是斜拉桥索力测试中广泛应用的方法。该方法的关键之一是采取合理的试验方案获取可靠的响应信号以识别出正确的索振动频率。本文通过试验,详细讨论了斜拉索模态测试中若干试验参数,如激振类型、激振位置、传感器位置以及频率分辨率对试验结果的影响;比较环境激励与锤击激励的效果,验证了环境激励的方法也可以准确识别索的频率;最后给出了较为合理的斜拉索模态试验方案,用该方案确定的索力与理论计算结果相差不超过 1. 5%。     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

Active Control of Shallow,Slack Cable Using the Parametric Control of End Tension

This study presents a practical, fieldable active controller applicable to shallow, slack cables. The controller is based upon an underlying theory that does not assume that the inertial forces are negligible in the direction of the cable chord. The theoretical framework is developed, and an experimental verification is presented. The experiments utilized a TV camera to track a small number of LED targets on the controlled cable, a commercial X-Y tracker and a PC to compute the cable's present position and velocity, and a linear actuator to vary the cable's tension. A Kalman filter was employed to smooth the position estimates and provide least-squares estimates of the cable velocity. The stability limits for the phase and amplitude of the control signal were established. An analytical estimate of the induced damping expected from the control algorithm is derived. Experimental measurements of the cable frequency and induced damping compared well with theoretical predictions. It is shown that he proposed active control approach is a simple, reliable, as well as an effective, method for vibration attenuation in lightly damped, shallow, slack cables.     

碳纤维复合材料拉索的非线性力学性能

采用解析法分析碳纤维复合材料(CFRP)拉索的非线性力学性能,通过与钢拉索对比分析,得出了CFRP拉索的受力特点.由于密度低,CFRP拉索自重应力、垂度约为钢拉索的1/5,承载效率也比钢拉索高许多,其承载极限长度为钢拉索的7倍,且随跨径增大,钢拉索的等效弹性模量下降非常快,而CFRP索仍保持较高值.由于CFRP线胀系数比钢材的线胀系数要小得多,约为其1/14,在温差与约束相同时,无垂度CFRP索温度应力仅为无垂度钢索的1/23.当有垂度时,钢索的温度应力降低,CFRP索的温度应力变化很小.     

Nonlinear interactions in the planar dynamics of cable-stayed beam

An analytical model is proposed to study the nonlinear interactions between beam and cable dynamics in stayed-systems. The integro-differential problem, describing the in-plane motion of a simple cable-stayed beam, presents quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions. Mainly studied are the effects of quadratic interactions, appearing at relatively low oscillation amplitude. To this end an analysis of the sensitivity of modal properties to parameter variations, in intervals of technical interest, has evidenced the occurrence of one-to-two and two-to-one internal resonances between global and local modes. The interactions between the resonant modes evidences two different sources of oscillation in cables, illustrated by simple 2dof discrete models.In the one-to-two global–local resonance, a novel mechanism is analyzed, by which cable undergoes large periodic and chaotic oscillations due to an energy transfer from the low-global to high-local frequencies.In two-to-one global–local resonance, the well-known parametric-induced cable oscillation in stayed-systems is correctly reinterpreted through the autoparametric resonance between a global and a local mode. Increasing the load the saturation of the global oscillations evidences the energy transfer from high-global to low-local frequencies, producing large cable oscillations. In both cases, the effects of detuning from internal and external resonance are presented.     

LT50m缩比模型悬索舱体系统的风振分析

在新一代大型射电望远镜（LT）的悬索舱体系统中增加了抑制风激振动的下拉结构控制索系，并分析了50m缩比模型的风振响应。首先，由非线性静力分析确定结构初始静态参考位形和初应力；其次，针对结构特点模拟作用在悬索和馈源舱上的随机风荷。在上述研究的基础上，在时域中模拟分析下拉索系对结构风振的抑制效果，同时通过典型节点时域响应的FFT变换，给出响应自功率谱图，进一步描述了振动控制的效果。     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

Hydroelastic vibration of two annular plates coupled with a bounded compressible fluid

A theoretical study on a linear hydroelastic vibration of two annular plates coupled with a bounded fluid is presented. The proposed method, based on the Rayleigh–Ritz method and the finite Hankel transform, is verified through a finite element analysis by using a commercial computer code, with an excellent accuracy. It is assumed that plates with an unequal thickness and with an unequal inner radius are clamped along their edges and an inviscid compressible fluid fills the space between the annular plates and the outer rigid vessel. When the two annular plates are identical, distinct in-phase and out-of-phase modes are observed. By increasing the difference in the plate thickness, the symmetric in-phase and out-of-phase modes with respect to the middle plane of the system are gradually shifted to pseudo in-phase and out-of-phase modes, and eventually they are changed to mixed modes. It is found that the natural frequencies decrease with an increase of the fluid compressibility, and additional modes due to a fluid concentration are observed when the plates are coupled with a compressible fluid. The fluid compressibility effect on the natural frequency is dominant in the out-of-phase modes and the higher modes. Also, the effects of the fluid thickness or the distance between the plates and the inner radius of the plates on the natural frequencies of the wet modes are investigated.     

The static and dynamic behavior of MEMS arch resonators near veering and the impact of initial shapes

We investigate experimentally and analytically the effect of initial shapes, arc and cosine wave, on the static and dynamic behavior of microelectromechanical systems (MEMS) arch resonators. We show that by carefully choosing the geometrical parameters and the initial shape of the arch, the veering phenomenon (avoided-crossing) among the first two symmetric modes can be strongly activated. To demonstrate this, we study electrothermally tuned and electrostatically driven initially curved MEMS resonators. Upon changing the electrothermal voltage, we demonstrate high frequency tunability of arc resonators compared to the cosine-configuration resonators for the first and third resonance frequencies. For arc beams, we show that the first resonance frequency increases up to twice its fundamental value and the third resonance frequency decreases until getting very close to the first resonance frequency triggering the veering phenomenon. Around the veering regime, we study experimentally and analytically the dynamic behavior of the arc beam for different electrostatic loads. The analytical study is based on a reduced order model of a nonlinear Euler–Bernoulli shallow arch beam model. The veering phenomenon is also confirmed through a finite-element multi-physics and nonlinear model.     

含等档的输电线面内自由振动理论

连续档导线运动方程包含平方和立方非线性项,倍频时会产生多模态耦合的复杂响应,因此研究连续档导线模态及共振的频率分布规律尤为重要．基于模态综合法获得了具有相等档距的连续档导线模态函数,基于动刚度理论得到了不同模态对应的频率理论公式,并应用有限元方法验证了模态及频率理论公式的准确性．研究了不同模态对应频率随几何参数的变化趋势,结果表明有相等档距的连续档导线的共振条件和单档导线有明显区别,连续档导线面内对称模态之间容易产生1:1共振,面内对称与反对称模态之间易产生1:2共振．本文研究内容可用来分析连续档导线内共振及其分岔行为．