谐波激励下斜拉桥面内非线性振动试验研究

斜拉桥拉索的振动问题一直是桥梁工程领域的研究热点。为揭示拉索大幅振动的力学机理，课题组建立了斜拉桥的全桥精细化模型，本文测试和研究了单频激励下的斜拉桥可能的非线性振动行为。首先，通过自由振动试验测试了模型的模态参数，并与两类有限元模型（OECS模型和MECS模型）进行对比，结果吻合良好。其次，试验研究了在单个竖向简谐激励下斜拉桥模型的非线性响应。研究发现：当激励频率与斜拉桥某阶全局模态频率接近时，主梁产生主共振，并引起多根长索产生大幅的参强振动；当激励频率与某根斜拉索面内一阶频率之比为1：2或者2：1时，可以观测到索中产生超谐波和亚谐波共振现象。     

不同端部激励下斜拉索的空间耦合振动

斜拉索在端部激励下将发生空间耦合振动.为探究不同端部激励下斜拉索空间耦合振动的特性,利用斜拉索非线性振动运动方程,采用数值方法研究了在第一、二主共振区,上端水平面内激励、下端竖向激励和下端面外激励三种常见情形下斜拉索的空间耦合振动响应.研究表明:因上端水平面内激励和下端竖向激励均是面内激励,对斜拉索空间耦合振动的影响具有相似性;重力的影响使斜拉索面内、外运动方程不同,从而导致斜拉索在面外激励下(下端面外激励)与面内激励下(上端水平面内激励及下端竖向激励)的空间耦合振动特性具有本质区别.     

端点位移激励下斜拉索非线性振动计算方法研究

考虑拉索不同阶模态大幅振动之间的耦合效应,根据拉索的振动理论,详细地推导了单根拉索在端点位移激励下发生大幅振动时的非线性振动方程。根据某实际斜拉桥拉索参数,讨论了不同垂跨比对拉索振动特性的影响。使用四阶Runge-Kutta法求解拉索的非线性振动方程,通过对比有限元模型的非线性动力时程积分数值计算结果,验证了理论模型的可靠性与适用性。     

端点位移激励下斜拉索非线性振动计算方法研究Investigation on computing method of nonlinear vibration of the stayed cable with displacement excitation at the end point

考虑拉索不同阶模态大幅振动之间的耦合效应,根据拉索的振动理论,详细地推导了单根拉索在端点位移激励下发生大幅振动时的非线性振动方程。根据某实际斜拉桥拉索参数,讨论了不同垂跨比对拉索振动特性的影响。使用四阶Runge-Kutta法求解拉索的非线性振动方程,通过对比有限元模型的非线性动力时程积分数值计算结果,验证了理论模型的可靠性与适用性。     

端部激励相位差下斜拉索的非线性振动

斜拉桥中拉索承受着多种端部激励，可激发大幅空间振动．以斜拉索为对象，探究不同端部激励间相位差对其非线性振动的影响．首先，推导斜拉索无量纲离散控制方程，引入考虑相位的三向端部激励得到一般化模型；然后，针对拉索下端存在的纵桥向、竖向和横桥向激励的两两组合，受大幅或小幅激励，及其在主共振区或主参数共振区几组因素，共计12种工况，采用数值分析法分别研究了各工况下不同激励相位差时的斜拉索稳态响应．研究发现：激励相位差能加剧与激励频率相近的面内、外模态振动；在任意端部激励组合下，激励相位差不仅可使斜拉索非线性振动出现定量变化，还可改变内共振的表现形式．面内、外激励组合下，相位差对拉索响应幅值的影响以π为周期变化，且当相位差趋于π/2 + kπ (k = 0, 1, 2…)时影响最为突出；而面内激励组合下，以2π为变化周期，当相位差为π + 2kπ (k = 0, 1, 2, …)时其对稳态幅值的影响最显著．其原因是：面外激励关于拉索所在的竖直面对称，故其本质上以π为周期；而面内激励无此对称性，仍以2π为周期．因此，有无面外激励参与决定了激励间相位差对斜拉索响应的影响规律．     

斜拉桥塔-索-桥面连续耦合参数振动特性分析

针对端部激励作用下斜拉索与桥塔、桥面协同振动问题,考虑拉索几何非线性、倾角、阻尼以及拉索重力弦向分力的影响,引入拉索高精度抛物线形,建立了桥塔-索-桥面连续非线性精细化振动模型,推导了拉索与桥塔和桥面共同在激励作用下的耦合振动方程组,研究了桥塔-索-桥面结构系统参数的振动特性,并用数值仿真方法分析桥面与拉索频率比、桥面激励幅值、索力及拉索阻尼对结构耦合振动特性的影响规律。结果表明:桥面与拉索的频率比分别为1∶2和2∶1时,拉索会发生不同模式的大幅振动;相比于超谐波共振模式,亚谐波共振模式的拉索振幅更大,但达到共振所需时间较长;拉索振幅随桥面激励幅值的增大呈非线性增大,桥面激励幅值越大,拉索积蓄共振能量所需的时间越短;拉索振幅随索力增大而减小;拉索自身阻尼对其振动的影响较小,增大拉索阻尼时,拉索振幅虽有减小趋势,但是减小幅度有限。     

考虑弯曲刚度斜拉索面内面外内共振分析

本文研究桥梁工程中含弯曲刚度斜拉索的面内面外内共振问题．描述了工程中斜拉索变形的三种状态,考虑弯曲刚度、大变形及垂度等因素,忽略斜拉索纵向惯性力的影响,运用Hamilton变分原理建立了含弯曲刚度的斜拉索面内面外耦合偏微分控制方程,采用Galerkin方法对偏微分方程离散,并运用多尺度摄动方法进行了求解,获得了斜拉索可能存在的内共振模式,以工程中一根斜拉索为例,运用有限元法对其进行动力特性分析,列出了斜拉索前10阶面内面外振动频率,找出面内面外可能产生内共振的模态,分别研究了主共振条件下斜拉索面内和面外1:1、2:1内共振情形,获得了有意义的结论．     

FREE VIBRATION OF TAUT CABLE WITH DAMPER AND PARALLEL SPRING

分析了拉索-并联弹簧-阻尼器系统的自由振动特性,由系统的运动方程及边界条件 得到其复特征值方程。进一步研究了系统的极限解,由此讨论了拉索-并联弹簧-阻尼器系统的模态变化分区现象。以拉索-并联弹簧-阻尼器系统的二阶模态解为例,给出了模态频率和阻尼比的变化分布区间及其对应振型的变化情况。讨论了系统分区中存在的模态交叉现象;同时也讨论了斜拉索垂度对于一阶振动模态变化规律的影响。研究表明拉索-并联弹簧-阻尼器系统的振动模态演化因并联弹簧-阻尼器的位置不同而存在的明确的分区现象;安装并联弹簧和阻尼器后拉索的模态阻尼比和模态频率均可明显提高。     

悬索的多重内共振研究

本文对谐波激励的悬索的非线性响应进行了研究,同时考虑了如下问题(1):面内第三阶对称模态的主共振:(2):面内第一阶、第三阶对称模态和面外第五阶模态之间的内共振.本方首先针对考虑大变形的悬索动力学方程,由线性理论求得各阶频率,考察可能出现的内共振.然后利用直接法对悬索的运动学方程和边界条件进行非线性求解.由多尺度法得到系统的平均方程和悬索响应的二阶近似解.随后利用Newton-Raphson 方法和弧长法对特定张拉索进行数值仿真计算,得到面内第一阶对称模态、面内第三阶对称模态和面外第五阶模态的稳态解,并分析了解的稳定性.绘制幅频响应曲线,发现了关于悬索响应的多种分叉现象,并且对各种分叉现象周期解、混沌解进行了讨论.     

大跨斜拉桥拉索面内参数振动简化模型及工程应用

斜拉索是斜拉桥的重要受力构件,当索梁耦合振动时,斜拉索易发生大幅的参数振动。本文提出了一个不计拉索垂度并忽略桥梁自重影响的单自由度力学模型,推出了斜拉索参数振动的简化分析方程。为了考察方程的可行性,利用计算机对该方程进行了仿真,并与另一考虑索桥耦合作用运动方程的仿真分析作了对照,得到了非常相近的结果。基于实验室24.2m的模型斜拉索,辨识了该拉索的固有频率和阻尼,通过试验实现了该斜拉索的一阶参数振动,并分析了发生一阶参数振动时激励频率与固有频率比值。另外运用本文提出的简化方程对实验室拉索作了仿真,得到了参数振动现象,与实验结果对比得到:利用简化分析模型对试验拉索的仿真分析与实验室拉索的实验结果比较吻合,表明不考虑桥梁质量的简化模型具有足够的准确性。最后,结合工程实例,对鄂东大桥的某一可能发生参数振动的拉索进行了仿真分析,表明在桥梁自重较重时本文所提出的简化方程具有较好的适用性;并且发现加装阻尼器可以有效抑制拉索的参数振动。     

In-plane non-linear dynamics of the stay cables

Stay cables used in cable-stayed bridge and cable-stayed arch bridge are prone to vibration due to their inherent susceptibility to external deflection. The present work is devoted to the mitigation of a stay cable from the point of view of its nonlinear dynamics. The Galerkin integral, multiple scales perturbation method, and numerical techniques are applied to analyze the primary and subharmonic resonances of the stay cable. The nonlinear dynamic response of the stay cable subjected to parametrical and forced excitations is investigated numerically. The effects of some key parameters of the stay cable, such as initial tension force, damping and inclination angle, and the excitation frequency and amplitude are discussed. The carbon fiber reinforced polymers (CFRP) cable is also studied to understand the effect of the material properties of cable. The results show that these parameters have a considerable effect on the dynamic behavior of the cable. In particular, unreasonable tension force and inclination angle of stay cable may cause excessive vibration. It is suggested that CFRP cable replaces steel cable, which can mitigate the vibration of a stay cable.     

斜拉桥塔-索-桥面耦合参数振动模型及响应分析

针对大跨度斜拉桥拉索与桥塔、桥面的协同振动问题,考虑拉索垂度、阻尼、倾角以及重力弦向分力的影响,引入拉索高精度抛物线形,建立了桥塔-索-桥面连续非线性精细化振动模型,推导了桥塔和桥面共同激励作用下斜拉索耦合振动方程,对比分析了2种激振模式下斜拉索的参数振动特性,并编制程序研究了桥面与拉索的频率比、桥面激励幅值、索力及阻尼对结构耦合振动特性的影响规律。结果表明：桥面与拉索频率比对系统振动的影响较大,频率比为1:2和2:1时拉索均产生强烈振动,但2:1激振模式下拉索振幅更大,达到共振时间较长;随着桥面激励幅值的增大,2:1亚谐波共振模式下的拉索振幅增长速率更快;拉索振幅随索力的增大呈非线性减小趋势;斜拉索阻尼超过2%时,继续提高自身阻尼不能有效减小其振动幅值,需要通过设置附加阻尼才能更好地抑制其振动。     

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.     

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.     

Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation

In this paper, the nonlinear vibration of a single-walled carbon nanotube conveying fluid is investigated utilizing a multidimensional Lindstedt–Poincaré method. Considering the geometric large deformation of the single-walled carbon nanotube and external harmonic excitation force, based on nonlocal elastic theory and Euler–Bernoulli beam theory, the nonlinear vibration equation of a fluid-conveying single-walled carbon nanotube is established. Analyzing the equation through the multidimensional Lindstedt–Poincaré method, and from the solvability condition of the nonlinear vibration equation, the cubic algebraic equation which indicates the amplitude–frequency relation is obtained. Based on the root discriminant of the cubic equation, the first order primary response of the pinned–pinned carbon nanotube is discussed. The relations among internal resonance, the amplitude and frequency of the external excitation force are analyzed in detail. When the external excite force frequency is around the first mode natural frequency, the first mode primary resonance occurs. If simultaneously the first two modes natural frequency ratio is around 3, internal resonance occurs and the internal resonance region depends on the amplitude of external excitation force.     

斜拉桥拉索在轴向窄带随机激励下的振动响应

导出了拉索在考虑垂度以及索张力沿索长变化时的参激随机微分方程,进一步给出了预测拉索在窄带随机激励下响应的近似理论解------用统计矩截断法求解矩方程,获得高斯闭合解和一阶非高斯闭合解. 以南京长江二桥约330米长的A20拉索为研究对象,对以上高斯闭合解和一阶非高斯闭合解进一步进行数值求解以获得拉索的响应,并采用Monte-Carlo数值方法对求解进行验证. 分析了拉索振动的一般特征,特别分析了激励中心频率和拉索频率比为1和2时的响应随激励带宽的变化特征,得到了一些新的结论.      

Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading-I. Single internal resonance

The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.     

长单索结构的多参数流固耦合分析

介绍了长索结构在风场中的流固耦合振动,总结已有的流固耦合基本理论,着重介绍弱耦合问题的具体分析过程.以100m长索在不同拉索倾角和风向角的多参数流固耦合风振为例,应用ANSYS-CFX软件分别在层流模型和湍流模型风场中进行数值分析,得出与力学概念分析基本一致的结果.