梁主共振情形下索梁结构1∶1内共振分析

研究梁产生主共振情形下索梁组合结构的1∶1内共振问题。基于斜拉桥中的索梁组合结构模型,忽略索梁纵向惯性力的影响,考虑弯曲刚度、几何非线性及垂度等因素,利用索梁连接处的变形协调条件,采用Hamilton变分原理建立了索梁结构面内耦合非线性偏微分方程,运用Galerkin离散和多尺度法研究了梁主共振情形下索梁的1∶1相互作用问题,获得了内共振时的平均方程和分叉响应曲线方程。以某斜拉桥中索梁结构参数为例,研究了内共振时索梁结构之间的相互影响及时程曲线。结果表明,索容易出现共振情形,并呈现出较强的非线性特点;梁振动对索振动影响显著,索振动对梁振动影响较小;索梁内共振时能量相互交换,索梁振幅呈现此消彼长的现象。     

有间隙折叠舵面的振动实验与非线性建模研究

针对折叠舵面内、外舵铰接处存在的间隙对地面振动响应的影响及间隙处的非线性建模方法展开研究.消除间隙,利用锤击法对线性折叠舵面进行模态实验,得到了前五阶模态参数;打开间隙,进行振动台扫频基础激励,实验结果表明间隙的存在会使结构的动力学响应产生非线性现象,如正反向扫描差异、跳跃、多谐波及频率漂移.非线性的影响主要体现在一阶弯曲模态上,激励量级的增大和间隙的减小均会使基频增大,且逐渐趋向于无间隙的结果,但对第二阶扭转模态的影响与第一阶相比较小.建立了折叠舵面的有限元模型.提出了一种适用于具有集中非线性的折叠机构的模型缩减方法,并对舵面进行了模态缩减.根据Hertz接触理论,用具有线性和3/2次刚度组合形式的非线性扭转弹簧来模拟铰接处的间隙和接触.通过比较锤击实验与数值计算得到的前四阶频率和振型对模型的线性部分进行验证.通过Bathe两子步隐式复合算法计算基础激励下非线性结构的动力学响应,得到的传递函数可以模拟实验中出现的频率变化特征,验证了连接处非线性建模方法的合理性.     

有间隙折叠舵面的振动实验与非线性建模研究

针对折叠舵面内、外舵铰接处存在的间隙对地面振动响应的影响及间隙处的非线性建模方法展开研究.消除间隙,利用锤击法对线性折叠舵面进行模态实验,得到了前五阶模态参数;打开间隙,进行振动台扫频基础激励,实验结果表明间隙的存在会使结构的动力学响应产生非线性现象,如正反向扫描差异、跳跃、多谐波及频率漂移.非线性的影响主要体现在一阶弯曲模态上,激励量级的增大和间隙的减小均会使基频增大,且逐渐趋向于无间隙的结果,但对第二阶扭转模态的影响与第一阶相比较小.建立了折叠舵面的有限元模型. 提出了一种适用于具有集中非线性的折叠机构的模型缩减方法,并对舵面进行了模态缩减.根据Hertz接触理论,用具有线性和3/2次刚度组合形式的非线性扭转弹簧来模拟铰接处的间隙和接触.通过比较锤击实验与数值计算得到的前四阶频率和振型对模型的线性部分进行验证.通过Bathe两子步隐式复合算法计算基础激励下非线性结构的动力学响应,得到的传递函数可以模拟实验中出现的频率变化特征,验证了连接处非线性建模方法的合理性.      

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．     

移动柔性梁作用下简支梁的动力响应

对移动结构作用下梁的响应问题进行了推广,采用柔性梁作为移动结构模型,在考虑结构柔性和悬挂连接的前提下对系统的耦合振动进行了分析.根据一般边界条件梁建立振动方程,通过量纲一参数以及模态叠加法处理系统动力学方程.以简支边界条件为例,得到了梁响应的数值结果,对系统主要参数即移动结构频率、移动速度及连接刚度对简支梁振动的影响进行了讨论.结果表明:考虑移动体的柔性频率对简支梁的振动会产生一定的影响.     

挠性航天器太阳翼全局模态动力学建模与实验研究

现代柔性航天器通常安装有大型太阳翼为其在轨运行提供所需动力. 航天器入轨后太阳翼展开并锁定成为铰链连接多板结构, 此类结构质量轻、跨度大、刚度低的特点使其低频振动和非线性振动问题越来越凸显. 分析和处理此类结构出现的复杂振动问题的关键在于建立系统精确的非线性动力学模型. 为此, 本文提出铰链连接多板结构解析全局模态的提取方法, 获取太阳翼的固有频率和解析函数表征的全局模态. 提出可变刚度的扭转弹簧等效模型, 考虑铰链非线性刚度及摩擦力矩等因素, 通过全局模态离散得到系统的低维高精度非线性动力学模型, 研究了太阳翼在周期激励作用下的非线性特性. 开展太阳翼地面振动实验研究, 采用锤击法获取系统模态, 利用振动台施加正弦扫频激励, 将物理实验结果与理论结果进行对比, 从而验证全局模态动力学建模方法的合理性与准确性. 结果表明, 铰链刚度等结构参数对系统固有特性的影响较大, 铰链的存在会使太阳翼的动态响应出现跳跃等非线性现象. 全局模态动力学建模方法能很好地解决多板结构在非经典边界下解析全局模态求解的困难, 系统全局模态反映的是系统各个部件弹性振动的真实模态, 所建立的动力学模型具有低维高精度的特点, 对于复杂组合结构非线性动力学建模具有重要的参考价值.      

多档输电线非线性建模及面内自由振动分析

考虑边界条件和耦合连接条件,基于Hamilton变分原理,建立了多档输电线结构的精细化动力学模型。对两档输电线系统的特征值问题进行了研究;根据面内特征值方程,确定结构的模态函数,分析了垂跨比、跨度比等参数对面内固有频率的影响。研究结果表明,随着跨度比和垂度比的增大,各档之间横向振动耦合增强,模态频率会发生频率穿越现象。本文结合模态局部化因子描述体系的局部模态、整体模态、混合模态行为,输电线档间通过连续条件耦合,产生混合模态。结果表明,在Veering区和频率穿越区附近,某些频率接近相等,存在1:1内共振和2:1内共振模式。     

温度效应对覆冰导线舞动特征的影响研究

以某耐张段单跨导线作为研究对象,基于哈密顿原理和增量热场理论,建立了考虑温度效应的覆冰导线非线性舞动模型并推导出其舞动方程。分别使用近似模态和复杂模态得到离散后的面内-面外耦合的有限维度舞动方程,接着采用多尺度法得到该方程的近似解。数值算例研究表明,温度对频率的影响与Irvine参数有关,对张力的影响与初始张力的范围有关,而且降温会增大导线舞动幅值;采用精确模态计算得到的舞动幅值显著小于使用近似模态计算得到的幅值。     

考虑损伤效应的悬索非线性共振响应分析

悬索在其施工、运营和维护阶段会不可避免地遭受损伤,导致振动特性发生改变。本文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,建立损伤效应影响下悬索面内动力学模型,并推导其无穷维的非线性动力学微分方程。利用高阶多尺度法得到系统发生主共振响应时的幅频响应方程及稳态解。数值算例表明,悬索线性和非线性共振响应特性与损伤效应密切相关。悬索一旦发生损伤,其张力减小,垂跨比增加,将形成新的静力构形。受损悬索的固有频率将下降,且随着损伤程度增加而进一步减小。损伤会导致悬索正/反对称模态频率的交点发生偏移,影响系统内共振响应特性;损伤会引发系统振动特性发生明显定量和定性改变,但是垂跨比不同,其共振响应特性受损伤影响会有明显区别;损伤甚至会直接改变系统稳态响应幅值以及稳定解的数量,导致系统产生明显大幅振动,影响结构安全。     

斜拉桥中斜拉索的面内外耦合内共振分析

研究了桥面侧振引起的斜拉索非线性振动问题。基于Hamilton原理建立了拉索的非线性振动控制方程,并利用多尺度法得到了斜拉索振动方程的二阶近似解。通过具体算例分析了斜拉索面内一阶模态与面外一阶模态相互耦合发生内共振的可能性,讨论了拉索倾斜角对拉索振动的影响,比较了在零初始条件和非零初始条件下拉索振动响应的区别。研究发现:拉索内共振发生在一定的激励频率和激励幅值区域内;改变倾斜角度,会影响拉索发生内共振时激励频率区域的大小;初始条件的不同,拉索的振动形式会相差很大。     

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。     

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.     

索-梁组合结构的建模及面内动力特性分析

利用哈密顿变分原理以及结构动静态构型的影响,建立了索-梁组合结构的约化运动学控制方程。考虑到边界条件和耦合连接条件,我们研究了体系的面内特征值问题。根据求解得到的面内特征值方程,并通过分段函数的引入,结构的模态函数可以被直接确定。随后,我们研究了参数垂跨比f,刚度比和质量比对面内固有频率的影响。研究发现从结构的频率谱图中可以看出频率跳跃现象是存在的,另外,频率穿越现象也是十分明显。随后 ,考虑到局部模态和整体模态,结合之前确定的特征值方程及分段振型函数,我们研究了索-梁组合结构可能的模态形状。最后,我们讨论了索-梁组合结构可能发生的内共振形式,比如面内1:1内共振形式以及1:2内共振形式。研究表明梁的静态构型不仅直接影响到耦合力连接条件,还将影响索-梁组合结构频率的确定。     

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．     

Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions

A nonlocal Euler beam model with second-order gradient of stress taken into consideration is used to study the thermal vibration of nanobeams with elastic boundary.An analytical solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to axial thermal stress.The effects of the nonlocal parameter,thermal stress and stiffness of boundary constraint on the vibration behaviors of nanobeams are revealed.The results show that natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises.The boundary-constrained springs have significant effects on the vibration of nanobeams.In addition,numerical simulations also indicate the importance of small-scale effect on the vibration of nanobeams.     

桩-土-独塔斜拉桥相互作用地震响应分析

在Penz ien模型的基础上,建立了独塔斜拉桥考虑桩土相互作用的计算模型,推导了其运动方程,分析了桩土相互作用对斜拉桥地震响应的影响,同时,对地震动自由场输入和桩端输入对独塔斜拉桥的影响进行了比较,分析结果表明:桩土相互作用对独塔斜拉桥地震响应有一定的影响,但它比相应的刚性简支梁小。     

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.     

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.     

Static and dynamic response of elastic suspended cables with thermal effects

Cable structures are often subjected to severe and variable environmental conditions, and their mechanical behavior is known to be particularly sensitive to different ambient factors. The paper analyzes temperature effects on the static and dynamic response of suspended inclined cables through a continuous monodimensional model including geometric nonlinearities. Uniform temperature changes are introduced through a non-homogeneous constitutive law for the material linear elasticity. Exact and approximate solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the temperature-dependent variation of tension and sag are parametrically investigated. The spectral properties characterizing the free dynamics are obtained in a closed-form fashion for shallow parabolic cables within the low frequency vibration range. The sensitivity of the linear frequencies to temperature changes is discussed, outlining two thermal effects, which are distinguished by their different origins, geometric or static. For a generic temperature change, the geometric effect produces a systematic increment or reduction of all the frequencies, for both symmetric and anti-symmetric modes. The static effect stiffens or softens only the symmetric modes, and may prevail over the competing geometric effect, depending on the cable Irvine parameter. Finally, the thermal effects on the frequency veering and modal hybridization phenomena, which characterize quasi-resonant shallow cubic cables, are analyzed.