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1.
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.  相似文献   

2.
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.  相似文献   

3.
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。  相似文献   

4.
研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程.  相似文献   

5.
研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应。对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在。首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程。  相似文献   

6.
悬索在考虑1:3内共振情况下的动力学行为   总被引:2,自引:0,他引:2  
研究了悬索在受到外激励作用下考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.  相似文献   

7.
推导了考虑温度变化影响的悬索非线性运动微分方程,利用Galerkin法得到离散后的多自由度方程;考虑一阶正对称模态,以悬索同时发生主共振和1/3阶次谐波共振为例,利用多尺度法求解幅频响应方程组,并判断稳态解的稳定性;选取三组垂跨比及两组温度变化,基于幅频响应曲线和调谐相位曲线,探究温度变化影响下的主/次谐波联合共振响应。数值算例结果表明:主/次谐波联合共振时,系统响应变得更加复杂,同时展现出主共振和次谐波共振响应特性;温度变化会定性和定量地改变联合共振特性,改变系统振动的软/硬弹簧特性及程度;联合共振响应的幅值大小、相位和共振区间与温度变化密切相关;相同温度变化对联合共振响应的幅值和相位影响有差异,通过研究联合共振响应的相位,可以区分系统的多个稳态解。  相似文献   

8.
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.  相似文献   

9.
对称性是振动理论中5大美学特征之一,然而对称性破缺又难以避免.本文以工程中常见的易损结构—悬索为例,探究当该系统遭遇非对称性损伤时,对称性破缺对其面内耦合振动特性影响.首先建立受损悬索面内非线性动力学模型,并采用Galerkin法得到离散的无穷维微分方程.利用多尺度法计算该非线性系统发生面内耦合共振响应的调谐方程.截取前9阶模态,利用数值计算方法得到无损和受损悬索的各类共振曲线及其稳定性,通过计算最大李雅普诺夫指数来确定系统的混沌运动.研究结果表明:已有研究常采用抛物线模拟悬索静态构形,然而一旦发生不对称损伤,采用分段函数更能准确描述悬索受损后的静态构形;对称性破缺会导致悬索固有频率之间的交点变为转向点,其正、反对称模态均变为非对称模态;受损后悬索的非线性相互作用系数会发生显著改变,其内共振响应会产生明显变化;当激励直接作用在高阶模态时,无损系统会呈现出单模态解和内共振解,而受损系统并没有呈现出明显的单模态解;受损系统的分岔和混沌特性会发生改变,系统将通过倍周期分岔产生混沌运动.  相似文献   

10.
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.  相似文献   

11.
Garg  Anshul  Dwivedy  Santosha K. 《Nonlinear dynamics》2020,101(4):2107-2129

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

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12.
针对载流导线的非线性振动问题,在以往只考虑安培力的载流导线振动方程中引入了气动荷载。在此基础上进一步引入了受迫激励荷载,以研究动态风或相邻档导线对载流覆冰导线非线性振动特征的影响,建立了一种新的气动力-安倍力-受迫激励联合作用下的载流覆冰导线系统。推导出非线性振动方程,利用Galerkin方法将该振动方程转变为有限维度的常微分方程,采用多尺度法求解得到系统的非线性受迫主共振和亚谐波共振的幅-频响应函数。通过数值计算,分析了参数变化对系统受迫共振响应的影响以及受迫主共振定常解的稳定性。结果表明,考虑气动力的振动幅值和系统非线性较未考虑气动力时更小和更弱;线路参数的变化对导线的响应幅值和系统的非线性都有一定程度的影响;主共振和亚谐波共振的响应幅值随着激励幅值的增大而增大,共振峰值向着调谐参数σ的负值方向偏移,呈现出软弹簧特征并伴随着多值和跳跃现象;主共振时,随着调谐参数的变化,响应幅值则出现同步和失步现象。  相似文献   

13.
In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.  相似文献   

14.
为研究间隙碰撞对系统动力学响应的影响,以理想简支梁的振型函数为Rayleigh--Ritz基函数建立了单侧约束简支梁系统的非线性离散动力学方程组,应用数值方法研究了系统在基础谐波激励下的动力学响应特征及其对共振频率线性等效方法适用性的影响。研究表明:非线性动力学系统间隙产生的局部碰撞,使得系统振动能量在系统各阶模态之间转移,使得线性等效方法失效;即使进行非线性分析,也需要考虑系统固有频率远大于激励频段上限的模态。  相似文献   

15.
The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin's method and the multi-scale method are used to obtain ordinary differential equations(ODEs) of the system and their modulation equations, respectively. Frequency-and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin's integration are conducted.The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.  相似文献   

16.
胡宇达  张晓宇 《应用力学学报》2020,(2):674-681,I0015
研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。  相似文献   

17.
韩维  金栋平  胡海岩 《力学学报》2003,35(3):303-309
研究两自由度参数激励系统的非线性动力学与控制问题.利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程,以多尺度方法获得一阶近似控制方程.然后,对系统受一阶摸态参激主共振与一、二阶模态间3:1内共振联合作用下的幅额响应及其稳定性,以及反馈参数对系统稳态行为的影响作了详细分析.结果表明,响应的稳定域位置和大小取决于位移反馈,位移立方反馈改变了系统的非线性程度,速度反馈类似于阻尼,可使系统呈现自激振动特性.  相似文献   

18.
The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.  相似文献   

19.
Kang  Houjun  Su  Xiaoyang  Pi  Zihao 《Nonlinear dynamics》2022,107(2):1545-1568

Support stiffness is one of important factors on structure dynamics. Considering the vertical support stiffness, a multi-cable-stayed shallow-arch model of the cable-stayed bridge is established. Its differential equation governing the planar motion of cables and the shallow arch and the boundary conditions are derived by Hamilton’s principle. Firstly, the in-plane free vibration of the system is explored in order to find the modal functions and the possible internal resonances of nonlinear dynamics. Then, the 1:2:2 internal resonance among the different modes of the shallow arch and two cables are investigated by the multiple time scale method and pseudo-arclength algorithm. Meanwhile, the frequency-/force–response curves are used to explore the nonlinear behaviors of the system, especially the influence of vertical support stiffness, excitation frequency and amplitude on the internal resonance of the system is considered. To a certain extent, the support stiffness can reduce the response amplitudes of members by absorbing some energy from excitation.

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20.
In this study, a unified nonlinear dynamic buckling analysis for Euler–Bernoulli beam–columns subjected to constant loading rates is proposed with the incorporation of mercurial damping effects under thermal environment. Two generalized methods are developed which are competent to incorporate various beam geometries, material properties, boundary conditions, compression rates, and especially, the damping and thermal effects. The Galerkin–Force method is developed by implementing Galerkin method into force equilibrium equations. Then for solving differential equations, different buckled shape functions were introduced into force equilibrium equations in nonlinear dynamic buckling analysis. On the other hand, regarding the developed energy method, the governing partial differential equation for dynamic buckling of beams is also derived by meticulously implementing Hamilton’s principles into Lagrange’s equations. Consequently, the dynamic buckling analysis with damping effects under thermal environment can be adequately formulated as ordinary differential equations. The validity and accuracy of the results obtained by the two proposed methods are rigorously verified by the finite element method. Furthermore, comprehensive investigations on the structural dynamic buckling behavior in the presence of damping effects under thermal environment are conducted.  相似文献   

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