非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.      

非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.     

轴向变速运动黏弹性梁稳定性渐近分析和数值验证

研究了轴向变速运动黏弹性梁参数振动的稳定性.对黏弹性本构关系采用物质时间导数,轴向速度用关于恒定平均速度的简单谐波变化来描述.发展浙近摄动法确定稳定性条件.应用微分求积法数值求解简支边界条件下的轴向变速运动黏弹性梁方程,并进而确定次谐波参数共振的稳定性边界.数值结果显示了梁的黏性阻尼和轴向平均速度的影响并验证了次谐波共振的解析结果.     

变速轴向运动三参数模型黏弹性梁的稳定性

本文研究了速度变化的轴向运动三参数模型黏弹性梁在主参数共振以及组合参数共振范围内的稳定性.轴向运动梁的黏弹性本构关系采用三参数模型并引入了物质时间导数.运用渐进摄动法,直接求解梁的控制微分方程并导出了当运动参数激励频率接近某一阶固有频率2倍或接近某两阶固有频率之和时主参数共振和组合参数共振的稳定性条件.在解谐参数和激励振幅平面上,可以找出由于共振而产生的失稳区域.数值结果给出了梁的刚度系数、黏弹性系数及轴向平均速度对失稳区域的影响.在发生组合共振和主共振时,随着刚度系数E1的变大,失稳区域变小;刚度系数E2的变大,失稳区域变大.随着黏弹性系数的变大,失稳区域变小.发生组合共振时,随着平均速度的变大,失稳区域变小;发生主共振时,随着平均速度的变大,失稳区域变大.     

黏弹性轴向运动梁非线性受迫振动稳态幅频响应

本文研究了黏弹性轴向运动梁横向受迫振动稳态幅频响应问题.在控制方程的推导中,对黏弹性本构关系采用物质导数.把多尺度法直接应用于梁横向振动的非线性控制方程,利用可解性条件消除长期项,得到系统稳态的幅频响应曲线.运用Lyapunov一次近似理论分析幅频响应曲线的稳定性.通过算例研究了黏性系数,外部激励幅值以及非线性项系数对稳态幅频响应曲线及其稳定性的影响.运用数值方法对两端固定边界下黏弹性轴向运动梁的控制方程直接数值解,分析梁横向非线性振动的稳态幅频响应,通过数值算例验证直接多尺度法的结论.     

轴向变速黏弹性Poynting-Thompson梁参数共振稳定性

本文研究了轴向变速黏弹性梁的组合参数共振和主参数共振稳定性．梁的材料黏弹性本构关系由Poynting-Thompson模型描述．使用多尺度法渐近展开求解，导出了其可解性条件．根据Routh-Hurwitz准则给出了组合参数共振和主参数共振稳定性条件．考虑Poynting-Thompson模型退化到Kelvin-Voigt模型的情况．通过数值算例对两个模型进行了失稳边界的比较．     

轴向变速运动正交各向异性板的动态稳定性

研究面内载荷作用下轴向变速运动正交各向异性薄板的横向振动及其稳定性。利用Galerkin法与平均法,在激励频率为2倍固有频率或为两阶固有频率之和附近时得到自治的常微分方程组;在参数激励频率和激励振幅平面上,分析由共振引发的失稳区域。数值算例验证了面内载荷、轴向速度、加速度参数对失稳区域的影响。     

轴向运动梁非线性振动内共振研究

采用多元L-P方法分析轴向运动梁横向非线性振动的内共振，首先根据哈密顿原理建立轴向运动梁的横向振动微分方程，然后利用Galerkin方法分离时间和空间变量，再采用多元L-P方法进行求解，推导了内共振条件下频率-振幅方程的求根判别式，理论分析发现内共振与强迫力的振幅有关，而且可以从理论上决定这一界乎不同内共振的强迫力振幅的临界值，典型算例获得了轴向运动梁横向非线性振动内共振复杂的频率一振幅响应曲线，揭示了很多复杂而有趣的非线性振动特有的现象，多元L-P方法的数值结果，在小振幅时与IHB法的结果一致。     

输送周期性脉动流体的管道稳定性分析的一种方法

本文基于Floquet理论推广了Bolotin方法,使之既可以确定参变共振失稳区域的边界,也可以确定组合共振失稳区域的边界。算例表明此法的精度与Bolotin方法相当,低阶近似所用机时Floquet-Liapunov是半解析法的1/5～1/3。     

黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究

黏弹性阻尼一直是轴向运动系统的研究热点之一.以往研究轴向运动系统大都没有考虑黏弹性阻尼的影响.但在工程实际中, 存在黏弹性阻尼的轴向运动体系更为普遍.本文研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性.首先, 采用广义Hamilton原理给出了轴向运动黏弹性Timoshenko梁的动力学方程组和相应的简支边界条件.其次, 应用直接多尺度法得到了轴速和相关参数的对应关系, 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似解析解.最后, 采用微分求积法分析了在有无黏弹性作用下前两阶固有频率和衰减系数随轴速的变化; 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似数值解, 验证了近似解析解的有效性.结果表明: 随着轴速的增大, 梁的固有频率逐渐减小.梁的固有频率和衰减系数随着黏弹性系数的增大而逐渐减小, 其中衰减系数与黏弹性系数成正比关系, 黏弹性系数对第一阶衰减系数和固有频率的影响很小, 对第二阶衰减系数和固有频率的影响较大.      

考虑气动力作用的载流覆冰导线主共振及谐波共振分析

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。     

梁的动力稳定性分析的有限元方法

提出了对梁进行动力学稳定性分析的有限元方法──给出了单元质量矩阵，抗弯刚度矩阵，几何刚度矩阵及相应的Ｍａｔｈｉｅｕ方程，通过坐标变换消除了方程的动力与静力耦合，然后说明了由这种具有参数激励耦合的多自由度系统的Ｍａｔｈｉｅｕ方程求得系统一般参数共振及组合参数共振的过渡曲线的约束参数方法与多尺度方法。最后作为算例求出了均匀简支梁受简谐轴向力作用时的过渡曲线。     

Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models

Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.     

分数阶达芬振子的超谐与亚谐联合共振

研究了含分数阶微分项的达芬(Duffing)振子的超谐与亚谐联合共振.采用平均法得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念.建立了联合共振定常解幅频曲线的解析表达式,并对联合共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,证明了求解过程及近似解析解的正确性.然后,将等效线性阻尼和等效线性刚度的概念与传统整数阶系统进行比较,证明分数阶微分项不仅起着阻尼的作用同时还起着刚度的作用.最后,通过数值仿真研究了不同的分数阶微分项系数和阶次对联合共振幅频曲线多值性和跳跃现象的影响,并与单一频率下超谐共振或亚谐共振进行了对比.研究发现,分数阶微分项系数与阶次不仅影响着系统的响应幅值、共振频率,同时还对系统的周期解个数、发生区域面积、发生先后等有重要影响.并且,在不同的基本参数下该系统分别表现出单独超谐共振、单独亚谐共振以及超谐共振和亚谐共振同时存在的现象.这些结果对系统动力学特性的研究具有重要意义.     

横向磁场中轴向运动铁磁梁的磁弹性主共振

研究在外激励力与磁场作用下轴向运动铁磁梁的磁弹性非线性主共振问题．基于弹性理论和电磁理论，给出梁的动能和弹性势能表达式，根据哈密顿原理，推导出磁场中轴向运动铁磁梁的磁弹性双向耦合非线性振动方程．通过伽辽金积分法进行离散，得出两端简支边界条件下铁磁梁磁弹性非线性强迫振动方程．应用多尺度法对方程进行求解，得出幅频响应方程．最后通过算例，给出铁磁梁的幅频特性曲线、振幅-磁感应强度和振幅-外激励力曲线并进行分析．结果显示，在幅频响应曲线中铁磁梁的轴向运动速度、外激励力、轴向拉力越大，共振振幅越大；而磁感应强度越大，振幅越小．     

拱型结构在参、强激励下的非线性振动分析

利用数值解析法研究了拱型结构在参数激励与强迫激励联合作用下的非线性振动特性。得到了轴向力与固有频率的关系及轴向力对发生主共振,１／２亚谱共振的影响,由于１／２亚谐共振是高频激振低频响应,是最危险区域,应得到足够的重视,为工程设计提供了可靠的理论依据。     

Sub-harmonic resonance in a nearly pre-loaded mechanical oscillator

In this paper, the sub-harmonic resonance of a single degree of freedom system with a nearly preloaded non-linearity (gain-changing clearance) is studied. First, a new perturbation approach is incorporated in the traditional multi-term harmonic balance method to calculate the sub-harmonic resonance. Our approach significantly reduces the work as it computes the sub-harmonic responses with just one run. Initial guesses in the vicinity of sub-harmonic regime are relaxed compared to prior approaches. Second, a parametric study is conducted to examine the occurrence and characteristics of sub-harmonic resonance. The possibility of the sub-harmonic occurrence increases with an increase of dynamic excitation or the stiffness ratio. For instance, our analysis shows that the sub-harmonic resonance typically occurs when the mean load is close to the stiffness transition point. In the extreme case, a very small excitation would generate a sub-harmonic resonance. With a higher mean load, the resonant peak appears at lower frequencies as a result of the reduced equivalent stiffness. Finally, our analytical formulation is successfully validated using numerical integration results.     

Nonlinear Vibration of Plane Structures by Finite Element and Incremental Harmonic Balance Method

A nonlinear steady state vibration analysis of a wide class of planestructures is analyzed. Both the finite element method and incrementalharmonic balance method are used. The usual beam element is adopted inwhich the nonlinear effect arising from longitudinal stretching has beentaken into account. Based on the geometric nonlinear finite elementanalysis, the nonlinear dynamic equations including quadratic and cubicnonlinearities are derived. These equations are solved by theincremental harmonic balance (IHB) method. To show the effectiveness andversatility of this method, some typical examples for a wide variety ofvibration problems including fundamental resonance, super- andsub-harmonic resonance, and combination resonance of plane structuressuch as beams, shallow arches and frames are computed. Most of theseexamples have not been studied by other researchers before. Comparisonwith previous results are also made.     

THE STABILITY OF AN AXIALLY ACCELERATING BEAM ON SIMPLE SUPPORTS WITH TORSION SPRINGS

The axially moving beams on simple supports with torsion springs are studied. The general modal functions of the axially moving beam with constant speed have been obtained from the supporting conditions. The contribution of the spring stiffness to the natural frequencies has been numerically investigated. Transverse stability is also studied for axially moving beams on simple supports with torsion springs. The method of multiple scales is applied to the partialdifferential equation governing the transverse parametric vibration. The stability boundary is derived from the solvability condition. Instability occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies or is two fold natural frequency of the unperturbed system. It can be concluded that the spring stiffness makes both the natural frequencies and the instability regions smaller in the axial speed fluctuation frequency-amplitude plane for given mean axial speed and bending stiffness of the beam.     

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is presented. The governing equation and simple support boundary condition are derived with the extended Hamilton’s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method (DQM) is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.