旋转简支梁的局部限制失稳分析

应用广义Hamilton变分原理,建立了任意位置限位器约束下绕轴线自转简支梁的非线性模型。取简支梁前两阶模态为Ritz基分析系统的稳定性,获得了限位器无摩擦情形下系统的限制失稳临界值、分岔模式、后屈曲解以及致稳限位器的最佳配置位置。扩充Ritz基对失稳临界值和限位器最佳配置位置的收敛性进行了验证;进一步分析了限位器夹紧力摩擦效应对系统稳定性的影响规律。研究表明:在限位器约束下,绕轴线自转简支梁存在临界转速,当转速超过临界转速时,梁的挠度和轴向位移分别发生叉式分岔和跨临界分岔而失去稳定性;限位器夹紧力摩擦效应将提升失稳临界值,使失稳后的系统在转速返回时出现明显的滞后效应,以比失稳临界值更低的转速回到原平衡位置;绕轴线自转简支梁系统致稳限位器的最佳配置位置在梁长的中间。这些结果对提升绕轴线自转简支梁的局部限制失稳性能的认识和指导限位器的配置具有实际意义。     

机电耦联系统余维3动态分岔研究

以r_{sl}, r_{f}以及x_{c}为分岔参数，对具有串补电容的单 机无穷大电力系统的失稳振荡问题，运用动态分岔理论进行了研究. 对系统同时出现有3对 纯虚根特征值的一类多参数高余维分岔情况，运用中心流行方法降维后得到约化方程，对此 强非线性约化方程的求解难点，运用多参数稳定性理论、谐波平衡法、归一化技术和Normal Form方法，得到了系统的解析解. 由分析得知，系统会出现3种Hopf分岔情况、二维环面 情况，以及三维环面分岔解，甚至会出现四维环面，或者更高维的环面分岔. 详细讨论 了系统各种分岔解的稳定性条件和稳定区域，并作了详细的数值分析加以验证.     

多孔介质中的双稳热对流

对矩形横截面多孔介质中热对流的复杂分岔行为──二次分岔进行研究．使用Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化并充分利用问题本身的对称性，研究了于最低的两个不同临界Ｒａｙｌｅｉｇｈ数处从平凡的静态传热解产生的热对流主分岔解之间的相互作用；揭示了主分岔解的二次分岔并给出了主分岔解及二次分岔解的渐近展开．稳定性分析表明从第二临界Ｒａｙｌｅｉｇｈ数产生的主分岔解经二次分岔后由不稳定变得稳定，从而与由最小临界Ｒａｙｌｅｉｇｈ数产生的主分岔解组成双稳定热对流．文中理论分析可较恰当地解释已有的数值模拟结果．     

基于Euler-Bernoulli梁理论研究热膨胀下梁的非线性振动问题

采用二次摄动法研究了弹性梁在热膨胀状态下的横向非线性振动问题。首先,根据能量原理建立了大挠度梁的非线性振动方程,并对其进行量纲归一化;然后,采用二次摄动法将非线性方程进行离散,得到各阶摄动方程,逐阶求解其渐进解。算例结果表明:应用传统的摄动法(KBM)法分析大幅振动问题的偏差比较大,远不及应用二次摄动法得到的解准确;温度和振幅对梁的固有频率影响显著。     

有界窄带激励下具有黏弹项的Duffing振子

讨论含线性黏弹项的Duffing振子对有界窄带随机激励的响应.用多尺度法分离了系统的稳态平均运动及随机摄动,讨论了系统的黏弹项对阻尼和刚度的效应,得到了系统随机均方响应的近似表达式.分析结果表明:在一定条件下,对应于同一解谐参数,系统有2个稳定的稳态平均运动,从而导致随机均方响应有2值解.数值模拟肯定了上述分析结果。     

机电耦联系统失稳振荡的动态分岔研究

应用机电分析动力学的理论建立了交流电机组的机电耦联振动方程组。运用微分动力学系统理论深入分析了交流电机的机电耦联失稳振荡问题。对于该系统出现的高余维分岔问题,通过中心流形定理、多参数稳定性理论和归一化方法得到了原系统的Normal Form形式,并详细讨论了系统的分岔情况以及分岔解的稳定性,并进行了详细的数值计算分析,很好地验证了理论分析结果。取得了交流电机失稳振荡更深入一步的研究成果。     

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.     

内共振条件下直线运动梁的动力稳定性

基于Kane方程,建立起了包含有耦合的三次几何及惯性非线性项大范围直线运动梁动力学控制方程．利用多尺度法并结合笛卡尔坐标变换,对所得方程进行一次近似展开,着重对满足一、二阶模态间3:1内共振现象的两端铰支梁参激振动平凡解稳定性进行了详尽的分析,得出了稳定性边界的解析表达式．采用中心流形定理对调制微分方程组进行降维处理,分析了相应Hopf分岔类型并通过数值计算发现了稳定的极限环存在．     

窄带随机激励下复合材料层合梁的非线性动力行为研究

基于von Kármán应力应变关系和Reddy高阶剪切变形理论,利用Hamilton原理导出了轴向激励下复合材料层合简支梁的非线性动力学方程。采用有界噪声理论,将窄带随机激励作为梁的参数激励模式,利用多尺度法获得了评价单模态主参激共振系统的平凡稳态响应稳定性的最大Lyapunov指数解析表达式,并表明了带宽γ的增大将有利于低激励幅值的稳定性,但同时也将扩大高激励幅值的不稳定区间。对表征上述系统稳态响应间随机跳跃与分岔的FPK方程中的联合概率密度函数进行了数值计算。对幅-频特性而言:当激励频率B越大,系统越有可能从围绕非平凡解支运动向围绕平凡解支运动跳跃;当B达到一定值后,系统主体的运动即为围绕平凡解支的小幅运动;窄带带宽γ越小,系统围绕非平凡解支运动的可能性越大。对力-幅特性而言:激励幅值减小,外扇型峰削弱而中心火山口峰增强,表明系统从围绕非平凡解支运动向围绕平凡解支运动跳跃。     

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.      

The codimension-two bifurcation for the recent proposed SD oscillator

In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α, are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α=1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.     

Chaotic oscillations in pipes conveying pulsating fluid

Chaotic motions of a simply supported nonlinear pipe conveying fluid with harmonie velocity fluetuations are investigated. The motions are investigated in two flow velocity regimes, one below and above the critical velocity for divergence. Analyses are carried out taking into account single mode and two mode approximations in the neighbourhood of fundamental resonance. The amplitude of the harmonic velocity perturbation is considered as the control parameter. Both period doubling sequence and a sudden transition to chaos of an asymmetric period 2 motion are observed. Above the critical velocity chaos is explained in terms of periodic motion about the equilibrium point shifting to another equilibrium point through a saddle point. Phase plane trajectories, Poincaré maps and time histories are plotted giving the nature of motion. Both single and two mode approximations essentially give the same qualitative behaviour. The stability limits of trivial and nontrivial solutions are obtained by the multiple time scale method and harmonic balance method which are in very good agreement with the numerical results.     

Postbuckling Analysis of Nonconservative Elastic Systems

Abstract

Previous work on the postbuckling and imperfection-sensitivity of elastic structures has concentrated on conservative systems. The results of Koiterand others have led to a general theory of nonlinear stability behavior for these systems. The theory must be modified when nonconservative forces are present, and this is the aim of the present paper.

Discrete, nonconservative, elastic systems which exhibit static (divergence) instability are considered. The nonlinear behavior in the neighborhood of a critical point is analyzed by means of a perturbation procedure. When the critical point is simple, the results are similar to those for conservative systems. When a coincident critical point exists, however, different types of behavior occur. In many cases there is no bifurcation at all, with only the fundamental (trivial) equilibrium path passing through the critical point. Imperfection-sensitivity is more severe than for the typical bifurcation points and can even occur when the perfect system has no bifurcation. The results are illustrated with the use of a nonlinear double pendulum model subjected to a partial follower load.     

DYNAMIC BEHAVIOR OF THIN RECTANGULAR PLATE ATTACHED TO MOVING RIGID

A nonlinear dynamic model of a thin rectangular plate attached to a moving rigid was established by employing the general Hamilton's variational principle. Based on the new model, it is proved theoretically that both phenomena of dynamic stiffening and dynamic softening can occur in the plate when the rigid undergoes different large overall motions including overall translational and rotary motions. It was also proved that dynamic softening effect even can make the trivial equilibrium of the plate lose its stability through bifurcation. Assumed modes method was employed to validate the theoretical result and analyze the approximately critical bifurcation value and the post-buckling equilibria.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

广义增量法在三维扁平桁架结构的弹塑性屈曲分析中的应用

本文利用广义逆矩阵,提出了解决非线性屈曲问题的广义增量法。并用该法分析了非线性荷载-位移关系曲线(其中包括极值点)。最后,给出一数值算例—三维扁平桁架结构的弹塑性屈曲分析,证明了本文方法的正确性和实用性。     

Dynamic and buckling analysis of a thin elastic-plastic square plate in a uniform temperature field

The nonlinear models of the elastic and elasticlinear strain-hardening square plates with four immovably simply-supported edges are established by employing Hamilton‘s Variational Principle in a uniform temperature field. The unilateral equilibrium equations satisfied by the plastically buckled equilibria are also established. Dynamics and stability of the elastic and plastic plates are investigated analytically and the buckled equilibria are investigated by employing Galerkin-Ritz‘s method. The vibration frequencies, the first critical temperature differences of instability or buckling, the elastically buckled equilibria and the extremes depending on the final loading temperature difference of the plastically buckled equillibria of the plate are obtained. The results indicate that the critical buckling value of the plastic plate is lower than its critical instability value and the critical value of its buckled equilibria turning back to the trivial equilibrium are higher than the value. However, three critical values of the elastic plate are equal. The unidirectional snap-through may occur both at the stress-strain boundary of elasticity and plasticity and at the initial stage of unloading of the plastic plate.     

Convection in a closed cavity heated from below when equilibrium conditions are disrupted

The equilibrium of a fluid is possible in a closed cavity in the presence of a strictly vertical temperature gradient (heating from below) [1]. There is a distinct sequence of critical Rayleigh numbers R_i at which this equilibrium loses its stability relative to low characteristic perturbations. The presence of different finite perturbations, unavoidable in an experiment, leads to the absence of a strict equilibrium when R < R₁. The problem of the influence of the perturbation on the convection conditions near the critical points arises in this context [2, 3]. The case in which the cavity is heated not strictly from below is investigated in [2] and the case in which the perturbation of the equilibrium is due to the slow movement of the upper boundary of the region is investigated in [3]. In [2, 3] the perturbation has the structure of a first critical motion and thus the results of these papers coincide qualitatively. The perturbation of the temperature in the horizontal sections of the boundary, which creates a perturbation with a two-vortex structure corresponding to the second critical point R₂, is examined in this paper. A similar type of perturbation is characteristic for experiments in which the thermal conductivity properties of the fluid and the cavity walls are different. The nonlinear convection conditions are investigated numerically by the net-point method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 203–207, March–April, 1977.The author wishes to thank D. B. Lyubimova, V. I. Chernatynskii, and A. A, Nepomnyashchii for their helpful comments.     

The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.