内压载荷作用下薄膜椭球的稳定性分析

超弹性橄榄状和南瓜状薄膜椭球在内压载荷作用下存在不同的分岔形式.对橄榄状薄膜椭球来说,细长比大于某一临界值时,在一定内压作用下会发生梨形分岔;小于该临界值时,薄膜椭球的分岔行为与圆管的局部起鼓现象相类似.对南瓜状薄膜椭球,无论圆扁,当内压达到某载荷值时都会发生梨形分岔.本文采用能量判据,分析了在压强控制和质量控制两种加载方式作用下,不同形状的薄膜椭球的均匀解及分岔解的稳定性.通过计算要考察的平衡状态及施加小扰动之后状态的能量差来判断当前状态是否稳定,结果表明,在压强控制下,P-V曲线下降段的均匀解和分岔解均为不稳定解.但在质量控制下,在P-V曲线下降段中只有均匀解出现时,均匀解为稳定解;而在均匀解和分岔解共存的区间内,均匀解为不稳定解,分岔解为稳定解.另外,P-V曲线两个上升段的均匀解则均为稳定解.     

非线性模态构造方法与机电耦合系统Hopf分岔

大型汽轮发电机组轴系与电网耦合次同步谐振（ＳＳＲ）现象是在某种参数条件下机电耦合系统产生Ｈｏｐｆ分岔的结果。在文献［１］中，作者提出了分析这种系统Ｈｏｐｆ分岔的非线性模态方法，得出了在固定参数下分岔解的结果。本文针对高维非线性动力系统（包括奇数维），提出新的非线性模态构造方法，并给出了机电耦合次同步振荡系统在辅助参数变化条件下分岔解的的变化规律。     

挤压油膜阻尼器—滑动轴承—刚性转子系统的稳定性及分岔行为

对挤压油膜阻尼器－滑动轴承－转子系统的稳定性及分岔行为进行了研究，由于该动力系统为一强非线性系统，具有复杂的非线性现象。本文采用Ｆｌｏｑｕｅｔ理论对其周期解的稳定性进行了计算分析：随着系统参数的变化，该系统将出现稳态周期解、准周期分岔、倍周期分岔。文中也对系统平衡点的稳定性进行了分析，讨论了Ｈｏｐｆ分岔行为。     

挤压油膜阻尼器－滑动轴承－刚性转子系统的稳定性及分岔行为

对挤压油膜阻尼器－滑动轴承－转子系统的稳定性及分岔行为进行了研究，由于该动力系统为一强非线性系统，具有复杂的非线性现象。本文采用Ｆｌｏｑｕｅｔ理论对其周期解的稳定性进行了计算分析：随着系统参数的变化，该系统将出现稳态周期解、准周期分岔、倍周期分岔。文中也对系统平衡点的稳定性进行了分析，讨论了其Ｈｏｐｆ分岔行为     

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．     

基于可观测状态的轴承-转子系统周期解计算及稳定性分析

分析了轴承-转子系统的稳定性和分岔,基于系统可观测状态信息给出1种求解系统周期解及识别周期解稳定性的方法,同时将该方法与Floquet理论相结合分析系统周期解的稳定性及失稳分岔形式,将转速作为分岔参数分析系统响应的周期、拟周期、多解共存和跳跃现象.结果表明,采用该方法计算系统周期解及稳定性时,利用系统可观测稳态和瞬态信息,即可求解出系统Jacobian矩阵而无需实时求解轴承非线性油膜力的Jacobian矩阵.与传统PNF方法相比,该方法不仅具有很高的精度而且可以节约计算量,同时可以预测追踪随控制参数变化的系统周期解及其稳定性,可用于指导轴承-转子系统的非线性动力学设计.     

轴对称转动粘弹性简支梁的稳定性分析

建立了轴对称转动粘弹性不可移简支梁的几何非线性动力学模型.应用Laplace变换和摄动法分析了超静定粘弹性杆的平衡解,得到了转动粘弹性梁的预应力平凡平衡态.应用Galerkin和摄动法得到了粘弹性梁平凡解的失稳临界值,分析了梁轴向伸长对失稳临界值的影响;通过极限分析获得了系统的后屈曲稳态近似解,讨论了平凡解二次分岔后的近似稳定吸引域,并数值仿真了系统平凡解失稳后初始挠动向稳态解的演变.本文的大范围稳定性分析发现了粘弹性系统叉式分岔失稳后的平凡态又经二次鞍结点分岔而稳定以及单向跳跃(突变)等不同于弹性系统的现象.     

Hooke材料的微孔形空穴分岔

研究Hooke材料在大位移下的轴对称平面应力问题的空穴分岔现象。根据Liapunov第一方法的基本思想,列出该非线性问题的线性化扰动方程,找到了这个线性化扰动方程封闭解。由于在分岔点扰动方程具有非零解,结合边界条件得到临界载荷满足的特征方程。用二分法搜索特征方程的最小正根,即临界载荷。其中得到Poisson比为1／2时微孔萌生临界载荷的精确解,计算了Poisson比从0到1／2变化时材料的微孔萌生和 微孔突变的临界载荷以及失稳模态。计算结果和超弹性材料中微孔萌生和微孔突变的某些现象一致。     

宏观交通流模型的余维2分岔分析

交通流特性是混合交通流建模的一个重要因素. 交通流模型中的分岔现象是导致复杂交通现象的因素之一. 交通流的分岔, 涉及复杂的动力学特征且研究较少. 因此, 提出了一个最优速度模型来研究驾驶员记忆对驾驶行为的影响. 基于带有记忆的最优速度连续交通流模型, 利用非线性动力学, 分析和预测了复杂交通现象. 推导了鞍结 (LP) 分岔存在条件, 并通过数值计算得到了余维1 Hopf (H) 分岔、LP分岔和同宿轨 (HC) 分岔以及余维2广义Hopf (GH) 分岔、尖点 (CP) 分岔和Bogdanov-Takens (BT) 分岔等多种分岔结构. 根据双参数分岔区域的特点, 研究了记忆参数对单参数分岔结构的影响, 分析了不同分岔结构对交通流的影响, 并用相平面描述了平衡点附近轨迹的变化特征. 选择Hopf分岔和鞍结分岔作为密度演化的起点, 描述了均匀流、稳定和不稳定的拥挤流以及走走停停现象. 结果表明, 驾驶员记忆对交通流的稳定性有重要影响; 动力学行为很好地解释了交通拥堵现象; 考虑余维2分岔的影响, 能更好地理解交通拥堵产生的根源, 并为制定有效抑制拥堵的方法提供一定的理论依据.      

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.     

Experimental investigation of a buckled beam under high-frequency excitation

In this paper, we investigate theoretically and experimentally dynamics of a buckled beam under high-frequency excitation. It is theoretically predicted from linear analysis that the high-frequency excitation shifts the pitchfork bifurcation point and increases the buckling force. The shifting amount increases as the excitation amplitude or frequency increases. Namely, under the compressive force exceeding the buckling one, high-frequency excitation can stabilize the beam to the straight position. Some experiments are performed to investigate effects of the high-frequency excitation on the buckled beam. The dependency of the buckling force on the amounts of excitation amplitude and frequency is compared with theoretical results. The transient state is observed in which the beam is recovered from the buckled position to the straight position due to the excitation. Furthermore, the bifurcation diagrams are measured in the cases with and without high-frequency excitation. It is experimentally clarified that the high-frequency excitation changes the nonlinear property of the bifurcation from supercritical pitchfork bifurcation to subcritical pitchfork bifurcation and then the stable steady state of the beam exhibits hysteresis as the compressive force is reversed. This work was partially supported by the Japanese Ministry of Education, Culture, Sports, Science, and Technology, under Grants-in-Aid for Scientific Research 16560377.     

Snapping of an elastica under various loading mechanisms

This paper studies the snap-through buckling of a hinged elastica subject to a midpoint force. The focus is placed on how different load models affect the deformation and snapping load. Three different load models are considered. In the first model, the point force is fixed onto a midpoint of the elastica. In the second model, the point force is fixed on a central line in space. In the third model, the external force is applied through a rigid bar, which is allowed to slide along the central line in space. When the loaded elastica deforms symmetrically, as in the case when the magnitude of the point force is small, there is no difference between these three load models. However, when the elastica deforms unsymmetrically, the three load models produce different results. Vibration method is used to determine the stability of the equilibrium configurations. It is found that the elastica may snap via either a sub-critical pitchfork bifurcation or a limit-point bifurcation. If the elastica snaps via a sub-critical pitchfork bifurcation, the pre-snapping deformation is symmetric. If the elastica snaps via a limit-point bifurcation, on the other hand, the pre-snapping deformation is unsymmetric. For a specified initial shape of the elastica, different load models predict different snapping loads and pre-snapping deformations. The theoretical predictions are confirmed by experimental observations.     

抛物线壁内细杆的摩擦平衡分析

细杆在抛物线壁内支承,平衡特性与杆长、倾角和摩擦因子相关.细杆在自身重力作用下可发生焦点下方的顺时针运动,焦点上方的逆时针运动以及两端同时下滑.基于端部支撑力达到摩擦锥边界的条件,可确定细杆状态为不平衡、稳定或不稳定的平衡和摩擦平衡.平衡集为具有宽度的叉式分岔.     

Bifurcation Control of a Parametrically Excited Duffing System

A linear time-delayed feedback control is used to delay the occurrenceof pitchfork bifurcations and to eliminate saddle-node bifurcations,which may arise in the nonlinear response of a parametrically excitedDuffing system under the principal parametric resonance. The feedbackgains and the time delay are chosen by analyzing the modulationequations of the amplitude and the phase. It is shown that by using anappropriate feedback control, the stable region of the trivial solutionscan be broadened, a discontinuous bifurcation can be transformed into acontinuous one, and the jump phenomenon in the resonance response can beremoved.     

Lateral-torsional buckling of simply supported beams under uniform bending and axial tensile force

In this paper, the effect of a centrally applied external axial tensile load on the lateral-torsional buckling resistance of simply supported I-beams under uniform bending acting in the plane of maximum rigidity is studied. A linear and a nonlinear analysis are performed. Following the linear analysis, an expression for the critical moment of lateral-torsional buckling is presented in which the influence of the axial tensile force is included. There is an upper limit of this force over which the equilibrium in the deformed state is not possible. In the nonlinear analysis, the nature of the critical state is studied, considering the initial part of the post-buckling path. It is concluded that this critical state is associated with a stable symmetrical bifurcation point. Nevertheless, the post-buckling path is very shallow; therefore, the beam cannot exhibit practically post-buckling strength. The paper is supplemented by a representative example.     

Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton's Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent. The project supported by the National Natural Science Foundation of China (10272002) and the Doctoral Program from the Ministry of Education of China (20020001032) The English text was polished by Yunming Chen.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.     

Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity

This paper aims at offering an insight into the dynamical behaviors of incommensurate fractional-order singularly perturbed van der Pol oscillators subjected to constant forcing, especially when the forcing is close to Andronov–Hopf bifurcation points. These bifurcation points are predicted thanks to the theorem on stability of incommensurate fractional-order systems, as functions of the forcing and fractional derivative orders. When the forcing is chosen near Andronov–Hopf bifurcation, the dynamics of fractional-order systems show a static-looking transient regime whose length increases exponentially with the closeness to the bifurcation point. This peculiar phenomenon is not common in numerical simulation of dynamical systems. We show that this quasi-static transient behavior is due to the combine action of the slow passage effect at folded saddle-node singularity and fractional derivation memory effect on the slow flow around this singularity; this forces the system to remain for a long time in the vicinity of its equilibrium point, though unstable. The system frees oneself from this quasi-static transient state by spiraling before entering relaxation oscillation. Such a situation results in mixed mode oscillations in the oscillatory regime. One obtains mixed mode oscillations from a very simple system: A two-variable system subjected to constant forcing.