Stability and Bifurcation of Longitudinal Vehicle Braking

The longitudinal braking dynamics of a two-wheel vehicle model on an incline are considered using techniques from nonlinear dynamics. The model is planar and incorporates the coupled dynamics of two independently braked wheels and the vehicle body, and takes into account the slip dynamics of each wheel. By using the wheel slip values and the vehicle speed as dynamic states, it is shown that the qualitative behavior of the system can be completely captured by studying a relatively simple phase plane problem described in terms of the slip values. A systematic bifurcation analysis is carried out in which the brake torques of the two wheels are varied, and it is shown how the system transitions from stable braking to the possibility of lockup in one or both wheels, to guaranteed lockup in both wheels. In this manner a quite complete picture of the dynamic behavior is obtained as a function of the two brake torques, including regions with multiple possible steady braking outcomes, depending on the initial conditions. This analysis provides new insights into the dynamics of vehicle braking, and it provides a correction to the standard result for the critical values of the brake torques at which the wheels undergo lockup. This approach may also prove useful for evaluating brake proportioning schedules, or for investigating anti-lock braking systems and other methods of traction control.     

Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

We consider a time-dependent free boundary problem with radially symmetric initial data: σ_t − Δσ + σ = 0 if and σ(r,0)=σ₀(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σ_S(r), R_S), where R_S depends only on the number . We prove that there exists a number μ_*, such that if μ < μ_* . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ_* then the stationary solution is unstable.     

随机ARNOLD系统的稳定性与分叉

本文详细讨论了当ｎ＝２时Ａｒｎｏｌｄ系统在小强度的随机参数激励扰动下，系统的运动稳定性及分叉。为了研究系统响应的统计特性，本文使用了Ｍａｒｋｏｖ近似技巧。在线性系统的情形，给出了系统矩稳定及样本稳定的充分必要条件。在非线性情形，本文的结果表明随机扰动可使系统的分叉点发生漂移     

叶片振动影响下双盘转子-轴承系统的稳定性与分岔

研究叶片与转子-轴承系统的耦合非线性振动,建立了一个带叶片的双盘转子-轴承系统的非线性动力学模型,其中包含一个弹性转轴、两个滑动轴承、两个刚性圆盘和两组弹性叶片.为了分析叶片的惯性影响,将其简化为单摆模型.采用4阶Runge-Kutta法进行了数值模拟,并利用分岔图、三维谱图、轴心轨迹和Poincaré映射图等方法分析了系统的非线性动力学特性.研究发现,随着转速的变化,系统响应演化出了倍周期运动、概周期运动、混沌运动和倍周期分岔等典型的非线性动力学行为.在与忽略了叶片振动的转子系统对比后发现,叶片振动使转子发生混沌运动的转速区域增大.在某些参数条件下,采用不同的叶片刚度,叶片振动可能引起转子系统产生混沌运动.     

热超弹性薄板非对称拉伸的分岔和稳定性

本文应用超弹性材料的有限变形理论分析了在面内等双向拉伸死载荷作用下不可压热超弹性方形薄板发生非对称变形的分岔及其稳定性问题.给出了方板变形的分岔曲线和临界载荷,发现对受面内等双向拉伸载荷作用的均匀方板,当拉伸载荷值较小时,方板双向等伸长变形,发生对称的拉伸变形;但当此载荷值大于某一临界值时,从方板的对称拉伸变形中分岔出非对称的变形,方板在两个方向的变形不再相等.通过变形发生分岔前后的能量比较发现,分岔后的对称变形是不稳定的,而非对称变形是稳定的.同时,给出了板中的应力分布曲线,并由不同温度下变形的分岔曲线和应力分布曲线讨论了温度对方板变形和板中的应力分布的影响.     

Hopf Bifurcation and Exchange of Stability in Diffusive Media

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized perturbations.     

扰动法在结构分枝失稳分析中的应用

对结构进行平衡路径的跟踪分析，是全面了解该结构的受力性能所必须进行的一项工作。目前的结构非线性稳定分析技术一般仅对极值点失稳型问题较为有效，而对分枝点失稳型问题则困难较多。对于具有缺陷敏感性的结构，如拱结构、壳体等，在普通荷载作用下，其失稳路径常包含分枝点。文献[1]提出并认为位移扰动法和力扰动法在分析结构分枝失稳时具有很好的效果。本文采用多个不同类型的算例，对扰动法在结构分枝失稳问题中的应用进行了分析比较，表明该方法具有较强的跟踪能力。最后，就扰动法在结构分枝失稳问题中的应用提出几点建议。     

Nonlinear Stability of Taylor Vortices in Infinite Cylinders

We consider the Taylor‐Couette problem in an infinitely extended cylindrical domain in the case when Couette flow is weakly unstable and a family of spatially periodic equilibria, called the Taylor vortices, has bifurcated from this trivial ground state. We show that those Taylor vortices which are not linearly unstable in the sense of Eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations. The main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the Taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis. Bloch‐wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime. (Accepted September 8, 1997)     

四神经元时滞网络的稳定性与分岔

本文研究具有长连接的四神经元构成的时滞网络的稳定性与分叉.网络系统可采用一组含有多个时滞的非线性微分方程描述.通过分析网络零平衡点处线性近似系统特征方程的根的分布情况,给出了网络零平衡点全时滞局部渐近稳定条件和与时滞相关的局部渐近稳定条件.讨论网络平衡点的数目及稳定性,得到了网络静态分叉产生条件.以时滞为分叉参数,给出了零平衡点失稳后网络出现由Hopf分叉引起的周期运动的存在性条件.分叉周期运动的性质由网络的非线性因素确定.借助中心流形定理和规范型理论,得到了确定Hopf分叉方向,分叉周期运动稳定性和分叉周期运动周期的公式.最后给出数值算例,验证了理论分析的结果.研究表明:网络中长连接的强度和性质对网络的动力学行为有着重要的影响.     

Stability and Hopf Bifurcation of Four-Wheel-Steering Vehicles Involving Driver's Delay

A mathematical model is presented for four-wheel-steeringvehicles, with the time delay in driver's response and the nonlinearityin lateral tyre forces taken into account. It is proved that thevehicle-driver system has a trivial steady state motion, as well aseight non-trivial steady state motions due to the nonlinearity of tyreforces. The asymptotic stability and Hopf bifurcation of the trivialsteady state are analyzed for two control strategies ofrear-wheel-steering. It is shown through the numerical simulations thatthe four-wheel-steering technique based on the bilinear control strategyworks better when the driver's response involves time delay.     

雨刮器非线性摩擦振动的稳定性与分岔特性

针对雨刮器建立2自由度非线性摩擦振动动力学模型,基于复模态理论计算复特征值并进行稳定性及其对刮刷速度的依赖性分析;通过数值计算分析摩擦振动对刮刷速度的分岔特性,并利用相轨迹、庞加莱映射、频谱特性分析不同刮刷速度下的非线性振动现象.研究发现:摩擦-速度特性的负斜率是导致系统不稳定的根本原因,增大刮刷速度有利于提高系统的稳定性;在高、低刮速区,随着刮刷速度的下降,系统振动形态遵循周期→准周期→混沌的演化规律,并会伴随显著的粘滑振动;仅高速区的周期振动和非振动条件下,刮刷时无附加的粘滑振动.     

涡轮泵转子-迷宫密封系统的非线性稳定性和分岔

研究迷宫密封对某一工程涡轮泵转子系统动力特性的影响,迷宫密封力采用Muszynska非线性力模型,应用有限元法建立转子系统的动力学方程,采取系统动力学方程中包含的高阶线性自由度和低阶的非线性自由度进行分块处理的方法,有效地缩短了求解时间。根据Floquet理论,判别系统的临界失稳转速,并由Floquet乘子来确定系统失稳后分岔方式。采用分块-Newmark法数值模拟了转子二涡轮盘的轴心轨迹。最后分析了涡轮泵转子二涡轮盘的质量偏心同相位和存在90度相位差时对转子系统运动特性的影响。     

Stability and Bifurcation of a Soap Film Spanning a Flexible Loop

The Euler–Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012), concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler–Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For a spanning surface with energy density σ and a bounding loop with length 2πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R ³ σ/a=3, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable.     

Bifurcation Corresponding to the Onset of Asymmetric Periodic Structures and a Self-Induced Pressure Gradient in a Modified Taylor Flow

With reference to the example of a modified Taylor flow, the bifurcation of the loss of flow symmetry with the onset of a self-induced pressure gradient is studied theoretically and numerically. A linear analysis shows that the bifurcation is supercritical. It is necessarily accompanied by the appearance of a longitudinal pressure gradient and takes place at values of the parameters for which the solution of the linear system for the perturbations satisfies the condition of zero mass flow. It is established that, as a result of the bifurcation, two asymmetric solutions with oppositely directed pressure gradients are simultaneously generated. In the supercritical region, the symmetric branch of the solutions is also retained but becomes unstable. Bifurcation of the loss of symmetry and a self-induced pressure gradient can occur only in nonlinear systems.     

Quasi-Periodic Stability of Subfamilies of an Unfolded Skew Hopf Bifurcation

In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. The simplest setting concerns rotationally symmetric diffeomorphisms of S ¹×R ². Their dynamics involve periodicity, quasi-periodicity and chaos, including mixed spectrum. The present paper deals with the persistence under symmetry-breaking of quasi-periodic invariant circles in this bifurcation. It turns out that, when adding sufficiently many unfolding parameters, the invariant circle persists for a large Hausdorff measure subset of a submanifold in parameter space. Accepted: September 3, 1999     

两系非线性悬挂车辆的运行稳定性与分叉

本文选取两系具有滞后非线性悬挂的车辆为目标，建立其数学模型和运动微分方程，用常微分方程稳定性理论对车辆蛇行运动进行理论分析，并应用分叉理论研究了整车在蛇行失稳后的动力学行为，得出蛇行运动的分叉解及稳定判据，得到防止车辆蛇行运动的充分条件，并研究了系统参数对临界速度的影响、分叉解振幅及稳定性的影响，为车辆设计和参数选取提供依据。     

高速客车横向稳定性及分岔研究

以一类比较典型的具有17个自由度的四轴铁道客车系统为研究对象.利用Vermeulen-Johnson蠕滑理论和一分段线性函数来分别计算轮轨滚动接触蠕滑力和轮缘力.应用数值方法并结合稳定性与分岔理论对该车辆系统运行于理想平直轨道上的横向稳定性与分岔问题进行研究,得到车辆系统的Hopf分岔点、鞍结分岔点及其稳定性转变过程,据此确定车辆系统的线性临界速度和非线性临界速度.同时也对该车辆系统在超高速情况下的摆振方式进行分析,结果表明系统首先经简单的单频率周期运动,逐渐演变成两个甚至多个频率互相耦合的拟周期运动,随着新的耦合频率不断出现,系统最终进入混沌运动状态.     

弹性支承滑动轴承不平衡转子系统非线性动力稳定性和分岔的研究

研究弹性支承滑动轴承不平衡转子系统的稳定性及分岔特性。建立了弹性支承-滑动轴承-转子非线性动力系统的力学模型，在油膜力非线性的情况下，应用数值模拟，采用打靶法计算了刚性转子系统的周期解，并与龙格-库塔法计算的结果进行了对比，依据Floquet理论，分析了周期解的稳定性，再结合龙格-库塔法、Poineare映射法作出了系统运动分岔图。最后，讨论了轴的柔性对转子系统运动稳定性的影响。     

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

In this paper we present a spectral technique for building asymptotic expansions which describe periodic processes in conservative and self-excited systems without assuming the oscillations to be weakly nonlinear. The small parameter of the expansion is connected with the ratio of the amplitudes of higher than the first harmonics in contrast to the traditional parameter connected with weak nonlinearity. In the case of an oscillator with power nonlinearity the frequency of the main harmonic and the complex amplitudes of higher harmonics are computed as the expansions of either integer (for weakly nonlinear oscillations) or algebraic (for strong nonlinearity) functions of the complex amplitude of the first harmonic depending on the character of the initial conditions and the maximum power of the nonlinear term in the equation. In the simplest case of weakly nonlinear oscillations the complete asymptotic expansion is shown to be valid in the whole domain of the periodic motions of definite type until the separatrix is reached. The expressions for the first terms of the expansion for concrete examples coincide with the expressions obtained both with the use of other methods and by expanding the exact solutions. For some special cases of the strongly nonlinear oscillations the comparison of the results with known exact solutions is carried out as well as the criteria of convergence of the expansions are determined.