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。     

求非线性转子－轴承系统周期响应的一种计算方法

本文把求非线性转子－轴承系统瞬态响应的分块Ｎｅｗｍａｒｋ方法与打靶法结合求系统的周期解，该方法利用了分块Ｎｅｗｍａｒｋ方法求解速度快的优点和Ｊａｃｏｂｉ矩阵求解时每步不需迭代的特点。本文首先给出周期解的求解公式，然后用一个算例，讨论了周期解的稳定性及失稳后的分岔行为。     

非线性系统周期强迫不平衡响应的稳定性分析

多自由度强非线性系统是工程实际中经常遇见的一类模型,利用非线性动力学分析中的打靶法求解该类系统的周期解,并对Flopuet主导特征值判断周期解的失稳方式,利用该方法对旋转机械中的一个具体模型;双盘县臂柔性转子-非同心型挤压油膜阻尼器（SFD）系统周期强迫不平衡响应的稳定性和分岔行为进行了分析,分析表明,在该系统中存在着第二Hopf分岔、倍周期分岔、鞍-结分岔三种分岔形式。     

具有局部非线性动力系统周期解及稳定性方法

对于具有局部非线性的多自由度动力系统，提出一种分析周期解的稳定性及其分岔的方法该方法基于模态综合技术，将线性自由度转换到模态空间中，并对其进行缩减，而非线性自由度仍保留在物理空间中在分析缩减后系统的动力特性时，基于Ｎｅｗｍａｒｋ法的预估－校正－局部迭代的求解方法，与Ｐｏｉｎｃａｒé映射法相结合，推导出一种确定周期解，并使用Ｆｌｏｑｕｅｔ乘子判定其稳定性及分岔的方法     

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.     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。     

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。     

转子－轴承系统的分叉行为研究

本文完善和改进了求解非线性常微分方程组周期解及分叉特性分析的ＰＮＦ方法，用以有效地分析谐波、次谐波运动和倍周期分叉行为。然后，应用该方法对一个单盘挠性转子－轴承系统的动力行为进行了研究。结果显示运动呈现拟周期分叉、倍周期分叉和切分叉等复杂动力学现象，并与一些理论和实验结论作了比较。     

齿轮－转子－滑动轴承系统时变非线性动力特性研究

本文应用求周期解的数值计算方法─—打靶法和判定周期解稳定性的Ｆｌｏｑｕｅｔ乘子研究了齿轮－转子－滑动轴承系统中齿轮啮合时变刚度，滑动轴承非线性特性对转子系统不平衡响应和失稳的影响，并比较了平衡位置失稳和不平衡响应周期解失稳，以及按双轴计算与单轴计算结果的差别，为工程设计理论计算提供基础。     

阻尼器—轴承—柔性转子系统的稳定性及分岔

本文对挤压阴尼器－滑动轴承－柔性转子系统的稳定性及分岔特性进行了理论分析，首先讨论了系统平衡位置的稳定性及共Ｈｏｐｆ分岔，然后讨论了不平衡响应的稳定性及分岔。分析表明：在一定参数条件下，系统的稳态响应将发生倍周期分岔、二次Ｈｏｐｆ分岔及鞍－结分岔。     

碰摩裂纹转子轴承系统的周期运动稳定性及实验研究

根据碰摩裂纹耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶法，研究了系统周期运动的稳定性。研究发现，小偏心量下系统周期运动发生Hopf分岔，大偏心量下系统周期运动发生倍周期分岔，偏心量的加大使周期解的稳定性明显降低；系统碰摩间隙变小，碰摩影响了油膜涡动的形成，使失稳转速有所提高；裂纹深度的加大降低了系统周期运动的稳定性。本文的研究为转子轴承系统的安全稳定运行提供了理论参考。     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

On the aeroelastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation

Dynamic behavior of panels exposed to subsonic flow subjected to external excitation is investigated in this paper. The von Karman’s large deflection equations of motion for a flexible panel and Kelvin’s model of structural damping is considered to derive the governing equation. The panel under study is two-dimensional and simply supported. A Galerkin-type solution is introduced to derive the unsteady aerodynamic pressure from the linearized potential equation of uniform incompressible flow. The governing partial differential equation is transformed to a series of ordinary differential equations by using Galerkin method. The aeroelastic stability of the linear panel system is presented in a qualitative analysis and numerical study. The fourth-order Runge-Kutta numerical algorithm is used to conduct the numerical simulations to investigate the bifurcation structure of the nonlinear panel system and the distributions of chaotic regions are shown in the different parameter spaces. The results shows that the panel loses its stability by divergence not flutter in subsonic flow; the number of the fixed points and their stabilities change after the dynamic pressure exceeds the critical value; the chaotic regions and periodic regions appear alternately in parameter spaces; the single period motion trajectories change rhythmically in different periodic regions; the route from periodic motion to chaos is via doubling-period bifurcation.     

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.     

Stability and bifurcation analysis of a nonlinear transversally loaded rotating shaft

The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.     

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.     

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.     

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.