两自由度机械臂λ^2=1下的Hopf分岔与混沌研究

研究了一类周期系数力学系统因周期运动失稳而产生Hopf分岔及混沌问题.首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据Floquet理论建立了其给定周期运动的Poincaré映射,根据该系统的特征矩阵有一对复共轭特征值从-1处穿越单位圆情况,分析该Poincaré映射不动点失稳后将发生次谐分岔、Hopf分岔、倍周期分岔,而多次倍周期分岔将导致混沌.并用数值计算加以验证.结果表明,随着分岔参数的变化,系统的周期运动可通过次谐分岔形成周期2运动,进而发生Hopf分岔形成拟周期运动,并再次经次谐分岔、倍周期分岔形成混沌运动.     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

线性碰振系统周期解擦边分岔的一类映射分析方法

擦边分岔是碰振机械系统的一种重要分岔行为. 以固定相位面作为Poincaré截面, 建立了线性碰振系统单碰周期$n$运动的Poincaré映射. 通过分析该映射,得到了系统 发生擦边分岔的条件和分岔方程,并以单自由度碰振系统为实例验证了分析结果的正确性. 该方法不仅可以计算线性碰振系统擦边分岔的参数值,还可以计算系统的任意周 期$n$解的分岔参数值.     

两空间耦合下齿轮传动系统多稳态特性研究

通过将系统参数定义为参数变量,构成参数空间,研究齿轮传动系统在参数空间和状态空间耦合下的非线性全局动力学特性,以及多参数、多初值和多稳态行为之间的关联特性.首先设计了一个两空间耦合下非线性系统多稳态行为的计算和辨识方法.其次,基于该方法并结合相图、Poincaré映射图、分岔图、最大Lyapunov指数、吸引域等,研究齿轮传动系统在不同参数平面上多稳态行为的存在区域和分布特性,以及多稳态行为在状态平面上的分布特性,揭示了参数平面和状态平面上系统可能隐藏的多稳态行为和分岔,并分析了多稳态行为的形成机理.结果发现,两空间耦合下系统在参数平面上存在大量多稳态行为并呈"带状"分布,状态平面上多稳态行为出现两种不同的侵蚀现象,即内部侵蚀和边界侵蚀.分岔点或分岔曲线对初值的敏感性导致多稳态行为的出现.当齿侧间隙和误差波动在较小的范围内变化时,系统全局动力学特性受间隙和误差扰动的影响较小,受啮合频率的影响较大.两空间耦合下系统全局动力学特性变得丰富和复杂.     

两空间耦合下齿轮传动系统多稳态特性研究

通过将系统参数定义为参数变量, 构成参数空间,研究齿轮传动系统在参数空间和状态空间耦合下的非线性全局动力学特性,以及多参数、多初值和多稳态行为之间的关联特性.首先设计了一个两空间耦合下非线性系统多稳态行为的计算和辨识方法.其次,基于该方法并结合相图、Poincaré映射图、分岔图、最大Lyapunov指数、吸引域等,研究齿轮传动系统在不同参数平面上多稳态行为的存在区域和分布特性,以及多稳态行为在状态平面上的分布特性,揭示了参数平面和状态平面上系统可能隐藏的多稳态行为和分岔,并分析了多稳态行为的形成机理. 结果发现,两空间耦合下系统在参数平面上存在大量多稳态行为并呈"带状"分布, 状态平面上多稳态行为出现两种不同的侵蚀现象, 即内部侵蚀和边界侵蚀.分岔点或分岔曲线对初值的敏感性导致多稳态行为的出现.当齿侧间隙和误差波动在较小的范围内变化时,系统全局动力学特性受间隙和误差扰动的影响较小,受啮合频率的影响较大.两空间耦合下系统全局动力学特性变得丰富和复杂.      

三自由度齿轮系统的双参数分岔与全局特性研究

以三自由度齿轮系统为研究对象,通过构造参数平面内不同运动类型的边界线算法,得到了系统在参数平面内的分岔曲线。为了判断分岔曲线的分岔类型,构造了三自由度齿轮系统Poincaré映射的Jacobi矩阵及Floquet乘子算法。结合系统的分岔图、最大Lyapunov指数图(TLE)、相图、Poincaré映射图和Floquet理论,讨论了双参数平面上系统的分岔特性以及参数平面内系统动力学特性的演变,并利用胞映射法对系统随啮合频率变化下的全局动力学特性进行了研究。结果表明:系统在参数平面k-ξ33内存在倍化分岔曲线、鞍结分岔曲线、Hopf分岔曲线等;阻尼系数越大,综合误差越小,系统运动越稳定;鞍结分岔对系统的全局稳定性影响较大,而Hopf分岔对系统的全局稳定性影响较小。研究结果可为齿轮系统设计和参数选择提供理论依据,研究方法也适用于其它非线性系统的双参数分岔分析。     

集成式动力总成机电耦合动态特性分析

综合考虑时变啮合刚度、齿轮啮合误差等因素,建立了斜齿轮六自由度非线性动力学方程;并根据集成式动力总成的机械系统和电机系统之间的耦合关系,推导出了集成式动力总成的非线性状态空间机电耦合方程,采用变步长Runge-Kutta法对机电耦合方程进行了数值求解。结合系统的分岔图、时域图、相图、Poincaré映射图和FFT频谱图研究了系统参数对系统振动的影响,分析了系统在电机电流、转子转速、时变刚度和齿轮啮合误差变化时的动力学特性。结果表明:随着电流变化,系统呈周期-混沌-周期的运动状态变化;随着时变刚度变化,系统呈周期倍化到混沌运动状态变化;随着齿轮啮合误差变化,系统由周期运动转变为混沌运动。研究为进一步改善集成式动力总成振动情况提供了依据。     

一类单侧碰撞悬臂振动系统的擦边分岔分析

与光滑动力系统不同,擦边分岔是非光滑动力系统中的一种特殊分岔行为.局部不连续映射是研究非光滑动力系统擦边分岔的一种有力工具.对一类单侧弹性碰撞悬臂振动系统进行了擦边分岔分析.首先建立了系统对应的局部不连续映射(ZDM)和全局Poincaré映射,进而在其他参数固定,碰撞间隙9为分岔参数时利用数值仿真的方法分别对原系统和对应的Poincaré映射进行擦边分岔分析,得到了该系统的两种不同类型的擦边分岔行为:周期1到周期2运动和周期1到混沌,这两种擦边分岔与刚性碰撞系统的情况是不相同的.由分析可知,对于含高阶非线性项的非光滑动力系统的擦边分岔,同样可以利用局部不连续映射的方法进行研究.     

系泊海洋平台周期运动倍周期分岔的胞映射分析

应用胞映射方法研究了系泊海洋生产平台的周期运动及其倍周期分岔。系泊运动的数学模型是一个具有指数回复力特性的非线性强迫振子 ,以波浪作用力为外激励。将波浪激励周期作为分岔控制参数 ,研究了周期系泊运动的倍周期分岔。胞映射方法用于寻找系统的稳定吸引子并确定其吸引域。时间历程、相图、功率谱和Poincar啨映射用于确定吸引子的具体类型芯糠⑾?,分岔参数处于不同的区域时 ,系统存在着相异的倍周期分岔特性。观察到了倍周期分岔的产生和突然消失 ,也找到了一个趋于吸引子的倍周期分岔序列。根据吸引域的胞映射分析结果解释了上述不同的倍周期分岔特征。发现其原因在于倍周期序列中的每个吸引子是否具有全局吸引性。     

一类含间隙振动系统的周期运动稳定性、分岔与混沌形成过程研究

建立了两自由度含间隙振动系统对称周期碰撞运动的Poincar啨映射方程 ,讨论了该映射不动点的稳定性与局部分岔 .通过数值仿真研究了含间隙振动系统对称周期碰撞运动经叉式分岔、倍化分岔、“擦边”奇异性向混沌转迁的全局分岔过程     

惯性式冲击振动落砂机周期倍化分岔的反控制

在不改变惯性式冲击振动落砂机系统平衡解结构的前提下,考虑碰撞振动系统的Poincaré映射的隐式特点以及经典的映射周期倍化分岔临界准则给反控制带来的困难,基于不直接依赖于特征值计算的周期倍化分岔显式临界准则,研究了落砂机系统周期倍化分岔的反控制。论文首先对落砂机系统施加线性反馈控制,得到受控闭环系统的Poincare映射,并应用不直接依赖于特征值计算的周期倍化分岔显式临界准则,获得了系统发生周期倍化分岔的控制参数区域。然后应用中心流形-正则形方法分析了周期倍化分岔的稳定性。最终采用数值仿真验证了在任意指定的系统参数点通过控制能产生稳定的周期倍化分岔解。     

Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics.Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.     

Autoparametric amplification of two nonlinear coupled mass–spring systems

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.