多种非线性力作用下不平衡弹性转子的分岔特性

研究了4自由度不平衡弹性转子在非线性油膜力、非线性内阻力和非线性弹性力联合作用下的动力学特性。结果表明，当只有非线性油膜力作用时，转子只存在由于油膜失稳而导致的倍周期分岔。而当非线性油膜力与非线性内阻力共同作用时，在油膜失稳后，转子产生低频振动。转速继续增加，还会诱发内阻失稳，产生概周期运动。在倍周期分岔中，存在分岔激变现象。本文发现的由于油膜涡动而导致的内阻失稳(概周期运动)是一种未见报道的转子失稳模式(组合失稳)，它与油膜失稳(倍周期运动)一起可作为转子故障诊断的典型失稳模式。     

谐激励作用下输流曲管的混沌振动研究

研究了谐激励作用下输流曲管在系统参数区域内的混沌振动．基于牛顿法导出了输流曲管模型的非线性控制方程,并利用微分求积法对此方程在空间域进行离散,导出了输流曲管的非线性动力学方程组．在此基础上,对输流管道的动力响应进行了数值模拟．采用分岔、相平面、时间历程和庞加莱映射图等手段分析发现,在流速和激励频率的参数区域内,系统将可能发生包括混沌振动在内的多种运动形式．系统可经由倍周期分岔或概周期运动通向混沌．分析结果为工程输流管道模型的合理设计提供了参考．     

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.     

强共振情况下冲击成型机的亚谐与Hopf分岔

通过理论分析与数值仿真研究了双质体冲击振动成型机的周期运动在强共振条件下的亚谐分岔与Hopf分岔，证实了此系统的1／1周期运动在强共振(λ0^4=1)条件下可以分岔为稳定的4／4周期运动及概周期运动．讨论了冲击映射的奇异性，分析了冲击振动系统的“擦边”运动对强共振条件下周期运动及全局分岔的影响。     

冲击消振器的概周期碰振运动分析

建立了冲击消振器对称周期运动的Poincar啨映射方程 ,讨论了对称周期运动的稳定性与局部分岔。通过数值仿真研究了冲击消振器在非共振、弱共振和强共振条件下的概周期碰振运动及其向混沌的转迁过程。     

参-强激励联合作用下输流管的分岔和混沌行为研究

研究输送脉动流的两端固定输流管道在其基础简谐运动激励下的分岔和混沌行为,考虑管道变形的几何非线性和管道材料的非线性因素,推导了系统的非线性运动方程,并应用Galerkin方法对其进行了离散化处理。通过采用数值模拟方法,对系统的运动响应进行仿真,重点探讨了流体平均流速、流速脉动振幅以及基础简谐运动激励振幅对系统动态特性的影响。结果表明,系统在不同的参数下会发生围绕不同平衡点的周期和混沌等运动,并在系统中发现了两条通向混沌运动的途径:倍周期分岔和阵发混沌运动。     

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”．     

悬臂输流管道在基础激励与脉动内流联合作用下的参激振动

首先建立了悬臂输流管道在基础激励与脉动内流联合作用下的运动方程;然后基于Galerkin法研究了该系统的非线性动力学行为,分析了系统运动状态随激励频率和相位差的变化,以及混沌百分比随频率比(基础激励频率与脉动频率之比)和相位差的变化。结果表明,无论以激励频率还是以相位差为分岔参数,系统都具有多种形式的动态响应,包括周期运动、概周期运动和混沌运动,但进入和脱离混沌的途径不同。相位差和频率比对系统的混沌百分比有重要影响:相位差为π/2时系统混沌百分比最大;频率比为1时系统混沌百分比最小,频率比较小或较大时系统混沌百分比与只有基础激励时接近。     

粘弹性轴向运动梁的非线性动力学行为

本文研究了带有小脉动的轴向运动粘弹性梁的分岔及混沌现象。建立了系统的动力学模型。通过二阶Galerkin截断，把描述系统运动的偏微分方程离散化。利用数值方法分别分析了几种运动脉动频率时，梁随轴向运动脉动幅值，平均速度及粘弹性系数等几个参数变化时的运动分岔行为。利用Lyapunov指数识别系统的动力学行为，区分准周期振动和混沌运动。     

A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid

In this paper, the non-linear dynamics of simply supported pipes conveying pulsating fluid is further investigated, by considering the effect of motion constraints modeled as cubic springs. The partial differential equation, after transformed into a set of ordinary differential equations (ODEs) using the Galerkin method with N=2, is solved by a fourth order Runge-Kutta scheme. Attention is concentrated on the possible motions of the system with a higher mean flow velocity. Phase portraits, bifurcation diagrams and power spectrum diagrams are presented, showing some interesting and sometimes unexpected results. The analytical model is found to exhibit rich and variegated dynamical behaviors that include quasi-periodic and chaotic motions. The route to chaos is shown to be via period-doubling bifurcations. Finally, the cumulative effect of two non-linearities on the dynamics of the system is discussed.     

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.     

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.     

Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control

The nonlinear dynamics of a differential system describing the motion of a vehicle driven by a pilot is examined. In a first step, the stability of the system near the critical speed is analyzed by the bifurcation method in order to characterize its behavior after a loss of stability. It is shown that a Hopf bifurcation takes place, the stability of limit cycles depending mainly on the vehicle and pilot model parameters. In a second step, the front wheels of the vehicle are assumed to be subjected to a periodic disturbance. Chaotic and hyperchaotic motions are found to occur for some range of the speed parameter. Numerical simulations, such as bifurcation diagrams, Poincaré maps, Fourier spectrums, projection of trajectories, and Lyapunov exponents are used to establish the existence of chaotic attractors. Multiple attractors may coexist for some values of the speed, and basins of attraction for such attractors are shown to have fractal geometries.     

Stability and Oscillations in a Time-Delayed Vehicle System with Driver Control

In this paper, linear stability and chaotic motion of a time-delayednonlinear vehicle system are studied. The stability is determined bycomputing the spectrum associated with a system of linear retardedfunctional differential equations, which reveals that a loss ofstability occurs following a Hopf bifurcation. Beyond the critical valuefor linear stability, the system exhibits limit cycle motions.Subharmonic, quasi-periodic and chaotic motions are observed for asystem excited by a periodic disturbance.     

Chaotic flow-induced vibration of a flexible tube with end mass

Experimental evidence for chaotic vibrations of a hanging tube with a rigid cylindrical end weight attached is presented. A series of end-mass weights is examined. Bifurcations subsequent to the initial flutter instability result in a variety of complex motions in various domains of the parameter space. The system generically follows a quasi-periodic route to chaos. The data for one particular case is examined in some detail, documenting the quasi-periodic route to chaos with delay-embedding reconstructions of the attractors, dimension calculations and spectral analyses.     

冲击渐进振动系统相邻基本振动的转迁规律

冲击振动现象广泛存在于动力机械系统中,使得系统表现出复杂的动力学响应.目前对冲击振动系统的p/1类基本振动的稳定性及分岔研究报道较少,而且已有的对冲击振动系统动力学的研究基本都是基于单参数分岔进行分析的.研究以小型振动冲击式打桩机为工程背景,建立了冲击渐进振动系统的力学模型.分析了激振器和缓冲垫发生碰撞的类型,以及滑块渐进运动的条件.给出了系统可能呈现的四种运动状态的判断条件和运动微分方程.通过二维参数分岔分析得到系统在(ω,l)参数平面内存在的各类周期振动的参数域和分布规律.详细分析了相邻p/1类基本振动的转迁规律.在5/1基本振动的参数域的右边区域,相邻p/1基本振动的参数域临界线上存在一个奇异点X_p,相邻p/1类基本振动的分岔特点以奇异点X_p为临界点.在l小于l_X_p的区域内,相邻p/1基本振动经实擦边分岔和鞍结分岔相互转迁,实擦边分岔线和鞍结分岔线之间存在迟滞域,迟滞域内,系统存在两个周期吸引子共存的现象.在l大于l_X_p的区域内,相邻p/l类基本振动的参数域之间存在一个中间过渡区域.中间过渡区域内,系统呈现(2p+2)/2和(2p+1)/2周期振动等.在5/1基本振动的参数域的左边区域,p/1基本振动经多重滑移分岔产生(P+1)/1基本振动.     

Hopf bifurcation and chaotic motions of a tubular cantilever subject to cross flow and loose support

The Hopf bifurcation and chaotic motions of a tubular cantilever impacting on loose support is studied using an analytic model that involves delay differential equations. By using the damping-controlled mechanism, a single flexible cantilever in an otherwise rigid square array of cylinders is analyzed. The analytical model, after Galerkin discretization to five d.o.f., exhibits interesting dynamical behavior. Numerical solutions show that, with increasing flow beyond the critical, the amplitude of motion grows until impacting with the loose support placed at the tip end of the cylinder occurs; more complex motions then arise, leading to chaos and quasi-periodic motions for a sufficiently high flow velocity. The effect of location of the loose support on the global dynamics of the system is also investigated.