Bifurcation and chaos study on transverse-torsional coupled 2K-H planetary gear train with multiple clearances

A new non-linear transverse-torsional coupled model was proposed for 2K-H planetary gear train, and gear's geometric eccentricity error, comprehensive transmission error, time-varying meshing stiffness, sun-planet and planet-ring gear pair's backlashes and sun gear's bearing clearance were taken into consideration. The solution of differential governing equation of motion was solved by applying variable step-size Runge-Kutta numerical integration method. The system motion state was investigated systematically and qualitatively, and exhibited diverse characteristics of bifurcation and chaos as well as non-linear behavior under different bifurcation parameters including meshing frequency, sun-planet backlash, planet-ring backlash and sun gear's bearing clearance. Analysis results show that the increasing damping could suppress the region of chaotic motion and improve the system's stability significantly. The route of crisis to chaotic motion was observed under the bifurcation parameter of meshing frequency. However, the routes of period doubling and crisis to chaos were identified under the bifurcation parameter of sun-planet backlash; besides, several different types of routes to chaos were observed and coexisted under the bifurcation parameter of planet-ring backlash including period doubling, Hopf bifurcation, 3T-periodic channel and crisis. Additionally, planet-ring backlash generated a strong coupling effect to system's non-linear behavior while the sun gear's bearing clearance produced weak coupling effect. Finally, quasi-periodic motion could be found under all above–mentioned bifurcation parameters and closely associated with the 3T-periodic motion.

Theoretical and experimental study on non-linear vibration characteristic of gear transmission system

In order to investigate the vibration of gear transmission system with clearance, a vibratory test-bed of the gear transmission system was designed. The non-linear dynamic model of the system was presented, with consideration of the effects of nonlinear dynamic gear mesh excitation, flexible rotors and bearings. Integration method was used to investigate the non-linear dynamic response of the system. The results imply that when the mesh frequency is near the natural frequency of gear pair, it is the first primary resonance, the bifurcation appears, and the vibration becomes to be chaotic motion rapidly. When the speed is close to the natural frequency of the first-order bending vibration, it is the second primary resonance, the periodic motion changes to chaos by period doubling bifurcation. The vibratory measurement of test-bed of the gear transmission system was performed. Accelerometers were employed to measure the high frequency vibration. Experimental results show that the vibration acceleration of the gear transmission system includes mesh frequency and sideband. The numerical calculation results of low speed can be validated by experimental results basically. It means that the presented non-linear dynamic model of the gear transmission system is right.

Nonlinear Dynamic Analysis of Planetary Gear Train System with Meshing Beyond Pitch Point

Nonlinear dynamic analysis was performed on a planetary gear transmission system with meshing beyond the pitch point.The parameters of the planetary gear system were optimized,and a two-dimensional nonlinear dynamic model was established using the lumped-mass method.Time-varying meshing stiffness was calculated by the energy method.The model consumes the backlash,bearing clearance,time-varying meshing stiffness,time-varying bearing stiffness,and time-varying friction coefficient.The time-varying bearing stiffness was calculated according to the Hertz contact theory.The load distribution among the gears was computed,and the time-varying friction coefficient was calculated according to elastohydrodynamic lubrication(EHL)theory.The dynamical equations were solved via numerical integration.The global bifurcation characteristics caused by the input speed,backlash,bearing clearance,and damping were analyzed.The system was in a chaotic state at natural frequencies or frequency multiplication.The system transitioned from a single-period state to a chaotic state with the increase of the backlash.The bearing clearance of the sun gear had little influence on the bifurcation characteristics.The amplitude was restrained in the chaotic state as the damping ratio increased.

NONLINEAR STABILITY OF BALANCED ROTOR DUE TO EFFECT OF BALL BEARING INTERNAL CLEARANCE

Stability and dynamic characteristics of a bail bearing-rotor system are investigated under the effect of the clearance in the ball bearing. Different clearance values are assumed to calculate the nonlinear stability of periodic solution with the aid of the Flo- quet theory. Bifurcation and chaos behavior are analyzed with variation of the clearance and rotational speed. It is found that there are three routes to unstable periodic solution. The period-doubling bifurcation and the secondary Hopf bifurcation are two usual routes to instability. The third route is the boundary crisis, a chaotic attractor occurs suddenly as the speed passes through its critical value. At last, the instable ranges for different internal clearance values are described. It is useful to investigate the stability property of ball bearing rotor system.

Bifurcation analysis of dual-sidestay landing gear locking performance considering joint clearance

Sidestay lock mechanism is an important part of landing gear system,and the locking performance can be analyzed based on changes in its stability.However,during numerical continuation analysis of fully-rigid dual-sidestay landing gear without clearance,it has been found that the appearance of bifurcation points does not necessarily imply that both sidestay links can be locked synchronously.This problem reveals the limitations of fully-rigid model with ideally-articulated in solving dual-sidestay mechanisms with extremely high motion sensitivity.Therefore,this study proposes a bifurcation analysis method for synchronous locking of dual-sidestay landing gears,which takes into consideration the joint clearance.For in-depth analysis of this problem,we initially build kinematic and mechanical models of a landing gear mechanism that consider joint clearance.Then,the models are solved based on continuation.The fundamental causes of synchronous locking are discussed in detail,and the number of bifurcation points is found to be closely related to whether the landing gear is completely locked.Finally,the effects of structural parameters on the synchronous locking are analyzed,and the feasible region of parameters satisfying synchronous locking condition is given,which agrees well with the test results.

结构间隙对起落架地面滑跑摆振特性影响分析

针对两种常见的飞机滑跑状态,即减摆状态和操纵状态,建立多自由度起落架摆振动力学模型,研究间隙对两者的影响机理,得到了两种不同形式的含间隙起落架摆振动力学数学模型。通过非光滑系统光滑化处理以及多维系统数值分析求解。结果表明,减摆状态下,起落架摆振频率会略微增大,结构库伦摩擦能够减弱结构间隙引发的持续摆振影响。对于操纵状态,通过快速傅里叶变换进行频域特性分析,发现支柱侧弯以及扭转之间耦合严重,两者的刚度直接决定起落架的摆振频率;存在结构间隙后,起落架整体的频率会变小,最终系统均会逐步趋于稳定的极限环震荡状态。

非圆齿轮行星轮系非线性动力学特性分析

为解决非圆齿轮行星轮系动力学模型不完善、非线性动力学特性难以获取等问题,提出了一种非圆齿轮行星轮系动力学建模方法,并聚焦于时变参数激励下系统的动态响应机制,对非圆齿轮行星轮系非线性动力学特性进行研究。为精确分析系统非线性振动,对非圆齿轮齿侧间隙函数进行了拟合,在综合考虑齿面摩擦、时变啮合刚度、黏弹性阻尼、静态传递误差的基础上,通过引入相对位移坐标消除非线性方程的变量耦合。在此基础上,建立非圆齿轮行星轮系传动系统的动力学模型,并利用四阶变步长Runge-Kutta数值方法对系统非线性动力学方程组进行求解。获取了分岔图、时域图、相轨迹以及Poincaré映射,得到阻尼、齿面摩擦、时变啮合刚度等控制参数激励下系统的动态响应分布规律。结果表明:随各激励参数取值不同,系统呈现出混沌和周期运动相互过渡状态;合理选取激励参数,可减小系统在混沌与周期运动之间的时间间隔,快速进入稳定运动状态。研究成果可为抑制非圆齿轮行星轮系传动系统非线性振动、预测系统的动力学行为提供理论依据。

磁液双悬浮轴承-单盘转子系统动力学行为研究

随着轴承系统高转速、高效率的发展需求,磁液双悬浮轴承转子与静子的装配间隙不断减小,导致碰摩事故经常发生。综合考虑转子偏心比、转速比、磁液双悬浮轴承与静子碰摩等多种耦合故障,建立磁液双悬浮轴承转子系统“间隙-碰摩”动力学方程,数值模拟转子运行规律。研究结果表明:随着偏心比的增加,转子由周期1运行演化出周期2、周期3、拟周期、混沌等多种运行规律;当偏心比ρ∈(0.28~0.4)及转速比w∈(1.2~1.7)时,转子位移波动剧烈,在此区域内转子极易发生分岔甚至混沌,且在碰摩区间内,转速比在1.5、1.67附近时,转子轴承处与转盘处碰撞力分别出现最大值。

高速重载人字齿轮传动非线性动力学分析

人字齿轮承载能力强,重合度大,可靠性高,多于高速、重载工况下使用.探究高速重载人字齿轮传动系统非线性动力学特性,可为其设计提供参考.首先,计算齿轮副时变啮合刚度;引入齿侧间隙、间隙非线性函数和综合传动误差,计算时变啮合力;引入轴承游隙,计算轴承受力.随后,建立高速重载人字齿轮传动系统非线性动力学方程,使用4阶Runge-Kutta法对方程求解.最后,探究不同因素对系统动态响应影响.保持系统其他参数不变,分别改变输入转矩、啮合阻尼、齿侧间隙、啮合刚度及激励频率,绘制系统时间−位移图像、时间−速度图像、空间相图、空间频谱图及分岔图,观察系统非线性动力学响应变化趋势,判别系统运动状态.结果表明:在一定范围内,系统稳定性与啮合阻尼、啮合刚度呈正相关关系,与齿侧间隙、输入转矩呈负相关关系;逐渐增大外部激励频率时,系统运动从单周期运动逐渐变为混沌运动,随后又回归单周期运动.为保证系统平稳运行,应合理选取外部激励频率.

含摩擦的风电齿轮传动系统非线性动力学分析

建立了含摩擦的风电齿轮传动系统的纯扭转非线性动力学模型,引入时变啮合刚度、综合啮合误差等因素,考虑了齿侧间隙与摩擦转矩等非线性因素的影响,进行量纲一化处理后得到了系统的动力学方程组;采用Runge-Kutta法对其进行求解,得出风电齿轮传动系统的非线性动态响应;计算了摩擦因数分别为0和0.07时系统随啮合频率变化的分岔图,并通过庞加莱截面、功率谱和关联维数等定性和定量工具对系统的非线性动力学特性进行了研究。结果表明,摩擦激励会使系统的周期响应中含有较多的次谐成分,诱发更多频率成分;摩擦激励会使得低频区域混沌区间减少,较快进入周期或拟周期运动状态;高频区域周期区间减少,拟周期区间、混沌区间加长,系统运动更为复杂。研究可为风电齿轮传动系统的故障机制研究、结构设计和更优工况的选择提供理论指导。

星型人字齿轮传动系统非线性分岔特性

为探究星型人字齿轮传动系统分岔动力学特性,课题组采用集中质量法建立了星型人字齿轮系统的纯扭转非线性动力学模型,通过Runge-Kutta法求解了系统非线性振动微分方程,利用相图、Poincaré截面法和分岔特性等分析手段研究了不同转速条件下啮合阻尼比对系统振动响应及分岔特性的影响。结果表明:系统在不同转速下会表现出丰富的非线性动力学行为;随着啮合阻尼比的增大,系统通过倒分岔从混沌状态进入倍周期状态,再由倍周期状态进入单周期状态。因此,在保证系统传动效率的前提下适当提高系统的啮合阻尼比,能够明显弱化系统的混沌运动,减小其振动响应,提高系统稳定性,对系统噪声的降低和寿命的延长具有一定的帮助。

啮合阻尼比对人字齿轮传动系统分岔的影响

为了研究啮合阻尼比对人字齿轮传动系统的非线性动态特性,文中建立含齿侧间隙、时变啮合刚度、啮合阻尼、综合传动误差等多种非线性因素的人字齿轮系统动力学模型。采用4阶变步长Runge-Kutta法对人字齿轮系统动力学方程进行数值求解。通过分岔图、相图、Poincaré映射图研究系统的非线性动力学特性。研究表明:系统经擦边分岔、跳跃分岔及连续的倍化分岔转迁为混沌运动,当啮合阻尼比逐渐增大时,系统跳跃分岔值逐渐滞后,混沌运动逐渐退化为单周期运动;当齿侧间隙增大时,系统经倍化、跳跃分岔进入混沌运动,当啮合阻尼比增大时,齿轮副双边冲击周期运动逐渐转换为单边冲击运动,运动周期数逐渐减小,系统跳跃分岔值随啮合阻尼比的增大而发生超前现象;在保证齿轮传动系统正常运行的前提下,适当增大啮合阻尼比能够改善齿轮系统的稳定性。

内外激励作用下含侧隙的齿轮传动系统的分岔和混沌

为研究支承刚度和齿侧间隙对齿轮-转子系统的分岔和混沌运动的影响,考虑齿侧间隙和内外激励的作用,建立齿轮-转子系统的非线性动力学模型。在对动力学方程进行数值仿真的过程中,为判断齿轮-转子系统的振动是否为混沌运动,采用混沌时间序列分析的方法计算齿轮-转子系统的高维非线性方程的最大Lyapunov指数。结果表明,随着支承刚度的增加,系统的弯扭耦合临界转速也会相应地增大,出现分岔和混沌的区域也随之改变。齿侧间隙对齿轮-转子系统的一阶弯曲临界转速附近的振动的影响较大,当齿侧间隙相对较小时,一阶弯曲临界转速附近的振动相对较好;当齿侧间隙增大到一定值时,一阶临界转速处的振动接近倍周期运动;当齿侧间隙继续增大时,一阶临界转速附近的振动从倍周期运动进入混沌;当齿侧间隙增大到较大值时,一阶临界转速附近的振动迅速转变为混沌运动。混沌时间序列分析方法能有效的计算高维非线性方程的最大Lyapunov指数。

行星齿轮系扭转非线性振动建模与运动分岔特性研究

建立行星齿轮系扭转非线性振动模型,模型中考虑了行星齿轮系各齿轮副间的时变啮合刚度、齿侧间隙以及综合传递误差等非线性因素;推导出系统的量纲一振动微分方程,采用数值积分方法研究行星齿轮的运动特性随转速以及齿侧间隙等参数的分岔特性,结合Poincaré图形分析,研究转速、啮合阻尼以及齿侧间隙等参数对系统分岔特性的影响。结果发现,随着转速的逐渐增大,系统会通过激变途径进入到混沌运动,而随着齿侧间隙的逐渐增大,系统会通过倍周期分岔途径进入到混沌。阻尼过小将会导致行星齿轮系统的稳态运动由短周期运动向复杂运动转变。齿侧间隙是影响系统运动分岔特性的重要因素,但是影响范围主要限于量纲一齿侧间隙大于3.5的大间隙区段。

含摩擦和间隙直齿轮副的混沌与分叉研究

齿面摩擦、齿侧间隙及时变啮合刚度是导致齿轮副产生复杂非线性振动的主要因素。以齿轮副啮合点间沿啮合线的相对位移为广义坐标,建立了计及摩擦、间隙及时变刚度等因素的直齿轮副非线性动力学模型,用5～6阶变步长Runge-Kutta法求得系统的各类周期响应和混沌响应。通过对位移响应的FFT分析,发现在计及摩擦时,系统的超谐和次谐响应成分增多;比较不同参数下的Poincare映射图得出,摩擦使混沌吸引子有所变大,但也可能使其转变为周期吸引子;由系统振动的分叉图及最大Lyapunov指数变化图看出,摩擦导致系统提前进入混沌,但混沌的程度有所降低。

考虑径向内间隙的滚动轴承平衡转子系统的非线性动力稳定性

研究滚动轴承平衡转子系统在不同轴承内间隙量,不同转速下系统的稳定性及其分岔特性和混沌.考虑Hertz接触力、滚动体通过振动和轴承径向内间隙等非线性因素建立数学模型,根据Floquet理论分析不同间隙量下滚动轴承转子系统的周期解稳定性,找到了3种导致周期解失稳的方式:倍周期分岔失稳、拟周期分岔失稳和边界激变导致混沌失稳.通过对各间隙量下转子系统拓扑特性变化和失稳区域的研究,表明滚动轴承间隙量是影响转子系统动力稳定性的一个重要因素.

螺旋锥齿轮间隙非线性系统的分岔与混沌

建立了含间隙的7自由度螺旋锥齿轮动力学方程,结合分岔图和最大Lyapunov指数曲线研究了分岔的演化过程及系统参数对系统分岔和混沌行为的影响。结果表明,分岔图与最大Lyapunov指数曲线从不同层面反映了系统随参数变化的全局特性,揭示出系统随参数变化在周期n、拟周期和混沌运动间反复演化复杂过程的全景;随参数变化系统经三种途径嵌入混沌:经倍周期分岔,由长周期运动嵌入混沌;经Hopf分岔产生拟周期运动,由"磕碰"运动嵌入混沌;拟周期运动经锁相嵌入混沌;负载的加重、阻尼系数的增大和刚度系数的减小有利于扩大系统的稳定区域,减缓、抑制分岔和混沌。

齿轮耦合的转子轴承系统的不平衡响应

在考虑时变啮合刚度及齿侧间隙的情况下,建立了齿轮耦合的转子-轴承系统的多自由度非线性动力学模型。在一定的转速范围内,用数字仿真法研究了系统的不平衡响应。当转速增加接近第二阶临界转速时,系统经过一系列分岔后,将由周期、准周期运动转变为混沌运动。系统做混沌运动时,振幅及平均变形远大于其他转速下的振幅及平均变形。要想避开混沌运动,系统应该在非共振区工作。当转速超过某值时,混沌运动将激变为周期运动。在工程实际应用中,除混沌运动状态外,可用线性动力学模型来代替非线性模型。

基于A-算符方法的齿轮系统的分岔与混沌

利用Adomian分解算法的思想,针对齿轮-转子-轴承系统的间隙非线性模型构造了求解系统解析解的A-算符方法(AOM),建立了基于AOM的符号-数值方法,并利用该方法研究了系统的分岔与混沌现象。研究表明,对于不同的啮合阻尼参数,随着激励频率的变化,系统的响应都经历了从稳态到混沌再到稳态的历程,且都发现了倍周期分岔的现象。同时,对于不同的激励频率,随着系统阻尼的变化,系统响应也经历了从稳态到混沌的历程,出现了倍周期分岔及激变现象。该方法不仅适用于对该系统的非线性模型进行研究,也为机械系统中一般的非线性模型的动力学分析提供了一个有力的工具。

转速激励下齿轮系统拍击振动的分岔特性

在考虑齿轮轴偏心质量、主动轴转速波动、轮齿时变啮合刚度及齿轮副齿侧间隙的情况下,建立了齿轮传动系统拍击振动分析的集中质量模型。计算了主动轴转速波动激励下齿轮传动系统振动性态随负载力矩变化的分岔规律。计算结果表明,随着负载力矩的减小,齿轮副依次出现三种振动状态,即:①完全啮合的振动状态; ②仅有齿面碰撞的脱啮碰撞振动状态;③同时具有齿面碰撞和齿背碰撞的脱啮碰撞振动状态。通过倍周期分岔, 系统振动由周期转变为混沌,而在混沌区域中还存在一些周期窗口。齿轮副在振动状态转变的过程中均出现了跳跃现象,而跳跃过程的振动周期并不改变。分岔过程揭示出系统具有复杂的非线性特征。