Effects of dynamic transmission errors and vibration stability in helical gears

Transmission error is an important reason for instability in helical gears. A six-degree-of-freedom dynamic model coupled flexional, torsional and axial motion of a helical gear transmission system, which includes time varying mesh stiffness, bearing supporting stiffness, mesh damping and backlash, is developed, after taking into account the dynamic characteristics and vibration responses of helical gear in three dimensions. Influences of involute contact ratio, bearing supporting stiffness, mesh damping and backlash on the dynamic transmission errors and vibration stability of the helical gear system are investigated using numerical simulation technique. The effects on dynamic transmission errors and stabilities by contact ratio, supporting stiffness and mesh damping as well as gear backlash are analyzed. The intrinsic relationship between above parameters and dynamic transmission errors and stabilities for helical gear system are presented. The stable and unstable regions under different parameters are given. The results in this paper can be helpful to the dynamic and stable design of a helical gear transmission system.     

具有动态侧隙的齿轮传动系统动力响应分析

综合考虑动态侧隙、时变啮合刚度、齿轮偏心和传动误差等非线性因素,建立齿轮 转子 轴承传动系统的非线性动力学模型。以分形理论为基础,引用尺寸一致性的Weierstrass-Mandelbrot函数模拟动态侧隙的变化趋势。利用4阶龙格 库塔法对系统的非线性动力学微分方程求解,探究动态侧隙对系统响应的影响。结果显示：综合对比定侧隙系统和动态侧隙系统,齿轮在扭转方向上振动幅值的波动较大,动态侧隙系统由准周期运动渐变为混沌运动,并且与动态侧隙曲线的复杂程度有关;齿轮在扭转方向相平面的轨迹随动态侧隙的标准差(standard deviation,简称STD)呈现出规律性变化;在系统振动响应频谱中,随着动态侧隙曲线的复杂程度的加剧,频率成分多样性增强,噪声频率的幅值逐渐增加,高倍频的幅值激增。     

齿轮系统动力学模型内部激励参数的优化设置研究

时变啮合刚度与齿侧间隙是齿轮传动系统的主要内部激励源,决定了齿轮系统动力学的基本特点和性质。啮合刚度的时变性影响齿轮系统的稳定性、引起系统的参数共振,齿侧间隙则引起系统强烈的非线特性。考虑时变啮合刚度、齿侧间隙等激励源,建立了齿轮系统非线性动力学模型,从模型参数设置合理性的新角度阐述时变啮合刚度、齿侧间隙对系统动态特性的影响。结果表明:在低速工运行况下,过度简化时变啮合刚度会扼杀由单双齿交替啮合而产生的振动冲击响应;此时齿轮处于单侧啮合状态,在建模时可以不考虑齿侧间隙的影响,以达到简化模型、提高求解效率的目的。而在较高速运行状态下,齿轮处于单边冲击或双边冲击状态,齿侧间隙引起系统强烈的非线性特性,建模时必须考虑齿侧间隙。     

间隙对含摩擦和时变刚度的齿轮系统动力学响应的影响

主要研究在考虑摩擦和时变刚度时,轮齿间隙对齿轮系统动力学响应的影响。建立常间隙、时变间隙和随机间隙三种不同的间隙形式。利用数值仿真的方法得到系统的幅频响应曲线和时间历程曲线。分析发现：①在低速时,随着摩擦因数的增大,系统响应的方均值和平均分量增大;并经过三次跳跃之后系统进入混沌运动状态;②时变间隙幅值增大导致系统提前进入混沌状态,而且随着时变幅值的增大跳转频率逐渐减小;时变间隙频率较小时,间隙对系统的影响较小;当时变间隙的频率较大时,系统在 =0.2,0.3时出现明显的共振响应,系统的响应以高频率分量4、5和6为主;③考虑随机间隙时,随着 的增大,齿轮系统响应的平均分量波动比较大。     

基于增量谐波平衡法的星型齿轮传动非线性动力学分析

建立了两级星型齿轮传动系统的非线性动力学分析模型,模型中考虑了系统的综合啮合误差、时变啮合刚度以及齿侧间隙。推导了多自由度多间隙系统的增量谐波平衡法计算公式,利用上述方法求解了系统非线性微分方程组,得到了两级星型齿轮传动的非线性频响特性。分析了阻尼系数、时变啮合刚度以及误差等参数对系统动态特性的影响。分析结果表明:间隙会使两级星型齿轮传动系统中出现多值解及跳跃现象的典型非线性特征;增大系统阻尼系数可以抑制系统的共振幅值;增大时变刚度幅值使得齿轮副传动误差的幅值增大;增大激励误差的幅值,使得系统各构件的振动幅值增大;多级星型齿轮传动系统有着比单级传动更丰富的非线性动态特性。     

Effects of eccentricity defect on the nonlinear dynamic behavior of the mechanism clutch-helical two stage gear

This paper investigates the nonlinear dynamic behavior of an automotive clutch coupled with a helical two stage gear system. The nonlinear dynamic model is simulated by twenty seven degrees of freedom and including three types of nonlinearity: dry friction path, double stage stiffness and spline clearance. The utility of the proposed nonlinear model is illustrated by the industrial need to clearly identify the dynamic behavior of mechanical elements (shafts, bearings, gears, flywheel, pressure plate, hub of the clutch…) and reduce vibration. The governing nonlinear time varying motion equation formulated is resolved by the analytic Runge Kutta method.Then the modeling of the eccentricity defect located on the gear and the flywheel of the clutch is done. The effect of this defect on the nonlinear dynamic behavior of the system is investigated.     

含间隙和时变啮合刚度齿轮传动系统的分岔和混沌

齿侧间隙和时变啮合刚度的存在,使齿轮传动系统中存在着丰富的非线性动力学行为。基于Poincare映射,建立了两自由度齿轮传动系统的数学模型,并通过数值仿真,说明了齿轮传动系统中存在着倍周期分岔、Hopf分岔和混沌等复杂的非线性现象。分析了系统参数对齿轮动力学行为的影响,进而为消除齿侧间隙和时变啮合刚度引起的非线性动力学行为指明了方向,为齿轮传动系统的优化设计奠定了基础。     

单级齿轮传动系统的Hopf分岔与混沌研究

为研究齿轮传动系统中齿侧间隙等非线性因素对系统振动特性的影响,综合考虑齿侧间隙、时变啮合刚度、综合啮合误差和轴承纵向响应,建立了三自由度单级直齿轮副传动系统的扭转振动非线性动力学模型;采用变步长4-5阶Runge-Kutta法,对系统运动的状态方程进行了数值求解;构建了系统的Poincaré截面,得到了系统的分岔图。结合系统相图、Poincaré映射图及FFT频谱图,分析了系统在激励频率变化时的动力学特性,发现系统在不同激励频率下会发生Hopf分岔、环面倍化、擦切分岔及倍化分岔。     

Dynamic behavior of the spur gear system with time varying stiffness by gear positions in the backlash

Gear backlash is a nonlinear effect of the gear system. In a spur gear system with the backlash, the initial position of gears with the backlash affects the impact force. This work conducted a dynamic analysis of the spur gear system with time-varying mesh stiffness and bearing stiffness with a focus on the initial gear position within the backlash. For this purpose, the time-varying stiffness of the gears and rolling bearings were calculated. Mesh force with the time-varying stiffness and the gear backlash was applied to four DOF equations of motion. The equations of motion were solved using the Newmark beta method and Newton-Raphson method. The dynamic characteristics of the spur gear system by the initial position of gears within the backlash were investigated along with the magnitude of the backlash. The results showed that as the backlash increased, the mesh and bearing forces increased as well. The mesh and bearing forces were highly dependent on the initial gear position within the backlash. Significant initial mesh and bearing forces by the initial gear position within the backlash can lead to cumulative damages to the gear system.     

大型能源装置齿轮传动间隙非线性时变动力学分析

针对某大型能源装置齿轮传动系统,应用拉格郎日方程,建立了系统的非线性动力学模型,模型中考虑了齿轮副的啮合综合误差、时变啮合刚度以及齿轮的啮合间隙。用数值解法获得了在有间隙非线性时变的情况下系统的动态响应,利用时间历程、相平面、Poincare映射以及Fourier频谱分析了该系统的基本性质,并研究了该装置中时变啮合刚度对动态特性的影响。     

含间隙和时变啮合刚度的弧齿锥齿轮传动系统非线性振动特性研究

齿面侧隙和时变啮合刚度等因素的存在,将导致弧齿锥齿轮传动系统在工作过程中呈现典型的非线性特性;置于转子上的弧齿锥齿轮传动系统被等效处理为8自由度动力学模型,借助动态相对传动误差,使两轮转动自由度合并,建立了7自由度的非线性振动方程。采用A算符算法获得了不同工况下弧齿锥齿轮系统的扭转、横向及轴向的振动位移和速度,发现随着啮合频率的变化,系统经倍周期分岔进入混沌,而随着支承刚度的变化,系统经拟周期分岔进入混沌振动,在啮合频率的变化过程中,系统存在跳跃现象。     

Effects of a one-way clutch on the nonlinear dynamic behavior of spur gear pairs under periodic excitation

Nonlinear behavior analysis was used to verify whether a one-way clutch is effective for reducing the torsional vibration of a paired spur gear system under periodic excitation. The dynamic responses were studied over a wide frequency range by speed sweeping to check the nonlinear behavior using numerical integration. The gear system with a one-way clutch showed typical nonlinear behavior. The oscillating component of the dynamic transmission error was reduced over the entire frequency range compared to a system without a one-way clutch. The one-way clutch also eliminated unsteady continuous jump phenomena over multiple solution bands, and prevented double-side contact, even with very small backlash. Installing a one-way clutch on both sides of the gear system was more effective at mitigating the negative effects of external periodic excitation and various parameter changes than a conventional gear system without a one-way clutch.     

齿面误差激励对二级齿轮系统脱啮行为的影响

在考虑齿轮时变啮合刚度、阻尼、齿轮误差及齿侧间隙的情况下,建立了具有5自由度的二级齿轮传动系统的动力学模型。为了便于用算法分解算法求解,用多项式拟合齿侧间隙,而将啮合刚度、齿轮综合误差用Fourier级数表示。利用Adomian分解算法的思想,用AOM得到了系统对齿轮啮合误差激励的动力学响应。仿真结果表明,在研究的转速范围内,齿面误差激励会导致强烈振动;在低速齿轮对中,可能同时伴随脱齿对象,并导致较大的动载荷。当脱啮现象发生时,响应频谱变得复杂起来,并导致系统产生准周期运动。     

弧面分度凸轮机构传动系统非线性动力学特性研究

对弧面分度凸轮机构传动系统,用集中参数法建立了较为精确的5自由度等效动力学模型,在该模型中,考虑了电动机动力特性、传动副间隙、阻尼、油膜挤压、构件弹性和负载特性的影响,并推导出了该传动系统的5自由度非线性微分方程组,使用有限差分法,用FORTRAN77语言编制了该传动系统动态模拟计算机应用软件RGCAMD（Roller gear cam dynamic）,并进行了大量的动态模拟计算,详细地分析了系统参数对动态响应的影响,总结出了各参数对动态响应的影响规律,得到了一些工程上很有应用价值的结论。     

齿轮系统动态传递误差和振动稳定性的数值研究

建立了计及轮齿时变啮合刚度、啮合阻尼、支承刚度和阻尼的齿轮系统扭转-横向振动耦合的3自由度动力学模型。用数值仿真方法,研究了重合度、支承刚度、啮合阻尼和支承阻尼对齿轮动态传递误差和动力稳定性的影响。研究结论对于高速、精密齿轮传动的动力学设计具有积极意义。     

转矩激励对二级齿轮系统脱啮行为的影响

在考虑齿轮时变啮合刚度、阻尼、转矩波动及齿侧间隙的情况下,建立了具有5自由度的二级齿轮传动系统的动力学模型.为了便于用算法分解算法求解,用多项式拟合齿侧间隙,而将啮合刚度、输入及输出转矩波动用Fourier级数表示.利用Adomian分解算法的思想,用AOM得到了系统对转矩激励的动力学响应.仿真结果表明,在某些转速区段,转矩激励会导致强烈振动;在低速齿轮对中,可能同时伴随脱齿现象,并导致较大的动载荷.     

履带车辆齿轮传动系统非线性振动特性研究

以齿轮系统动力学和非线性振动理论为基础,针对具有齿侧间隙和时变啮合刚度的某履带车辆齿轮传动系统,建立单自由度齿轮系统非线性振动模型,通过数值仿真方法求解并分析在不同档位下的振动特性,并对其在某些变量参数下进行了振动特性研究,所得结果既反映了动力学性能,又为下一步进行多自由度齿轮系统的非线性振动研究提供了有力的依据.     

Comparisons of gear dynamic responses with rectangular mesh stiffness and its approximate form

The mesh stiffness is close to rectangular stiffness, and the first harmonic approximate term of rectangular stiffness is generally adopted in the nonlinear gear dynamic analysis. The differences between the rectangular stiffness and its approximate form are analyzed in detail. The frequency response and dynamic factor are calculated by a numerical method, to illustrate the dynamic characteristics of the gear nonlinear system with different mesh stiffness forms. The results show that: The trends of frequency response of gear dynamic system with rectangular stiffness and its approximate form are identical. The jump phenomena are detected in both cases. Without the effect of static transmission error, the dynamic factor with rectangular mesh stiffness is larger than that with approximate mesh stiffness. Under design power and speed condition, the result with approximate mesh stiffness function may deduce reasonless suggestions for a designer. The static transmission error will enlarge the vibration amplitude and dynamic factor when the approximate mesh stiffness is adopted, but the effects on the response of gear system with rectangular mesh stiffness are fractional. The mesh stiffness may excite the odd subharmonic resonance, and the static transmission error may excite the even sub-harmonic resonance respectively.     

基于齿背接触刚度的高速斜齿轮瞬态振动特性

为了分析基于齿背接触刚度的高速斜齿轮瞬态振动放大特性,针对高转速瞬态工况下斜齿轮齿面啮合-脱啮-齿背接触的齿面实际承载接触状态,建立了同时考虑啮合时间与齿面振动位移耦合机理的斜齿轮动态啮合刚度。在细化考虑齿背啮合机理、基于齿背实际啮合刚度的模型基础上,进一步建立斜齿轮啮合型瞬态振动模型,并在此基础上展开不同齿侧间隙以及齿背接触对系统瞬态振动特性影响分析研究。搭建封闭功率流式斜齿轮瞬态扭转振动测试试验台,对基于齿背接触刚度的斜齿轮瞬态振动特性进行了验证。该研究具有较好的理论研究意义,有利于斜齿轮传动系统在航空传动、新能源传动系统上的应用推广,进一步提升高转速齿轮系统的瞬态振动噪声品质。     

含侧隙齿轮副的动载荷分析

以振动理论为基础，提出一种考虑齿轮拍击振动的齿轮动载荷的数值计算方法。建立计算动载荷的齿轮冲击模型，在模型中考虑了齿轮正、反冲击时实际的啮合刚度，并给出啮合柔度的计算方法。分析在考虑静态传递误差、啮合刚度、侧隙、摩擦力及外部扭矩变化等多种激励时，作用在轮齿上的动态载荷以及整个齿轮上的综合动态载荷的计算公式。最后通过实例分析作用在轮齿上的动态载荷、综合动态载荷变化规律以及相关激励参数对动态载荷的影响。