Calculation of time dependent mesh stiffness of helical planetary gear system using analytical approach

Time-dependent mesh stiffness is a most important reason of vibration and dynamic excitation in gear sets. In this research, analytical formulas of the helical gear set and the planetary gear system are combined to calculate the time-dependent mesh stiffness of the helical planetary gear system. For this purpose, at the first step, the analytical equations are derived for the spur gear pair. Then by dividing a helical tooth into the several independent thin spur tooth slices, the helical gear pair mesh stiffness is extracted. Finally, these equations are extended to the helical planetary gear system. The suggested analytical results and those which obtained by the finite element method (FEM) are compared and are in good agreement when the helix angle is less than 15 degrees. Also, the helical planetary gear system mesh stiffness in different cases such as fixed carrier, fixed sun gear and fixed ring gears is calculated. These results show that the value of mesh frequency ratio in each case scales the mesh stiffness shapes in the rotation angle direction. In other words, mesh frequency ratio parameter determines the number of meshing period in each rotation of planets.     

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。     

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.     

时变刚度渐开线直齿轮啮合冲击响应的研究

由于渐开线齿轮啮合刚度的时变性，传动齿轮的振动具有非线性振动的性质。文章将啮合冲击区分成若干微区段，视各微区段啮合刚度为定值，采用时域方法辨识出相应模态参数，用直线拟合固有频率分布曲线，从而提出一种新的计算啮合冲击响应的方法，其计算结果与实验值相符，并给出了用固有频率作敏感因子诊断轮齿裂纹故障的例子。     

计入齿圈柔性的直齿行星传动动力学建模

为揭示齿圈柔性对传动系统动态性能的影响,在系杆随动坐标系下建立计入齿圈柔性的直齿行星传动精细化动力学模型,建模中计入时变啮合刚度、支承刚度、陀螺效应和齿圈柔性等诸多影响因素。采用有限段单元的离散化建模方法,将连续体的柔性齿圈离散成由等效虚拟弹簧连接的刚性齿圈段,并推导齿圈段与行星轮的啮合判据。通过分析各构件间的相对位移关系及其受力,最终建立系统的运动微分方程。以三行星轮的NGW型直齿行星传动为例对其进行固有特性分析,得出系统的低阶固有频率分布及相应的振动模式,并进一步分析齿圈柔性和安装方式对系统固有特性的影响。研究结果表明,直齿行星传动的振型可归结为中心构件(太阳轮、系杆)扭转振动和中心构件平移振动两种模式,两种模式下齿圈和行星轮均做复杂平面振动;齿圈的柔性会降低系统的低阶固有频率,且其影响程度与齿圈安装方式有关,齿圈完全固定时其影响最小,径向浮动或周向转动时影响较明显。     

A hybrid finite element and analytical model for determining the mesh stiffness of internal gear pairs

Journal of Mechanical Science and Technology - This work developed an efficient model for calculating the mesh stiffness of spur/helical internal gear pairs by combining the finite element method...     

大兆瓦级风电齿轮箱耦合动态特性及结构噪声分析

风电齿轮箱是风电机组的重要组成部分,其动态性能的好坏直接影响整个机组的性能。建立了具有两级行星加一级平行轴齿轮传动的大兆瓦级风电齿轮箱齿轮-传动轴-轴承-箱体系统耦合非线性动力学有限元模型,采用Lanczos法对齿轮箱系统进行耦合模态分析。在综合考虑直斜齿轮时变啮合刚度、齿轮误差及齿轮啮合冲击等内部激励因素综合作用影响下,运用直接积分法对整个风电齿轮箱系统进行了动态响应求解,从而获得齿轮箱各点的振动位移、速度及加速度动态评价指标,并且对系统结构噪声进行了分析。研究结果可为大兆瓦级风电齿轮箱的动态性能优化提供参考。     

多圆柱齿轮液压马达的性能

针对普通单对圆柱齿轮液压马达单位体积排量小、排量脉动率大、低速稳定性差的缺点,对多圆柱齿轮液压马达的性能指标进行理论研究与探讨.通过分析多圆柱齿轮液压马达的工作原理与结构特点,得到多圆柱齿轮液压马达的构成形式,推导该类型液压马达的排量、单位体积排量及排量脉动率等主要性能指标的计算公式.分析各齿轮齿数的选取对多圆柱齿轮液压马达排量、排量脉动率的影响,得出提高多圆柱齿轮液压马达单位体积排量及降低多圆柱齿轮液压马达排量脉动率的各齿轮齿数的选取方法,并给出在保证多圆柱齿轮液压马达密封可靠性前提下中心齿轮齿数应满足的条件.通过设计计算实例,比较在相同排量及相同齿轮宽度条件下,多圆柱齿轮液压马达与普通单对圆柱齿轮液压马达的具体性能指标,验证了多圆柱齿轮液压马达具有单位体积排量大、脉动率小等优点.     

内啮合斜齿轮高精度三维有限元自动建模方法

为提高内啮合斜齿轮有限元接触分析的建模速度和模型精度,提出了一种齿轮高精度三维有限元模型的自动建模方法。基于齿轮插刀齿廓方程,利用齿廓法线法,得到包括齿根过渡曲线的内、外斜齿轮端面齿廓,建立了内、外齿轮参数化粗网格有限元模型。开发了表层六面体网格剖分方法,自动识别齿面接触带单元,进行分级剖分细化,保证了有限元模型的建模精度和网格密度。进行了齿面接触分析,得到了内啮合斜齿轮的弯曲应力、接触应力、接触印痕、传动误差、时变啮合刚度和载荷分配率。粗细网格有限元模型计算结果对比分析表明,该方法提高了内啮合斜齿轮有限元建模效率和计算精度,缩短了计算时间,为快速准确的齿轮接触分析奠定了基础。     

Effects of different coupling models of a helical gear system on vibration characteristics

Time-varying mesh stiffness (TVMS) and the dynamic coupling between the helical gears have a great influence on the vibration characteristics of a helical gear rotor system. Considering the effects of TVMS and adopting two coupling models (lateral-torsional coupling model and lateral-torsional-axial-swing coupling model), the dynamic behavior of a helical gear system was studied. First, an analytical model was used to analyze TVMS of a helical gear pair where the helical tooth is simulated by many spur tooth slices along the direction of the tooth width and the mesh stiffness of each slice is calculated using the energy method. Then, considering the effects of the TVMS excitation, the finite element model of a helical gear rotor system was established. Gear mesh was simulated by the above-mentioned two coupling models to investigate the effects of coupling forms on the system vibration characteristics. The strain energy was used to distinguish the dominant mode and dominant shaft of a gear system in natural characteristics analysis. The results show that the full coupling model can analyze accurately the vibration characteristics of the system and the axial and swing motions cannot be ignored in vibration analysis. Finally, the effects of helix angle on TVMS and vibration responses of a helical gear system were also studied.     

3K-Ⅱ型直齿行星齿轮传动的固有特性

为揭示3K-Ⅱ型直齿行星齿轮传动的固有特性,建立该类传动系统在系杆随动参考坐标系下的平移-扭转耦合动力学模型.模型设定系统中每个构件均拥有3自由度,并计入各构件的支承刚度、轮齿时变啮合刚度和陀螺效应等影响因素.通过分析各构件间的相对位移关系,推导出系统的运动微分方程,进而求解其特征值问题即可获知系统的固有频率和相应振型.固有特性分析表明,3K-Ⅱ型行星齿轮传动具有与2K-H行星齿轮传动类似的三种典型振动模式,即扭转振动模式、平移振动模式和行星轮振动模式.列出三种典型振动模式下传动系统的运动特征.     

内、外啮合斜齿轮三维有限元网格自动生成原理及其程序实现

对于不同的齿轮参数,斜齿轮副啮合面上的接触线位置也不相同,这就使得斜齿轮的三维有限元网络划分产生很大困难。为此,建立了一套斜齿轮三维有限元网络自动方法,并编制了相应的微机通用计算程序。使用者只需输入一对斜齿轮的齿数、模数、齿宽、分度圆螺旋角、齿轮变位系数和小齿轮旋向等基本参数,计算机就可按要求自动生成一对外啮合或内啮合斜齿轮副的三维有限元网格。实际应用表明,所提出的方法及程序使用方便灵活和准确,为内、外啮合斜齿轮副及齿轮传动系统的深入研究提供了可靠的基础。该程序还可以用来划分内、外啮合直齿圆柱齿轮副的三维有限元模型网格,并且其网格划分原理可方便地推广到其他齿轮传动中。     

Experimental measurement of gear mesh stiffness of cracked spur gear by strain gauge technique

In this paper, an experimental technique based on strain gauge has been proposed to measure the gear mesh stiffness of healthy spur gear as well as of cracked spur gear pair system. Calculation of mesh stiffness of healthy and cracked spur gear tooth are based on strain energy and strain energy release rate respectively. The location of strain gauge plays an important role in calculation of strain energy stored in gear tooth. The locations of strain gauge on gear tooth are illustrated for healthy and cracked gear. Stress intensity factor (SIF) has been calculated by strain gauge technique for calculating the stiffness of cracked pinion tooth. The effect of crack length on mesh stiffness has been investigated by strain gauge technique and results are compared with the established analytical method.     

Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis

Tooth pitting is a common failure mode of a gearbox. Many researchers investigated dynamic properties of a gearbox with localized pitting damage on a single gear tooth. The dynamic properties of a gearbox with pitting distributed over multiple teeth have rarely been investigated. In this paper, gear tooth pitting propagation to neighboring teeth is modeled and investigated for a pair of spur gears. Tooth pitting propagation effect on time-varying mesh stiffness, gearbox dynamics and vibration characteristics is studied and then fault symptoms are revealed. In addition, the influence of gear mesh damping and environmental noise on gearbox vibration properties is investigated. In the end, 114 statistical features are tested to estimate tooth pitting growth. Statistical features that are insensitive to gear mesh damping and environmental noise are recommended.     

基于温度效应的多自由度齿轮系统的非线性动力学分析

齿轮在工作时,由于功率损耗和环境温度等原因,引起齿轮轮体及箱体的温度发生变化,影响齿轮传动性能。以直齿圆柱齿轮传动系统为研究对象,考虑齿面摩擦、齿侧间隙和时变啮合刚度等非线性因素,引入温度变化的影响,建立六自由度的齿轮系统非线性动力学模型,并采用4~5阶龙格-库塔算法对模型进行求解,结合分岔图、相图和Poincare映射图,分析温度变化和激励频率对齿轮系统动力学的影响。结果表明,温度变化对系统的影响与激励频率取值有关;系统随着激励频率的变化会表现出不同的动态特性响应,包括单周期响应、多周期响应以及分岔和混沌响应。相关结论为进一步改善齿轮系统的设计和安装提供了参考。     

用激光透射法测量双圆弧齿轮齿面油膜厚度

本实验首次采用激光透射法,通过测量齿轮箱外伸出轴端上安装的一对渐开线直齿轮的侧隙的变化,间接测量了齿轮箱内双圆弧齿轮的齿面中心油膜厚度。测试过程中采用了逐次标定的方法,由此可以完全消除测量仪器本身精度以外的任何误差。此方法测量精度分别达到了0.1μm（动态）和0.01μm（静态）,测试结果与理论分析基本吻合。本实验方法也适用于其他圆柱真齿轮和斜齿轮的齿面中心油膜厚度测量。     

含裂纹故障齿轮系统的非线性动力学研究

考虑时变啮合刚度、间隙非线性及传动误差的影响,针对试验齿轮箱中的单对齿轮传动建立齿轮副扭转振动的参数化动力学模型,对裂纹故障的非线性动力学机理进行研究。采用平均法分析齿轮裂纹模型的主共振及1/2亚谐共振的动力学响应;给出裂纹演化过程对齿轮系统啮合刚度及动力学行为的影响;通过幅频特性曲线、时域图、相轨迹图、Poincaré截面图及频谱图综合分析含有裂纹故障齿轮的振动特征;通过奇异性理论分析裂纹程度及传动误差所产生的内部激励与系统动力学分岔的关系,从而揭示了不同裂纹程度和传动误差所引起的不同分岔模式;最后通过试验提取含有裂纹故障齿轮的振动特征,试验结果验证了理论分析的结果,从而为齿轮系统裂纹故障的识别提供理论依据。     

M-DOF dynamic model for load sharing behavior analysis of PGT

A Multi-degree-of-freedom (M-DOF) nonlinear dynamic model for n-pinion Planetary gear train (PGT) is presented in this paper to investigate load sharing behavior of planet gears. In this dynamic model, manufacturing and assembly errors, elastic deformation and time-varying mesh stiffness are considered. Two sets of elastic compatibility equations are proposed to describe compatibility relationship between displacements, errors and elastic deformations. By means of Ishikawa formula, time-varying mesh stiffness of the gear pair is determined. The dynamic motion equations are solved with Runge-Kutta numerical integral method, which yields the displacements and deformations of each component. With the model, dynamic load sharing behavior of planet gears is evaluated. An example of 3-pinion PGT dynamic modeling is included, for which the influence of floating sun gear and adding flexible planet pin on the load sharing characteristics is analyzed.     

Study of Influence of Helix Angle on Friction Characteristics of Helical Gears

The minimizing of friction loss of helical gears is one of today's most important problems. Though helical gears are very widely used in power transmissions of automobiles and ships, basic research on helical gears is comparatively scarce. Under these circumstances, the authors studied the friction loss of helical gears in view of various helix angles, loads, and rotational speeds. Further the relationship between temperature rise of teeth and friction loss was made clear. The results were discussed from the viewpoint of reduction of friction loss and were related to the results of the authors' spur gear tests. Friction loss and temperature rise of helical gear teeth were also investigated while the helix angles were widely varied. All these results, based on experimental investigations, were summarized and general formulas were established in view of gear design.     

发动机激励下行星传动非线性啮合力特性

建立了发动机波动转矩激励下的行星传动纯扭非线性振动模型,模型中考虑了齿侧间隙、时变啮合刚度及其相位差、综合啮合误差以及各行星轮位置相角时变性。根据信号调制原理,分析了行星排啮合力边频带的产生原因。把发动机转速、转矩分成4种典型工况,在时域和频域内对行星排啮合力进行深入分析。研究表明,啮合力以单边冲击为主,啮频倍频与部件转频倍频调制普遍存在,受发动机工作状态影响相对较小,啮频倍频与波动转矩频率倍频调制则受工作状态影响很大,高速重载时作用最显著。该结论为行星齿轮传动设计提供了依据。