Planetary gearbox with localised bearings and gears faults: simulation and time/frequency analysis

Planets bearings of planetary gear sets exhibit high rate of failure; detection of these faults which may result in catastrophic breakdowns have always been challenging. The objective of this paper is to investigate the planetary gears vibration properties in healthy and faulty conditions. To seek this goal a previously proposed lumped parameter model (LPM) of planetary gear trains is integrated with a more comprehensive bearing model. This modified LPM includes time varying gear mesh and bearing stiffness and also nonlinear bearing stiffness due to the assumption of Hertzian contact between the rollers/balls and races. The proposed model is completely general and accepts any inner/outer race bearing defect location and profile in addition to its original capacity of modelling cracks and spalls of gears; therefore, various combinations of gears and bearing defects are also applicable. The model is exploited to attain the dynamic response of the system in order to identify and analyze localized faults signatures for inner and outer races as well as rolling elements of planets bearings. Moreover, bearing defect frequencies of inner/outer race and ball/roller and also their sidebands are discussed thoroughly. Finally, frequency response of the system for different sizes of planets bearing faults are compared and statistical diagnostic algorithms are tested to investigate faults presence and growth.     

RV减速器主支撑角接触球轴承混合润滑分析

针对RV减速器角接触球轴承承受预紧力、轴向力和径向力等联合外载荷作用的工况,分析得出了内、外圈滚道接触界面的接触区几何参数和接触载荷.在此基础上,综合考虑了角接触球轴承的接触区宏观几何、接触载荷、真实表面粗糙度、瞬态效应等因素,建立了角接触球轴承混合润滑数学模型,分析了在不同工况下角接触轴承的润滑状况及表面以下应力分布.结果表明:随着载荷的不断增加,钢球与内圈沟道之间的油膜厚度会不断减少,导致干接触面积迅速扩大,接触点表面以下最大应力增大;转速的增加会使油膜变厚,干接触面积缩小.该结果对角接触球轴承的实际工程应用具有重要借鉴意义.     

预紧方式对弹流润滑下角接触球轴承内部力学特性的影响

油膜厚度预测在评估弹流润滑(EHL)下角接触球轴承的性能和耐久性方面发挥着重要的作用. 耦合拟静力学理论和自旋下椭圆接触弹流模型,以干接触角接触球轴承拟静力学分析方法为基础,建立了定压和定位预紧方式下考虑弹流润滑和钢球自旋运动的角接触球轴承的拟静力学分析模型. 采用快速傅里叶变换(FFT)计算椭圆接触的弹性变形,运用Gauss-Seidel迭代方法求解Reynolds方程,得到自旋弹流模型的完全数值解,将其代入轴承拟静力学模型中迭代,得到轴承内部接触载荷、三维接触压力及三维膜厚分布. 对采用不同预紧方式的SKF7210型角接触球轴承进行分析,结果表明:富油润滑下,当轴承转速从0增大到15 000 r/min时,定压预紧时内圈轴向位移减小17.83%,而定位预紧时内圈承受的轴向载荷增大23.17%;定压预紧方式下球与内外滚道间膜厚均略大于定位预紧. 此外,不同预紧方式下,外圈上的中心膜厚大于内圈10%. 与干接触相比,定压下考虑弹流润滑内圈上接触载荷略大0.64%.      

Influences of wear on dynamic characteristics of angular contact ball bearings

Wear between balls and races has significant effects on the dynamic characteristics of bearing, which is the main reason to cause bearing failure. Some existing contact stiffness models were established to study the dynamic characteristics of bearing. However, the wear of bearing has been rarely investigated due to the complexities of contact load and wear mechanism. This paper presents a new dynamic wear simulation model of angular contact ball bearings mounted in pairs to solve this problem. A final contact stiffness model is established based on the wear model. The effects of running distance, horizontal load, preload, initial contact angle, number and diameter of balls on wear performances are analyzed. A generalized time-varying and piecewise-nonlinear dynamic model of angular contact ball bearings is established to perform an accurate investigation on its dynamic characteristics, especially considering the coupling effects of wear and rolling contact. The effects of wear on the contact stiffness and nonlinear dynamic characteristics are analyzed according to the dynamic model. Additionally, the variations of the contact stiffnesses and frequency responses with different preloads are discussed and the results indicate that parameter selection has significant effects on the wear and nonlinear response.

     

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding–crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider–crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimisation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.     

混凝土材料的层裂特性

利用Hopkinson压杆作为实验设备,通过试件后面吸收杆上应变波形研究了混凝土材料的层裂特性。不同强度混凝土在不同加载率下的实验结果表明,混凝土层裂强度与其压缩强度以及加载速率有关,给出了层裂强度和压缩强度以及加载率之间关系的经验公式,并指出不同加载速度下混凝土裂纹扩展方式不同是层裂强度率效应的主要原因。加载压缩波损伤的影响、重复加载实验以及多次层裂的顺序都表明,损伤对混凝土的层裂过程有重要影响。     

圆锥滚子轴承润滑与动力学耦合研究

为准确分析圆锥滚子轴承润滑和动力学耦合性能,建立了基于油膜刚度与阻尼的圆锥滚子轴承动力学耦合方程,并对方程进行了验证和数值求解. 数值结果表明:与不考虑润滑相比,考虑润滑后轴承内圈轴向运动更加稳定,轴向位移变小;在不同的滚子端面球半径和挡边倾角下,润滑效应能够使内圈径向振动加速度级减小1.71到2.07 dB;同时滚子个数的增加会使轴承内圈滚道和内圈挡边的平均最小油膜厚度分别增加7.97%和4.43%.      

NONLINEAR THERMAL ANALYSIS FOR THE INTER-SHAFT BEARING UNDER THE DYNAMIC LOAD 1)

中介轴承作为双转子系统高低压转子重要的支承部件,其内圈和外圈均随着低压转子和高压转子高速旋转, 其传热问题更加突出.本文研究中介轴承在非线性动载荷作用下的非线性热行为.基于双转子系统动力学响应定义中介轴承动载荷,考虑中介轴承的径向游隙、分数指数非线性和参数激励等非线性因素,中介轴承动载荷会出现跳跃和双稳态等非线性行为. 考虑润滑剂的黏温关系,根据Palmgren经验公式建立动载荷作用下中介轴承的热传递模型,通过数值求解得到中介轴承稳态温度,发现动载荷的非线性行为导致中介轴承温度出现跳跃和双稳态等非线性热行为.分析转速比、偏心距、中介轴承径向游隙、Hertz接触刚度和滚子数目对中介轴承温度及非线性热行为的影响,表明偏心距、径向游隙和刚度只影响非线性热行为,而转速比和滚子数目对两者都有重要影响. 本文研究表明,动载荷相较于静载荷更适合描述中介轴承的实际载荷,由于双转子系统具有非线性振动特性, 中介轴承的热行为也表现出复杂的非线性行为.     

动载荷作用下中介轴承的非线性热行为研究

中介轴承作为双转子系统高低压转子重要的支承部件,其内圈和外圈均随着低压转子和高压转子高速旋转, 其传热问题更加突出.本文研究中介轴承在非线性动载荷作用下的非线性热行为.基于双转子系统动力学响应定义中介轴承动载荷,考虑中介轴承的径向游隙、分数指数非线性和参数激励等非线性因素,中介轴承动载荷会出现跳跃和双稳态等非线性行为. 考虑润滑剂的黏温关系,根据Palmgren经验公式建立动载荷作用下中介轴承的热传递模型,通过数值求解得到中介轴承稳态温度,发现动载荷的非线性行为导致中介轴承温度出现跳跃和双稳态等非线性热行为.分析转速比、偏心距、中介轴承径向游隙、Hertz接触刚度和滚子数目对中介轴承温度及非线性热行为的影响,表明偏心距、径向游隙和刚度只影响非线性热行为,而转速比和滚子数目对两者都有重要影响. 本文研究表明,动载荷相较于静载荷更适合描述中介轴承的实际载荷,由于双转子系统具有非线性振动特性, 中介轴承的热行为也表现出复杂的非线性行为.      

滚动体几何相似条件在轴承边界元法中的应用

滚动轴承作为多物体接触系统,具有高度的非线性。为了求解滚动轴承的三维载荷分布,考虑所有滚动体的形状和边界条件相同,利用几何相似条件通过初始位置滚动体的离散模型,得到其他滚动体的离散数据。将所有滚动体看作是一个物体,轴承系统可以用3个物体无摩擦接触边界元法进行模拟计算。考虑全部滚动体作为一个整体的特殊性,叙述了边界积分方程和耦合的矩阵方程的建立,编制了基于滚动体几何相似条件的轴承边界元法Fortran源程序。利用该程序对轧机四列圆锥滚子轴承进行数值模拟,得到轴承的接触压力和载荷,滚动体接触宽度和接触滚动体个数,并说明了方法的有效性。     

Study for ball bearing outer race characteristic defect frequency based on nonlinear dynamics analysis

The characteristic defect frequencies are widely used for diagnosing the local defect of the ball bearing. The varying compliance (VC) frequency of a fault-free rotor–bearing system equals to the BPFO (ball bearing outer race defect frequency) due to the internal kinematic relationship of a bearing assembly. In order to indicate this issue, a semi-analytical method—the harmonic balance method with alternating frequency/time domain technique—is exploited to obtain the solutions of rotor–ball bearing systems with /without an outer race defect. The solutions and the features of a rotor–ball bearing system with essentially nonlinear parametric excitation are analyzed. We prove the VC frequency equals the BPFO and explain the reasons that the harmonics of the characteristic defect frequency generally appear in the frequency domain. The VC, BPFO as well as their harmonics affected by the primary and super-harmonic resonance of the system are found out. Finally, a test rig of a rigid rotor–bearing system is established to verify the theoretical analysis qualitatively by presenting the performance of VC, BPFO and their harmonics in the frequency domain. In addition, the tests are accomplished in a cycle of running up and down to reveal the primary and super-harmonic resonance characteristics. On the basis of the theoretical and experimental results, the basic BPFO is not enough to judge an outer race defect. The discussion on frequency spectrum, the primary and super-harmonic resonance provides a more reliable way to elucidate the characteristic defect frequencies.     

滑动轴承二维动态摩擦接触的分析研究

应用边界单元法研究了滑动轴承动态摩擦时的接触力学性质,揭示了接触压力分布的不对称性;讨论了载荷,摩擦系数等因素对不对称性的影响,认为摩擦系数可使峰值接触压力降低和接触面积增加,增大载荷会使接触压力分布的不对称性更为明显,接触压力分布的不对称意味着磨损分布的不对称。同时分析了实际工作中滑动轴承摩擦副间的当量摩擦系数,指出在滑动试验研究中应采了当量摩擦系数。     

Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes

The paper deals with the interaction of a pair of outer cracks on a central crack situated at the interface of two dissimilar orthotropic half-planes. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved numerically by using Jacobi polynomials. The analytical expressions for stress intensity factors at the central crack tip and the expression of the strain energy release rate have been derived for general loading. Numerical values of the interaction effects of the outer cracks on the central crack have been calculated through stress magnification factors. It is seen that the interaction effects are either shielding or amplification depending on the size of the outer cracks and their spacing from the central crack.     

An approach for calculating the dynamic load of deep groove ball bearing joints in planar multibody systems

This study is focused on dynamic modeling of planar multibody systems with multiple deep groove ball bearing joints, in which the radial clearance, contact deformation, and bearing kinematics are included. By using the approach presented, the variation of the joint reaction force and the dynamic load on each ball element in bearings can be simulated. The deep groove ball bearing joints are modeled by introducing a nonlinear force system, which takes into account the contact elastic deformations between the ball elements and the raceways. The contact force is calculated by the Hertzian contact deformation theory that accounts for the geometrical and material properties of the contacting bodies. A planar slider-crank mechanism with two deep groove ball bearing joints is chosen as an example to demonstrate the application of the methodologies presented in this paper. In this model, one bearing locates at the joint between the ground and crank, while the other one locates at the joint between the crank and connecting rod. By numerical calculation, the dynamic load distribution characteristics of bearings under real mechanism movement conditions are simulated. From the results, it can be concluded that the dynamic load on each rolling element varies differently and belongs to a variable load with the change of mechanism configuration. Load characteristic analysis is the foundation of developing research on the fatigue life and reliability of bearings. This study will provide a key mechanical support for the performance evaluation, dynamic design, and geometrical parameter optimization of the joint rolling element bearings.     

混合式陶瓷球轴承在液氮中的摩擦学性能研究

研究了由ＳｉａｌｏｎＳｉ３Ｎ４陶瓷球,９Ｃｒ１８内外环以及由ＰＴＦＥ和玻璃纤维制成的保持架组合成的混合式陶瓷球轴承在液氮气氛和重负荷条件下的摩擦磨损性能,结果表明,通过９０ｍｉｎ的－１９６℃超低温台架试验后,混合式陶瓷球轴承仍可以正常工作,对试验后的陶瓷球进行显微镜观察未发现大面积的接触疲劳破坏,但在内外环滚道表面有轻微疲坑,混合式陶瓷球轴承的摩擦功耗明显低于钢制球轴承。     

Super ellipse as tyre-ground contact area

In tyre-ground interaction studies there is a need to describe the contact area between a tyre and the ground. Tyre load and contact area give the ground pressure, which is important in evaluating environmental impact. The super ellipse provides an excellent means for describing, in a simple mathematical way, the shape and size of widely varying contact areas ranging from circles over ellipses to squares or rectangles. In the most complicated case, eight parameters are used, but usually only five are needed. The suggested mathematical expression gives greater flexibility in designing contact areas. It provides explicit border values for integration of ground pressure over the contact area. Thus a more accurate location of ground pressure centre is expected.     

The 3-D boundary element method of roller bearing by plate element analogue

The mill roller bearing is made up of an internal ring, middle rolls and an external ring, the analysis of which is a multi-bodies contact problem. In this paper, based on the three-dimensional elastic contact BEM without friction, and using the structural characteristics of roller bearings, middle rolls are described by elastic plate units of different shapes, which is placed on the internal ring. The discontinuous traction problem of the contact element can be dealt with by the traction sub-element method. Therefore the contact problem can be changed from one of multi-bodies to one of two-bodies. As an elastic deformation of the plate element, it can be calculated by an elastic contact formula and is regarded as an equivalent gap to be put into the total matrix equality. The face-to-face contact condition of displacement and traction on the element will serve as the convergence judging condition. So convergence solution can be obtained before any geometry inconsistent on the contact element may take place. Therefore, the three-dimensional distribution of load of the mill roller bearing can be achieved. It has been proved that the calculating model of this method is visualized and simple, has fast convergence solution and higher accuracy, making it an effective numerical method of designing and analyzing load characteristics of the mill roller bearing. Project supported by the National Natural Science Foundation of China (50075075).     

一种基于空穴聚集的层裂模型

基于如下假设，给出了层裂过程中的应力松弛方程，并建立了一种基于空穴聚集的延性层裂模型:在层裂早期，微空穴的主要效应是减小应力作用面积；在层裂后期，应力按空穴聚集时的应力-空隙度依赖关系减小。把P.F.Thomason、D.L.Tonks等及S.Cochran等给出的依赖于应力（压力）的层裂空隙度方程分别耦合于守恒方程、计及损伤的状态方程及本构方程，建立了求解所有变量包括损伤的封闭方程组。这种基于空穴聚集的层裂模型已被应用于一维层裂试验的数值模拟。模型中的层裂强度及临界损伤度初始可以估计，最终的确定将使数值模拟结果与实测的速度（或应力）剖面以及观测的层裂面上的损伤基本一致。分别基于D.L.Tonk等及S.Cochran等给出的依赖于压力的层裂空隙度方程所作的一维层裂试验数值模拟结果基本一致，而与基于P.F.Thomason给出的依赖于应力的层裂空隙度方程所作的相应数值模拟结果有明显差异。     

水化硅酸钙纳米压痕分子动力学模拟方法优化

针对水化硅酸钙纳米压痕模型忽视了压头与基底之间相互作用的问题,由尺寸差异引起的金刚石压头难以计算的问题,以及Wittmann模型无法得到实际接触面积的问题,提出了新的模型与计算方法．结合分子动力学方法,采用金刚石压头-Wittmann模型基底的组合方式构建无定形态水化硅酸钙纳米压痕试验模型．在建模阶段,考虑到压头模型与基底模型粒子间尺寸差异,提出了等比例替换模型,通过公式推导并就不同尺寸模拟结果验证了等比例替换模型的可行性．在计算阶段,提出了局部前处理的弛豫方法进行模拟．确定最大荷载位置处的接触面积为546 nm2,进而求出水化硅酸钙模型硬度H为0.84 GPa、折合模量Er为30.52 GPa．并通过纳米压痕试验,验证了模拟结果的准确性,证明了模型的科学性,对今后水化硅酸钙(C-S-H)纳米层面的模拟具有重要借鉴意义．     

基于荧光法的滚动轴承内部润滑油层分布研究

润滑油在轴承内的分布及其变化规律对轴承的润滑性能有显著影响. 在本文中搭建了滚动轴承模拟试验台,基于激光诱导荧光方法实现了滚动轴承内钢球-外圈接触区附近润滑油分布的观察与测量,获得了润滑油供给油层分布的三维形貌图,研究了不同供油量和转速对轴承内部供给油层分布的影响规律. 试验结果表明充分润滑条件下相邻钢球-外圈接触区供给油池之间会形成相互连接的油带;在高速情况下,钢球-外圈接触区供给油层厚度受前一个接触区尾部空穴影响而减小;供油量的增加会增大表观油池,但并不意味着入口有效供油层的增加.