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.     

Nonlinear Dynamic Response of a Rotor Bearing System Due to Surface Waviness

In this paper, the radial vibrations of a rigid rotor supported by ball bearings are studied. In the analytical formulation, contacts between the balls and races are considered as nonlinear springs whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The implicit type numerical integration technique Newmark- with the Newton-Raphson method is used to solve the nonlinear differential equations iteratively. The effect of bearing running surface waviness on the vibrations of rotor is investigated. The formulation predicts discrete spectrum with specific frequency components for each order of waviness. Numerical results obtained from the simulation are validated with respect to those of prior researchers.     

Study of milling stability with Hertz contact stiffness of ball bearings

This present work examines the stability and nonlinear responses of a spindle milling system supported by ball bearings. A shaft finite element based on Timoshenko beam theory is employed to model the spindle, and modal reduction method is therefore adopted for saving the numerical calculating time. The issues of evaluating the effects of the ball bearing Hertz contact stiffness are consequently addressed. It is found that suitable constant bearing stiffness can be adopted to replace the nonlinear nonsmooth Hertz stiffness in prediction of the critical cutting depth of the milling system in certain bearing configuration conditions. For the constant bearing stiffness can be obtained by experiment, this replacement will undoubtedly simplify the spindle-bearing milling system. But with the increase in the bearing clearance, the spindle milling system will present obvious nonlinear behaviors, and the nonlinear Hertz contact bearing stiffness will take over. Isolated islands of chatter vibration, which are induced by the nonlinear nonsmooth bearing Hertz stiffness, can be found exist in milling processes in large bearing clearance conditions.     

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.Nomenclature O–xyz rectangular coordinate system - , _x, _y inclination angle of a shaft and its projections on the xz- and yz-planes - _x, _y inclination angles in rotating coordinates - , polar coordinates - I _p polar moment of inertia of a rotor - I diametral moment of inertia of a rotor - i _p ratio of I _p to I - dynamic unbalance of a rotor - rotating speed (angular velocity) - F magnitude of a dynamic unbalance force, F = (1 – i _p )² - c external damping coefficient - h internal damping coefficient - t time - D _x , D _y internal damping terms in stationary coordinates - D _x , D _y internal damping terms in rotating coordinates - N _x , N _y nonlinear terms in restoring forces     

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.     

滚动轴承-偏置转子系统动力特性数值分析与实验研究

应用有限元法建立偏置转子的计算模型,采用考虑轴承Hertzian接触力和内间隙等非线性因素的二自由度滚动轴承模型,建立了滚动轴承-偏置转子系统的非线性动力学模型.通过数值仿真和实验研究分析了转子系统的非线性动力特性.实验数据和有限元模型计算结果是一致的,证实了所建立滚动轴承-转子系统非线性模型的合理性.发现由于滚动轴承非线性因素的影响,当转速达到系统共振转速的两倍附近时,激起了系统亚谐共振.     

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.

     

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.      

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.     

Subharmonic resonance of a symmetric ball bearing–rotor system

The performance of a ball bearing–rotor system is often limited by the occurrence of subharmonic resonance with considerable vibration and noise. In order to comprehend the inherent mechanism and the feature of the subharmonic resonance, a symmetrical rotor system supported by ball bearings is studied with numerical analysis and experiment in this paper. A 6DOF rotordynamic model which includes the non-linearity of ball bearings, Hertzian contact forces and bearing internal clearance, and the bending vibration of rotor is presented and an experimental rig is offered for the research of the subharmonic resonance of the ball bearing–rotor system. The dynamic response is investigated with the aid of orbit and amplitude spectrum, and the non-linear system stability is analyzed using the Floquet theory. All of the predicted results coincide well with the experimental data to validate the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is in the vicinity of twice synchroresonance frequency, while the rotor system loses stability through a period-doubling bifurcation and a period-2 motion i.e. subharmonic resonance occurs. It is found that the occurrence of subharmonic resonance is due to the together influence of the non-linear factors, Hertzian contact forces and internal clearance of ball bearings. The effect of unbalance load on subharmonic resonance of the rotor system is minor, which is different from that of the sliding bearing–rotor system. However, the moment of couple has an impact influence on the subharmonic resonances of the ball bearing–rotor system. The numerical and experimental results indicate that the subharmonic resonance caused by ball bearings is a noticeable issue in the optimum design and failure diagnosis of a high-speed rotary machinery.     

推力轴承对轴承－转子系统的耦合作用研究

研究了推力轴承对轴承－转子系统的耦合作用。在传递矩阵法的基础上，提出一种研究考虑推力轴承影响的轴承－转子系统的动力学的通用方法。研究中考虑了如下几个因素：（１）推力盘的静态倾斜；（２）转子的静变形；（３）径向轴承中负荷的重新分配；（４）偏载对径向轴承性能的影响；（５）推力轴承对系统稳定性的影响。研究结果表明，在某些情况下，推力轴承对径向轴承的动特性、转子的静挠度、系统稳定性等具有显著的影响     

Equations of Motion of an Asymmetric Timoshenko Shaft

The equations of motion of an asymmetric Timoshenko shaft, that is having unequal principal moments of inertia, are derived within the framework of the Lagrangian formulation for continuous systems and fields. The Lagrangian density of the system is calculated in a moving frame, that is a rotating frame attached to the deformed shaft, and proves to depend on the four Lagrangian variables (fields) of the system and their first derivatives w.r.t. space and time.On account of general results of the theory of continuous systems and fields, the four Lagrange's equations of motion are derived from the Lagrangian density and are successively reduced to the two usual equations in the displacements.The procedure described in this work is compared with both a different Lagrangian formulation, based on the use of a floating frame, that is a rotating frame attached to the undeformed shaft, and the well-known Newtonian approach adopted by Dimentberg. Sommario. Si applica la formulazione lagrangiana per i sistemi continui e i campi per ricavare le equazioni del moto di un albero di Timoshenko asimmetrico, la cui sezione presenta cioé momenti principali dinerzia diversi. La densità di lagrangiana è calcolata in un sistema di riferimento rotante solidale allalbero deformato e risulta essere funzione delle quattro variabili lagrangiane (campi) del sistema e delle loro derivate prime rispetto allo spazio e al tempo.In accordo con i risultati generali della teoria dei sistemi continui e dei campi, si ricavano, a partire dalla densità di lagrangiana, le quattro equazioni di Lagrange successivamente ridotte alle due classiche equazioni rispetto ai soli spostamenti.Il procedimento proposto viene messo a confronto con una diversa formulazione lagrangiana, basata sulluso di un sistema di riferimento rotante solidale allalbero indeformato, e con la ben nota formulazione newtoniana adottata da Dimentberg.     

Experimental Study of the Influence of Changes in Load Direction on the Performance of a Crown Bearing

Laboratory tests have been carried out in order to investigate the influence of small changes in load direction on the performance of a steadily loaded hydrodynamic crown journal bearing. Hydrodynamic pressures, temperature distribution on the bush internal surface, shaft temperature, oil flow rate and bush torque were measured for fixed sets of operating conditions, at three different groove locations () with respect to the line of load ( = 0, –30, and +30 degrees). For each groove location quantitative information is provided concerning to the influence of applied load and shaft rotational speed on the performance characteristics. Changing the location of the groove around the load line did affect flow rate, bush torque and maximum hydrodynamic pressure, but only had a slight effect on bush maximum temperature.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Stress-strain state of an ice sheet subjected to a moving load under shallow-water conditions

The following two classes of problems of determining the stress-strain state of an ice sheet under a moving load are considered: determination of the resonant velocity for a load moving over a continuous ice field and calculation of the deflections of an ice field with a bounded ice-free zone subjected to a moving load. The problems are solved in a dynamic formulation. The algorithm of solution is based on the finite-clement method and finite-difference methods. Examples of calculations are given. Komsomol'sk-on-Amur State Technical University, Komsomol'sk-on-Amur 681013. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol 41, No. 4, pp. 206–210, July–August, 2000.     

精密测试转台空气静压轴承动态特征

由于气体的可压缩性，惯导测试设备上的空气静压轴承，在工作中常发生气膜振荡而严重影响主轴刚度和运动精度。本文深入研究了在阶跃载荷及交变载荷作用下，轴承气膜的过渡特性及动刚度特性，为分析空气静压轴承的动态特性提供了理论根据，并成功地解决了转台轴承的稳定性问题。     

Bending vibration analysis of rotating machineries using multibody dynamics technology — development of computer program RotB

This paper examines a computer program developed to analyze the vibration of rotating machineries based on theories of vibration and multibody dynamics (MBD). Bending vibration problems of rotating machineries have generally been categorized as either linear or nonlinear. Linear problems can be formulated by standard methods and nonlinear problems can be formulated by MBD methods. In our study, nonlinear problems are treated by the use of a general-purpose computer program, RecurDyn (RD). In the program we developed, rotor bending vibration analysis (RotB) structural properties such as shafts, rotating rotary disks, unbalanced masses and foundation structures are modeled as multibody elements. Also, nonlinearities such as contact, non-symmetrical shaft effects, bearing characteristics, nonlinear restoring and damping characteristics in the bearings are taken into account. The computational results demonstrate the validity of RotB.     

四瓦可倾瓦径向滑动轴承动力特性的实验研究

采用在线分析方法考察了300MW汽轮发电机可倾瓦轴承的静、动力特性,探讨了载荷、轴颈转速以及动态激振力频率等参数变化对可倾瓦轴承的静、动态性能的影响,比较试验结果与理论计算结果发现,在不同频率激振力作用下,理论计算所得到的可倾瓦轴承动特性参数与试验结果有较大的差异。     

Nonlinear response and stability of a spindle system supported by ball bearings

In this effort, the nonlinear responses and stability of a spindle system supported by ball bearings are presented. The dynamics of this system is described by a set of second order differential equations with a nonlinear piecewise smooth force. The Floquet theory is applied to investigate the stability of the periodic solution. Due to the loss of contact between the raceways and balls in the ball bearing, the bending of the frequency response curves switch to the left at the weak resonance region, which is similar to the frequency response curves of a system with a soft spring. With the decrease of the bearing clearance, the bending of the frequency response curves switch to the right, which is similar to the frequency response curves of a system with a hard spring. Increase of the frequency ratio, the bending of frequency response curves transforms from left to right. The route to chaos through a period doubling process is also observed in this spindle-bearing system.