Effect of rotational speed fluctuations on the dynamic behaviour of rolling element bearings with radial clearances

The behaviour of rolling element bearings is of major importance to the vast majority of rotating machines. Due to the varying assembly compliance and the internal clearance existing in the bearing in most cases, a highly non-linear behaviour of the rotor bearing system appears. Thus, a dynamic model of a horizontal rotor supported on ball bearings with radial internal clearance is considered, taking into account contact forces between the balls and the races, as well as the effect of varying compliance and of the internal radial clearance. However, since rotational speed fluctuations of various levels practically appear in all rotating equipment, their effect is now additionally considered. Their effect is examined in two characteristic rotor bearing cases. The analysis includes all the characteristic states of the dynamic behaviour of the rotor, such as periodic, unstable periodic and chaotic responses, using methods like frequency spectra, phase spaces, higher-order Poincare maps and Lyapunov exponents. All results presented show a dominant stabilization effect of the speed fluctuations with respect to the system behaviour. From the analysis performed, it is concluded that even a minimum fluctuation of the rotor speed may result in major changes of the system dynamics, indicating that speed fluctuations of the rotor are a governing parameter for the dynamic behaviour of the system.     

Stability analysis of a rotor bearing system due to surface waviness and number of balls

In the paper, the stability analysis of a rigid rotor supported by ball bearings has been studied. In the analytical formulation, the contacts between 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-β with Newton–Raphson method is used to solve the nonlinear differential equations iteratively. The effects of surface waviness and the varying number of balls on stability of rotor bearing system are observed. All results presented in form of fast fourier transformations show that the vibration characteristics of the rotor and its bearings change, when the bearings operate in different regions of their nonlinear load deflection characteristics. From the analysis, it is implied that the number of balls and number of waves in the ball bearing are two important governing parameters affecting its dynamic behavior.     

碰摩故障转子-滚动轴承耦合系统非线性动力学研究

针对滚动轴承支承下的转子碰摩故障机理分析和故障诊断问题,考虑滚动轴承非线性赫兹接触和轴承径向间隙,建立了含碰摩故障的转子-滚动轴承系统动力学模型。应用数值积分方法得到系统的非线性响应,利用时间波形图、分叉图、频谱图以及Poincaré映射图,研究了系统响应随转速、轴承间隙、碰摩刚度、偏心量以及碰摩间隙的变化规律,为有效诊断滚动轴承支承下的转子碰摩故障提供了理论依据。     

Effect of Disk Flexibility on Rocking Mode Frequencies of a Disk Drive Spindle Motor System

The lowest natural frequency of a hard disk drive related to a spindle motor is the so-called “rocking mode,” the frequency at which the rotor of the motor is whirling conically. A traditional rotordynamics theory with thick/rigid disks is not able to predict the rocking mode frequencies correctly. In this paper, the rocking mode frequencies of a single-disk rigid-rotor motor were solved analytically, including the effects of disk flexibility and ball bearing contact angle. In addition, the disk flexibility model was integrated to a finite element-based (FEM) flexible-shaft rotordynamics computer program. The closed-form solution of the analytical model provides design engineers a tool to identify critical design parameters. It was found that the rocking mode frequency, which is mainly affected by the clamped condition at the inner radius of the disk, is the flexible disk one-nodal-diameter mode frequency reduced by the compliance of ball bearings. The results obtained from both the analytical model and the FEM computer program matched the experimental results with an error less than two percent. Neglecting the effect of ball bearing contact angle increases the error three times. The effects of design parameters were also investigated.     

角接触球轴承支承的轴系振动频率计算方法

建立了角接触球轴承支承的轴系有限元模型,开发了相应的振动频率分析软件,计算了轴系静频,分析了在一定工况下转速对动频的影响和幅频特性,与实测试验结果吻合较好。结果表明:频域信号的最大峰值处的频率与轴系的静频相关。     

Prediction of the friction torque in grease lubricated angular contact ball bearings using grey system theory

The development of special grease makes it possible for angular contact ball bearings to operate at high speed and temperature; however, as an important performance parameter, friction torque of bearings lubricated with grease is much greater than that of bearings lubricated with oil-air, and heat generation due to frictional loss is also greater, so it is necessary to predict the friction torque occurring in grease lubricated angular contact ball bearings. Based on grey system theory, a new prediction methodology for bearing friction torque is proposed which capitalizes on the notion that the information about friction torque of angular contact ball bearing is generally poor, incomplete and uncertain. A grey prediction model, GM (1, N) model, is presented to predict the friction torque in grease lubricated angular contact ball bearings. Several experiments on the friction torque of grease lubricated angular contact ball bearings were conducted to model and validate the effectiveness of the GM (1, N) model through on-line and off-line approaches. Experimental results show that about 90% of bearing friction torque under varying speed can be predicted in the on-line prediction; above 85% of bearing friction torque under varying speed and different loads can be predicted in the off-line prediction. Comprehensive analysis shows that, the GM (1, N) model performs very well for both modeling data and model validation data under different loads, varying bearing speed and work cycles, the proposed methodology can be used to predict bearing friction torque with good accuracy and robustness.

     

滚动轴承支承的柔性转子系统的非线性动力学分析

研究滚动轴承支承的柔性转子系统的非线性动力学特性。建立柔性转子系统的有限元模型,研究不同轴承间隙系统的稳定性及分岔行为。在转子系统中考虑Hertz接触力,以及由转子质量偏心引起的不平衡力,用数值积分的方法得出系统间隙量在不同区域内的轴心轨迹、Poincaré映射图以及时域图。研究结果表明:在间隙较小时,系统表现出丰富的非线性现象,随着间隙的增大,系统的失稳区域也随之增大。     

高速电主轴球轴承——转子系统动力学性能分析

结合高速电主轴的结构特征,论述了球轴承支承的高速电主轴转子系统动力学性能的基本规律和一般特点,在分析角接触球轴承高速时动刚度的基础上,应用有限元法,建立了高速电主轴球轴承———转子系统动力学的分析模型和数值计算方法,并结合具体算例,分析了有关球轴承内部结构、外部工况条件以及转子系统结构等方面的参数对电主轴转子系统动力学性能的影响,并得出相关结论。     

Quasi-static Analysis of Thrust-loaded Angular Contact Ball Bearings Part Ⅱ:Results and Discussion

Ball bearings are widely employed mechanical components characterized by high precision and quality,and usually play important roles in various rotary machines and mechanisms.Many advanced applications require a deep understanding of their various kinematic and tribological characteristics that are essential to predict the fatigue endurance,relieve the vibration and minimize the power dissipation of ball bearings in particular applications.An angular contact ball bearing under a specified operating condition is simulated with the quasi-static/creepage analytical model proposed in the preceding article.The results demonstrate that the ball bearing is a statically determinate system.That the balls spin on both inner and outer races means the ball is controlled by neither the inner nor the outer raceway.The friction between the ball and raceway renders the inner and outer contact angles unequal.The larger the coefficient of friction is,the larger the angle deviation.The tangential traction perpendicular to the rolling direction due to the spin induces a gyro-like rotation of the ball with respect to the raceway even if no inertial effects are considered.The tangential elastic compliance of contacting surfaces gives rise to locked areas within the contact patch and transforms the sliding lines from circles into spirals.The differential slip due to the close conformity of the ball and raceway makes the sliding and traction distributions asymmetric,which will influence the location of the spinning center of the ball with respect to the raceway.The quasi-static/creepage model can be used to reveal the operating behaviors of ball bearings running under steady conditions and to optimize the design of ball bearings for specific applications.     

滚动轴承支承下的不平衡转子系统非线性动力响应分析

建立了滚动轴承支承下的转子系统非线性动力学模型。在滚动轴承模型中,充分考虑了滚动轴承的间隙、滚动轴承的滚珠与滚道的非线性赫兹接触力及由滚动轴承支撑刚度变化而产生的VC（varying compliance）振动,在转子系统中考虑了由于转子质量偏心而产生的不平衡力。运用数值积分方法获取了系统的非线性动力响应,分析了转子旋转速度、滚动轴承间隙对系统动力响应的影响,并运用分叉图、相平面图、频谱图及Poincaré映射进行了系统分叉与混沌特征分析,发现了通往混沌的倍周期分叉、拟周期环面破裂和阵发性分叉途径。     

Stability Analysis of Floating Ball Bearing

Conventional disk drive motors supported on ball bearings (BBs) cause nonrepeatable runout (NRRO) due to the surface imperfections on balls and raceways. NRRO is a source of track misregistration between head and disk that inhibits high track density in a hard disk drive. Fluid dynamic bearings with herringbone grooves either on the rotating or stationary surfaces are a suitable replacement for conventional ball bearings. Herringbone grooved bearings have considerably lower noise level than ball bearings and have better stability compared to plain journal bearings at concentric operating position. However, herringbone-grooved patterns are difficult to manufacture because groove depth is of the order of bearing clearance. The major limitation of the BBs is the direct contact between the rotating and stationary parts and also lack of damping effects. This present work attempts to overcome these drawbacks in BBs by eliminating the metal-to-metal contact using a layer of fluid film, and a theoretical analysis of stability characteristics of a floating BB is presented. Results indicate that there is an improvement in the stability of floating BB rotor systems with increase in outer to inner film clearance ratio (β) from 0.7 to 1.3, and with decrease in ratio of outer race radius to inner race radius (δ) from 3.0 to 1.2.     

挤压油膜阻尼器-滚动轴承-转子耦合系统的非线性响应分析

为了研究转子系统非线性动力学响应,建立了挤压油膜阻尼器-滚动轴承-转子耦合系统动力学模型。在转子系统模型中,考虑了转子、滚动轴承及挤压油膜阻尼器之间的相互耦合作用,并充分考虑了滚动轴承的间隙、非线性赫兹接触和挤压油膜阻尼器非线性油膜力等。运用数值积分方法分析了转子转速、支承刚度以及挤压油膜阻尼器油膜间隙对系统动力响应的影响,并结合分岔图、频谱图、Poincaré映射图和轴心轨迹图分析了转子系统的非线性动力学响应。结果表明:转子系统当转速较高、支承刚度较大或挤压油膜阻尼器油膜间隙较大时,转子系统容易出现拟周期运动。     

基于ROMAX的球轴承转子系统动态性能分析

滚动轴承转子系统在旋转机械系统中得到广泛应用,其动态性能对旋转机械有着重要的影响。基于Romax软件建立球轴承转子系统的动力学分析模型,通过算例分析了工况参数和结构参数对球轴承转子系统动态特性的影响规律,并通过实验结果验证了Romax对轴承性能预测的准确性。研究发现球轴承的接触角、内外圈沟曲率系数与安装位置对轴承的刚度、寿命与转子系统临界转速等动态性能有较大影响。工程中需要选择合适的结构与工况参数以满足转子系统的性能要求。     

Application of an envelope technique in the detection of ball bearing defects in a laboratory experiment

To judge the diagnostic capabilities of an envelope technique, based on enveloping the resonance frequency of a transducer,

1 Bearcon Signature™, Carl Schenck AG, Germany

artificially damaged ball bearings (SKF 6207) were run under oil lubrication at different radial loads and speeds in a laboratory experiment. Damage to the outer race, inner race and ball of different magnitudes was introduced by spark erosion and the response of the measurement system was analysed. The results showed that vibration spectrum components of outer race defects showed up in the envelope spectra of undamaged bearings, and that the detectability of defects was primarily limited by the speed of the bearing; on occasion the automatic scaling facility of the analyser was a limiting factor. Inner race and ball defects showed the well-known effects of load modulation in the defect contact zone, which can become dominant in the envelope spectra at high loads and speeds. This phenomenon could be confirmed by computer simulation. The detection of ball damage was hampered by the fact that overrolling of the defect in purely radially loaded bearings takes place incidentally. This drawback could be mitigated by applying a trigger technique to the envelope signal.     

转子在线自动平衡研究

研究了转子在工作转速下实现在线自动平衡的方法。以电磁执行器作为施力装置,运用傅里叶分析和线性系统的叠加原理,对不停车的情况下转子不平衡振动的识别和抑制的算法进行了推导,并在一滚珠轴承支承的转子系统上进行了试验研究。结果证明该方法和装置能够使旋转工作频率下转子的不平衡振动大大减小。     

球轴承涡轮增压器轴承-转子系统动力学分析与应用

相比于普遍使用的浮动轴承,在涡轮增压器中使用球轴承具有机械效率高和加速响应快的优势。以车用球轴承涡轮增压器为研究对象,用有限元法对轴承-转子系统进行了转子动力学特性的研究,对轴承-转子系统的临界转速进行了计算与分析,这是判断转子工作转速是否稳定和涡轮增压器工作是否可靠的重要依据;建立了增压器模型,并对比了计算结果和试验结果,证明了方法的可行性。通过整机试验表明,球轴承涡轮增压器能够满足当前车用发动机的需求,能够提高发动机的工作性能。     

DESIGN AND EVALUATION OF AN ULTRAHIGH SPEED MICRO-MACHINING SPINDLE

A new ultrahigh speed micro-spindle has been developed for micro-milling that can be used at rotational speeds approaching 500,000 rpm. Since conventional ball bearings or fluid lubricated journal bearings cannot be used at such speeds, the new micro-spindle uses a set of journal and thrust foil bearings. Prior to fabrication of the micro-spindle, rotordynamic analysis of the rotor with an attached cutting tool confirmed that the rotor would be stable at the desired speeds. The cutting tool was then attached to the rotor using a shrink-fit approach. The micro-spindle was integrated with a 3-axis micro-milling machine. Cutting experiments were performed on an aluminum alloy at speeds greater than 300,000 rpm using 125 and 300 μm end-mills. Vibration spectra for free rotation and during cutting confirmed the dynamic stability of the micro-spindle. The vibration spectrum was dominated by the rotational frequency and was free of deleterious vibrations. The increase in rotational speed to 450,000 rpm in micro-milling of aluminum alloy allowed an increase in feed rate to nearly 750 mm/min, thus increasing the material removal rate by more than two orders of magnitude. The dimensional accuracy of several straight cuts made at different feed rates was measured.     

Acoustical properties of hydrodynamic journal bearings

Results of theoretical investigations on acoustical properties of hydrodynamic journal bearings are presented. Nonlinear analysis including rotor imbalance is performed for a rotor-bearing system in order to obtain acoustical properties of the bearing. Furthermore, a cavitation algorithm, implementing the Jakobsson–Floberg–Olsson boundary condition, is adopted to predict cavitation regions in a fluid film. Acoustical properties are investigated through frequency analysis of pressure fluctuation calculated from the nonlinear analysis. Results show that the acoustical frequency spectra of the fluid film are pure tone spectra, containing the frequency of the shaft rotation and its super-harmonics. The analysis also shows that super-harmonics are predominant at the neighborhood of the fluid film reformation and rupture regions.     

角接触球轴承流固耦合仿真分析

角接触球轴承在运行过程中的润滑状况至关重要,润滑油直接影响滚动轴承的接触状态。为分析角接触球轴承的润滑状况以及考虑润滑时轴承的机械特性,基于有限元和晶格玻尔兹曼方法,建立了双向流固耦合轴承仿真模型,对角接触球轴承进行动力学有限元仿真和润滑流体仿真,并与轴承拟静力学理论计算结果进行对比,验证模型的准确性。分析结果表明,保持架与滚珠接触并撞击,在轴承腔内油膜压力最大,滚珠与内圈、外圈滚道接触区分别为第二、第三大油膜压力区。润滑油受滚珠公转影响,沿着滚珠转动方向流动,实现对滚珠与内圈、外圈和保持架之间的润滑。滚珠运动和最大接触应力仿真结果与轴承拟静力学理论求解结果一致,即流固耦合仿真模型计算轴承机械特性具有较高的准确性。     

Thermal characteristics of a CNC feed system under varying operating conditions

In high-speed and high-precision feed systems, thermal positioning errors are mainly caused by the non-uniform temperature variations and resulting time-varying thermal deformations under different operating conditions. The research presented here ultimately aims to develop a generic method capable of evaluating the thermal characteristics (such as temperature rise of heat sources, thermal positioning error) of the feed system induced by varying operating conditions (feed speed, cutting load and preload of ball screw). The thermal contact resistance between the balls and the inner and outer rings of supporting bearing is calculated using the Hertzian theory and JHM method. Experiments were carried out on a high-speed feed system experimental bench, and the influences of operating conditions on temperature rises of supporting bearings and ball screw nut were analyzed. Based on a WNN-NARMAL2 model, the relationship between temperature rise of supporting bearings and operating conditions was established. Furthermore, with the temperature of the ball screw nut set to be a moving heat source load, the temperature and thermal deformation distributions of the ball screw were simulated. The work described lays a solid foundation for thermal error prediction and compensation of a feed system under varying operating conditions.