Optimization of the double sided spiral groove thrust bearing: A comparison to approximate analytical solutions

Reliable high performance bearings are essential in rotordynamics. As speeds and loads are continuously increased the demands on bearing technology grow simultaneously. Optimization of bearing characteristics becomes more important under these circumstances. For hydrodynamic thrust bearings, recently a special design has been presented, which is supposed to enhance the load carrying capacity by 60% compared to the best known thrust bearing type. The bearing consists of two spiral grooved surfaces and is therefore called a double sided spiral groove bearing. Early calculations of the bearing performance base on approximate analytical methods. In order to verify those results, a detailed numerical model of the bearing is presented by the authors. With the help of a particle swarm optimization method, optimal bearing parameters are determined and a comparison is drawn to other thrust bearing designs. It is shown that previously published results overestimate the performance of the double sided spiral groove bearing severly. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two classes of sub-optimal shapes for one dimensional slider bearings with couple stress lubricants

Shape optimization of slider bearings operating with couple stress lubricants is performed here for the first time by using a novel direct optimal control approach, which defines the gradient of the film height as a control. The bearing load is maximized. One dimensional Reynolds and energy equations are used. Several constraints are taken into consideration. They avoid the occurrence of cavitation and ensure the validity of the Reynolds equation. The model is validated against a known analytical solution (the Rayleigh step bearing). Two simple design rules are inferred, which yield two different classes of sub-optimal shapes: the multi-stepped bearings and the multi-sloped bearings, respectively. Multi-stepped bearings consist of several steps and the couple stress parameter may affect the constant value of the film height between steps. Multi-sloped bearings consist of several inclined regions and the couple stress fluid parameter may affect the constant value of the film height between regions. The slider bearings operation under variable load is stable. A sensitivity analysis identified the design parameters which have the highest impact on bearing performance. The optimal slider bearing shapes obtained for Newtonian lubricants do not change when most common couple stress fluids are used. Isothermal models may be used successfully at lower values of the couple stress parameter.     

The influences of longitudinal surface roughness on sub-critical and super-critical limit cycles of short journal bearings

By applying the stochastic model of rough surfaces by Christensen (1969–1970, 1971)  and  together with the Hopf bifurcation theory by Hassard et al. (1981) [3], the present study is mainly concerned with the influences of longitudinal roughness patterns on the linear stability regions, Hopf bifurcation regions, sub-critical and super-critical limit cycles of short journal bearings. It is found that the longitudinal rough-surface bearings can exhibit Hopf bifurcation behaviors in the vicinity of bifurcation points. For fixed bearing parameter, the effects of longitudinal roughness structures provide an increase in the linear stability region, as well as a reduction in the size of sub-critical and super-critical limit cycles as compared to the smooth-bearing case.     

螺旋槽球形轴承的承载能力

本文运用J·H·Vohr和C·H·T·Pan的理论和方法,建立了广义坐标系下的螺旋槽轴承的压力雷诺方程.然后在球轴承的边界条件下采用参数摄动法导出了动压螺旋槽球轴承润滑油膜的雷诺方程的近似解析解.由此,对各轴承槽型参数关于承载能力的影响作了计算和讨论,给出的最佳槽型参数与实验结果是一致的,与当前已发表的国内外资料相比较也是一致的.由于目前业已发表的文章均为计算机数值解,因此,本文对螺旋槽球轴承的特性研究提供了一个新的方法和途径.本文承中国科学院力学研究所林同骥,付仙罗同志及上海651研究所丁世德,蔡建中同志审阅,并提出了宝贵意见,作者谨在此表示衷心的感谢.     

Wear and fluid-solid-thermal transient coupled model for journal bearings

In this study, the transient interactions between the sliding wear behaviour and fluid–solid–thermal (FST) characteristics of journal bearings are revealed using an established mathematical model. The calculated temperature distribution is validated by a comparison with experimental results from the literature. Furthermore, a wear test for lubricated journal bearings is conducted to verify the predicted wear rate. The time-varying wear and FST performances of the journal bearing, including the wear rate, wear depth, fluid pressure, contact pressure, and maximum temperature are calculated numerically. Through numerical simulations, the effects of the boundary friction coefficient and surface roughness on the wear and FST performances are evaluated. To demonstrate the importance of considering the three-dimensional (3D) thermal effect during the wear analysis of lubricated journal bearings, the numerical results predicted by the isothermal model and the thermal model are compared systematically within a wide range of operating conditions. The numerical results reveal that the worn surface profile slightly decreases the maximum temperature. Additionally, the worn region is primarily located at both edges of the bearing, and the time-varying worn surface profile may be beneficial for improving the hydrodynamic effect. Furthermore, the effect of the 3D thermal characteristics on the wear prediction of journal bearings cannot be ignored when the external load, boundary friction coefficient, surface roughness are relatively large.     

Approximation of quasi-periodic solutions of a rotor in two-lobe bearings with time-varying geometry

Improving the dynamic behaviour of rotor systems in journal bearings represents an ongoing topic of research. The pressure distribution within journal bearings is described by the Reynolds equation, whereby unwanted oscillations can be caused by the fluid-solid interaction within the bearings. An approach of a two-lobe bearing with time-varying geometry is suggested to suppress or at least to reduce occurring oscillations. In order to systematically analyse the system, a spectral reduction is performed, allowing to handle also quasi-periodic behaviour by means of numerical continuation algorithms. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling of the Reynolds Fluid-Film Equation with the 2D Navier-Stokes Equations

The pressure field in thin fluid films can quite precisely be calculated by Reynolds fluid-film equation. In some problems, it may be useful to couple thin fluid-films with general 2D or 3D fluid flows. In the current work, we analyze the fluid flow, pressure and temperature field in a hydrodynamic journal bearing with a rectangular oil groove. Pressure and temperature in the fluid gap are calculated by means of the Reynolds equation and the 2D energy equation. Cavitation effects are taken into account by incorporating a 2-phase cavitation approach. In order to calculate the velocity and pressure field in the oil groove, the 2D Navier-Stokes equations are used; the temperature distribution in the oil groove is computed by means of the 2D energy equation. Appropriate coupling conditions for velocity, pressure and temperature are formulated in order to couple the flow in the fluid gap with the flow in the oil groove. Thermal expansion of journal shaft and bearing housing are also taken into account, since the bearing clearance changes with increasing temperature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dynamics of a nonlinear rotor-floating ring bearing system

Today, in high speed applications the rotors are commonly supported by hydrodynamic journal bearings. One typical configuration of journal bearings incorporated in automotive turbochargers is the floating ring bearing. Rotors supported by floating ring bearings have many advantages, regarding costs and power consumption for example. However, they might become unstable with increasing speed of rotation. At the onset of instability both the perfectly balanced and unbalanced rotor undergo self-excited vibrations which could cause the mechanical breakdown of the system. The “oil whip”-phenomenon, very well known from the investigations of the plain journal bearing occurs here in a different form. At the stability limit the rotor begins either oscillating with about the half of the ring speed or the half of the ring speed plus the half of the journal speed depending on the system parameters. For this reason a rotor-floating ring bearing model is presented showing the mentioned characteristics. By applying the nonlinear equations of motion the limit cycles of the system are determined and its loss of stability is investigated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Active magnetic damper in a power transmission system

In rotor dynamics, the bearing characteristics exerts a decisive influence on dynamics of the rotating shaft. The research and application experience have led to active magnetic bearings (AMBs), which allow for unique applications in rotating systems. The paper presents the investigations concerning optimization of the magnetic bearing construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line (aircraft industry applications) and modifies its dynamic properties. In the developed concept of AMBs for aviation purposes, a necessity of increasing its bearing load capacity and damping has occurred. The second important criterion is a weight reduction. This advanced problem leads to specific requirements on the design and materials for the AMB. To achieve these goals, some simulations have been performed. The experimental results are presented as well.     

Structural design and analysis for a sensor-integrated ball bearing

The structural design issues associated with the development of a sensor-integrated ball bearing were investigated. A finite element model was established to analyze the structural integrity of a ball bearing whose outer ring was modified to accommodate a miniaturized load sensor module. The sensor module was designed to monitor dynamic load variations within the bearing, which is an important indicator of a bearings safe operation. This structural integration of “intelligence” is significant in that it bring's automated, on-line, self-diagnosis capability to a dynamic system for which reliable, safe operation is of critical importance. A design tool was developed to evaluate the effect of outer ring modification on the dynamic load-carrying capability of the bearing. The tool also provided quantitative guidelines for the design of load and other types of sensor modules that will be structurally embedded into the outer ring of a conventional bearing for industrial applications.     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Fully Coupled Thermo-Hydrodynamic Simulation Model for Journal Bearings

The isothermal form of Reynolds fluid film equation is used to predict the pressure generation in hydrodynamic journal bearings if temperature effects are neglected. Often, however, temperature effects may be important and cannot be neglected, because oil viscosity significantly varies with temperature. Also, thermal expansion of journal shaft and bearing housing must be taken into account since the bearing clearance changes with increasing temperature. Hence, the Reynolds pressure field equation, the energy equation for the fluid film and the heat transfer equations for journal and bearing housing have to be solved simultaneously. The coupled thermo-hydrodynamic fluid flow problem is mathematically defined by a system of nonlinear integro-differential equations. The governing equations are discretized and solved by a finite element approach. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic characteristics for wide magneto-hydrodynamic slider bearings with a power-law film profile

The magneto-hydrodynamic (MHD) dynamic characteristics of a wide power-law film-profile slider bearing lubricated with an electrically conducting fluid under the application of transverse magnetic fields has been proposed. A closed-form solution is obtained for the MHD power-law film-shape slider bearings, in which special bearing characteristics of the inclined-plane shape and the parabolic-film profile can also be included. Comparing with the non-conducting-fluid power-law film-shape bearing, the MHD bearing provides an increase in the load capacity, and the stiffness and damping coefficients. Comparing with the MHD inclined-plane slider bearing, the MHD parabolic-film bearing signifies an improvement in the steady performances and the dynamic characteristics.     

Optimum design of one-dimensional journal bearings

When two surfaces in relative motion are separated by a thin layer of viscous fluid, they are kept apart by the pressure generated in the fluid film due to viscous forces. The magnitude of the resultant of this pressure is the load capacity of the bearing. We find the shape of a one-dimensional journal bearing which supports the greatest load.     

转动与摆动复合运动下脂润滑关节轴承的数值分析

建立了球面轴承的三维润滑模型,该模型将内圈的转动运动、轴颈倾斜引起的内圈倾斜和内圈的摆动运动等因素纳入考虑,推导出球坐标下适用于非Newton(牛顿)流体润滑的Reynolds(雷诺)方程.应用该模型,并考虑使用润滑脂的Ostwald流变模型,对向心关节轴承的润滑问题进行了数值计算,研究了在不同的幂律指数、内圈倾斜角度和摆动角速度下,脂润滑膜的压力分布、最大压力、承载力和流量.结果表明:在合适的操作条件下,脂润滑能产生明显的流体动压效应;在其它参数不变时,幂律指数对脂润滑膜的最大压力和承载能力影响显著,相对于Newton流体,剪切稠化流体可提高润滑膜的最大压力和承载能力,并增加周向流量,而剪切稀化流体的影响效果则相反;内圈倾斜角度对脂润滑膜最大压力和承载能力的影响较小,内圈摆动角速度的影响则较为明显.     

Unexpectedly low angular extent of journal bearing pressures: experiment and theory

Journal bearings have been studied for a long time. Pressure solutions for the same, as presented in textbooks, typically have angular extents exceeding 150°. Here, for a bearing with a relatively larger clearance ratio (0.01 as opposed to, say, 0.001), our experiments show an angular extent of about 50° only. Such small angular extents cannot be predicted, even approximately, by the existing simple theories for journal bearing pressures. However, such theories are based on assumptions whereby only the relative speed between bearing and journal surfaces enters the governing equations. We discuss how these same assumptions motivate some new combinations of boundary conditions that allow reasonably simple numerical treatment. In this paper, the resulting families of possible solutions are computed semi-numerically using a Fourier series expansion in one direction and finite differences and numerical continuation in the other. We find that one such solution family contains small-extent solutions similar to those observed experimentally.     

Couple stress fluid improve rub-impact rotor-bearing system – Nonlinear dynamic analysis

This study performs a dynamic analysis of the rub-impact rotor supported by two couple stress fluid film journal bearings. The strong nonlinear couple stress fluid film force, nonlinear rub-impact force and nonlinear suspension (hard spring) are presented and coupled together in this study. The displacements in the horizontal and vertical directions are considered for various non-dimensional speed ratios. The numerical results show that the dynamic behaviors of the system vary with the dimensionless speed ratios, the dimensionless unbalance parameters and the dimensionless parameter, l^∗. Inclusive of the periodic, sub-harmonic, quasi-periodic and chaotic motions are found in this analysis. The results of this study contribute to a further understanding of the nonlinear dynamics of a rotor-bearing system considering rub-impact force existing between rotor and stator, nonlinear couple stress fluid film force and nonlinear suspension. We also prove that couple stress fluid used to be lubricant do improve dynamics of rotor-bearing system.     

A study of pressure distribution for a slider bearing lubricated with a second‐grade fluid

An infinitely wide lubricated bearing consisting of connected surfaces of second‐grade fluid is analyzed in the present study. The velocity and pressure fields are obtained by a homotopy analysis method (HAM). Two cases, namely, inclined and parabolic slider bearings are considered. The viscoelastic effects play an important role even under the assumption of the order of the magnitude of the variable. However, inertial term does not contribute. Graphical results are interpreted. Comparison between the numerical values for the inclined and parabolic slider bearing is also presented. In addition to that for a meaningful solution, the graphs of convergence parameter, residual error, and numerical comparison are also presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1231–1241, 2011     

Chaotic response and bifurcation analysis of a flexible rotor supported by porous and non-porous bearings with nonlinear suspension

This study presents numerical work investigating the dynamic responses of a flexible rotor supported by porous journal bearings. Both porous and non-porous bearing types are taken into consideration in this study. The rotating speed ratios and imbalance parameters are also presented and proved to be important control parameters. Many non-periodic responses to chaotic and quasi-periodic motions are found, too. From the bifurcation diagrams in this paper, it is also evidenced that the vibration behaviors would be improved by porous bearings. The modeling result obtained here can be employed to predict the dynamics of bearing–rotor systems, and undesirable behavior of the rotor and bearing orbits can be avoided. Also, this could help engineers and researchers in designing and studying bearing–rotor systems or some turbo-machinery in the future.