Analysis of stiffness and damping characteristics of externally pressurized gas-lubricated porous bearings under conical mode of vibration

This work gives a theoretical analysis of the dynamic behaviour of externally pressurized gas-lubricated porous bearings with journal rotation and with regard to the stiffness and damping in the conical mode of vibration. The theoretical analysis has been obtained by the simultaneous solution of the continuity equation of flow through a porous medium (derived from a modified version of Darcy's law) and the modified Reynolds equation (governing the pressure in the bearing clearance) by the finite difference method (successive over relaxation, central difference) satisfying the appropriate boundary conditions. The effects of journal speed, feeding parameter, supply pressure, porosity parameter and length-to-diameter ratio on the stiffness and damping coefficients have been investigated. A one-dimensional flow model without velocity slip is considered.     

圆轴承温粘热效应有限元方法分析

构筑了有限宽圆轴承温粘热效应二维有限元模型。依据圆轴承温粘热效应的三维模型计算结果，油膜温度场变化沿轴向可以忽略，在轴瓦和轴颈与油膜界面使用绝热边界条件，用有限元方法联立求解周向和径向二雏能量方程、双雷诺边界条件广义雷诺方程、周向速度方程和温粘方程，给出中心对称面上油膜温度和速度分布，轴承特性系数与工程上应用数据吻合。研究结果表明，不考虑轴瓦的导热使油膜力计算结果偏高，同时，将进油槽处温度作为油膜的始温叉使油膜力的计算结果偏低，这一高一低相互抵消，证实该模型合理，符合工程应用。     

Stiffness and damping coefficients of an inclined journal bearingSteifigkeit und dämpfungskoeffiziente eines schrägen gleitlagers

In the determination of the dynamic behaviour of a rotating shaft, the fluid film stiffness and damping coefficients of the bearings play an important role. The general practice is to ignore the rotational stiffnesses and damping coefficients due to the tilt of the journal in the bearing. This paper presents the stiffness and damping coefficients of such journal bearings. Using the expression for film thickness, the modified Reynolds' Equation for the tilted finite journal bearing is set up. The solution of this equation for the film pressure is obtained by using Fedor's proportionality hypothesis. The results obtained are presented in the form of non-dimensional charts.     

The influence of variation of viscosity with temperature on the steady state characteristics of journal bearings — simplified analysis

Investigation of the influence of temperature on journal bearing characteristics, through the strong dependence of viscosity on temperature, is rendered difficult by the complicated nature of the thermal boundary conditions, the flow in the cavitated region, etc. and by the computational requirements for the simultaneous solution of the Reynolds equation and the energy equation. It was found that the judicious use of the experimentally observed fact that the variation of temperature in the axial direction is negligible can drastically simplify the investigation without impairing its accuracy. The analysis is presented briefly, in dimensionless form, together with a note of caution on the erroneous use of isoviscous bearing characteristics for the estimation of the effective temperature of lubricant film.     

Stability analysis of rough journal bearings under TEHL with non-Newtonian lubricants

The combined effects of surface roughness and lubricants rheology on stability of a rigid rotor supported on finite journal bearing under thermal elastohydrodynamic lubrication have been investigated using the transient method. The newly derived time dependent modified Reynolds and the adiabatic energy equations were formulated using a non-Newtonian Carreau viscosity model. The simultaneous systems of modified Reynolds equation, elasticity equation, energy equation, and the rotor motion equation with initial conditions were solved numerically using multigrid multi-level method with full approximation technique. From the characteristic equation, the instability threshold is then obtained with various surface roughness parameters and the elastic modulus of the bearing liner materials. The results show that stability of the bearing system deteriorates with decreasing both the power law exponent and the elastic modulus of bearing liner material. The rough surface journal bearing with transverse pattern under TEHL regime exhibits better stability when compared with the rough surface journal bearing with longitudinal pattern.     

Comparative study of the performance of six-pocket and four-pocket hydrostatic/hybrid flexible journal bearings

This paper compares the performance of a six-pocket capillary compensated hydrostatic/hybrid flexible journal bearing to that of a simi ar four-pocket journal bearing. The comparison is based on theoretically computed results. The finite-element method has been used to obtain simultaneous solutions of the three-dimensional elasticity equations and the Reynolds equation. It is observed that the six-pocket journal bearing may be more efficient from a stability point of view as compared to a similar four-pocket journal system.     

圆轴承流体润滑热效应模型及其应用

提出了对轴向解析、周向有限元压力分布的有限宽圆轴承模型进行流体动力润滑热效应分析的方法。该方法采用绝热边界条件，能量方程无需与雷诺方程迭代求解，即可确定出油膜沿周向方向的平均温度场和粘度场分布；基于变分不等方程理论，雷诺方程采用一维直接解法求解。应用提出的方法计算了流体润滑热效应对进油槽位于水平方向两侧的圆轴承承载系数、偏位角和压力分布的影响，与工程上的标准数据计算结果吻合，方法简单适用，节约计算量。     

基于PDE的气体润滑轴承特性的数值分析

针对雷诺方程求解困难的问题,应用Matlab PDE工具箱提出了一种比较方便的数值离散求解方法,同时提出了一种对剖迭代的逼近算法来处理压力边界条件。并采用该算法通过改造PDE工具箱对小孔节流静压气体径向轴承的静态性能进行了研究,将其计算结果与其它分析方法的计算结果进行了比较,表明该方法的计算结果精度很高而且非常稳定,证明将该方法应用在流体润滑领域的可行性。     

Dynamic stiffness and damping coefficients of aerodynamic tilting-pad journal bearings

The dynamic gas–film forces of aerodynamic bearing often can be characterized by eight linear stiffness and damping coefficients. How to theoretically predict these coefficients is a very difficult issue for tilting-pad gas bearing design because of its structural complexity. The current study presents a novel and universal theoretical analysis method for calculating the dynamic stiffness and damping coefficients of aerodynamic tilting-pad bearing. The gas–film pressure within the bearing is expressed in the form of dimensionless compressible gas-lubricated Reynolds equation, which is solved by means of the finite element method. With the assumption that the journal and the pads are disturbed with the same frequency, the dynamic coefficients of tilting-pad gas bearing are computed by using the partial derivative method and the equivalent coefficient method. Finally, the investigations are conducted about the effects of bearing number, perturbation frequency of the journal and the pads, eccentricity ratios, preload and length-to-diameter ratio of the bearing on the dynamic coefficients of aerodynamic tilting-pad journal bearing. The numerical results indicate that the dynamic stiffness and damping coefficients of tilting-pad gas bearing are closely related with these factors. The proposed analytical method provides a valuable means of predicting dynamic performances of tilting-pad gas bearing. The solution can be used for the purpose of prediction of dynamic behavior of the rotor systems supported by aerodynamic tilting-pad bearings.     

Numerical simulation of piston ring lubrication

This paper describes a numerical method that can be used to model the lubrication of piston rings. Classical lubrication theory is based on the Reynolds equation which is applicable to confined geometries and open geometries where the flooding conditions are known. Lubrication of piston rings, however, fall outside this category of problems since the piston rings might suffer from starved running conditions. This means that the computational domain where the Reynolds equation is applicable (including a cavitation criteria) is unknown. In order to overcome this problem the computational domain is extended to include also the oil film outside the piston rings.The numerical model consists of a 2D free surface code that solves the time dependent compressible Navier–Stokes equations. The equations are cast in Lagrangian form and discretized by a meshfree moving least squares method using the primitive variables u, v, ρ for the velocity components and density, respectively. Time integration is performed by a third order Runge–Kutta method. The set of equations is closed by the Dowson–Higginson equation for the relation between density and pressure. Boundary conditions are the non-slip condition on solids and the equilibrium of stresses on the free surface. It is assumed that the surrounding gas phase has zero viscosity. Surface tension can be included in the model if necessary. The contact point where the three phases solid, liquid, and gas intersect is updated based on the velocity of the solid and the angle between the normals of the solid and the free surface.The numerical model is compared with the results from an analytical solution of the Reynolds equation for a fixed incline slider bearing. Then results from a more complicated simulation of piston ring lubrication are given and discussed.     

径向轴承考虑压粘效应的雷诺方程式及短轴承的近似解

本文针对滑动轴承压强低的特点，将压粘方程线性化，提出了线性化的压粘方程式，在此基础上，建立了考虑压粘效应的有限长径向轴承的雷诺方程式，此方程可作为研究有限长径向轴承的基础，此外，在本文中还介绍了用近似法解考虑压粘效应对矩径向轴承计算的主要结果，并用算例将它与考虑压粘效应的短轴承的解析解作计算对比，结果表明近似解表达式的结构简单，更方便应用，而且误差很小，数值计算说明了压粘效应对最小油膜厚度的影响是使它的厚度有一定程度的增厚，润滑计算计算时应当考虑。     

Steady-state thermo-hydrodynamic analysis of cylindrical fluid film journal bearing with an axial groove

A steady-state thermohydrodynamic analysis of an axial groove journal bearings in which oil is supplied at constant pressure is performed theoretically. Thermohydrodynamic analysis requires simultaneous solution of Reynolds equation, energy equation and heat conduction equations in the bush and the shaft. From parametric study it is found that the temperature of the fluid film raises due to frictional heat thereby viscosity, load capacity decreases. Increased shaft speed resulted in increased load carrying capacity, bush temperature, flow rate and friction variable. It is difficult to obtain the solution due to numerical instability when the bearing is operated at high eccentricity ratios.     

Thermal Shaft Effects on Load-Carrying Capacity of a Fully Coupled,Variable-Properties Cryogenic Journal Bearing

The purpose of this work was to perform a rather complete analysis for a cryogenic (oxygen) journal bearing. The Reynolds equation required coupling and simultaneous solution with the fluid energy equation. To correctly account for the changes in the fluid viscosity, the fluid energy equation was coupled with the shaft and bearing heat conduction energy equations. The effects of pressure and temperature on the density, viscosity, and load-carrying capacity were further discussed as analysis parameters, with respect to relative eccentricity and the angular velocity. The isothermal fluid case and the adiabatic fluid case represented the limiting boundaries. The discussion was further extrapolated to study the Sommerfeld number dependency on the fluid Nusselt number and its consequence on possible total loss of load-carrying capacity and/or seizure (catastrophic failure). Finally, an attempt was made to discuss the change in entropy in the bearing during one given cycle as a function of the fluid pressure and temperature changes.     

Analysis of externally pressurized porous gas journal bearings —II

A theoretical analysis of load-carrying capacity generated by vibration of the journal (the squeeze-film effect) of an externally pressurized porous gas journal is made. The flow of gas in the porous media is in three dimensions and obeys Darcy's law. The solution has been obtained by the perturbation method. The journal is assumed to vibrate either by translatory motion or by rotational motion around the transverse axis. The dimensionless squeeze load, moment, load- and moment-phase angles are given for various squeeze numbers, feeding parameters and a porosity number.     

The elastohydrodynamic lubrication of line contacts with pseudoplastic fluids

A new Reynolds equation, obtained by using the pseudoplastic model to express the non-newtonian fluid property, is derived for line contact elastohydrodynanic lubrication problems. The Reynolds equation results in a non-linear equation of high order which is derived by using an approximate method. The Newton-Raphson technique is utilized to solve the simultaneous system of modified Reynolds and elasticity equations. The results in this paper show that on increase in the coefficient of non-linearity, the pressure spike and film thickness decrease, and they approach a hertzian situation. With respect to the various loads and/or velocity values, the behaviours of the pressure spike and film thickness for non-newtonian fluids are almost the same as those of a newtonian fluid.     

A thermohydrodynamic analysis of journal bearings operating under turbulent conditions

An estimate of temperatures and pressures of a hydrodynamic journal bearing operating in the turbulent range has been obtained by the simultaneous numerical solution of the energy and Reynolds equations accounting for the variation of lubricant properties. The method has been applied to a particular bearing geometry and lubricant, and it is demonstrated that large reductions in load-carrying capacity result which are accompanied by, to a lesser extent, reduced bearing power loss.     

雷诺方程的数值计算方法概述

作为流体力学中的基本公式,雷诺方程的数值求解一直是流体润滑领域研究的重要方向之一。以雷诺方程的基本形式为基础,分别介绍有限差分法、有限元法和有限体积法求解雷诺方程的过程,讨论各自的特点及存在的问题;介绍多重网格法同上述方法结合在求解雷诺方程中的应用,指出多重网格法在求解雷诺方程的高效性方面有了很大程度的突破,但在求解精度上并未有显著改善;讨论等几何分析方法在求解雷诺方程上的应用,指出等几何分析方法求解雷诺方程具有较高的效率和求解精度,但仍存在如样条基函数的不可插值性和IGA在雷诺方程求解方面的普适性等问题,探讨IGA的研究方向,如针对特定雷诺方程引入适于IGA的新型样条表征求解空间、修改IGA理论与雷诺方程的离散模式引入新型边界条件加载模式等;因数值求解雷诺方程时在边界处理、复杂求解域等问题上仍没有一个稳定适用面广的方法,建议可尝试联合IGA与多重网格法来求解雷诺方程,以进一步提高求解效率和精度。     

The influence of non-newtonian oil film short journal bearings on the stability of a rigid rotor

The stability behaviour of rotors supported in plain cylindrical cavitated non-Newtonian fluid film journal bearings is investigated. Expressions for the journal force due to the fluid film are developed for the short bearing approximate solution to the modified form of Reynolds' equation. Linearised fluid film coefficients are used for the stability analysis. The results show that the dynamic fluid film coefficients for non-Newtonian lubricants are different from Newtonian coefficients and that they have a strong influence on the stability of rotor-bearing systems.     

Non-linear analysis of two-lobe bearings in turbulent flow regimes

By adopting the non-linear turbulent theory proposed by Elrod and Ng, the classical Navier-Stokes equations were modified. These modified momentum equations and the continuity equation were solved by the finite element method using Galerkin's technique and a suitable iterative scheme.The performance characteristics of a finite two-lobe hydrodynamic bearing operating in a fully developed turbulent regime were studied in terms of the load-carrying capacity, the stiffness and damping coefficients and the non-dimensional critical journal mass at various eccentricities for Reynolds' numbers up to 12000. Computed results were compared with the results obtained using linearized theory.