Analysis of aerostatic porous journal bearings using the slip velocity boundary conditions

A theoretical analysis of the static characteristics of aerostatic porous journal bearings is presented using the slip velocity boundary condition at the interface of the lubricating film and porous surface of the bearing. With thin wall bearings, the pressure in the porous bushing is obtained in closed form and the modified Reynolds equation is solved numerically using the finite difference method. The load capacity and the mass rate of flow are expressed non-dimensionally and are calculated numerically for different operating conditions. Variation in $\text{W}\text{?}$ and $\text{G}\text{?}$ with the feeding parameter and other parameters is shown graphically. The effect of velocity slip on $\text{W}\text{?}$ and $\text{G}\text{?}$ for different operating conditions was studied. The theoretical results are compared with a similar available solution using a three-dimensional flow model in the bushing. There is near-perfect agreement.     

The effect of axial pinch on the squeeze film action between curved annular plates

A theoretical study is presented of the effect of pinch induced by an axial current on the behaviour of the squeeze film of an electrically conducting lubricant between curved annular plates. The film thickness is assumed to be an exponential function of the radial coordinate. The expressions for the pressure, load-carrying capacity and squeeze time as functions of film thickness are obtained. With a convex plate configuration, axial-current-induced pinch decreases the pressure and load capacity. For a concave plate configuration the pressure increases with increase in current for large values of the dimensionless curvature parameter $\text{}\text{?}$gb while the trend is reversed for small values of $\text{}\text{?}$gb.     

Estimation of the performance of a narrow porous journal bearing with hydrodynamic turbulent lubrication considering slip velocity and arbitrary wall thickness

A closed-form mathematical analysis is presented for the hydrodynamic lubrication of a 360° short porous metal journal bearing with arbitrary wall thickness which is press fitted in a solid housing and works with a turbulent film of newtonian lubricant. A new pressure equation is used. The bearing is assumed to be narrow, and therefore circumferential flow of the lubricant in the clearance region is negligible in comparison with that in the axial direction which makes the governing differential equation simpler to solve. However, this simplification is not applicable to darcian flow in the porous matrix so that a three-dimensional Laplace equation is required to describe the continuity of flow in the pores. The film curvature is included by retaining terms containing $\text{C}\text{R}{}_1$ in the expression for film thickness. The curvature of the permeable bearing matrix, which allows it to have an arbitrary wall thickness, is taken into account by a direct approach. Infinite Fourier series and their orthogonal properties are utilized for the determination of the turbulent hydrodynamic pressure distribution from which the load-carrying capacity and attitude angle are calculated. All the results of interest are simple and fully analytical in nature permitting easy and economical calculation of numerical data over a very wide range of parameters.     

Unsteady flow between rectangular plates and between circular plates

An analysis is presented of the unsteady flow of a viscous incompressible fluid between two parallel infinitely long rectangular plates and between two parallel circular plates. The lower plate is fixed and the upper plate moves towards the lower plate. Full Navier-Stokes equations are used to obtain the pressure distribution as a function of the film thickness h(t) and the velocity $\text{h}\text{?}\text{(t)}$ of the upper plate. The inertia is taken into account and, for a given load on the upper plate, the sinkage relation between h and t is determined for various values of the Reynolds number. The departure from the classical inertialess solution is exhibited for various values of the two dimensionless parameters involved, one characterizing the load and the other gravity.     

Stability limits of elliptical journal bearings supporting flexible rotors

The stability of a system consisting of a flexible shaft with a single central rotor supported by two finite elliptical journal bearings was considered. The Reynolds equation was solved numerically for several values of the eccentricity ratio, the $\text{L}\text{D}$ ratio and the dimensionless velocity of the journal centre. The resulting pressure profiles are used to determine the load capacity and the spring and damping coefficients. Limiting speeds of stable operation are obtained from the roots of characteristic equations for the corresponding bearing-rotor system. The operating load, ellipticity, $\text{L}\text{D}$ ratio and shaft flexibility significantly affect the limit of stable operation. Elliptical bearings are suitable for stiff and moderately flexible rotors.     

Study on the dynamic characteristics of a new type externally pressurized spherical gas bearing with slot-orifice double restrictors

This study proposes a new type externally pressurized spherical gas bearing with slot-orifice double restricted restrictors, used in a new type gas gyroscope. Based on the spherical Reynolds equation and cylindrical Reynolds equation, small perturbation theory is employed to analyze the dynamic characteristics of the new type externally pressurized spherical gas bearing. The slot gas film and the spherical film are united by means of compatibility condition and a weak solution formula, based on Galerkin residual method, is erected. The nonlinear coupled weak solution equations are solved with FEM and the dynamic coefficients are obtained. Numerical simulation is operated and the change laws that the dynamic coefficients change with dimensionless perturbation frequency f, eccentricity ratio E_h and other structure parameters are analyzed. This paper provides a reference and preparation for analysis on the dynamic characteristics of the similar gas bearings.     

The dynamic behaviour of squeeze films in one-dimensional porous journal bearings lubricated by a micropolar fluid

The generalized Reynolds equation governing the pressure distribution for a micropolar lubricant in a dynamically loaded porous journal bearing is derived and applied to one-dimensional squeeze film journal bearings operating under a cyclic load. The analysis indicates how the microstructure in the lubricant, the permeability of the bearing material and the bearing wall thickness influence the operating eccentricity ratio.     

Squeeze films in full porous metal bearings

The behaviour of squeeze films in porous journal bearings under a steady load is analysed. A full journal bearing with a non-rotating journal is considered. The Reynolds equation governing the pressure in the film region is modified to take account of the mass exchange of the lubricant between the film and the bearing matrix. In the case of thin-walled bearing (H?R), the modified Reynolds equation takes a simpler form. However, since this equation cannot be solved directly as applied to a finite bearing, solutions in a closed form are found both for the long bearing and narrow bearing approximations. It is found that the permeability of the bearing matrix and the wall thickness of the bearing adversely affect the load capacity and greatly reduce the time needed for the journal to attain a given eccentricity. With a permeable bearing, there is the possibility of the journal coming into contact with the bearing (ε=l) in a finite time. Generally the performance of a long bearing is better than that of a narrow bearing.     

The effect of variable permeability on the performance characteristics of porous bearing

An analytical solution for the performance characteristics of a short bearing having an axial and radial permeability of $\text{k}{}_{\mathrm{z}}\text{[1 + cos(}\text{πz}\text{L}\text{)]}$ and $\text{k}{}_{\mathrm{z}}\text{[1 + (}\text{2}\text{π}\text{)]}$, respectively, is obtained. Results are presented which relate the eccentricity ratio and coefficient of friction as functions of load number for various design variables. These results are compared to the results obtained for an isotropic bearing having a permeability of $\text{k}{}_{\mathrm{z}}\text{[1 + (}\text{2}\text{π}\text{)]}$.     

Analysis of hydrostatic thrust bearings under non-axisymmetric operation

Analyses of the pressure distribution, the load capacity and the lubricant flow rate for a laminar flow hydrostatic thrust bearing in a symmetrical operation are presented. In a symmetrical operation the supported surface rotates about an axis parallel to but offset from the bearing axis. These bearings are commonly used as circular saw guides in the forest products industry. Bearing pressure and load and the lubricant flow are strong functions of a non-dimensional bearing number λ, the film thickness variation ? and the bearing offset $\text{L}\text{?}$ from the rotation axis.     

The load capacity of a partial journal bearing with a non-Newtonian film

Lubricating oils with viscosity index improver additives exhibit non-Newtonian behaviour. Starting from the most general type of fluid flow equation, connecting cubic shear stress to rate of shear for non-Newtonian lubricants, a modified form of Reynolds' equation has been derived and solved for the steady state load capacity for finite width full cylindrical journal bearings. The method has been extended to partial journal bearings both for the centrally loaded film and for the film with minimum thickness at the trailing edge. Numerical results are given for bearing arcs of 180° and 120° and length-to-diameter ratios of 1 and $\text{1}\text{2}$.     

A fast method for the numerical solution of thermal-elastohydrodynamic lubrication problems

The paper presents a method for the numerical solution of the elastohydrodynamic lubrication problem in line contacts, which includes thermal and rheological effects. The method consists in alternating repeatedly the solution of a generalised Reynolds equation and of the energy equation. In both cases use is made of a finite-element technique. Since the method proves to be very fast, a computer program has been developed which runs on common personal computers. Results (presented for two values of load parameter and for a range of values of slip/roll ratio up to 50%) show that thermal and non-Newtonian effects can strongly influence the film thickness and are essential for a reasonable evaluation of traction force. In this connection, the dimensionless parameter K (which depends on the equivalent curvature radius as well as on thermal conductivity and other material parameters) appears to be determinant.     

Performance analysis of high-speed spindle aerostatic bearings

The methods adopted to derive the pressure distribution and precision of bearing rotation are fundamental issues in the arena of gas bearing design. The current study presents a detailed theoretical analysis of bearing performance, in which the gas flow within the bearing is initially expressed in the form of simplified dimensionless Navier Stokes equations. Adopting the assumption of mass flow continuity between the bearing clearance and the orifice, the nonlinear dimensionless Reynolds equation is then derived and subsequently discretized using the Newton method. Finally, the modified Reynolds equation is solved by means of the iterative rate cutting method. The current numerical models are valid for the analysis of the film pressure distribution, friction effects, loading capacity, rigidity, lubricating gas flow rate, and eccentricity ratios of a variety of static and dynamic pressure aerostatic bearings, including high-eccentricity ratio journals, high-speed non-circular journals, thrust bearings, and slider bearings, etc. The proposed analytical models provide a valuable means of analyzing the static and dynamic performance of a high-precision rotating gas bearing, and allow its design to be optimized accordingly.     

Theoretical analysis and experimental investigation of a porous metal bearing

To improve the tribological behaviour of porous bearings a new type of self-lubricated porous metal bearing is introduced with special non-uniform distribution of permeability. It has been manufactured using a practical production process, its local permeability varying regularly along its circumference. Theoretical analysis and experimental investigation show that: the new bearing has lower friction and higher load capacity than that of ordinary porous bearings, and there is no initial temperature rise and friction increase with the new bearing. The limiting pV(tested) value of the new bearing is about 200% that of an ordinary porous bearing. Considering non-uniform permeability and the effects of curvature, cavitation and velocity slip, a modified Reynolds equation for the oil film and lubrication equation for the porous matrix are derived. A numerical solution for the equations is obtained. Some aspects of the lubrication mechanism in porous bearings and ways of improving load capacity are also discussed.     

Hydrodynamic analysis of rough curved pivoted porous slider bearings with couple stress fluid

Abstract

The purpose of this paper is to study the effect of surface roughness on the performance of curved pivoted porous slider bearings lubricated with couple stress fluid. The modified B–J slip boundary condition is utilised at the porous/fluid film interface to derive the Reynolds type equation for the problem under consideration. To mathematically model the surface roughness due to non-uniform rubbing of bearing surfaces, a stochastic random variable with non-zero mean, variance and skewness is considered. The closed form solution is obtained for the averaged Reynolds equation, and the compact expressions for the mean fluid film pressure mean load carrying capacity, frictional force and the centre of pressure are obtained. The numerical computations of the results show the improved performance due to the couplestresses and the presence of negatively skewed surface roughness. However, the presence of porous facing and positively skewed surface roughness affects the performance of the pivoted porous slider bearing.     

Analysis of normal stress effects in a squeeze film porous bearing

An analytical study of a porous bearing lubricated by a second-order fluid is considered. This investigation explains the working of general porous bearings and, in particular, describes the lubrication aspects of synovial joints. An approximate method for the solution of the governing fluid film equation and Darcy's equation for a porous region is used. Exact expressions for dimensionless pressure, load capacity and response time are obtained. The load capacity and response time for the diseased joint decrease compared with the healthy joint. The decrease in permeability of cartilage enhances the load capacity.     

Elastohydrodynamic analysis of a cylindrical journal bearing with a flexible bearing shell

The effect of the elastic deformation of a bearing shell was considered in the determination of the performance characteristics of a hydrodynamic journal bearing. The finite element method with an iteration scheme was employed to solve the Reynolds equation governing flow in the clearance space and the three-dimensional linear elasticity equations representing the displacement vector field in the bearing shell. For design convenience a nondimensional deformation coefficient ψ relating μ, E_m, U₀ , C, R_j and tis defined. The performance characteristics were obtained in terms of load-carrying capacity, fluid flow, power loss and attitude angle for an aspect ratio $\text{L}\text{D = 1}$, eccentricity ? = 0.6 and for a wide range of deformation coefficients. The results are compared for bearing materials having Poisson's ratio v equal to 0.3 and 0.4.     

气浮主轴轴向闭式支承系统静特性数值分析

求解气体润滑问题的最大难点在于如何获得雷诺方程的精确解。为了提高气浮主轴轴向支承系统的设计精度,在建立小孔节流气体静压止推轴承润滑数学模型的基础上,运用有限元方法对雷诺方程进行离散化。基于MATLAB7.0工具平台,采用超松弛迭代法,开发求解气体静压止推轴承压力方程与静特性的数值仿真软件。以某气浮主轴轴向闭式支承系统静特性参数设计为例,通过数值仿真,获得单侧止推轴承润滑气膜的压力分布,及不同节流孔直径与不同供气压力下的静特性性能;以刚度最大为优化目标,确定该气浮主轴止推轴承的结构参数与操作参数,并由此获得闭式支承系统的静特性。     

Optimum clearance fits for journal bearings in relation to B.S.1916 and to lubrication theory

The tolerances given in B.S. 1916 Parts I and II “Limits and Fits for Engineering” have been compared with the optimum bearing clearance as calculated on a basis of maximum value for the minimum film thickness. It was found that the optimum design for a journal bearing when the effect of bearing tolerance is least is that with a diameter/length ratio of 3. A general approach to the problem showed optimum conditions for other $\text{D}\text{L}$ ratios.