Performance characteristics of a capillary-compensated hydrostatic journal bearing

Results are presented of an extensive experimental investigation into the performance of a six-pocket hydrostatic journal bearing with pockets of non-uniform depth subjected to a range of static loads. Lubricating oil was supplied, at pressures of up to 8.957 MN m^?2 (1300 Ibf in^?2), to capillarytype restrictors, connected to the six pockets in the bearing. The bearing had an $\text{L}\text{D}$ ratio of unity and operated with a journal rotation ranging from zero to 2000 rev min^?1.The measured load capacity and the stiffness characteristics indicated that bearings of this type had a significant load capacity which was limited by the bending of the shaft. The load capacity increased with speed at low supply pressures, but decreased slightly with increasing speed at higher supply pressures. The stiffness increased with eccentricity ratio except at low supply pressures. Film pressure surveys aided the elucidation of some of the bearing performance characteristics.     

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 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 $\text{1}\text{2}$.     

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.     

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.     

Micropolar fluid lubrication of one-dimensional journal bearings

A study of the characteristics of one-dimensional journal bearings shows that the load capacity increases and the coefficient of friction decreases as the parameter $\text{μ}{}_1$, which characterizes the microstructure of the base oil due to the presence of additives, increases. The time of approach increases as the parameter $\text{μ}{}_1$ increases.     

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.     

Lubrication of finite porous journal bearings

An analytical solution is attempted for a finite porous bearing, press-fitted into the housing and working with a full film of lubricant. The pressure distribution is determined by a simultaneous solution of Reynolds equation for the film and the Laplace equation for the bearing material while maintaining the continuity of pressure at the film-bearing interface.The pressure in the bearing material and film are taken in series form so that all the boundary conditions are satisfied. Using a suitably truncated series, the modified Reynolds equation is then solved by the Galerkin method. It is shown that two dimensionless parameters—(1) the permeability parameter and (2) the thickness to breadth ratio ($\text{H}\text{b}$)—adequately describe the operation of these bearings. Numerical calculations for all the bearing characteristics were carried out by digital computer. The results are tabulated so that designers can interpolate values required.The results indicate the progressive reduction in Sommerfeld number with permeability parameter; the dimensionless coefficient of friction and the attitude angle progressively increase with permeability parameter.     

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.     

The effect of rotational inertia on the squeeze film load between porous annular curved plates

The squeeze film behaviour between rotating annular plates was analysed theoretically for the case when the curved upper plate with a uniform porous facing approached normally the impermeable flat lower plate. Expressions for the pressure and the load capacity were obtained. The load capacity decreased when the speed of rotation of the upper disc increased up to a certain value $\text{B}\text{?}{}_0$ of the curvature parameter $\text{B}\text{?}$. This trend was reversed for $\text{B}\text{?}\text{?}\text{B}\text{?}{}_0$. The load capacity could be increased without altering the speed of rotation by increasing $\text{B}\text{?}$.     

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.     

The effect of lubricant inertia in externally pressurized bearings using a viscoelastic lubricant

The inertia effects in an externally pressurized bearing with a viscoelastic lubricant were considered using the methods of averaged inertia and of iteration. The inertia forces reduce the load-bearing capacity at a given flow rate. The effect of the elasticity of the liquid is to increase the pressure and the load-bearing capacity at any point. The effects of the elastic number S, the volume flow rate Q, the ratio σ of the inner radius of the bearing to the outer radius of the bearing and $\text{H =}\text{h}\text{a}$ on the pressure and the loadbearing capacity are presented graphically. The results obtained by the two methods are in good agreement.     

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 $\text{k}{}_{\mathrm{z}}\text{[1 + (}\text{2}\text{π}\text{)]}$.     

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.     

Analysis of step bearings with a micropolar lubricant

Squeeze film step bearings lubricated with a micropolar fluid were analysed. It was found that the load capacities increase as the micropolar parameter $\text{}\text{?}\text{gm}{}_1$ increases or as $\text{M =}\text{h}{}^2\text{μ}\text{4γ}$ decreases. The load capacity also decreases as the step height increases.     

Stability analysis of orthogonally displaced bearings

Journal bearings operating at high speeds often exhibit vibrational instability. Circular bearings are known to be prone to vibrations and, in general, a non-circular bearing geometry enhances shaft stability. Among the several possible configurations of non-circular bearings, elliptical, offset, three-lobe and tilted three-lobe bearings have attracted attention. The present analysis is a theoretical prediction of stability for a hybrid two-lobe bearing obtained by displacing the lobe centres of an elliptical bearing. It has been found that an orthogonally displaced bearing is more stable than the hitherto known bearings and at the same time it is easier to manufacture. Numerical results have been presented for nine combinations of horizontal and vertical displacements of lobes and three $\text{L}\text{D}$ ratios so that a suitable profile and $\text{L}\text{D}$ ratio can be selected by a designer in a specific application.     

Rotation effects on hybrid air journal bearings

Rotation effects of hybrid air journal bearings with multi-arrays of orifice feedings are investigated numerically. Porous air bearings are also solved for comparison. The results show that bearing load capacity W increases faster with eccentricity ratio than with rotation speed, i.e. bearing number Be. There are optimum orifice diameters, i.e. optimum feeding parameters λ_o, which give maximum load capacity W for orifice feeding; but for porous feeding, load capacity W increases with feeding parameters λ_p. It was found that the load capacity increases with feeding arrays of orifices and five rows of orifice feedings can approximate the operations of porous bearings very well. It was also found that load capacity W does not increase further when air supply pressure exceeds 5 atm because there is a critical pressure ratio through orifice (e.g. (P_o/P_s)<0.53).     

Design and analysis of mechanical face seals

An analytical study of the characteristics of mechanical face seals with misalignment is presented. The complete pressure equation was solved without using the narrow bearing assumption. Results are presented and it is concluded that simplified treatments overestimate the load and the moments particularly at touch. Design charts are given to calculate the separating force, the leakage rate, the moment and coefficient of friction for different tilt parameters ? and the ratio $\text{r}{}_{\mathrm{i}}\text{r}{}_{\mathrm{o}}$ of the inner radius to the outer radius.     

Experimental study of journal bearings with undulating journal surface

A journal bearing test rig was designed and constructed to test the behaviour of journals with wavy surfaces, the circumferential undulations being varied both in amplitude and in number. Results show that wavy journal surfaces may well enhance the load carrying capacity of a bearing. Moreover, surface undulations are shown to move the journal centre locus closer to the load line, ie cause a lower attitude angle. These effects are found to be more pronounced with larger wave amplitudes, and with higher numbers of waves around the journal circumference. In general, friction is found to be reduced with increase in surface wave amplitude.Good agreement is shown to exist between test results and a computer aided analysis conducted by the authors to predict wavy journals performance¹. It has been established that a wavy journal surface may, under certain conditions, display higher load capacity, lower friction and permit safer running of journal than bearings with perfectly smooth surfaces.