Characteristics of partial porous journal bearings of finite length considering curvature and slip velocity

A self acting, finite length, partial porous journal bearing with arbitrary load angle has been analysed to obtain the characteristics assuming the active film of a homogeneous single phase liquid lubricant extends over the whole bearing arc span. The adhesion condition used in earlier porous bearing analysis is no longer valid, due to the presence of the thin layer of fluid moving streamwise just beneath the surface of the porous bearing material, a new modified lubrication equation is used for the present analysis. Film curvature has been considered. The curvature effect of the thick porous bearing matrix is taken into account by a direct but new approach which makes possible the use of the separation of variables for the first time in such a problem. Collocation technique is utilised to determine the hydrodynamic pressure from which characteristics are evaluated as usual. The results are fully analytical in nature, simple, yet exact and accurate, permitting easy and economical calculation of numerical data over a very wide range of parameters.     

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.     

Hydrodynamic load capacity of a full porous journal bearing of finite length in the turbulent regime considering slip flow and curvature

The hydrodynamic lubrication of a 360° finite porous metal journal bearing of arbitrary wall thickness press fitted in a solid housing and working with a turbulent film of newtonian lubricant was analysed in a closed form using a new generalized pressure equation. Bearings of undefined axial length and narrow bearings have been considered previously. For finite bearings the flow of lubricant in the axial clearance space is of the same order as in the azimuth. Therefore the governing differential equation is three dimensional and difficult to solve. However, Warner's method was used to reduce this equation to two second-order ordinary linear differential equations with analytically known variable coefficients which are simpler to solve. Analytical solutions of these were obtained as a boundary value problem in terms of the end conditions defining the start and termination of the load-supporting film. The half Sommerfeld boundary conditions were used for the azimuthal film extent since they are mathematically simpler, and the results obtained in the laminar regime were close to those obtained using the more complex (but realistic) Reynolds boundary conditions. Film curvature effects were included by using C/R₁ in the expression for film thickness. The curvature effect of the thick porous bearing matrix, which allows it to have an arbitrary wall thickness, was taken into account by a new direct approach which makes it possible to use the separation of variables. The collocation technique was utilized to determine the hydrodynamic pressure from which the bearing characteristics were evaluated. The results are fully analytical in nature, simple and yet exact and accurate; they permit easy and economical calculation of numerical data over a very wide range of parameters.     

A lubrication equation including slip velocity for hydrodynamic porous bearings in the lower turbulent regime — a perturbation approach

A first attempt is made to analyse pressure development in the form of a new lubrication equation including slip velocity for finite self-acting hydrodynamic porous metal bearings operating in the turbulent regime (fully developed) with a single phase Newtonian incompressible lubricant. The derivation is based on the analogy of boundary layer theory wherein lubricant motion is treated as a generalized turbulent channel flow in which the impermeable wall is in motion and the porous wall is stationary. The type of flow varies from pure couette flow to shear flow coupled with codirectional and transverse pressure flow. A linearization (or perturbation) technique is used to decouple the two orthogonal flows by assuming that the shear stress in a finite bearing is a small perturbation of the shear stress valid for couette flow. Using Boussinesq's eddy viscosity formulation and the wellestablished power law as a universal law of wall, the governing pressure distribution equation is obtained from considerations of the conservation of momentum and continuity. The surfaces are considered to be hydrodynamically smooth. The whole treatment is approached from the viewpoint of fluid film design rather than from a fundamental fluid mechanics approach. No slip model has been used. The lubrication equation is fully analytical and can be applied to a number of particular bearing problems by using the simplifying restrictive conditions. The lubrication equation derived can be used to predict the bearing performance characteristics even in situations where the permeability of the bearings is anisotropic and the Poiseuille flow in the porous matrix does not obey Darcy's law.     

The solution of Reynolds equation under natural boundary conditions for hydrodynamic journal bearings

The solution of Reynolds equation for hydrodynamic pressures of the load-carrying lubricating layer in journal bearings is discussed.The accepted boundary conditions are shown to be helpful in explaining more accurately the physical meaning of the lubricant flow process and in assuming the boundary curvature for the load-carrying lubricant film in the convergent-divergent zone of the bearing clearance.New algorithms for determining the above curve boundary in the process of the numerical solution of a basic differential equation are suggested. The technique used for the theoretical study of the bearings made it possible to provide results of greater accuracy for their principal operating characteristics such as load-carrying capacity, frictional losses and lubricant consumption.     

Analysis of squeeze film performance between rough porous infinitely long parallel plates with porous matrix of non-uniform thickness under presence of magnetic fluid lubricant

Abstract

The performance of a magnetic fluid based squeeze film between infinitely long porous rough parallel plates with porous matrix of non-uniform thickness has been investigated. The bearing surfaces are considered to be transversely rough. The stochastic film thickness characterising the random roughness is assumed to be asymmetric with non-zero mean and variance. A magnetic fluid is used as a lubricant and the external magnetic field is oblique to the lower plate. With usual assumptions of hydrodynamic lubrication the associated Reynolds' equation is solved with suitable boundary conditions. Then expressions for pressure distribution, load carrying capacity and response time are obtained. It is observed that the load carrying capacity increases nominally due to magnetic fluid lubricant resulting in improved performance. But it is also seen that the composite roughness of the bearing surfaces introduces an adverse effect which gets more compounded due to the thickness ratio. However, the negative effect can be compensated to certain extent by the magnetic fluid lubricant in the case of negatively skewed roughness. This compensation further enhances when negative variance is involved. This study tends to suggest that the thickness ratio may play a crucial role for a better performance of the magnetic fluid based bearing system besides providing an additional degree of freedom.     

Elastic and damping properties of partial porous journal bearings of finite length and arbitrary wall thickness taking film curvature and slip flow into account

A self-acting porous journal bearing of finite length has been analysed in order to obtain the dynamic characteristics when an active film of a homogeneous single-phase liquid lubricant extends over the whole bearing arc span. A new governing lubrication equation which considers slip flow, film curvature and unsteady motion of the journal has been derived and solved analytically to yield results in a closed form for a dynamic load, taking into account the curvature of the porous bearing matrix and thus allowing the porous wall thickness to be arbitrary. Closed form results obtained in this way are valid for any arbitrary clearance ratio C/R₁ and wall thickness ratio H/R₁. The dynamic characteristics are evaluated as for impermeable bearings. The pertinent results are fully analytical in nature, simple yet exact and accurate, permitting easy and economical calculation of numerical data over a very wide range of parameters involved.     

The effect of inlet film boundary conditions on the steady state characteristics of hydrodynamic journal bearings

The well-known solutions for the pressure distribution in the lubricating film of a hydrodynamic journal bearing, satisfying the Reynolds boundary conditions, show a sudden change in the pressure gradient at the position of maximum film thickness, which is possible only if the lubricant is added precisely at this position. Since the pressure develops smoothly because of hydrodynamic action, a correction in the Reynolds boundary conditions is proposed and a new solution for the pressure distribution is obtained. The steady state characteristics of a hydrodynamic journal bearing using the new boundary conditions are compared with the well-known characteristics using the Reynolds boundary conditions.     

Numerical study on foil journal bearings with protuberant foil structure

A numerical model coupling the hydrodynamic pressure of lubricant film with the deformation of foil structure was developed for a type of foil journal bearing with protuberant foil structure. An isothermal and isoviscous lubricant was used in the fluid model, and a perturbation method was applied to linearize the Reynolds equation. The top foil was modeled as a strip of rectangular thin plate supported at a rigid point. The distributions of film pressure, film thickness, and foil deformation were solved by the finite element method (FEM). The effects of eccentricity ratio, bearing number, and number of protuberances on the characteristics of bearings were analyzed.     

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.     

Influence of balancing of internal combustion engines on the operating conditions of hydrodynamic bearings

We studied the influence of balancing internal combustion engines on the performance of hydrodynamic plain bearings. A non-linear approach makes it possible to calculate the forces of pressure generated by the lubricant film. This approach is coupled with a dynamic calculation, which determines the inertia forces of the rod. The counterweight to balance the engine is applied to the heads of rods and not to the crankshaft. We chose three models of connecting rod (rod of an engine in series, rod with partial and rod with complete counterweight). To determine the lubricant pressure field in the bearing, the modified Reynolds equation was solved using the finite difference method, taking into account the boundary conditions of Reynolds. Since the bearing is subjected to a variable load, the mobility method was used to facilitate the resolution of the Reynolds equation. The proposed numerical simulation allowed us to analyze the influence of counterweight applied to the connecting rod head on the variation of the lubricant pressure field, the minimum film thickness, the axial flow and the friction torque in the big end bearing during the operating cycle.     

The effect of modified inlet boundary conditions on the stiffness and damping characteristics of finite hydrodynamic journal bearings

The well-known solutions for the pressure distribution in the lubricating film of a hydrodynamic journal bearing, satisfying the Reynolds boundary conditions, show a sudden change in the pressure gradient at the position of maximum film thickness, which is possible only if the lubricant is supplied precisely at this location. Since the pressure develops smoothly because of hydrodynamic action, a correction in the boundary conditions is applied and a modified solution for the pressure distribution is obtained. The stiffness and damping characteristics of a finite hydrodynamic journal bearing using the new boundary conditions are compared with the wellknown characteristics incorporating the Reynolds boundary conditions. There is a significant change in the dynamic characteristics particularly at low values of the eccentricity ratio which can influence the characteristics of rotor-bearing systems.     

Hydrodynamic lubrication of short porous bearings

A theoretical analysis is presented for the hydrodynamic lubrication of short porous metal bearings that are press-fitted into a solid housing. It is customary with such bearings to neglect the circumferential flow of the lubricant compared to the axial flow and consequently the modified Reynolds equation has a simpler form. However, this device is not resorted to while considering the flow of the lubricant within the bearing material, and consequently the three dimensional Laplace equation describing such flow is solved. The pressure continuity at the bearing-film interface is maintained and the modified Reynolds equation is solved by the Galerkin method. Numerical results obtained on a digital computer indicate a progressive reduction in the load capacity and increment in the friction parameter and attitude angle as the permeability parameter is increased. These results significantly aid the rational design of short porous bearings.     

Porous inclined stepped composite bearings with micropolar fluid

Abstract

This paper describes the theoretical analysis of the effect of micropolar fluid on the lubrication characteristics of porous inclined stepped composite bearing. The lubricant with additives in the film region and also in the porous region is modelled as Eringen's micropolar fluid, which is characterised by the presence of suspended rigid particles with microstructures. The generalised Reynolds type equation is derived for the most general porous bearing configuration (porous composite bearings) lubricated with micropolar fluid. The closed form expressions are obtained for the fluid film pressure, load carrying capacity, frictional force and coefficient of friction. These expressions can be utilised to obtain the performance characteristics of four different bearing systems, namely, porous plane inclined slider, porous composite tapered land bearing, porous stepped bearing and composite porous tapered concave bearing. It is observed that the micropolar fluid lubricants provide an increased load carrying capacity and decreased coefficient of friction as compared to the corresponding Newtonian case.     

Hybrid journal bearings with particular reference to hole-entry configurations

There is a spectrum of pressure-fed journal bearings ranging from the purely hydrostatic bearing characteristics, ie zero speed operation, to the purely hydrodynamic bearing characteristics which depend completely on speed. Between these two extremes, hybrid bearing characteristics rely on mixed modes of external pressurisation and speed-dependent pressurisation. Large high speed hydrodynamic bearings require the lubricant to be pumped under pressure for temperature control. It is therefore attractive to use this external source of pressure to enhance the start-up performance by reducing wear and improving stability. Hybrid bearings offer the possibility of improving on both the zero-speed characteristics of hydrostatic bearings and on the whole range of speed characteristics of hydrodynamic bearings. It is concluded that hole-entry bearings may be particularly effective when compared with other bearing configurations for good load support and low energy consumption, when used in any of the four modes of operation including: zero-speed hydrostatic mode; high-speed hydrodynamic mode; zero and high-speed hybrid mode; and jacking mode where areas are pressurised for start-up. A modification to the procedure for solving the Reynolds equation is introduced to cope with cavitated regions. The technique presented for solving the bearing pressures and cavitation boundaries is efficient and has relevance to any type of liquid film bearing     

Effects of lubricant additives on the performance of hydrodynamically lubricated journal bearings

A non-Newtonian rheological model to investigate theoretically the effects of lubricant additives on the steady state performance of hydrodynamically lubricated finite journal bearings is introduced. In this model, the non-Newtonian behavior resulting from blending the lubricant with polymer additives is simulated by Stokes couple stress fluid model. The formed boundary layer at the bearing surface is described through the use of a hypothetical porous medium layer that adheres to the bearing surface. The Brinkman-extended Darcy equations are utilized to model the flow in the porous region. A stress jump boundary condition is applied at the porous media/fluid film interface. A modified form of the Reynolds equation is derived and solved numerically using a finite difference scheme. The effects of bearing geometry, and non-Newtonian behavior of the lubricant on the steady-state performance characteristics such as pressure distribution, load carrying capacity, side leakage flow, and coefficient of friction are presented and discussed. The results showed that lubricant additives significantly increase the load carrying capacity and reduce both the coefficient of friction and the side leakage as compared to the Newtonian lubricants.     

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.     

柴油机滑动轴承热流体动力润滑仿真研究

根据径向滑动轴承热流体动力润滑理论,基于JFO理论提出的质量守恒边界条件,建立同时包含油膜完整区和空 穴压力变化的单缸柴油机滑动轴承热流体动力润滑模型,采用有限差分法求解模型方程,仿真分析滑动轴承的油膜厚度、油膜压力、润滑油流量和温度等参数对润滑性能的影响,分析内燃机滑动轴承润滑特性,为轴承润滑可靠性设计提供一定的理论依据.     

Linear stability performance analysis of finite hydrostatic porous journal bearings under the coupled stress lubrication with the additives effects into pores

A theoretical investigation into the stability performance characteristics of a finite externally pressurized porous journal bearing has been undertaken with the slip flow of coupled stress fluid. The analysis deals with the modified Darcy-type equation incorporating the additive effect in the porous bush. A modified form of Reynolds equation is obtained using Stokes micro-continuum theory of coupled stress fluids as lubricant. Applying the first-order perturbation of the film thickness and steady-state film pressure, the stability parameters are obtained under various parametric conditions. The results reveal that stability deteriorates with increase in coupled stress parameter for a value of percolation factor under full slip condition.     

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.