Numerical Calculation of Lubrication: Methods and Programs

By Ping Huang

Focusing on basic lubrication problems this book offers specific engineering applications. The book introduces methods and programs for the most important lubrication problems and their solutions. It is divided into four parts. The first part is about the general solving methods of the Reynolds equation, including solutions of Reynolds equations with different conditions and their discrete forms, such as a steady-state incompressible slider, journal bearing, dynamic bearing, gas bearing and grease lubrication. The second part gives the ‘energy equation solution’. The third part introduces methods and programs for elasto-hydrodynamic lurbication, which links the Reynolds equation with the elastic deformation equation. The final part presents application lubrication programs used in engineering.

• Provides numerical solution methodologies including appropriate software for the hydrodynamic and elasto-hydrodynamic lubrication of bearings
• Offers a clear introduction and orientation to all major engineering lubrication problems and their solutions
• Presents numerical programs for specific applications in engineering, with special topics including grease-lubricated bearings and gas bearings
• Equips those working in tribology and those new to the topic with the fundamental tools of calculation
• Downloadable programs are available at the companion website

With an emphasis on clear explanations, the text offers a thorough understanding of the numerical calculation of lubrication for graduate students on tribology and engineering mechanics courses, with more detailed materials suitable for engineers. This is an accessible reference for senior undergraduate students of tribology and researchers in thin-film fluid mechanics.

• Language: English
• Publisher: Wiley
• Release date: Jul 19, 2013
• ISBN: 9781118451229

Ping Huang

Huang Ping is a Professor at the South China University of Technology. He has been awarded for his research and teaching. His work focusses on tribology and he has written several books on the topic in both Chinese and English.

Book preview

1

Reynolds Equation and its Discrete Form

1.1 General Reynolds Equation and Its Boundary Conditions

1.1.1 Reynolds Equation

The general form of the Reynolds equation is

(1.1)

equation

where U = U 0–U h; V = V 0–V h. If we assume that the fluid density does not change with time, we have .

1.1.2 Definite Condition

The definite conditions of the Reynolds equation usually include the boundary conditions, the initial conditions and the connection conditions.

1.1.2.1 Boundary Condition

In order to solve the Reynolds equation, the pressure boundary conditions should be used to determine the integration constants. There are commonly two forms of pressure boundary conditions, namely

equation

where s is the boundary of the solution domain; and n is the normal direction of the border.

Usually, the inlet and outlet pressure boundaries for an oil film can be easily determined according to its geometry and the situation of the oil supply. However, such as a journal bearing which has both a convergence clearance and a divergence clearance, the position of the outlet cannot be determined in advance. Therefore, it can be assumed that both pressure and pressure derivative are equal to zero at the same time to determine the location of the outlet. Such a boundary condition is known as the Reynolds boundary condition, which is in this form

equation

Here are two examples of boundary conditions.

One-dimensional boundary conditions in the region of 0 ≤ x ≤ l

If the boundaries are known, we have and .

If the outlet is unknown, we have , and , where x ′ is the outlet boundary to be determined.

Two-dimensional boundary conditions in the rectangular area of (0 ≤ x ≤ l, −b /2 ≤ y ≤ b /2)

If the boundaries are known, we have and

If the outlet is unknown, we have , , and .

1.1.2.2 Initial Condition

For the nonsteady-state lubrication problem in which the velocity and/or the load change with time, such as the fluid hydrodynamic lubrication of a crankshaft bearing in the internal combustion engine, the Reynolds equation contains the squeeze term at the right end of Equation 1.1. The lubrication film thickness changes with time, so we need to give some initial conditions for solving the Reynolds equation. The general forms of the initial condition are as follows.

Initial film thickness:

Initial pressure:

If the lubricant viscosity and density also vary with time, their initial conditions should also be given.

1.1.2.3 Connection Condition

If the film thickness varies abruptly in several parts, like a ladder, the lubrication region also needs to divide into several subregions to solve the problem because the film thickness derivatives at the right end of Equation 1.1 do not exist at the abruptly changing positions. Therefore, the connection conditions should be given. The commonly used connection conditions are the continuity conditions of pressure and flow. If a film thickness changes abruptly at x ′, its connection conditions will be as follows.

Continuous pressure condition:

Continuous flow condition:

1.1.3 Computation of Lubrication Performances

After obtaining the pressure distribution from the Reynolds equation, we can calculate the static performances of lubrication, including the load carrying capacity, the friction and the flow.

1.1.3.1 Load Carrying Capacity w

The load carrying capacity of the lubricating film can be obtained by integrating the pressure p (x,y) in the entire lubrication domain, that is

(1.2) equation

1.1.3.2 Frictional Force f

The frictional forces of the lubricating film on a solid surface can be obtained by integrating the shear stress over the whole boundary of the lubricating film. The fluid shear stress is equal to

(1.3)

equation

For surfaces of z = 0 and z = h, to integrate the shear stress on both surfaces we have

(1.4) equation

where f 0 and f h are the frictional forces respectively on surfaces z = 0 and z = h.

After the frictional forces have been obtained, we then can determine the friction coefficient μ = f/w as well as the frictional power loss and the heat due to the friction.

1.1.3.3 Lubricant Flow Q

The side leaking flows of lubricant can be obtained by integrating the flow rates through the lubricating film boundary.

(1.5) equation

By summing all leaking flows over all boundaries we can obtain the total flow, which gives us the amount of lubricant needed to fill the clearance. At the same time, the total leaking flow will influence the extent of convection so that we can calculate the balanced thermal temperature according to leaking flow and friction power loss.

1.2 Reynolds Equations for Some Special Working Conditions

In Section 1.1, we have given the general form of the Reynolds equation. However, for some specific engineering problems, the general Reynolds equation can be simplified, which may make solving much easier. In the following, some forms of the Reynolds equation for different conditions are given.

1.2.1 Slider and Thrust Bearing

A wedge slider is the simplest problem of lubrication design. If the geometry of the slider is not very complicated, we can obtain an analytical solution. In addition, through analysis of the slider problem, it will not only help us to understand the basic characteristics of lubrication, but will also be useful for the thrust bearing lubrication design.

Because the side leakage of lubricant need not be considered for solving an infinitely wide slider, its Reynolds equation then can be simplified into a one-dimensional ordinary differential equation:

(1.6) equation

The common two-pressure boundary conditions of a slider are as follows.

; (x ′ is the outlet boundary, x ′ = b; and b is the slider width).

; and (x ′ is the outlet boundary to be determined, x ′ ≤ b).

If the film thickness or its derivative is discontinuous, we should divide the lubrication region into two parts at the discontinuous line so that the number of the integral constants will correspondingly increase. Therefore, the connection conditions must be used at the discontinuous line. If the discontinuous line is at x *, the connection conditions will be:

(1.7)

equation

(1.8)

equation

1.2.2 Journal Bearing

By spreading the journal bearing along the circumferential direction, we can transform x into Rθ so that the general form of the Reynolds equation is

(1.9)

equation

The corresponding shape of the clearance can be expressed as:

(1.10) equation

where e is the eccentricity, c is the clearance of the radii of the bearing and the journal, = e/c is the eccentricity ratio and θ is the circumferential coordinate starting from the maximum film thickness position.

1.2.2.1 Infinitely Narrow Bearing

If the axial width of a bearing along the y direction is much less than the circumferential length along the x direction, we have so that we can set . Because the film thickness h is only related to θ, but independent of y, the Reynolds equation becomes

(1.11) equation

The above Reynolds equation has only side boundary conditions. They are p = 0 at and at y = 0 due to symmetry.

1.2.2.2 Infinitely Wide Bearing

We can approximately choose for an infinitely wide bearing because the side leakage can be ignored. Therefore, the Reynolds equation changes into an ordinary differential equation.

(1.12) equation

Its boundary conditions usually are , and (where θ 2 is the outlet boundary to be determined, θ 2 ≤ 2π).

1.2.3 Hydrostatic Lubrication

The oil film for hydrostatic lubrication is formed by a fluid forced in under pressure from the outside. Therefore, even if two lubricating surfaces have no relative motion, a thick enough lubricating film can be achieved. The advantages of hydrostatic lubrication are: (1) its load carrying capacity and the oil film thickness have no relationship with the sliding velocity; (2) the film stiffness is so strong that it has a very high accuracy; (3) its friction coefficient is so low that we can ignore the influence of the static friction. The main disadvantages of hydrostatic lubrication are: its structure is complex and a pressure oil supply system must be required which often affects the working life and reliability of hydrostatic lubrication.

Substituting the condition of no relative sliding velocity into the Reynolds Equation 1.1, we have the Reynolds equation for hydrostatic lubrication as follows

(1.13) equation

For a rectangular region, the outer pressure boundary conditions are usually ; ; ; and the boundary pressure condition in the oil chamber is: p = p s, where p s is the supplied oil pressure.

For a journal hydrostatic bearing, Reynolds Equation 1.13, the film thickness equation and the boundary conditions can be solved easily in the form of cylindrical coordinates. For solving the above equations, we can determine the variation relationship between the load and the film thickness. Furthermore, if we consider the working conditions, such as equal film thickness, incompressibility or isoviscosity, the Reynolds Equation 1.13 can be further simplified.

1.2.4 Squeeze Bearing

The relative sliding between two bearing surfaces is usually assumed to be zero when analyzing squeeze lubrication, so that the Reynolds Equation 1.1 can be written as follows

(1.14)

equation

Usually, for a rectangular region, the boundary conditions are ; ; and . To solve the above equation we can determine the variation relationship between the load and the film thickness.

1.2.5 Dynamic Bearing

Most actual bearings withstand a varying load whose direction, rotational speed or other parameters change with time. Such bearings are collectively referred to as dynamic bearing or nonstable load bearing. Obviously, the axis or the thrust plate of a dynamic bearing moves along a certain trajectory. If the working parameters are periodic functions of time, the trajectory of the axis is a complex and closed curve.

The working principles of the dynamic bearing can be divided into two types. The first is where the journal does not rotate around its central axial, that is, there is no relative sliding. Therefore, the journal axial moves along a certain trajectory under the load. In this case, the journal and the bearing surfaces move mainly in the direction of the film thickness so that the film pressure is generated by the squeeze effect. The other type is where the journal rotates around its own center and the journal center also moves. Therefore, the film pressure originates from the squeeze effect of these two movements, that is, the journal rotation and the axis movement.

The general Reynolds equation for incompressible and dynamic lubrication is the basic equation to analyze dynamic bearings. It can be written as follows

(1.15)

equation

In Equation 1.15 the first term on the right is the hydrodynamic effect; the second term represents the squeeze effect; and when the Reynolds Equation 1.15 is applied to a stable bearing, the term of the squeeze effect can be omitted, that is,

equation

The problem of calculating the axis trajectory of a dynamic bearing by Equation 1.15 belongs to an initial value problem. The stepping method is usually used to determine the axis of the trajectory according to the given initial position of the axis.

1.2.6 Gas Bearing

The main feature of gas lubrication is that a gas is compressible. Therefore, the density of the gas must be treated as a variable, that is, by using the Reynolds equation for a variable density

(1.16)

equation

Because the gas density varies with temperature and pressure, the ideal gas equation can be expressed as follows

(1.17) equation

where T is the absolute temperature, and R is the gas constant which does not change for a certain gas.

For a usual gas lubrication problem, gas lubrication can be regarded as an isothermal process and this assumption has an error less than a few percent. For such a problem, Equation 1.15 becomes

(1.18) equation

where k is a proportional constant.

In addition, if a gas lubrication process is so fast that the heat cannot be conducted in time, the process can be thought to be adiabatic. The gas state equation of the adiabatic process is as follows

(1.19) equation

where n is the gas specific heat ratio relate to the atomic number of the gas molecules. For air, n = 1.4.

For an isothermal process, the Reynolds equation becomes

(1.20)

equation

Equation 1.20 is the basic equation for solving gas lubrication problems.

1.3 Finite Difference Method of Reynolds Equation

If the boundary conditions are given for solving a differential equation, this is known as a