Numerical Calculation of Elastohydrodynamic Lubrication

Methods and Programs

Ping Huang

• Publisher: Wiley
• Year: 2015
• Print ISBN: 9781118920961
• eBook ISBN: 9781118920992
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

The book not only offers scientists and engineers a clear inter-disciplinary introduction and orientation to all major EHL problems and their solutions but, most importantly, it also provides numerical programs on specific application in engineering. • A one-stop reference providing equations and their solutions to all major elastohydrodynamic lubrication (EHL) problems, plus numerical programs on specific applications in engineering
• offers engineers and scientists a clear inter-disciplinary introduction and a concise program for practical engineering applications to most important EHL problems and their solutions
• brings together a number of case studies in one text, each being solved using solution methods which share common features and methods

1
Basic equations of elastohydrodynamic lubrication

1.1 Basic equations

1.1.1 One-dimensional Reynolds equation of elastohydrodynamic lubrication

One-dimensional isothermal Reynolds equation of elastohydrodynamic lubrication (EHL) is given as follows [1, 2]:

(1.1)

Let us assume that the surfaces do not stretch and the density of lubricant does not change with time, the general form of the one-dimensional Reynolds equation of hydrodynamic lubrication can be written as follows:

(1.2)

where, the last term is obtained from

as we assume that the density of lubricant does not change with time.

In order to transform the general form of Reynolds equation (1.2) into the EHL Reynolds equation (1.1), let us analyze the problem of two surfaces with the rolling speed, see Figure 1.1a. First, expand the surfaces as shown in Figure 1.1b, where we set the down surface to be horizontal and the up surface to be declined.

Figure 1.1 Velocity analysis of rolling problem: (a) two rolling cylinders and (b) simplified model

In order to substitute the velocities into the Reynolds equation (1.2), let us analyze the velocities given in Figure 1.1b. If set u is the velocity in the x direction and w in the z direction, we can see that on the down surface, u₀ = u₁ and w₀ = 0. The velocity on the up surface can be decomposed into the horizontal and the vertical two components as shown in Figure 1.2.

Figure 1.2 Velocity decomposition of rolling problem

After decomposition, the two velocity components can be written as follows:

(1.3)

where, α is the declined angle of the up surface.

Because the declined angle α is very small, we can approximately take

and

. Therefore, we have

(1.4)

Substituting u₀, u_h, w₀, and w_h into the Reynolds equation (1.2), we have:

(1.5)

Let u₁ + u₂ = 2u...

Table of contents

• Cover
• Title page
• Table of Contents
• Preface
• Introduction
• Nomenclature
• 1 Basic equations of elastohydrodynamic lubrication
• 2 Numerical calculation method and program of elastic deformation
• 3 Numerical calculation method and program for energy equation
• 4 Numerical calculation method and program for isothermal EHL in line contact
• 5 Newton–Raphson method and programs to solve EHL problems in line contact
• 6 Numerical calculation method and program for isothermal EHL in point contact
• 7 Numerical calculation method and programs of multigrid method for isothermal EHL
• 8 Numerical calculation method and program for isothermal EHL in ellipse contact
• 9 Numerical calculation method and program for isothermal EHL in elliptical contact with two-dimensional velocities
• 10 Numerical calculation method and program for thermal EHL
• 11 Numerical calculation method and program for grease EHL
• 12 Numerical calculation method and program for EHL considering effect of electric double layer
• 13 Numerical calculation method and program for time-dependent EHL in line contact
• 14 Numerical calculation method and program for isothermal EHL with rough surface
• 15 Numerical calculation method and program for micropolar fluid EHL
• References
• Index
• End User License Agreement