3 Numerical Methods of Lubrication Calculation
3.3 Numerical Solution of Elastohydrodynamic Lubrication
3.3.1 EHL Numerical Solution of Line Contacts
Petrusevich gave an isothermal EHL numerical solution in line contacts first, and proposed a thickness formula. Although the formula is limited, the characteristics of the typical EHL pressure distribution and film shape are clear. Since then, Dowson and Higginson have given a series of systematic numerical calculations on the isothermal EHL in line contacts [3]. Based on the results, they proposed a more accurate formula for the thickness, which has been verified by experiments and is widely used.
3.3.1.1 Basic Equations
The equations to solve EHL problems in line contacts are as follows.
1. Reynolds equation d
dx (𝜌h3
12𝜂 dp dx
)
=Ud(𝜌h)
dx , (3.77)
whereUis the average velocity,U=(u1+u2)/2;his the film thickness;𝜂is the viscosity of lubricant;𝜌is the density of lubricant;h,𝜂and𝜌are functions ofx.
k k The boundary conditions of Reynolds equation are
At the inlet, p|x=x1 =0
At the outlet, p|x=x2 =0; 𝜕p
𝜕x||
||x=x2=0
wherex1 is the inlet position. The inlet position is based on the lubricant supply, usually x1=(5−15)bis chosen;bis the half width of the contact region;x2 is the outlet position, and it will be determined in the solution process.
2. Film thickness equation
As shown in Figure 3.12, for the contact of an elastic cylinder and a rigid plane, the film thickness is expressed as
h(x) =hc+ x2
2R+v(x), (3.78)
wherehcis the center thickness without elastic deformation;Ris the equivalent radius. For two cylinders, 1/R=1/R1+1/R2;v(x) is the elastic deformation generated by pressure.
3. Elastic deformation equation
For the line contact problems, the length and radius of a contact body are always much larger than the width of the contact region so that the problem can be considered as a plane strain state. Such an elastic deformation is shown in Figure 3.13. According to the theory of elasticity, the elastic displacement along the vertical direction can be derived as
v(x) = − 2 𝜋E∫
s2 s1
p(s)ln(s−x)2ds+c, (3.79)
Figure 3.12 Film shape.
Figure 3.13 Elastic deformation.
k k wherep(s) is the load distribution, or pressure;s1ands2are for the starting-point and the
ending-point coordinates ofp(s);Eis the equivalent modulus of elasticity, 1/E=1/E1+1/E2; cis a constant to be determined in calculation.
4. Viscosity–pressure relationship
Barus viscosity–pressure formula is commonly used for convenience:
𝜂=𝜂0e𝛼p, (3.80)
where𝜂0is the viscosity of the lubricant atp=0.
5. Density–pressure relationship
Fitting with the experimental data, a density–pressure relationship can be obtained as 𝜌=𝜌0
(
1+ 0.6p 1+1.7p
)
, (3.81)
where𝜌0is the density of the lubricant underp=0.
3.3.1.2 Solution of the Reynolds Equation
From Equation 3.77, it can be seen that the pressure distributions are influenced by𝜂,h0and𝜌. Because the maximum increment of the density𝜌with pressurepis about 33%, the density variation has little influence on solutions. Therefore, the lubricant is usually considered as an incompressible fluid, or a simple density–pressure relationship may be used for convenience.
However,𝜂 having an exponential relation withpwill be dramatically changed, and the film thicknesshhas a cubic form in the Reynolds equation. Therefore, the visco-pressure effect and the elastic deformation have very significant influences so more attention must be paid to them in EHL.
In addition, from an EHL pressure distribution we can see that pressurepand its derivative dp/dxrapidly vary in a very narrow range. In order to solve the process stably, a parameter transformation normally needs to be used so that pressure varies slightly.
One variable transformation commonly used is the induced stressq(x)=1/𝛼(1−e–𝛼p). If we consider the viscosity–pressure effect, the Reynolds equation of EHL becomes
d dx
( 𝜌h3dq
dx )
=12𝜂0Ud(𝜌h)
dx . (3.82)
After having obtainedq(x), we can use an inverse parameter transformation to obtainp(x), that is
p(x) = −1
𝛼ln[1−𝛼q(x)].
In EHL calculation, the Vogelpohl transformation to setM(x)=p(x)[h(x)]3/2is also often used.
If so, the Reynolds equation will be d
dx (𝜌h3∕2
𝜂 dM
dx )
−3 2
d dx
(ph1∕2 𝜂
dh dx
)
=12Ud(𝜌h)
dx . (3.83)
3.3.1.3 Calculation of Elastic Deformation
If the pressure distributionp(x) has been obtained, the deformationv(x) can be obtained to integrate Equation 3.79. However, the deformation equation is singular at points=x. This is
k k one difficulty for calculating elastic deformation
I=∫
s2 s1
p(s)ln(s−x)2ds. (3.84)
To avoid the singularity, a simple way is to take sectional integrations. As the integral function is continuous except fors=x, it can be treated as
I=∫
x−Δx s1
p(s)ln(s−x)2ds+∫
s2 x+Δx
p(s)ln(s−x)2ds. (3.85)
However, the difficulty of this approach is how to determineΔxproperly. If incorrect, it may cause a considerable calculation error.
Another way to overcome singularity is to use a discrete integration method, referring to reference [4]. The main steps are as follows.
Divide the integral region [x1,x2] into a number of sub-regions and express pressure distri- butionp(x) approximately as a polynomial function ofx:
p(x) =c1+c2x+c3x2. (3.86)
The coefficientsc1,c2andc3can be determined according to the known pressure at the nodes.
For example, on the interval [xi,xi+1], the pressure distribution is expressed as
pi(x) =c1i+c2ix+c3ix2. (3.87) Therefore, the deformation integration becomes
Ii=∫
xi+1 xi
(c1i+c2is+c3is2)ln(x−s)2ds
=2 [
c1i
∫
xi+1 xi
ln|x−s|ds+c2i
∫
xi+1 xi
sln|x−s|ds+c3i
∫
xi+1 xi
s2ln|x−s|ds ]
.
(3.88)
The analytical integral formula, such as∫ lnsdsand∫ slnsds, can be used in the calculation ofIiof Equation 3.88.
Furthermore, in the above calculation,xis the coordinate and should be selected in the three intervals,x≤xi,xi<x<xi+1andxi+1≤x. Except forxi<x<xi+1, the singularity will appear in the other two intervals, that is,x=xiorx=xi+1. For example, whenx≤xi, if set DX=xi+1–xi andX=xi−x, we have
Ii
2 = (c1i+c2ix+c3ix2)[(X+DX)ln(X+DX) −XlnX−DX]
+ (c2i+2c3ix) [
(X+DX)2ln(X+DX) −X2lnX− 2XDX+DX2 2
] / 2 +c3i
[
(X+DX)3ln(X+DX) −X3lnX−3XDX(X+DX) +DX3 3
] / 3.
(3.89)
As long asX≠0,Iican be obtained. IfX=0, that is,x=xi,Iiis a singular integral. In this case, we can use limt→0+tlnt=0 to obtainIi. Therefore, Equation 3.89 becomes
k k Ii
2 = (c1i+c2ix+c3ix2)[DXlnDX−DX] + (c2i+2c3ix) [
DX2lnDX−DX2 2
] / 2 +c3i
[
DX3lnDX−DX3 3
] / 3.
(3.90)
Forxi<xi+1≤x, using the same method above can overcome the singularity atx=xi+1, and a similar formula may be obtained as well.
3.3.1.4 Dowson–Higginson Film Thickness Formula of Line Contact EHL
Based on a large number of systematically numerical calculations, Dowson and Higginson pro- posed twice the minimum film thickness formula of EHL in line contacts. Their experimental results showed that their formula results are very close to the most measured values of film thickness. The dimensionless formula of 1967 is
Hmin∗ =2.65G∗0.54U∗0.7
W∗0.13 . (3.91)
The dimensional form of the above formula is hmin= 2.65𝛼0.54(𝜂0U)0.7R0.43l0.13
E0.03W0.13 , (3.92)
whereH*,G*,U*andW*are the dimensionless parameters given in Section 2.4.
As we can see from the above formula, the minimum film thicknesshminof line contact EHL increases significantly with the initial viscosity𝜂0and the average speedU, but the load effect is very weak, that is, a substantial increase of load decreases the film thickness very little. This is one of the basic special features of EHL.
It should be pointed out that the Dowson–Higginson formula is used to calculate the min- imum necking thicknesshmin, but the Grubin formula is used to calculate the thicknessh0at the inlet of the contact zone, that is,x= −b. Dowson and Higginson showed that numerical calculated contact center thicknesshcis very close to the calculated results of Grubin formula.
The ratio of the minimum film thickness and the central film thicknesshmin/hc=3/4.
It should be noted that both the Dowson–Higginson formula and the Grubin formula have their applications. When the material parameterG* <1000, that is, the low elastic modulus of the solid materials and a low viscosity pressure coefficient of a lubricant, or when the load parameter W*<105, corresponding to a light load condition, the calculation errors of Equation 3.91 are quite large. In addition, the above formulas are derived under the condition that the lubricant supplication is sufficient for an isothermal EHL. If the oil supply is short, the film thickness will reduce, while in high-speed conditions when heat causes the viscosity to decrease significantly, the film thickness will decrease.