3 Numerical Methods of Lubrication Calculation
3.3 Numerical Solution of Elastohydrodynamic Lubrication
3.3.2 EHL Numerical Solution of Point Contacts
Generally, point contact problems include two spherical bodies forming an elliptical contact area. This is more complexthan line contact problems. In 1965, Archard and Cowking pro- posed the first Grubin approximate solution for circular contact EHL [5]. In 1970, Cheng gave a solution for an elliptic contact EHL problem [6]. Later, Hamrock and Dowson proposed the formula for calculating the minimum film thickness according to their numerical results of the elliptical contact EHL problems [3]. Wen and Zhu Dong presented a full numerical solution for elliptical contact EHL problem [7]. Below, we will briefly introduce its main points.
k k 3.3.2.1 The Reynolds Equation
If the surface speed is not along the contact zone axis, the Reynolds equation should be written as
𝜕
𝜕x (𝜌h3
𝜂
𝜕p
𝜕x )
+ 𝜕
𝜕y (𝜌h3
𝜂
𝜕p
𝜕y )
=12 (
U𝜕𝜌h
𝜕x +V𝜕𝜌h
𝜕y )
. (3.93)
Figure 3.14 expresses the coordinates and the solution region.xis the short axis of the ellipse contact zone. If the velocity components of the two surfaces in thexandydirections are respec- tivelyu1,u2,v1andv2, the average velocities are
u= 1
2(u1+u2) v= 1
2(v1+v2). (3.94)
The boundary conditions of Equation 3.93 are: the inlet and the side pressures at the bor- ders are equal to zero, that is,p=0 wherex=x1andy= ±B/2. At the outlet, we use Reynolds boundaries conditions, that is,p=0 and𝜕p/𝜕x=0 wherex=x2.
As this is the same situation as the line contact EHL, the induced pressureq(x,y) can be introduced as
q(x,y)≡ 1
𝛼[1−e−𝛼p(xy)] (3.95)
because
𝜕q
𝜕x =e−𝛼p𝜕p
𝜕x, 𝜕q
𝜕y =e−𝛼p𝜕p
𝜕y. (3.96)
Substituting them into Equation 3.93 gives
𝜕
𝜕x (
𝜌h3𝜕q
𝜕x )
+ 𝜕
𝜕y (
𝜌h3𝜕q
𝜕y )
=12𝜂0 [
u𝜕
𝜕x(𝜌h) +v𝜕
𝜕y(𝜌h) ]
. (3.97)
Equation 3.97 is the two-dimensional Reynolds equation with the viscosity–pressure relationship of lubricant considered.
Figure 3.14 Point contact solution region.
k k 3.3.2.2 Elastic Deformation Equation
According to the theory of elasticity, if the surface pressure isp(x,y), the surface deformation 𝛿(x,y) can be described as
𝛿(x,y) = 2 𝜋E∫ ∫Ω
p(s,t)
√(x−s)2+ (y−t)2
dsdt, (3.98)
wheresandtare the integral variables in thexandydirections;Ωis the solution region.
Obviously, whens=x,t=y, Equation 3.98 is singular. To overcome this, similar approaches as the line contact EHL are adopted. Moving the coordinate origin to𝜉=x−sand𝜁=y−t, Equation 3.98 becomes
𝛿(x,y) = 2
𝜋E∫ ∫Ω p(𝜉, 𝜁)
√𝜉2+𝜁2d𝜉d𝜁. (3.99)
For the polar coordinates, setx=rcos𝜃,y=rsin𝜃, then we have 𝛿(x,y) = 2
𝜋E∫ ∫Ωp(r, 𝜃)dr d𝜃. (3.100)
Usually, the calculation task of elastic deformation is excessive. A very effective way to over- come this difficulty is to use a deformation matrix. The steps are as follows.
First, divide the solution region into a mesh, for example, m nodes in the x direction and nin the ydirection, that is, i=1, 2, …, m andj=1, 2,…, n. Define Dklij as the elastic deformation of nodekandlcaused by pressure pij; the total deformation of nodekandlis equal to
𝛿kl= 2 𝜋E
∑n
i=1
∑m
j=1
Dklijpij. (3.101)
Therefore,Dklij only need to be calculated once and stored up to be used repeatedly in the iterative process. Thus it may reduce a large amount of computation work.
Because the total number of matrixDklij is (m×n)2, a uniform mesh will save much more storage. If the mesh is uniform in theydirection, we haveDklij =Dklis, wheres=|j−l|+1. So the total number ofDklij is reduced tom2×n. If the uniform mesh is used in thexdirection, the number will be further reduced tom×n.
When all the deformations are obtained, the film thickness will be h(x,y) =h0+ x2
2Rx+ y2
2Ry +𝛿(x,y), (3.102)
whereRxandRy are the equivalent radius in thexandydirections, respectively. Then, sub- stituting Equation 3.102 into the Reynolds equation, the pressure distribution can finally be obtained.
3.3.2.3 Hamrock–Dowson Film Thickness Formula of Point Contact EHL
Hamrock and Dowson proposed the following film thickness formula for isothermal point con- tact EHL after carrying out numerical analysis [3]:
k k Hmin∗ =3.63G∗0.49U∗0.68
W∗0.073 (1−e−0.68k) (3.103)
Hc∗=2.69G∗0.53U∗0.67
W∗0.067 (1−0.61e−0.73k) (3.104)
where Hmin∗ =hmin∕Rx is the dimensionless minimum film thickness; Hc∗=hc∕Rx is the dimensionless central film thickness; G*=𝛼Eis the dimensionless material elastic module;
U*=𝜂0u/ERx is dimensionless speed; W∗ =w∕ER2x is dimensionless load; k=a/b is the ellipticity, which is approximately equal tok=1.03(Rx/Ry)0.64.
From the above formula, it can be known that if other parameters except the ellipticity are kept unchanged, the film thickness rapidly decreases with increase of the ellipticity. Ifk>5 the film thickness changes slowly withk. With comparison, it is easy to know that whenk>5, the central film thickness of the point contact EHL is approximately equal to the film thickness of the line contact EHL.
In Figures 3.15 and 3.16, there are the pressure distribution and film shape of point contact EHL by Ranger [8]. They are much more complex than those of the line contact EHL.
Figure 3.15 shows that in the contact area of the point contact EHL, oil film gas appears as a horseshoe-shaped depression. The minimum film thickness appears at both sides of the neck.
We can see from Figure 3.16 that pressure distribution of point contact EHL has a crescent of the secondary pressure peak region, but the pressure peak in the center of this region is the highest, and is far from the contact center.
Figure 3.15 Film thickness contours.
Figure 3.16 Pressure contours.
k k