Lubrication

Source: Hori (2006)

Observations:

• Frictional resistance almost constant regardless the bearing load
• The friction coefficient is very small, around 1/1000
• Frictional resistance increases with sliding speed
• Frictional resistance decreases with temperature

Source: By Wizard191 - Own work, CC BY-SA 3.0, Wiki commons: 10279899

Given:

• Radius of the bearing / journal

• Properties of the oil

• Rotation speed / angular velocity

• Force / weight of the journal

Wanted:

• Eccentricity of the journal

• Oil film thickness

• Pressure distribution

• Radii \(R_b\) and \(R_f\)

• Radial clearance: \(c = R_b - R_f\)

• Eccentricity \(e\), \(k = \frac{e}{c}\)

• Angular / polar coordinate \(x = R_b \phi\)

• Oil film thickness: \(h(\phi) \approx c + e \cos (\phi)\)

• Sommerfeld boundary conditions: \(p(0)=p(2 \pi) = 0\)

• Reynold’s equation

\[ \frac{d}{dx} \left( h^3 \frac{d}{dx} p \right) = 6 \mu U \frac{d}{dx} h \]

Source: Hori (2006)

Sommerfeld pressure distribution (\(R_b \approx R_f:= R\)):

\[ p(k,\phi) = \frac{6 \mu U R}{c^2} \underbrace{\frac{k ( 2 + k \cos \phi) \sin \phi}{(2+k^2)(1+k \cos \phi)^2}}_{\bar p (k,\phi)} = \frac{6 \mu U R}{c^2} \bar p (k,\phi) \]

Source: Hori (2006)

The hydrodynamic oil film force \(P\) has to balance the bearing load \(P_1\)

This gives rise to the following componentwise force balance relations in eccentricity direction and perpendicular to it:

\[ L R \int_0^{2 \pi} p(k,\phi) \cos \phi d \phi + P_1 \cos \theta = 0 \]

\[ L R \int_0^{2 \pi} p(k,\phi) \sin \phi d \phi - P_1 \sin \theta = 0 \]

in which \(L\) denotes the length of the bearing.

Source: Hori (2006)

The hydrodynamic oil film force \(P\) has to balance the bearing load \(P_1\)

This gives rise to the following componentwise force balance relations in eccentricity direction and perpendicular to it:

\[ \underbrace{L R \int_0^{2 \pi} p(k,\phi) \cos \phi d \phi}_{=0} + P_1 \cos \theta = 0 \]

This means

\[ P_1 \cos \theta = 0 \quad \Rightarrow \theta = \frac{\pi}{2} \]

Source: Hori (2006)

The hydrodynamic oil film force \(P\) has to balance the bearing load \(P_1\)

This gives rise to the following componentwise force balance relations in eccentricity direction and perpendicular to it:

\[ L R \int_0^{2 \pi} p(k,\phi) \sin \phi d \phi - P_1 \sin \theta = 0 \]

\[ \Rightarrow P_1 = \mu U L \left( \frac{R}{c} \right)^2 \frac{12 \pi k}{(2 + k^2) (1 - k^2)^{1/2}} \]