Lubrication - Background |

**1. Introduction**

These notes will introduce some of the theory and describe the calculation procedure for an fluid lubricated plain journal bearing design.

In a dry bearing, the shaft tends to 'climb' up the bearing, whereas in a lubricated bearing, the lubricant being drawn into the bearing tends to push the shaft away from the bearing on the entry side, see Fig.L1.

**2. Newtons Law of Viscous Flow:** The shear stress in a fluid is proportional to the rate of change of velocity with respect to y, ie:

where is the dynamic or absolute viscosity

If it is assumed that the shear rate is constant, then: du/dy = U/h and

Units of absolute viscosity are Pa.s or N.s/m^{2}.

The ASTM method for determining viscosity uses a 'Saybolt Universal Viscometer' and involves measuring the time taken by a volume of fluid to descend a specific distance down a tube of a certain diameter.

**2. Petroff's Law:** If a shaft radius, r, is rotating in a bearing, length, l, and radial clearance, c at N revs per s, then the surface velocity is: m/s The shearing stress is the velocity gradient x viscosity:

The torque to shear the film is force x lever arm length:

If a small force, w, is applied normal to the shaft axis, the pressure in N/m^{2} is:

p = w/2rl The frictional force is fw, where f is the coefficient of friction, so the frictional torque is:

T = fwr = (f)(2rlp)(r) = 2r^{2}flp

Equating the two expressions for T and solving for f gives:

which is Petroff's Law.

and are dimensionless groups. The bearing characteristic or Sommerfeld Number is defined as: This is a key quantity in bearing design.

While the above equation is the original form of the Sommerfeld Number, it has since been realised that the performance of a hydrodynamic bearing is not only dependent upon the shaft or journal rotation, but also upon any rotation of the load vector and of the bushing. Hence the value of N which is important for bearing performance is:

where N_{j} is the journal or shaft speed of rotation

N_{b} is the bush speed of rotation

N_{w} is the load vector speed of rotation

Converting this to dimensionless form gives:

and it is the S

**3. Assumptions: **

- 1 The lubricant obeys Newton's laws of viscous flow.
- 2 Intertia effects of the lubricant are neglected.
- 3 The lubricant is incompressible.
- 4 The viscosity of the lubricant is constant throughout the film.
- 5 The pressure does not vary in the axial direction.
- 6 The curvature of the bearing can be ignored.
- 7 There is no flow in the axial (z) direction.
- 8 The film pressure is constant in the '
*y*' direction, and depends upon '*x*'. 9 The velocity of a lubricant particle depends on its x and y coordinates.

From the free body diagram of the forces acting on a small cube of lubricant:

and as then

Assuming there is no slip at the boundary, with *x* held constant, integrate twice with respect to *y* which gives:

which is the velocity distribution as a function of *y* and the pressure gradient, *dp/dx*. see Fig.L3.
The velocity distribution accross the film is obtained by superimposing a parabolic distribution (the first term) onto a linear distribution (the second term). When the pressure is a maximum, dp/dx = 0 and the velocity is u = - Uy/h.

If Q is the quantity of fluid flowing in the x direction per unit time: In practice these integrations have to be modified to include the effects of end leakage etc.

Next page - Bearing Design

David J Grieve. Revised: 22nd February 2010. Original: 14th November 2005.