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/m2.

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/m2 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) = 2r2flp

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:

N = Nj + Nb - 2Nw

where Nj is the journal or shaft speed of rotation
Nb is the bush speed of rotation
Nw is the load vector speed of rotation

Converting this to dimensionless form gives: S' = S(1 + (Nb/Nj) - 2(Nw/Nj))
and it is the S' value that is used to enter the Raimondi and Boyd charts along the abscissa.

Fig. L2, below, shows some bearing nomenclature

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.