Following methods are used in designing the sliding contact bearing design.

**1. PETROFF’S EQUATION**

Petroff’s equation is used to determine the coefficient of friction in journal bearings. It is based on the following assumptions:

- (i) The shaft is concentric with the bearing
- (ii) The bearing is subjected to a light load

In practice, such conditions do not exist. However, Petroff’s equation is important because it defines the group of dimensionless parameters that govern the frictional properties of the bearing. It is given by,

$$f=2 \pi^{2}\left(\frac{r}{c}\right)\left(\frac{\mu n_{s}}{p}\right)$$

Petroff’s equation indicates that there are two important dimensionless parameters, $\frac{r}{c}$ and $\frac{\mu n_{s}}{p}$ that govern the coefficient of friction and other frictional properties like frictional torque, frictional power loss and temperature rise in the bearing.

**2. MCKEE's EQUATION**

Bearing Modulus is a dimensionless parameter on which the coefficient friction in a bearing depends fig. In the region to the left of Point C, operating conditions are severe and mixed lubrication occurs as shown in the fig below.

A small change in speed or increase in load can reduce $Z \mathrm{N'}/ \mathrm{p}$ and a small education in $\mathrm{ZN'} / \mathrm{p}$ can increase the coefficient of friction drastically. This increases heat which reduces the viscosity of the lubricant. This further reduces $\mathrm{ZN}^{\prime} / \mathrm{p}$ leading to further increase in friction.

**3. REYNOLD'S EQUATION**

The theory of hydrodynamic lubrication is based on a differential equation derived by Osborne Reynolds. This equation is based on the following assumptions:

- (i) The lubricant obeys Newton’s law of viscosity
- (ii) The lubricant is incompressible
- (iii) The inertia forces in the oil film are negligible
- (iv) The viscosity of the lubricant is constant
- (v) The effect of the curvature of the film with respect to film thickness is neglected. It is assumed that the film is so thin that the pressure is constant across the film thickness
- (vi) The shaft and the bearing are rigid
- (vii) There is a continuous supply of lubricant

The Reynold's equation is as follows,

$$\frac{\partial}{\partial_{x}}\left(h^{3} \frac{\partial p}{\partial_{x}}\right)+\frac{\partial}{\partial_{z}}\left(h^{3} \frac{\partial p}{\partial_{z}}\right)=6 \mu U\left(\frac{\partial h}{\partial_{x}}\right)$$

There is no exact analytical solution for this equation for bearings with finite length. Theoretically, exact solutions can be obtained if the bearing is assumed to be either Infinitely long or very short. These two solutions are called Sommerfeld's solutions. Approximate solutions using numerical methods are available for bearings with finite length.

**4. RAIMONDI AND BOYD METHOD**

There is no exact solution to Reynold's equation fora journal bearing having a finite length. However, AA Raimondi and John Boyd of Westinghouse Research Laboratory solved this equation on a computer using the iteration technique. The results of this work are available in the form of charts and tables (**PSG7.36-7.39**).

In the Raimondi and Boyd method, the performance of the bearing is expressed in terms of dimensionless parameters.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

L/D | $\epsilon$ | $2h_o/C$ | S | $\phi$ | $\mu$ D/C | 4q/DCn’L | $\frac{q_{s}}{q}$ | $\frac{\rho C^{\prime} \Delta t o}{P}$ | $\mathrm{P} / \mathrm{Pmax}$ |

The parameters are explained below.

Length to diameter ratio (L/D ratio)

It is the ratio of bearing length to journal diameter. (Assume 1 as the default value)

If L/D is > 1, the bearing is said to be a long bearing.

If L/D is < 1, the bearing is said to be a short bearing.

If L/D is = 1, the bearing is said to be a Square bearing.

Attitude or eccentricity ratio ($\epsilon$)

It is the ratio of the eccentricity to the radial clearance. $\epsilon = 2e/C = 1 - 2h_o / C$

Film thickness variable

It is the ratio of film thickness to the radial clearance. $FTV = 2h_o / C= h_o / C/2$

Sommerfeld number (S)

The Sommerfeld number is also a dimensionless parameter used extensively in the design of journal bearings.

$$S=\frac{Z^{\prime} n^{\prime}}{P}\left[\frac{D}{C}\right]^{2}$$

Attitude angle $\phi$

It is the angle between the load axis and line of intersection of centers of journal and bearing.

Coefficient of friction variable (CFV)

It is a dimensionless parameter used to calculate the coefficient of friction between bearing and journal.

Flow variable (FV)

It is a dimensionless parameter used to calculate the flow rate of oil required for lubrication of bearing.

Side flow variable (SFV)

It is a dimensionless parameter defined as the ratio of side flowrate of oil to the total flow rate of oil.

Temperature variable (TV)

It is a dimensionless parameter used to calculate the temperature rise of the oil film of bearing.

Pressure variable (PV)

It is a dimensionless parameter used to calculate the pressure of oil inside the bearing.