Friction and Lubrication in Mechanical Design Episode 1 Part 3 docx

30 Chapter 2

Figure 2.6 Two cylindrical bodies in contact.

RI, R2 = radii of cylinders (positive when convex and negative when concave)

111

E, El +g -=-

El, E2 = modulus of elasticity for the two materials

Case 7: General Case of Contact Between Elastic Bodies with

Continuous and Smooth Surfaces at the Contact Zone

Analysis of this case by Hertz can be found in Refs 1 and 2. A diagrammatic

representation of this problem is shown in Fig. 2.7 and the contact area is

expected to assume an ellipitcal shape. Assuming that (RI, R;) and (R2, R;)

are the principal radii of curvature at the point of contact for the two bodies

respectively, and $ is the angle between the planes of principle curvature for

the two surfaces containing the curvatures l/R1 and l/R2, the curvature

consants A and B can be calculated from:

These expressions can be used to calculate the contact parameter P from the

relations hip:

The Contact Between Smooth Surfaces 31

Figure 2.7 General case of contact.

B-A cos0 = -

A+B

The semi-axes of the elliptical area are:

where

m, n = functions of the parameter 8 as given in Fig. 2.8

P = total load

1 - U: 1 - u2

nEl nE2

kl =- k2 = -

ul, u2 = Poisson’s ratios for the two materials

El, E2 = corresponding modulii of elasticity

Case 8: Beams on Elastic Foundation

The general equation describing the elastic curve of the beam is: