Page 109 - Tribology in Machine Design

Elements of contact mechanics 95

cylinders of radii r and R 1? and on the outer race the contact is between the
roller of radius r and the concave surface of radius (R^ +2r).
For involute gears it can readily be shown that the contact at a distance s
from the pitch point can be represented by two cylinders of radii,
K lj 2 sin^is, rotating with the angular velocity of the wheels. In this
expression JR represents the pitch radius of the wheels and i// is the pressure
angle. The geometry of an involute gear contact is shown in Fig. 3.14. This
form of representation explains the use of disc machines to simulate gear
tooth contacts and facilitate measurements of the force components and the
film thickness.
From the point of view of a mathematical analysis the contact between
two cylinders can be adequately described by an equivalent cylinder near a
plane as shown in Fig. 3.15. The geometrical requirement is that the
separation of the cylinders in the initial and equivalent contact should be
the same at equal values of x. This simple equivalence can be adequately
satisfied in the important region of small x, but it fails as x approaches the
radii of the cylinders. The radius of the equivalent cylinder is determined as
follows:

Using approximations
Figure 3.14

and

For the equivalent cylinder

Hence, the separation of the solids at any given value of x will be equal if

Figure 3.15
The radius of the equivalent cylinder is then

If the centres of the cylinders lie on the same side of the common tangent at
the contact point and R a > R b, the radius of the equivalent cylinder takes the
form

From the lubrication point of view the representation of a contact by an