Page 197 - Tribology in Machine Design

182 Tribology in machine design

force developed. The squeeze effect may occur on surfaces of all shapes,
including shapes that are flat and cylindrical. For an easy example, the case
of a flat circular bearing ring and shaft collar is chosen and the relationship
between the applied force, velocity of approach, film thickness and time is
determined. The case being analysed is shown in Fig. 5.8.
In the Reynolds equation, all surface velocities except V 2 will be zero, and
Figure 5.7
by symmetry dp/8& =0. With the upper surface approaching at a velocity
V, V 2 =dh/dt = — V. Thickness h is independent of r and 0 but a function of
time t. Equation (5.8) becomes

Thus (d/dr}(rdvldr\ = — 12u Vr/h* and bv inteeratine twice with resoect to i

Figure 5.8

whence

The boundary conditions are p =0 at r = D/2 and r=d/2. Substitution, and
simultaneous solution for Ci and C 2, and resubstitution of these values
gives for the pressure

The total force developed at a given velocity and a given film thickness is
found by integration over the surface of the force on an elemental ring, or

If the force is known as a function of time, the time for a given change in the
film thickness may be found from eqn (5.19) by the substitution of — d/j/dr
for V, the separation of variables, and integration between corresponding
limits t', t" and h', h", thus

If P is a constant of value W, such as obtained by a weight

The boundary condition for a solid circular plate at r=0 is different,
namely, dp/dr=Q. Use of this, beginning with the equation preceding eqn