Page 141 - Tribology in Machine Design

Friction, lubrication and wear in lower kinematic pairs 127

where </> is the angle of friction for the contact surfaces.

Solution

Referring to Fig. 4.31, EA and CA are the common normals to the contact
surfaces at the points of contact E and C respectively. RI and K 2 are the
resultant reactions at E and C inclined at an angle <p to the common
normals in such a manner as to oppose rotation of the shaft.
If R l and R 2 intersect at B, it follows that the points A, B, C, D, E lie on a
circle of diameter AD. Hence

Figure 4.31
The resultant of/?! and R 2 must be parallel to the line of action of the load
Q, so that

where BZ = .B£)sin(/> = y4Dcos(/>sin</> and r = AD sin a. Thus

where/' is the virtual coefficient of friction defined previously, and so

4.10. Drives utilizing In higher pairs of elements there is incomplete restraint of motion.
friction force Therefore, force closure is necessary if the motion of one element relative to
the other is to be completely constrained. In higher pairing, friction may be
a necessary counterpart of the closing force as in the case of two friction
wheels (Fig. 4.32). Here, the force P not only holds the cylinders in contact,
but must be sufficient to prevent relative sliding between the circular
elements if closure is to be complete.
Now, consider the friction drive between two pulleys connected by a belt,
Fig. 4.33, then for the pair of elements represented by the driven pulley and
Figure 4.32 the belt (case (b) in Fig. 4.33), the belt behaves as a rigid body in tension
only. If the force T 1 were reversed, or the belt speed V were to fall
momentarily below ra, this rigidity would be lost. Hence, force closure is
incomplete, and the pulley is not completely restrained since a degree of
freedom may be introduced. A pulley and that portion of the belt in contact
with it, together constitute an incompletely constrained higher pair which is
kinematically equivalent to a lower pair of elements.
Assuming that the pulleys are free to rotate about fixed axes, complete
kinematic closure is obtained when an endless flexible belt is stretched
tightly over the two pulleys. The effects of elasticity are for a moment
neglected, so that the belt behaves as a rigid body on the straight portions
and the motion can then be reversed. This combination of two incomplete