Page 250 - Tribology in Machine Design

Friction, lubrication and wear in higher kinematic pairs 235

or

where / r is defined as the coefficient of the rolling resistance. Thus the
resistance to rolling of bodies of imperfectly elastic materials can be
expressed in terms of their hysteresis loss factor. This simple theory of
rolling friction is due to Tabor. Using the same calculation for an elliptical
contact area given the result

where a is the half-width of the contact ellipse in the direction of rolling. For
i
J
a sphere rolling on a plane, a is proportional to (WR) so that the effective
4 3 ^
rolling resistance F T = M y/R should be proportional to W*R . This
relationship is reasonably well supported by experiments with rubber but
less well with metals.
There are basically two problems with this simple theory. First, the
hysteresis loss factor a is not usually a material constant. In the case of
metals it increases with strain (a/R), particularly as the elastic limit of the
material is approached. Second, the hysteresis loss factor in rolling cannot
be identified with the loss factor in a simple tension or compression cycle.
The deformation cycle in the rolling contact, illustrated in Fig. 6.2, involves
rotation of the principal axes of strain between points 2, 3 and 4, with very
little change in total strain energy. The hysteresis loss in such circumstances
cannot be predicted from uniaxial stress data.
The same deformation cycle in the surface would be produced by a rigid
sphere rolling on an inelastic deformable plane surface as by a frictionless
sphere sliding along the surface. In spite of the absence of interfacial friction
the sliding sphere would be opposed by a resistance to motion due to
hysteresis in the deformable body. This resistance has been termed the
deformation component of friction. Its value is the same as the rolling
resistance F r given by eqn (6.9).

6.5. Rolling friction Rolling motion is quite common in higher kinematic pairs. Ideally it should
not cause much power loss, but in reality energy is dissipated in various
ways giving rise to rolling friction. The various sources of energy dissipation
in rolling may be classified into:
(i) those which arise through micro-slip and friction at the contact
interface;
(ii) those which are due to the inelastic properties of the material;
(iii) those due to the roughness of the rolling surfaces.
Free rolling has been defined as a motion in the absence of a resultant
tangential force. Resistance to rolling is then manifested by a couple M y
which is demanded by the asymmetry of the pressure distribution, that is, by
higher pressures on the front half of the contact than on the rear. The