Page 251 - Tribology in Machine Design

236 Tribology in machine design

trailing wheels of a vehicle, however, rotate in bearings assumed to be
frictionless and the rolling resistance is overcome by a tangential force T x
applied at the bearing and resisted at the contact interface. Provided that
the rolling resistance is small (T x <^ W) these two situations are the same
within the usual approximations of small strain contact stress theory, i.e. to
first order in (a/R). It is then convenient to write the rolling resistance as a
non-dimensional coefficient f r expressed in terms of the rate of energy
dissipation P, thus

The quantity P/V is the energy dissipated per unit distance travelled.

Energy dissipated due to micro-slip
Energy dissipation due to micro-slip occurs at the interface when the rolling
bodies have dissimilar elastic contacts. The resistance from this cause
depends upon the difference of the elastic constants expressed by the
parameter /? (defined by eqn (6.11)) and the coefficient of sliding friction/

The resistance to rolling reaches a maximum value of

when fi/fx 5. Since, for typical combinations of materials, /? rarely exceeds
0.2, the rolling resistance due to micro-slip is extremely small. It has been
suggested that micro-slip will also arise if the curvatures of two bodies are
different. It is quite easy to see that the difference in strain between two such
surfaces will be second-order in (a/R) and hence negligible in any small
strain analysis. A special case is when a ball rolls in a closely conforming
groove. The maximum rolling resistance is given by

The shape of the contact ellipse (b/a) is a function of the conformity of the
ball and the groove; where the conformity is close, as in a deep groove ball-
bearing, b $> a and the rolling resistance from this cause becomes significant.
In tractive rolling, when large forces and moments are transmitted
between the bodies, it is meaningless to express rolling resistance as T x or
M y/R. Nevertheless, energy is still dissipated in micro-slip and, for
comparison with free rolling, it is useful to define the effective rolling
resistance coefficient f r = P/VW. This gives a measure of the loss of
efficiency of a tractive drive such as a belt, a driving wheel or a continuously
variable speed gear.