Page 249 - Tribology in Machine Design

234 Tribology in machine design

approximately in the x-y plane. Therefore

and

When the bodies have closely conforming curved surfaces, as for example in
a deep-groove ball-bearing, the contact area is warped appreciably out of
the tangent plane and the expressions for M x and M y, eqn (6.4), have to be
modified to include terms involving the shear tractions t x and t y.

6.4. Hysteresis losses Some energy is always dissipated during a cycle of loading and unloading
even within the so-called elastic limit. This is because no solid is perfectly
elastic. The energy loss is usually expressed as a fraction a of the maximum
elastic strain energy stored in the solid during the cycle where a is referred to
as the hysteresis loss factor. For most metals, stressed within the elastic
limit, the value of a is very small, less than 1 per cent, but for polymers and
rubber it may be much larger.
In free rolling, the material of the bodies in contact undergoes a cycle of
loading and unloading as it flows through the region of contact deform-
ation (Fig. 6.2). The strain energy of material elements increases up to the
centre-plane due to the work of compression done by the contact pressure
acting on the front half of the contact area. After the centre-plane the strain
energy decreases and work is done against the contact pressures at the back
of the contact. Neglecting any interfacial friction the strain energy of the
material arriving at the centre-plane in time dt can be found from the work
done by the pressure on the leading half of the contact. For a cylindrical
Figure 6.2 contact of unit width

where CD = V/R is the angular velocity of the roller. Taking p(x) to be given
by the Hertz theory

where Wis the contact load. If a small fraction a of this strain energy is now
assumed to be dissipated by hysteresis, the resultant moment required to
maintain the motion is given by equating the net work done to the energy
dissipated, then