Page 83 - Tribology in Machine Design

70 Tribology in machine design

scale times the volume under the hemispherical pressure plot, or

and the peak pressure has the value

Substitution of eqn (3.7) and the value of B below eqn (3.2) gives to eqns
(3.5) and (3.6) the forms shown for case 1 of Table 3.2. If both spheres have
the same elastic modulus E 1 =£ 2 = E, and the Poisson ratio is 0.30, a
simplified set of equations is obtained. With a ball on a plane surface,
R 2 = x, and with a ball in a concave spherical seat, R 2 is negative.
It has taken all this just to obtain the pressure distribution on the
surfaces. All stresses can now be found by the superposition or integration
of those obtained for a concentrated force acting on a semi-infinite body.
Some results are given under case 1 of Table 3.2. An unusual but not
unexpected result is that pressures, stresses and deflections are not linear
functions of load P, but rather increase at a less rapid rate than P. This is
because of the increase of the contact or supporting area as the load
increases. Pressures, stresses and deflections from several different loads
cannot be superimposed because they are non-linear with load.

3.4. Contact between Equations for cylinders with parallel axes may be derived directly, as shown
cylinders and between for spheres in Section 3.3. The contact area is a rectangle of width 2b and
bodies of general shape length /. The derivation starts with the stress for line contact (case 2 of Table
3.1). Some results are shown under case 2 of Table 3.2. Inspection of the
equations for semiwidth b, and peak pressure p 0, indicates that both
increase as the square root of load P. The equations of the table, except that
given for 6, may be used for a cylinder on a plane by the substitution of
infinity for R 2. The semiwidth b, for a cylinder on a plane becomes
l.l3[(P/l)(r]i + 772)^1]*- All normal stresses are compressive, with <r y and a.
equal at the surface to the contact pressure p 0. Also significant is the
maximum shear stress T VZ , with a value of 0.304p 0 at a depth 0.786/>.
Case 3 of Table 3.2 pictures a more general case of two bodies, each with
one major and one minor plane of curvature at the initial point of contact.
Axis Z is normal to the tangent plane XY, and thus the Z axis contains the
centres of the radii of curvature. The minimum and maximum radii for
body 1 are/?, and R\, respectively, lying in planes YjZand X {Z. For body
an m
2, they are R 2 d ^2^ lying planes Y 2Z and X 2Z, respectively. The angle
between the planes with the minimum radii or between those with the
maximum radii is \l/. In the case of two crossed cylinders with axes at 90°,
such as a car wheel on a rail, i// — 90° and R\ = R 2 = oo. This general case was
solved by Hertz and the results may be presented in various ways. Here, two
sums (B + A) and (B — A), obtained from the geometry and defined under
case 3 of Table 3.2 are taken as the basic parameters. The area of contact is
an ellipse with a minor axis 2b and a major axis 2a. The distribution of
pressure is that of an ellipsoid built upon these axes, and the peak pressure is