64
6 Herz’s Task
b
a
Fig. 6.5 Contact of barrel-shaped bodies with intersecting axes
M =
1
2R 1
+
1
2ρ 2
, N =
1
2ρ 1
+
1
2R 2
.
(6.34)
6.5.2 Case of Intersecting Axes
The contact of barrel-shaped rotation bodies with intersecting axes is shown
in Fig. 6.5a. Let us introduce two auxiliary rectangular systems of coordinates
Ox 1 y 1 z 1 and Ox 2 y 2 z 2 , where the axis Ox 1 is directed along the tangent line to
the generatrix of the body 1, and the axis Ox 2 along the tangent line to the meridian
of the body 2 going through the contact point O (Fig. 6.5a). Then the equations of
the surfaces of adjoining bodies in the vicinity of the point O can be written as
F 1 (x 1 , y 1 ) =
x 2
1
2R 1
+
y 2
1
2ρ 1
, F 2 (x 2 , y 2 ) = −
x 2
2
2R 2
−
y 2
2
2ρ 2
.
(6.35)
Now let us direct the axis Ox of the primary coordinate axis at the angle β 1 to
the axis Ox 1 , (Fig. 6.5b). In this case, the coordinates x 1 and y 1 , x 2 and y 2 are
expressed through x and y using formulas
x 1 = x cos β 1 − y sin β 1 ,
y 1 = x sin β 1 + y cos β 1 ,
x 2 = x cos(β 1 + β) − y sin(β 1 + β),
y 2 = x sin(β 1 + β) + y cos(β 2 + β).
(6.36)
By substituting transformations (6.36) into surface equations (6.35), we obtain