6.4 Compression of Orthogonal Cylinders
61
6.4.2 Primary Case
Let us consider two cylinders whose radii are R 1 and R 2 , (R 2 R 1 ). Let us
designate the contact points as O. When selecting the coordinate system shown
in Fig. 6.3, the equations of surfaces of these cylinders near the contact point will be
F 1 (x, y) =
1
2R 1
y
2 , F 2 (x, y, ) = −
1
2R 2
x
2 .
Formula (6.8) in the considered case gives
(k 1 + k 2 )(x, y) = δ − Mx
2
− Ny
2 ,
(6.24)
where
M =
1
2R 2
, N =
1
2R 1
.
(6.25)
A continuous solution of Eq. (6.24) will be found as follows
p(x, y) = c
1 −
x 2
a 2 −
y 2
b 2 ,
(6.26)
where the constant values a, b, and c are to be determined. Assume for the purpose
of certainty that
M < N.
(6.27)
In what follows, we will show that in the case of (6.27) a > b. To determine the
integral operator , one can use formula (5.38). By substituting this formula to
Eq. (6.24) and equaling the coefficients with the same degrees of x and y, we obtain
δ = πbc(k 1 + k 2 )K(e),
M = πbc(k 1 + k 2 )
K(e) − E(e)
a 2 e 2
,
N = πbc(k 1 + k 2 )
E(e) − (1 − e 2 )K(e)
a 2 e 2 (1 − e 2 )
,
(6.28)
where the elliptical integrals E(e) and K(e) are determined earlier by formulas
(5.39). For the convenience of calculations of formula (6.28), it is reasonable to
make it as follows: