A
joint of a mechanism can be considered as a journal bearing. As shown in Fig. 1,
the radius of the bearing 
is
greater than that of the journal 
.
Their difference is called a clearance, and 
As
indicated in Fig. 2, the position 
of
the center of the journal relative to the center of the bearing is within a
circle of radius 
.
This circle is called a clearance circle.
Fig. 1 Joint clearance
Fig. 2 Clearance circle
Assume
that the clearance is known and that the position
is uniformly distributed within the clearance circle 
.
Determine
(1) The joint PDF and CDF of X and Y
(4) The variances of X and Y
(5) The coefficient of correlation between of X and Y
Solution
(1) The joint PDF and CDF of X and Y
The position 
is uniformly
distributed within the clearance circle 
, and the PDF is
constant within the clearance circle .
where K is a constant.
Therefore
(2) The PDFs of X and Y
Therefore
Similarly,
(3) the mean of X is
Similarly,
(4) The variances of X and Y
Similarly,
(5) The coefficient of correlation between of X and Y
