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

(2) The PDFs of X and Y

(3) The means of X and Y

(4) The variances of X and Y

(5) The coefficient of correlation between of X and Y

Solution

(1)

(2)

(3)  

 (4)

  (5) 0