Example 8

The motion of the collar on beam CD is controlled by a motor at A. The distance between the collar and point D is , which follows a normal distribution of  due to the uncertainty in measurement. The collar moves upward at a velocity of . The length of  also follows a normal distribution of . At this instant the allowable velocity of a point on the cable is , use FOSM method to find the probability of failure. Assume that  and are independent.

Solution:

Assume that a point P lies in the AB part of the cable and the distance between this point and point B is , as shown in the figure below. Let  denote the distance from the point to the collar along the cable.

According to the figure, we have 

Taking the time derivative, we obtain the velocity of the point on the cable.

Let , and then the probability of failure is given by

Since the random variables are independent and follow normal distributions, the mean and standard deviation of  are calculated by

Thus, we have

Ans.