Example 3

The position of a slider-block linkage is required to be  m when . A failure occurs if the actual position  is outside the tolerance range  m. The design variables are independent and follow normal distributions of , . Since uncertainty exists in the measurement of , the angle also follows a normal distribution with . The measurement uncertainty is independent of that in the mechanism dimensions. Use the First Order Second Moment (FOSM) method to calculate the probability of failure.

Solution:

The vector of random variables is . The actual position of the slider is given by

Since the tolerance range is , failure occurs when  or . Let , and . Then, the probability of failure of the mechanism is given by

Since the ransom variables are independent and follow normal distributions, according to the FOSM method, the mean and standard deviation of  are calculated by

Thus, we have

Similarly, the mean and standard deviation of  are calculated by

We have

The probability of failure of this mechanism is then given by

Ans.