Since the test data under each stress level obey the lognormal distribution, when the normal probability density function is known, the distribution function f (XP) of the normal variable can be obtained, that is, the probability p (x < PX) that the normal variable x is less than a certain value PX:

Where f (XP) = P (x < XP), the geometric meaning is: from – ∞ to PX, the area surrounded by the curve and the x-axis, as shown in the shadow on the left in Figure 1. When the logarithmic fatigue life conforms to the normal distribution, then 1 − f (XP) is equivalent to reliability. It can be seen from the figure that the value of F (XP) is completely determined by XP, which is 0.841px.

According to the data in Figure 2, the fatigue life estimation of the test gear under different stress levels is obtained when the reliability is 84.1%.

When the stress level is lower than a critical value, the fatigue life is greater than 1 × 107 times, the critical value is called the fatigue limit of the gear. When the reliability is 84.1%, the fatigue limit is 232.68mpa.

The bending fatigue test of low-speed and heavy-duty gear is carried out by using the bending fatigue testing machine. The test data show that each group of test data obey the lognormal distribution, and the fatigue life increases with the decrease of stress level; When the reliability is 50%, the median fatigue life is longer, but the reliability is lower; When the reliability is 99.9%, the reliability is high, but the safety life is short. Through the application of mathematical statistics knowledge, when the reliability is 84.1%, the fatigue life is long and the reliability is relatively high; When the number of cycles is greater than 1 × When the reliability is 84.1%, the fatigue limit is 232.68mpa.