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A Preliminary Study on Single-Hole Pin Floating (Part 1)

Summary: This article compares theoretical calculation results with simulation calculation results using a simple case of single-hole pin floating, verifying the accuracy of the simulation calculation, and providing a theoretical verification step for our tolerance simulation calculation.

Question: Assuming the diameter tolerance of the pin is negligible, what is the vertical fluctuation of the pin when it floats tangentially?

We used DTAS3D to create a virtual assembly of the pin holes and perform virtual measurements along the vertical direction. We simulated 5000 times using the Monte Carlo method. The simulation animation is shown in the image above. The results of various statistical parameters are shown in the image below. The maximum and minimum values are ±5, the mean is close to 0, and the variance is 12.517. The histogram fit distribution curve has an peculiar shape and is not normally distributed. (The simulation results may vary slightly depending on the initial random seed).

Simulation Animation:

Simulation results:

This paper attempts to compare the differences between simulation results and theoretical calculations through strict mathematical theoretical derivation, mainly involving knowledge related to probability theory and calculus. The main knowledge points are as follows:

1. The probability density function (pdf) is the CDF of the cumulative distribution function, where pdf is the derivative of the CDF.

2. Expectation of a continuous random variable X 

3. The distribution, expected value, and variance of a function of a random variable, such as the expected value of Y=g(X). 

4. Differential and integral formulas (including trigonometric and inverse trigonometric functions, etc.), such as the derivative of arcsinx, are... 

5. Double - angle formulas of trigonometric functions.