DTAS invites you to explore the mysteries of floating single-hole pins (Part 2). Come and join us!

A Preliminary Exploration of Single-Hole Pin Floating (Part Two)

Summary: This paper uses a simple case of single-hole pin floating to compare theoretical calculation results with simulation calculation results, verifying the accuracy of the simulation calculation. It also provides a theoretical verification step for our tolerance simulation calculations. We not only focus on tolerance simulation modeling but also explore the underlying theoretical knowledge.

In the previous section, we answered the following question:

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

We used DTAS 3D to create a virtual assembly of the pin and a virtual measurement along the vertical direction. We simulated it 5000 times using the Monte Carlo method. The animation is shown in the figure above, and the results of various statistical parameters are shown in the figure below. The maximum and minimum values are ±5, the mean is close to 0, the variance is 12.517, and the histogram fit distribution curve has an peculiar shape, not a normal distribution. (The simulation results may vary slightly depending on the initial random seed).

Simulation animation

Simulation Results

In this section, we disregard the diameter tolerance of the pin and change the pin to uniform floating to further explore the pin's fluctuation in the vertical direction.

After changing to uniform floating, the mean is almost zero and the variance is 4.176, indicating better data dispersion. Next, we will theoretically derive how 4.176 is derived.

I. Mathematical Model

The practical problem in this case is transformed into the following mathematical model:

The probability density function (pdf) of the random variable is known to be:

The random variable R follows a uniform distribution, and its probability density function (pdf) is:

So what are the statistical parameters of the distribution of the random variable Y = R * sinθ?

II. Theoretical Calculation of Mean, Variance, and Standard Deviation

Mathematically, it is essentially the distribution of the product of two random variables, a two-dimensional joint probability distribution.

From the previous section, we know that the expected value of R is 5/2, and its variance is 0. sinθ has an expected value of 0 and a variance of 1/2.

Since "R" and "θ" are independent,

Therefore, the theoretical value of the standard deviation is 10.png, and the simulation calculation result is 2.044, which meets the engineering requirements.

If the problem becomes that the diameter of the pin has a deviation of ±1, and assuming the diameter tolerance is 6σ and tangentially floating, then R becomes a normal distribution. Using the same method, we can deduce that the theoretical value of the standard deviation is 3.539.

III. Reflections on Engineering Applications

1. The case in this article is simple, but it provides a theoretical verification step for our other tolerance simulation calculations. That is, to establish a mathematical model, then use mathematical knowledge to solve for the cumulative distribution function, probability density function, expected variance, etc., of the new random variables, and then compare the results with the calculation results. Using the same method, we can also deduce why, in three-dimensional tolerance simulation analysis, when we use two random quantities, amplitude and angle, to simulate position, the amplitude is usually set to a skewed distribution. Those interested can try to deduce and explain this. Of course, as the model becomes more complex, establishing the mathematical model becomes difficult, at which point specialized software is needed. Most numerical simulations have certain usage conditions or assumptions; having certain theoretical knowledge is very helpful in identifying the reasonableness of the calculation results.

2. In this case, to simulate the floating of the pin in the hole, we assumed that the floating subvalue was uniformly distributed within the range of R, thus simulating the uniformity of the pin's position in the hole. But is this really the case? We will discuss later how to truly simulate the uniformity of the pin's position in the hole to get as close to reality as possible.