[Exploration] Single‑Hole Pin Float – Part 3: Uniform Circular Distribution with DTAS

Summary
Common methods that sample radius *r* uniformly in [0, R] and angle θ uniformly in [0, 2π) do not achieve true uniform distribution in a circle. This article explains how to correctly implement uniform pin‑float distribution, covering both simulation modeling and theoretical principles.

I. Practical Context & Analytical Approach

Pin‑hole fits are used for part positioning, with the hole slightly larger than the pin to allow float. Previous discussions focused on the extreme case of tangential contact (maximum float). In practice, high‑precision automated assembly may align parts without chamfer guidance, so the pin and hole may not reach tangency.

With reduced guidance, pin‑hole offset can be modeled as a normal distribution. However, real production—considering fixture positioning and fastening—often results in a distribution between the normal and tangential extremes, such as a uniform distribution within a circle (see Figure 1).

Figure 1: Single pin hole fit conforms to a uniform distribution within a circle

II. Simulation Method 1

Since the pins in Figure 1 are uniformly distributed within the circular tolerance zone, a natural approach is to use polar coordinates:

•Radius r is uniformly sampled over [0, R] 

•Angle θ is uniformly sampled over [0, 2π)

2.1 Simulation Verification
We built a DTAS simulation model using the pin‑hole dimensions from Figure 1, with assembly defined by the above distribution. The results are shown below.

Figure 2: Radius and Angle Sampling with Uniform Distribution

Observing the final point distribution in Figure 2, the density is higher near the center—clearly not meeting the target of uniform circular distribution.

2.2 Probability Distribution of Simulation Results

We defined a vertical virtual measurement for the pin center and ran 5,000 virtual assembly cycles. The resulting distribution (Figure 3) has a standard deviation of 0.205.

Figure 3. Simulation results of vertical fluctuation at the pin center.

III. Simulation Method 2

For a true uniform distribution within a circle of radius R, the joint probability density is:

This requires that θ remain uniformly distributed, with marginal density:

Since r and θ are independent, their joint distribution function satisfies:

In terms of probability density, this is equivalent to:

From the equation above, it can be deduced that r² follows a uniform distribution over [0, R²]. (Readers interested in the derivation may contact us for further discussion.)

3.1 Implementation & Model Verification

To sample r² uniformly in [0, R²] as described, the pin‑float type must be configured as shown in Figure 4.

Figure 4 Random Floating 2

The fluctuation trajectory shows that the pin's fluctuation within the hole exhibits good uniformity.

Figure 5 Uniformly Distributed Sampling of Radius Square

3.2 Probability Distribution of Simulation Results

A virtual measurement in the vertical direction is established for the pin center. After 5000 virtual assembly cycles, the simulation measurement results shown in Figure 6 are obtained. The standard deviation is 0.252.

Figure 6 Simulation Results of Vertical Fluctuation at the Pin Center

Compared with Method 1, this standard deviation is significantly larger. Method 1 produces more points near the center, reducing overall fluctuation and yielding lower variance.

For reference: the theoretical standard deviation is 0.25 (probability density shown in Figure 7). The simulation results in Figure 6 closely match this value.

Figure 7. Theoretical Distribution of Vertical Fluctuation at the Pin Center

IV. Conclusion

1.For true uniform circular distribution in polar coordinates:

• Angle θ must be uniform.

• r² (not r) must be uniformly sampled over [0, R²], so sampling density increases with radius.

2.Uniform circular sampling supports key assembly functions—such as random pin‑center placement in multi‑hole fits—ensuring correct simulation of various pin‑hole floating behaviors.

3.In DTAS 3D, users can select either method based on their actual working conditions.