[Exploration] Implementation and Verification of Uniform Distribution of DTAS within a Circle!

[Exploration] Single-Hole Pin Floating (Part 3): Implementation and Verification of Uniform Distribution of DTAS within a Circle!

Summary: In previous articles, to achieve uniform distribution, we typically sampled the radius r uniformly within the interval [0, R] and the angle θ uniformly within the interval [0, 2π). However, this method is not truly uniform. This article will continue to explore how to truly achieve uniform distribution of the pin within the hole. We not only focus on tolerance simulation modeling but also delve into the underlying theoretical knowledge.

I. On-Site Conditions and Analytical Value

As is well known, pin-hole fits are generally used for relative positioning between hand parts. To ensure smooth assembly of the pin-hole fit, the nominal dimensions of the two parts are generally different, and the lower limit of the hole is slightly larger than the upper limit of the pin. Therefore, the pin's relative position floats within the hole.

Considering single-hole pin fits again, in the previous article we discussed the maximum floating condition of pin-hole tangency, but other conditions exist in reality. For example, in automated production line assembly, when the robot's positioning accuracy is high enough, the proper alignment of parts does not require the guidance of the pin head's arc, and the pins may not even be tangentially in contact after assembly. By weakening the guiding effect, the relative positions of the pin centers can be simulated using a normal distribution. However, in actual production, due to subsequent fixture positioning and fastening, pin floating generally does not follow the ideal normal distribution described above. Instead, it seeks an intermediate state between the two extremes of normal distribution and tangential floating, such as assuming a uniform distribution within a circle, as shown in Figure 1.

Figure 1. Single pin hole fit conforms to uniform distribution within the circle.

II. Simulation Method 1

The pins are uniformly distributed within the circular tolerance zone shown in Figure 1. Our first thought is to use a polar coordinate system to define the radius variable as r and the angle variable as θ. We sample the radius r uniformly within the interval [0, R] and the angle θ uniformly within the interval [0, 2π).

2.1 Simulation Animation Verification

We established a DTAS simulation model based on the single-hole pin dimensions shown in Figure 1 and assembled the pins within the hole according to the distribution defined by the method. The simulation results are as follows.

Figure 2. Radius and angle sampled according to a uniform distribution.

Carefully observing the final positions of the point distribution in Figure 2, the density at the center is higher, clearly failing to meet the target requirement of uniform distribution within the circle.

2.2 Probability distribution of simulation results

We further established a virtual measurement in the vertical direction for the pin center, and after 5000 virtual assembly cycles, we obtained the simulation measurement results shown in Figure 3, with a standard deviation of 0.205.

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

III. Simulation Method Two

Because the uniform distribution within a circle of radius R, the joint probability density of r and θ, f(r, θ) = 1/πR², requires the angle to also be uniformly distributed. Therefore, the probability density of θ is:

f(θ) = 1/2π

Since variables r and θ are independent, the distribution function F(r, θ) and the marginal distribution functions FR(r), Fθ(θ) of the two-dimensional random variable (r, θ) satisfy:

F(r, θ) = FR(r) / Fθ(θ)

The above equation can also be expressed using probability density, which is equivalent to the joint probability density:

f(r, θ) = f(r) * f(θ) = 1/πR²

Careful observation of the above equation reveals that r² is uniformly distributed from 0 to R². Those interested can theoretically derive the above conclusions or contact us for further discussion.

3.1 Implementation and Verification of Simulation Model

If it is necessary to sample the pin radius position r squared in the interval [0,R²] in a uniform distribution as described above, the pin floating type needs to be set as shown in Figure 4.

Figure 4 Random Floating 2

As can be seen from the fluctuation trajectory, the pin's fluctuation within the hole exhibits excellent uniformity.

Figure 5. Sampling of the squared radius according to a uniform distribution

3.2 Probability distribution of simulation results

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

Figure 6. Simulation results of vertical fluctuation at the center of the pin.

Comparing the standard deviation with Method 1, ours is significantly larger. This is because Method 1 generates more random points at the center of the pin, resulting in smaller fluctuations and thus a larger variance. Those interested can try to deduce the difference between the simulated and theoretical standard deviation values. Theoretical derivation shows a theoretical standard deviation of 0.25, with its probability density shown in Figure 7. Comparing the simulation results in Figure 6, the simulated standard deviation matches the theoretical value.

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

IV. Conclusion

1. To achieve uniform distribution within a circle using the polar coordinate method, uniform distribution must be satisfied in the angular direction, and the square of the radius must also satisfy a uniform distribution, not just the radius itself. That is, the sampling density of the radius increases with distance from the center. We can rigorously verify its correctness using relevant mathematical knowledge of probability theory and calculus.

2. Uniform sampling within a circle is fundamental to some assembly functions, such as the random uniform generation of the pin center position in two/multi-hole pin assemblies. The correct implementation of this function ensures the accuracy of random sampling in various types of hole pin assemblies, guaranteeing the correctness of software simulation results.

3. In DTAS3D, users can choose between the two methods mentioned above based on actual working conditions.