1. Introduction

2. Fundamentals of Bayesian Theory

3. Application of Bayesian Statistics in Predicting Carburizing Distortion

3.1. Establishing a Dimensional Distribution Model Using Bayesian Statistics

Consider a simple case where data $D=\{x_1,x_2,...,x_n\}$ are drawn from a normal distribution with known variance σ² and unknown mean θ. Assume a prior distribution for θ as a normal distribution with mean μ₀ and variance τ².

(1) Prior Distribution: $\theta ~N({\mu }_0,{\tau }^2)$

(2) Likelihood Function: $x_i~N(\theta ,{\sigma }^2)$

The likelihood function is: $P(D|\theta )=\prod _{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-{\displaystyle \frac{1}{2}}{({\displaystyle \frac{y_i-\theta }{\sigma }})}^2}$

(3) Posterior Distribution: The posterior distribution of θ given D is also a normal distribution, obtained by combining the prior and the likelihood:

$$P(\theta |D)\propto P(D|\theta )P(\theta )$$

Through some algebraic manipulation, it can be shown that:

$$P(\theta |D)~N({\displaystyle \frac{\frac{{\mu }_0}{{\tau }^2}+\frac{n\overline{x}}{{\sigma }^2}}{\frac{1}{{\tau }^2}+\frac{1}{{\sigma }^2}}},{\displaystyle \frac{1}{\frac{1}{{\tau }^2}+\frac{n}{{\sigma }^2}}})$$

where $\overline{x}$ is the sample mean of the data D.

To apply Bayesian statistics for analyzing the dimensions of torsion bars, it is essential to determine the prior distribution. A prior distribution, such as a normal distribution, incorporates initial beliefs about the parameters before observing any data. In this study, we assume that the total length of the torsion bars follows a normal distribution before carburizing. The average length of the torsion bars before carburizing is 347 mm according to the designed dimension, with a deviation not exceeding ±0.3 mm. According to Bayesian theory, dividing this deviation (0.3 mm) by 6 yields a prior standard deviation of 0.05 mm. Consequently, we adopt a normal distribution as the prior distribution for Bayesian analysis.

After precision machining and carburizing heat treatment, the total lengths of the torsion bars were measured. These measurements are presented in Table 1 and serve as the observed data for Bayesian statistical analysis. Assuming the observation samples follow a normal distribution, the likelihood function represents the probability of the observed data given the parameters.

Combining the prior distribution and the likelihood function via Bayes' Theorem yields the posterior distribution, which updates our belief about the parameter after observing the data. According to Bayesian theory, the posterior distribution of the torsion bar dimensions is proportional to the product of the prior distribution and the likelihood function. The conjugate normal posterior distribution provides a straightforward method to update our beliefs about a parameter when new data is observed. This is particularly useful in applications where parameters are continuously updated as more data becomes available, such as real-time quality control in manufacturing processes or iterative improvement in machine learning algorithms.

Using the sample standard deviation of 0.068 mm as the observational standard deviation in the likelihood function, we assume the likelihood follows a normal distribution with this standard deviation. Given a normal prior distribution, the resulting posterior distribution is also normal. Consequently, the Bayesian posterior distribution for the total length of the torsion bars before carburizing is calculated as N(347.405, 0.0122²). For the dimensional distribution after carburizing, accounting for unavoidable dimensional expansion, we use a normal distribution N(348, 0.05²) as the prior distribution. Under the same conditions as before carburizing, the posterior distribution of the total length after carburizing is N(347.973, 0.0112²).

The Bayesian statistical distribution curves of the total lengths of the torsion bars before and after carburizing are shown in Figure 3. The posterior distributions are notably more concentrated and have reduced deviations compared to the prior distributions. This improvement highlights the enhanced statistical credibility of Bayesian statistics, which integrates both prior experience and actual measurement data. With the increase in observed data, the precision of the posterior distribution further improves. By integrating Bayesian methods with machine learning and online quality monitoring systems, the control over the dimensional accuracy of torsion bars can be significantly enhanced. This synergy can lead to better prediction and management of carburizing-induced dimensional changes, ensuring higher assembly precision and product reliability. By employing Bayesian statistical methods, the study demonstrates the potential to achieve superior control over the dimensional properties of torsion bars, ultimately contributing to more reliable and efficient mechanical systems.

3.2. Establishing a distortion Model Using Bayesian Statistics

The relative expansion rates of torsion bars before and after carburizing heat treatment were shown in Table 1. Preliminary experimental results suggest that the prior knowledge of the expansion rate (%) due to carburizing heat treatment falls within the range of 0.1% to 0.3%. Based on the current data on the dimensional changes of the torsion bars, we select a normal prior distribution for the expansion rate. Using the sample standard deviation of 0.02 as the observational standard deviation for the likelihood function, the posterior distribution for the relative expansion rate is calculated as , as shown in Figure 4. The figure illustrates that for a torsion bar with an average total length of 348 mm, the mean relative expansion rate after carburizing heat treatment is 0.145%, corresponding to an absolute expansion mean of 0.538 mm.

Using Bayesian statistical methods, we calculate the 95% credible interval for the relative expansion rate of the torsion bars after carburizing heat treatment as 0.1372% to 0.1528%, as shown in Figure 5.

Additionally, Bayesian statistics can predict the distribution of future observations based on the results of past experiments, known as the predictive distribution. Suppose his predictive distribution also follows a normal distribution N(m, s²), where m is the posterior mean of the observed results, and s² is the sum of the posterior variance and the likelihood variance. Substituting the data, we establish the predictive distribution for the expansion rate of the next batch of torsion bars as N(0.145, 0.045²). The credible interval and predictive distribution of the relative expansion rate are shown in Figure 6. The distribution of expansion rates after carburizing heat treatment conforms to a normal distribution, with all data points falling within the fitted normal distribution curve which is shown in Figure 6b.

This detailed Bayesian analysis provides robust predictions and credible intervals, ensuring high precision in understanding and controlling the dimensional changes due to carburizing heat treatment. By integrating prior knowledge and observed data, Bayesian statistics offer a powerful tool for predicting and managing manufacturing processes, leading to improved quality and performance of critical components such as torsion bars.

4. Conclusion

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