Application of Bayesian Statistics in Analyzing and Predicting Carburizing-Induced Dimensional Changes in Torsion Bars

Abstract

This study investigates the application of Bayesian statistical methods to analyze and predict the dimensional changes in torsion bars made from 20CrMnTi alloy steel during carburizing heat treatment. The process parameters, including a treatment temperature of 920 °C followed by oil quenching, were selected to optimize surface hardness while maintaining core toughness. The dimensional changes were measured pre- and post-treatment using precise caliper measurements. Bayesian statistics, particularly conjugate normal distributions, were utilized to model the dimensional variations, providing both posterior and predictive distributions. These models revealed a marked concentration of the posterior distributions, indicating enhanced accuracy in predicting dimensional changes. The findings offer valuable insights for improving the control of carburizing-induced deformations, thereby ensuring the dimensional integrity and performance reliability of torsion bars used in high-stress applications such as pneumatic clutch systems in mining ball mills. This study underscores the potential of Bayesian approaches in advancing precision engineering and contributes to the broader field of statistical modeling in manufacturing processes.

1. Introduction

Bayesian statistics provide a probabilistic framework that combines prior knowledge with observed data to improve parameter estimation and decision-making accuracy [1,2,3,4,5]. This approach has shown great potential in manufacturing, particularly for predicting process-induced variations [6,7,8,9,10]. In this study, we apply Bayesian methods to model the dimensional expansion of 20CrMnTi alloy steel torsion bars caused by carburizing heat treatment. Our goal is to enhance the prediction accuracy of heat treatment-induced deformation and provide a statistical foundation for real-time process control in industrial applications.

As a burgeoning school of thought, Bayesian methods have been extensively applied in various domains. In medical research, Bayesian statistics play a crucial role in enhancing the precision of clinical trials and patient diagnoses [11,12,13]. By integrating prior studies with real-time patient data, Bayesian methods enable the development of more nuanced and personalized treatment plans. This approach allows researchers to update their models as new data become available, leading to continuous improvement in patient care and treatment efficacy. For instance, in oncology, Bayesian models can be used to predict tumor growth and response to therapy, enabling oncologists to tailor treatments to individual patient’s needs and improving outcomes [14,15,16,17]. In the realm of risk assessment, Bayesian methods provide a robust framework for decision-making under uncertainty. Industries such as finance and insurance leverage Bayesian approaches to update risk models with incoming data, allowing for better prediction and mitigation of potential risks [18,19,20]. For example, in financial markets, Bayesian models can be used to forecast stock prices and assess investment risks by incorporating both historical data and current market trends. This dynamic updating process enhances the reliability of risk assessments and supports more informed decision-making.

Bayesian methods also significantly contribute to advancements in machine learning. By integrating prior knowledge about data distributions, Bayesian approaches improve the performance and reliability of predictive models [21,22]. This is particularly valuable in applications such as natural language processing (NLP) and image recognition, where understanding the underlying data structure can greatly enhance algorithm accuracy. For example, in NLP, Bayesian models can improve the understanding of context and semantics, leading to more accurate language translation and sentiment analysis [23,24]. Similarly, in image recognition, Bayesian methods can enhance the detection and classification of objects by effectively managing uncertainties in the data. In the field of engineering, Bayesian statistics have also become an indispensable tool for quality control of component parts. By continuously updating probability distributions of key parameters with new data, engineers can more accurately predict and manage product quality, ensuring higher reliability and performance [25,26,27].

The application of Bayesian statistics in engineering and material sciences has gained significant traction due to its ability to integrate prior knowledge and refine predictions with new data. Bayesian statistics combined with machine learning methods have been increasingly applied to complex engineering problems, such as fatigue life prediction under multiaxial loading conditions, which demonstrates the potential of data-driven models in addressing mechanical reliability issues [28,29,30]. Although the Bayesian framework offers a robust approach to probabilistic reasoning, its practical applications extend well beyond theoretical constructs. In this study, Bayesian statistical methods are employed to analyze and predict the dimensional changes in torsion bars subjected to carburizing heat treatment—a crucial process aimed at improving surface hardness and wear resistance in low-alloy steel components.

Carburizing is a well-established heat treatment technique used to introduce carbon into the surface layer of ferrous alloys, significantly improving their mechanical properties [31,32,33]. This process is commonly followed by quenching, which further alters the microstructure by transforming the surface into a hard martensitic phase [34,35]. Carburizing heat treatment involves the diffusion of carbon into the steel surface at elevated temperatures, followed by quenching. This process induces volumetric changes due to phase transformations, especially the formation of martensite, which has a higher specific volume than austenite. The main mechanisms contributing to dimensional expansion include carbon concentration gradients, phase-induced lattice expansion, and thermal stresses. For 20CrMnTi alloy steel torsion bars, these factors lead to non-uniform expansion, significantly affecting the final dimensional accuracy. For torsion bars, which are key elements in high-stress environments such as pneumatic clutch systems in mining ball mills, maintaining dimensional accuracy is crucial for ensuring proper function and requires durability. The torsion bar is a critical component in pneumatic clutch systems used in mining ball mills, where it is subjected to high rotational speeds and significant torque transmission. During operation, torsion bars endure complex stress states, including bending, wear, and shear forces. To ensure the smooth and reliable performance of clutches, torsion bars must exhibit excellent toughness and wear resistance, along with stringent dimensional and assembly precision. Achieving these properties typically involves using low-alloy steel combined with surface carburizing heat treatment. However, the carburizing process, followed by quenching, induces expansion and distortion, leading to potential dimensional deviations and affecting assembly precision. Therefore, accurately understanding the expansion and distortion patterns during the carburizing heat treatment process is crucial for coordinating the dimensional relationship between hot and cold processes and ensuring the assembly precision of torsion bars.

Historically, dimensional control of torsion bars has relied on conventional measurement techniques, such as micrometry and coordinate measuring machines (CMMs). These methods provide direct measurements of physical dimensions but may lack the capability to predict and adjust for process-induced variations. Conventional methods offer limited insight into the underlying causes of dimensional deviations and often require extensive post-processing adjustments. Those methods primarily focus on individual measurement accuracy and process adjustments but often lack comprehensive integration with predictive modeling and real-time process control. This study aims to address these gaps by applying Bayesian statistical methods to provide a more nuanced understanding of dimensional variations, ultimately contributing to improved manufacturing precision and process control.

Despite the broad applicability of Bayesian methods, previous studies have primarily focused on their theoretical aspects, machining processes or uncertainty quantifications, often overlooking their practical implementation n in manufacturing processes, particularly in relation to heat treatment-induced deformations [36,37,38]. This study aims to address these challenges by applying Bayesian statistical methods to analyze and predict the dimensional changes induced by carburizing heat treatment. By integrating prior knowledge with empirical data, we develop a robust model that accurately forecasts the expansion behavior of torsion bars, thereby providing valuable insights for improving process control and ensuring the dimensional integrity of critical components.

2. Fundamentals of Bayesian Theory

Bayesian statistics is a probabilistic approach to statistical inference, which provides a coherent framework for updating beliefs in the light of new data. It fundamentally differs from frequentist statistics by incorporating prior knowledge or beliefs into the analysis, leading to more refined and contextually relevant conclusions [39]. When dealing with Bayesian statistics, a common practice is to use a conjugate prior distribution, which simplifies the computation of the posterior distribution. For a normal prior and likelihood, the resulting posterior distribution is also normal, a property that significantly eases analytical and computational efforts.

2.1. Basics of Bayesian Statistics

Bayesian statistics rely on Bayes’ Theorem, which relates the conditional and marginal probabilities of random events. The theorem is expressed as

$$P(\theta |D)={\displaystyle \frac{P\left(D\right|\theta \left)P\right(\theta )}{P\left(D\right)}}$$

(1)

where

• $P\left(\theta \right|D)$ is the posterior probability, representing the updated belief about the parameter θ after observing data D.
• $P\left(D\right|\theta )$ is the likelihood, representing the probability of observing data D given the parameter θ.
• $P\left(\theta \right)$ is the prior probability, representing the initial belief about the parameter θ before observing the data.
• $P\left(D\right)$ is the marginal likelihood or evidence, which normalizes the posterior distribution and ensures that it sums to one.

2.2. Prior Distribution

The choice of the prior distribution $P\left(\theta \right)$ is a crucial step in Bayesian analysis [40,41]. Priors can be informative or non-informative. Informative priors incorporate specific prior knowledge about the parameter θ. For example, if previous studies provide strong evidence about the likely range of θ, this information can be encoded into the prior distribution. Non-informative priors or vague priors are used when little prior information is available, and they aim to exert minimal influence on the posterior distribution. The prior distribution can be selected from uniform distributions or Jeffreys priors.

Uniform distributions assume that all values within a specified range are equally likely. They are often used when there is no prior knowledge to suggest that any particular value is more likely than another. For example, a uniform prior over the interval [0, 1] means that any value between 0 and 1 is equally probable. Jeffreys priors are non-informative priors that are designed to be invariant under transformations of the parameter space. They are particularly useful when little is known about the parameter. The Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix. This type of prior provides a default choice that does not overly influence the posterior distribution, especially when prior information is sparse. Additionally, conjugate prior distributions can also be selected. The primary advantage of using a conjugate prior lies in its algebraic convenience, which allows for a closed-form expression of the posterior distribution, significantly reducing the computational load. Therefore, in this study, a conjugate prior distribution was chosen to establish the relevant model.

2.3. Likelihood Function

The likelihood function $P\left(D\right|\theta )$ quantifies how likely the observed data D is given the parameter θ. It is derived from the underlying statistical model of the data [42]. Suppose that D₁, D₂, …, D_n are random variables, and let θ be a parameter characterizing the underlying conditions under which these random variables are generated. Then, the joint probability of observing D₁, D₂, …, D_n simultaneously is given by $P\left(D_1\right|\theta )\times ···\times P(D_n\left|\theta \right)$. For instance, if the data are assumed to follow a normal distribution with mean θ and a known variance σ²—either estimated from a large dataset or assumed based on prior process knowledge—the likelihood function would be

$$P(D|\theta )=\prod _{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-\frac{1}{2}{\left(\frac{y_i-\theta }{\sigma }\right)}^2}$$

(2)

2.4. Posterior Distribution

The posterior distribution $P\left(\theta \right|D)$ is obtained by combining the prior distribution with the likelihood function via Bayes’ Theorem [43]. The computation involves the following steps:

(1)

Determine the likelihood function based on the observed data and the assumed model.

(2)

Combine the likelihood with the prior distribution.

(3)

Divide by the marginal likelihood to ensure the posterior distribution integrates to one.

The posterior distribution is given by

$$P(\theta |D)={\displaystyle \frac{P\left(D\right|\theta \left)P\right(D)}{{\int }_{-\infty }^{+\infty }P\left(D\right|{\theta }^{\prime }\left)P\right({\theta }^{\prime })d{\theta }^{\prime }}}$$

(3)

The denominator, ${\int }_{-\infty }^{+\infty }P\left(D\right|{\theta }^{\prime }\left)P\right({\theta }^{\prime })d{\theta }^{\prime }$, is the marginal likelihood and ensures the posterior distribution is a valid probability distribution.

2.5. Example Calculation

Suppose we have a prior distribution N(347, 0.05²) for the length of a torsion bar. After observing data with a length mean of $\overline{x}=347.43$ mm and a sample size of n = 30 with known variance σ² = 0.068², the posterior parameters of length are calculated as follows:

(1)

Posterior Mean (μ_n):

$${\mu }_n={\displaystyle \frac{\frac{347}{{0.05}^2}+\frac{30\times 347.43}{{0.068}^2}}{\frac{1}{{0.05}^2}+\frac{30}{{0.068}^2}}}\approx 347.405$$

(4)

(2)

Posterior Variance (σ²):

$${{\sigma }_n}^2={\displaystyle \frac{1}{\frac{1}{{0.05}^2}+\frac{30}{{0.068}^2}}}\approx {0.0122}^2$$

(5)

Thus, the posterior distribution is

$$P\left(\theta \right|D)=N(347.405, {0.0122}^2)$$

(6)

This detailed approach ensures accurate predictions and control over manufacturing processes, as demonstrated by the torsion bar dimensional analysis.

3. Application of Bayesian Statistics in Predicting Carburizing Distortion

3.1. Material and Sample Preparation

The study utilized torsion bars made from 20CrMnTi alloy steel, which is known for its high strength and toughness, making it suitable for applications requiring high wear resistance and carburizing heat treatment. The torsion bars had a nominal length of 348 mm, as depicted in Figure 1. Prior to the carburizing treatment, the bars were machined to precise dimensions to ensure uniformity in the experimental results. Length dimensional changes before and after the carburizing heat treatment were precisely measured using a vernier caliper. The vernier caliper manufactured by Mitutoyo Corporation (Kawasaki, Japan) features a resolution of 0.01 mm ensuring high precision in the measurement of small dimensional variations. These measurements were foundational for the Bayesian statistical analysis conducted in this research.

The torsion bars were carburized at a temperature of 920 °C, chosen based on the optimal conditions for 20CrMnTi steel to ensure efficient carbon diffusion into the steel matrix. The carburizing process was sustained for 8 h, encompassing three key stages: uniform heating, carbon enrichment, and diffusion. This duration was chosen to achieve a sufficient depth of carbon penetration and a uniform distribution within the surface layer. The process was conducted in a methanol carbonaceous atmosphere to facilitate the diffusion of carbon atoms into the steel. The furnace atmosphere was carefully controlled to maintain a consistent carbon potential. Following the carburizing treatment, the torsion bars were first cooled slowly to 850 °C and then rapidly quenched in oil to induce martensite formation. The detailed sequence of the carburizing heat treatment, followed by the controlled cooling and quenching stages, is illustrated in Figure 2, which provides a clear schematic of the process parameters.

After precision machining the total lengths of the torsion bars were measured before and after carburizing heat treatment. These measurements are presented in

Table 1. Length size statistics of torsion bars.

Sample No.	1	2	3	....	28	29	30	Mean	Standard Deviation
Machined Size (mm)	347.50	347.44	347.46	....	347.5	347.5	347.34	347.43	0.068
Size After Carburizing (mm)	347.96	347.9	347.9	....	348	347.94	347.96	347.94	0.062
Expansion (%)	0.132	0.132	0.127	....	0.144	0.127	0.179	0.145	0.02

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.

Based on the data presented in Table 1, the statistical distribution of the lengths before and after carburizing heat treatment is illustrated in Figure 3.

From Figure 3, it is evident that the specimens undergo significant dimensional expansion and deformation after carburizing treatment. This dimensional increase can be attributed to two primary factors. Firstly, during the carburizing process, carbon atoms diffuse into the lattice structure of the base metal, causing lattice expansion and, consequently, an increase in overall dimensions. Secondly, after carburizing, the specimens are rapidly cooled to form martensite structures, which impart high surface hardness. The martensite transformation itself involves volume expansion. Due to these combined effects, torsion bars inevitably experience dimensional changes after carburizing treatment. To ensure the dimensional accuracy of the torsion bars and thereby maintain their performance and longevity, it is crucial to understand the patterns of dimensional changes during the carburizing process. Mastering these patterns allows for precise control over the manufacturing process and provides essential technical support and data references for subsequent machining and treatment of torsion bars. This understanding is pivotal for mitigating the adverse effects of dimensional changes, thus ensuring the torsion bars meet the stringent requirements of dimensional and assembly precision necessary for their application in high-stress environments, such as pneumatic clutch systems in mining ball mills. The dimensional control achieved through understanding carburizing-induced changes not only enhances the reliability of the torsion bars but also contributes to the overall efficiency and safety of the machinery in which they are employed.

3.2. Dimensional Analysis Before and After Carburizing

In this study, a conjugate prior in the form of a normal distribution was selected for modeling the unknown mean expansion. This choice is motivated by its analytical convenience and compatibility with the assumption that observed dimensional changes follow a normal distribution. The prior parameters were estimated based on historical production data and expert knowledge, ensuring that the posterior distribution would remain in the same distributional family, simplifying computation. The dimensional measurements of the torsion bars were taken using a precision vernier caliper, with a reported accuracy of ±0.05 mm. The length of each sample was recorded both before and after the carburizing process. The data collected from these measurements were processed using Bayesian statistical methods. The prior distribution, based on historical data, was updated with the new measurements to produce a posterior distribution that more accurately reflects the actual dimensional changes observed. This approach allowed for a more precise prediction of the final dimensions, accounting for the inherent variability introduced by the carburizing process.

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

$$P(D|\theta )=\prod _{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-\frac{1}{2}{\left(\frac{y_i-\theta }{\sigma }\right)}^2}$$

(7)

(3)

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

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

(8)

Prior distribution with mean μ₀ and variance τ². Data normal distribution with known variance σ² and unknown mean θ. Through some algebraic manipulation, it can be shown that

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

(9)

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 N(347, 0.05²) as the prior distribution for Bayesian analysis.

To accurately analyze and understand the dimensional variation patterns of torsion bars, this paper employs Bayesian statistical methods to analyze the deformation patterns and establish predictive models. 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 are 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 4.

From Figure 4, it is evident that the posterior distributions, both before and after carburizing, are significantly more concentrated and exhibit 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. The concentration of the posterior distribution reflects reduced uncertainty in the estimated expansion value. A narrower posterior variance indicates that the model has successfully integrated the prior belief and observed data to provide a more confident estimate. This is particularly beneficial in precision manufacturing, where tighter control over dimensional variation is essential.

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.

3.3. Establishing a Distortion Model Using Bayesian Statistics

The relative expansion rates of torsion bars before and after carburizing heat treatment are 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 in the torsion bars, we select a normal prior distribution N(0.15, 0.032²) 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 N(0.145, 0.0042²), as shown in Figure 5. 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 6.

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 Bayesian predictive distribution provides not only a point estimate for future observations but also a full probability distribution accounting for parameter uncertainty. This can be used to set probabilistic control limits and thresholds in Statistical Process Control systems for future batches, enabling proactive interventions when deviations are likely to occur. The credible interval and predictive distribution of the relative expansion rate are shown in Figure 7. The distribution of expansion rates after carburizing heat treatment conforms to a normal distribution, with all data points falling within the 95% fitted normal distribution curve which is shown in Figure 7b.

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. Compared to traditional deterministic or frequentist methods, Bayesian approaches incorporate prior process knowledge and can continuously update predictions with new measurements. This allows for dynamic, data-informed estimates of dimensional changes, enabling more accurate predictions even with limited sample sizes. Moreover, the probabilistic nature of Bayesian models quantifies uncertainty, offering confidence intervals rather than single-point estimates, which is crucial for risk-sensitive manufacturing environments.

4. Conclusions

The application of Bayesian statistical methods in analyzing the dimensional changes in torsion bars after carburizing has yielded substantial improvements in prediction accuracy and process control. The study demonstrates that integrating prior knowledge with actual measurement data leads to a posterior distribution with reduced variability, thereby providing more reliable predictions of dimensional changes. This improvement directly translates to better compliance with design specifications and reduces the risk of dimensional deviations during manufacturing.

Moreover, the methodology presented offers a robust framework for applying statistical analysis to other heat treatment processes, potentially transforming how dimensional control is managed in high-precision manufacturing environments. The ability to predict and account for such changes with higher accuracy enhances the overall quality and reliability of the manufactured components. Future work may focus on extending this approach to include more complex geometries and additional process parameters, further broadening its applicability and impact.