Enhancing Manufacturing Processing Stability and Efficiency with Linear-Regression Analysis: Modeling on a Flow-Drill Screw (FDS) Joining Process

Abstract

The instability (in processing time) in the flow-drill screwing process is undesired but inescapable due to variations in material property, gauge, and process parameters. A substantial number of materials and lab labor need to be used to test and control the variability of the real manufacturing joining process. To enhance the stability and efficiency of the screwing process, this study seeks multi-disciplinary collaboration by applying linear-regression modeling. Six hundred and forty-eight data points were collected and split into an 80% training set for model building and a 20% test set for model validation. A multiple linear-regression model was built. The results indicated that, compared to variable base level (6000 rpm rotational speed and 1100 N downforce), higher rotational speed (8000 rpm, 7000 rpm), greater downforce (1200 N, 1300 N), and their interaction were significantly associated with passage (processing) time, while the switch point did not significantly affect passage time. The interaction plot and effect size were adopted to provide measurements of the effect magnitude on processing time. The coefficient of determination indicated that 86% of the variability in the passage time can be explained by this model. Statistical analysis, such as data visualization, statistical modeling, and other data-driven analysis methods, can be used to detect underlying relationships between variables, investigate variations, and make predictions in the manufacturing process. The outcomes from the data-driven analysis can benefit from improving the economical manufacturing system, refining the processing setting, and reducing test material costs, labor, and lead time.

1. Introduction

1.1. Mechanical Joining Process

Light-weighting joining technologies have become a hot topic in joining mixed-material stack-ups in body-in-white, closure, and other applications for the automotive industry. To improve the quality and avoid some issues of joining mixed material by traditional welding techniques, multiple mechanical joining technologies involve strong candidates joining various material stack-ups, such as aluminum with aluminum, aluminum with steel, magnesium with steel, and even composite with different metal parts, etc. [1,2].

Mechanical joining can be catalogized into two types in terms of accessibility. As shown in Figure 1, self-piercing riveting (SPR) [3], mechanical clinching [4], and friction element welding (FEW) [5] require double-sided access to finish the joining process. Most of the joining equipment with the C-Frame configuration needs double-sided access to complete the joining process. Though space limits the application of these joining technologies, they are still the most cost-effective options for automated methods of joining dissimilar materials compared to manual operations.

One-side or single-sided access mechanical joining technologies, including flow-drill screw (FDS) [6], TACK [7], and blind rivets [8], are the typical joining methods for joining some body structures with hollow structures and extrusion parts. Unlike the SPR and other popular double-side mechanical joining methods, single-sided access joining technologies require either having or not having a pre-hole on the top sheet to form a competitive mechanical joint with a similar cycle time and joining performance as a double-side.

Figure 1. Dissimilar material mechanical joining technologies in terms of accessibility [9].

Figure 1. Dissimilar material mechanical joining technologies in terms of accessibility [9].

Flow-drill screw is the automotive industry’s primary choice of single-sided access joining technique that can provide a reliable and consistent joining process for various mixed-material stack-ups [6]. The joining process can be divided into four stages, as shown in Figure 2. First of all, the flow-drill screw starts to penetrate the material stacks with the highest rotational speed across four stages. Meanwhile, a pre-set downforce is applied on the screw to form a hole in the joined material, typically within one second. Then the relatively low speed and lesser force are used to continuously drive screws to form the threads on joined material stack-ups. Both threads-forming and screwing-in processes are accomplished under this rotational speed and downforce setting. Before the screw fully contacts with the top sheet material, the speed and force will be adjusted to even lower values to build up to the final tightening torque target. Finally, the process stops when the system detects that the target final torque value has reached. Based on the joined material thicknesses, the flow-drill screwing process generally takes 2 to 3 s. The hole-forming and thread-forming stages consume most of the time in the overall cycle. The rotational speed and downforce combination during these two stages can mainly determine the finish duration. The screw movement distance can decide the switch point between different stages. For example, the screw will maintain the specific rotational speed and amount of downforce provided by the system during the hole-forming stage. Once the screw goes into the material with the pre-set depth, the joining process will automatically switch to the next thread-forming stage. Then, the pre-set speed and force will be applied to the screw to keep executing the thread-forming process.

1.2. Linear Regression Analysis

The “artificial intelligence” (AI) concept emerged in the 1950s. In 1948, one of the computer science pioneers, Alan Turing, devoted himself to creating a “thinking machine” that could imitate human logic to compute and judge [10]. After years of development, current AI studies how to make a computer have human intelligence in learning, reasoning, and rational behaviors [11]. Machine learning (ML) is a subset of AI; it applies computer algorithms to identify patterns and learn the underlying knowledge of input data [11]. In recent years, researchers have put much effort into improving the accuracy of certain machine-learning algorithms in performing complex text in computer vision, prediction, data analysis, language processing, and information retrieval [12].

Statistical modeling methods are now widely applied in many domains, including medicine [13], biology [14], astronomy [15], traffic, and the automotive field [16]. Medical researchers have built machine-learning models to predict/classify patient classes based on diverse features from clinical data [13]. Statistical ML modeling has been used for cancer prognosis and prediction [14]. In the aerospace industry, engineers use ML methods for reproducible manufacturing in predictive assembly, process control, part standardization, and inspection; data-driven aerospace engineering can greatly contribute to faster design and testing cycles with digital models [15]. In addition, ML methods can benefit automotive engineers by identifying the most significant risk factors associated with injury severity from crash data [16].

Among the variety of ML algorithms, the widely used ones are linear regression, logistic regression, Naïve Bayes, Bayesian network, support vector machines, decision tree, random forest, classifier, k-nearest neighbor, and different neural networks (ANN, CNN, and DNN) [12]. Especially in manufacturing joining related fields, previous studies in additive manufacturing for aerospace found that regression can be used for defect detection, quality prediction, and quality assurance, and clustering learning can help with cost estimation [17]. Linear regression has also been applied to calculate the apparent activation energy for friction-welded Cu/Al joints [18]. Order polynomial (regression) and the analysis of variance (ANOVA) techniques have been utilized to analyze the process parameters in the aluminum alloy joining process and find the optimal welding conditions [19]. A study analyzed the relationship between grain size and hardness of aluminum alloy joints by applying both linear-regression and neural-network models [20].

1.3. Challenge and Research Question

The variations (of processing time) in the flow-drill screwing process are inevitable in real manufacturing due to variations in material property, gauge, and process parameters. A substantial number of materials and lab labor need to be used to test and control the variability of the real manufacturing joining process. Applications for solving real manufacturing problems through statistical modeling will greatly benefit manufacturing engineering. By deeply analyzing the features and patterns of the data, advanced modeling algorithms investigate the underlying relationship, effect, and causations through inductive/analytical learning. The results from this data-driven study can guide and improve the economical manufacturing system by saving test materials, costs, labor, and lead time.

To enhance the stability and efficiency of the screwing process in real manufacturing, this study seeks multi-disciplinary collaboration by applying linear-regression algorithms to analyze machine processing (passage) time. Specifically, with the above objective and experimental design, this study investigates the following research questions:

• How do variables affect the machine passage time (parameter estimation) (RQ1)?
• Can the model help predict the passage time based on the given conditions, and how good is the result (prediction and evaluation) (RQ2)?

2. Materials and Methods

2.1. Materials and Experimental Procedures

The testing material stack-up is composed of 1.5 mm A1008 steel, 3.2 mm AW-6061-T6 aluminum alloy, and 1.5 mm carbon steel ASTM-A1008 as the top sheet with 7.0 mm pre-hole. The main chemical composition of A1008 steel is Fe, 99.31–99.70%; Mn, 0.30-0.50%; C, 0.10%; S, 0.05%; P, 0.04% (in wt%). 3.2 mm EN AW-6061 T6 aluminum alloy as the bottom sheet is mainly composed of 6061-T6: Al, 98.27%; Fe, 0.50%; Mn, 0.03%; Cu,0.05%; Mg,0.65%; Si, 0.50% (in wt%). The mechanical properties of both types of steel and aluminum coupons can be found in

Table 1. Mechanical properties for aluminum AW-6061-T6 and ASTM-A1008 steel.

Variable	UTS (MPa)	YS (MPa)	Elongation (%)	Hardness (HRB)
6061-T6	310	276	12	60
ASTM-A1008	350	213	45	55

. The Flowform^® (Arnold Fastening Systems, Rochester Hills, MI, USA) M5x21 DS tip with autocert external drive-head Flowform^® screws were picked for the entire test matrix. A total of 12 screws were set on each 50 mm × 150 mm coupon as the maximum quantity, as shown in Figure 3.

The system used to install the Flowform^® screw was the K-Flow system (Atlas Copco, Auburn Hills, MI, USA) from Atlas Copco. The K-Flow system with a straight piercing configuration was used to install flow-drill screws, as indicated in Figure 3. The system is capable of creating 9000 rpm maximum rotation speed and 3000 N maximum downforce. Each coupon was fixed on the base during the joining process, and the screw was fed manually for each joint.

As the most critical part of the joining process, studying the stability of the hole-forming stage is crucial to affecting the FDS joint quality and joint manufacturability. Rotational speed, down-force, and switch point as the three main variables of the hole-forming stage were utilized to investigate the effect on hole-forming time. For rotational speed, from 6000 rpm to 8000 rpm, the typical speed range was used for aluminum joints. A similar rule can also be applied on down-force with an 1100 N to 1300 N range and switch point with a 5 mm to 7 mm range.

The orthogonal experimental design rule was followed since the experiment contains three variables, each with three sub-level conditions. This study examined results based on all 27 treatment conditions ($3^3$), as listed in

Table 2. Three-factorial orthogonal experimental design.

Treatment No.	Factor (Variable)
	Rotational Speed (rpm)	Down-Force (N)	Switch Point (mm)
1	6000	1100	5
2	6000	1100	6
3	6000	1100	7
4	6000	1200	5
5	6000	1200	6
6	6000	1200	7
7	6000	1300	5
8	6000	1300	6
9	6000	1300	7
10	7000	1100	5
11	7000	1100	6
12	7000	1100	7
13	7000	1200	5
14	7000	1200	6
15	7000	1200	7
16	7000	1300	5
17	7000	1300	6
18	7000	1300	7
19	8000	1100	5
20	8000	1100	6
21	8000	1100	7
22	8000	1200	5
23	8000	1200	6
24	8000	1200	7
25	8000	1300	5
26	8000	1300	6
27	8000	1300	7

. For each treatment condition, data were collected from 24 replicates.

2.2. Modeling on Joining Processing

Following the experimental procedure above, 648 data points were collected based on 27 treatment combinations (with 24 replicates of each combination). The independent variables included three variables: rotational speed, downforce, and switch point. Since each variable only contains three level records, all levels were coded as categorical variables with the level name to consist of the data feature. The dependent variable is the passage time, which refers to the hole-forming time during the joining process. The information on category and coding detail for the three independent variables and dependent variables for the model is given in

Table 3. Variables for linear regression models (base levels are noted by “*”).

Variable	Description	Recorded Level	Variable in Model	Variable Type
Rotational speed	The machine’s rotational speed (rpm) to penetrate the material stacks	6000 *	$R_{6000}$	Independent variable (categorical)
		7000	$R_{7000}$
		8000	$R_{8000}$
Down-force	The pre-set downforce (N) applied on the screw to form a hole in joined material	1100 *	$D_{1100}$	Independent variable (categorical)
		1200	$D_{1200}$
		1300	$D_{1300}$
Switch point	Screw movement distance (mm) for the switch to different process stages	5 *	$S_5$	Independent variable (categorical)
		6	$S_6$
		7	$S_7$
Passage time	The hole-forming time (ms) during the process	Continuous	y	Dependent variable (Numerical)

.

According to the literature review, regression models have been used for manufacturing joining-related studies [18], especially for optimal process parameter setting [19] and variable relationship analysis [20]. Since this data sample’s dependent variable (passage time) is continuous (360–1190 ms), a linear-regression model was built to accommodate the data feature.

Linear regression is one of the most widely used statistical techniques for modeling the relationships between variables and predicting the value of a dependent variable based on given variables [21]. The number of independent variables differentiates the model into linear-regression models (only one independent variable) or multiple linear-regression models (two or more independent variables). Since the data of this study contains three independent variables, a multiple linear-regression model was made. The model’s coefficients estimate the linear relationship between y (dependent variable) and independent variables [22]. The equation of linear regression with multiple variables and related interaction is as follows [23]:

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots +{\beta }_k{x}_k+{\beta }_{12}{x}_1{x}_2+{\beta }_{1k}{x}_1{x}_k+\dots +{\beta }_{k-1,k}{x}_{k-1}{x}_k+\mathit{\epsilon }$$

where y is the dependent variable to be predicted, $x_k$ is the independent variable (predictor), k is the number of independent variables, ${\beta }_0$ is the intercept indicating the value of y when all predictors are 0, ${\beta }_1...{\beta }_k$and ${\beta }_{12}...{\beta }_{k-1k}$ are the estimated coefficients of the independent variables and related interaction terms, and ε is the error term. For example, ${\beta }_k$ describes the expected changes per unit in y for a unit increase in the related independent variable $x_k$ while holding the other variables constant [23].

The scientific modeling analysis was performed in RStudio software (Version 2022.12.0+353). Detailed modeling procedures follow the common guidelines for modeling building [21]:

(1)

Collecting data from the experiment. A total of 648 data points were collected.

(2)

Splitting the data into an 80% training set for modeling building (518 data) and a 20% test set for model prediction and evaluation (130 data).

(3)

Employing a multiple linear-regression model from RStudio; the model is fitted with all the training set data.

(4)

Checking the required assumptions for multiple linear-regression models to ensure the model is validated. Adjustments were made as needed [21].

(5)

Performing parameter estimation, variation analysis, prediction, and evaluations.

Variables with p-value < 0.05 (significant p-value) were considered significant predictors. In addition, effect sizes were calculated according to Cohen’s (1988) recommendations [24] using d measures as the standardized differences between the two means of different groups (Cohen, J., 1988). Cohen’s (1988) recommendations classified effect size as small (d ≥ 0.2), medium (d ≥ 0.5), and large (d ≥ 0.8) [25].

3. Results

3.1. Descriptive Statistics

To learn more about the data, they were plotted based on the three variables. As shown from the three-column sections in Figure 4, the passage time decreases significantly as the rational speed increases from 6000 rpm to 8000 rpm. The color differences reflect the data from different levels of downforce; the passage time decreases as the downforce increases from 1100 N to 1300 N, while other conditions stay the same. According to the data pattern from Figure 4, the data does not show a significant change regarding the different levels of switch points.

The rotational speed is one of the main factors in deciding how much heat can be generated during both the hold-forming and thread-forming stages. The higher speed can lead to more heat generation at the same downforce level. The material can be softened at a higher speed than the condition at a lower speed level. The passage time at the hole-forming stage was decreased by increasing the rotational speed. In addition, passage time reduction was observed at the higher downforce level. Because the downforce was directly applied to the screw, the higher force can lead to a faster pushing speed on the screw. The passage time at hole-forming undoubtedly can be decreased.

3.2. Linear Regression Model-Effect Parameter Estimation

Based on the feature of the data (discussed in Section 2.2), a multiple linear-regression model was built to analyze the effect of the three independent variables (rotational speed, downforce, and switch point) on the length of passage time compared to the base level set for each variable. The base levels are noted by “*” in Table 3, and they will not be shown in the model result summary since they are the reference level for effect comparison.

First, a full model was built, including all the independent variables and related interaction terms, since examining the effects from all aspects is one of the common starting points of modeling. The results from the full model indicated that among the 12 interaction terms generated, only the interactions from rotational speed and downforce significantly affected the passage time (p < 0.05). Therefore, removing all the insignificant interaction terms, a second model was built. As mentioned in the method (Section 2.2), the model building is an iterative process; adjustments are made to ensure the model is validated and reliable. After checking with all the required assumptions for the multiple linear-regression model, the QQ–plot (a graph that checks the model normality by comparing the quantile distribution of samples to the standard normal distribution) suggested there might be some extreme values or outliers (non-linear) existing in passage time; since logarithmic transformation is a convenient and common means for analysis when a non-linear relationship exists [26,27], the value of passage time was logged to transform the data to log-normal distribution. After the above data processing, a second round of (iterative) modeling process was conducted. The final model was built by establishing a new full model with logged passage time, adding interactions and removing detected insignificant terms.

Table 4. Summary of multiple linear-regression models for significant factors.

Variables	Coef.	Exp (Coef.)	S.E.	p-Value	95% C.I. of Exp (Coef.)		Effect Size
					2.5%	97.5%	Cohen’s d	95% C.I.
Intercept	6.82	913.79	1.02	<0.01	891.95	936.16	1.74 ‡	[1.63, 1.85]
Rotational_Speed 7000	−0.25	0.78	1.02	<0.01	0.76	0.80	−1.10 ‡	[−1.23, −0.97]
Rotational_Speed 8000	−0.42	0.66	1.02	<0.01	0.64	0.68	−1.87 ‡	[−2.00, −1.73]
Down_Force 1200	−0.20	0.82	1.02	<0.01	0.79	0.84	−0.90 ‡	[−1.04, −0.77]
Down_Force 1300	−0.35	0.71	1.02	<0.01	0.69	0.73	−1.54 ‡	[−1.68, −1.41]
Switch_Point6	−0.02	0.98	1.02	0.09	0.97	1.00	−0.07	[−0.15, 0.01]
Switch_Point7	0.01	1.01	1.02	0.52	0.99	1.02	0.03	[−0.05, 0.10]
Rotational_Speed 7000: Down_Force 1200	0.04	1.04	1.02	0.08	0.99	1.08	0.17	[−0.02, 0.36]
Rotational_Speed 8000: Down_Force 1200	0.05	1.05	1.02	0.02	1.01	1.10	0.23	[0.04, 0.42]
Rotational_Speed 7000: Down_Force 1300	0.01	1.01	1.02	0.69	0.97	1.05	0.04	[−0.15, 0.23]
Rotational_Speed 8000: Down_Force 1300	0.03	1.03	1.02	0.22	0.98	1.07	0.12	[−0.07, 0.31]

(Note: Coef. = coefficient with logged passage time; Exp (coef.) = exponentiated coef.; SE = standard error; Significant p-value, a = 0.05; Significant variables are presented in bold font. Effect sizes were labeled following Cohen’s (1988) recommendations. ‡ large effect size, d ≥ 0.8.; C.I. (confidence interval) = sample mean ± margin of error).

shows the results summary from the final model.

According to the results of the multiple linear-regression model (Table 4), higher rotational speed and greater downforce are associated with shorter passage time, while changes in switch points do not significantly affect passage time; in addition, the interaction between rationale speed 8000 rpm with 1200 N downforce slightly increases the passage time. Specifically, compared to the base level of rotational speed (6000 rpm), rotational speed 7000 rpm (coef. = −0.25; p < 0.01; d = −1.10) showed a negative impact on the passage time; similarly, a rotational speed of 8000 rpm is associated with a greater negative impact on passage time (coef. = −0.42; p < 0.01; d = −1.87). However, the effect of rotational speed on passage time depended on the level of downforce and vice versa. Table 4 suggests that the interaction between rotational speeds at 8000 rpm and 1200 N downforce significantly affects passage time (coef. = 0.05; p = 0.02). Therefore, a three-way interaction plot was made, as shown in Figure 5, indicating that the interactions between an 8000 rpm rotational speed and 1200 N downforce positively affected passage time. In terms of downforce, compared to the base level of downforce (1100 N), 1200 N downforce (coef. = −0.20; p < 0.01; d = −0.90) and 1300 N downforce (coef. = −0.35; p < 0.01; d = −1.54) were identified as significant variables in decreasing the passage time. In contrast, the results from multiple linear regressions suggested that the level changes in the switch point are insignificant for the passage of time. In addition, following the Tukey pairwise showed significant differences between each rotational speed level, each downforce level, and different pairwise treatment combinations of rotational speed and downforce, but the Tukey pairwise test did not detect any significant difference between different levels of switch points, which is consistent with the results from the multiple linear-regression model.

Based on the model results, the estimation equation of passage time can be written as below (the information on category and coding detail is given in Table 3):

$$\begin{array}{c}\log \left(\mathrm{y}\right)=6.82+(-0.25)R_{7000}+(-0.42)R_{8000}+(-0.20)D_{1200}+(-0.35)D_{1300}+(-0.02)S_6+\left(0.01\right)S_7\\ +\left(0.04\right)R_{7000}\times D_{1200}+\left(0.05\right)R_{8000}\times D_{1200}+\left(0.01\right)R_{7000}\times D_{1300}+\left(0.03\right)R_{8000}\times D_{1300}\end{array}$$

3.3. Variation Analysis

Since the model detected significant interactions between variables, an interaction plot was created in Figure 5. This plot mainly reflects the relationships/trends that vary across levels of the three independent variable changes. Since we have 648 data points in total, with three variables, each has three sublevels. There are 27 (variable) treatment combinations. Therefore, there are 27 points plotted in Figure 5 (across three subplots), each reflecting the average passage time level based on 24 replicates for a unique treatment condition (under a certain rotational speed, downforce, and switch point level).

As shown from the interaction plots (Figure 5), when the switch point stayed the same, the interaction lines were not parallel, indicating that the interaction between rotational speed and downforce had a significant effect on passage time (logged). The vertical bars on different levels are the 95% confidence intervals for the parameters, which reflects the effect variabilities on the passage time. However, due to the plot scale, the differences between confidence intervals are not clearly shown in this plot.

The p-value from the regression model and the interaction plot indicate whether a significant relationship exists. Still, more measurements regarding the size of the effect for each term may be more helpful. Effect size is a useful statistical tool for effect/variance analysis, which measures the magnitude of the difference between each treatment group [28,29]. Cohen’s d is one of the most common methods for reporting the effect size, which calculates and reflects the difference in standard deviation units/scores [24,25,28]. According to the last two columns from Table 4, among all the three independent variables and related interaction terms, a rotational speed of 8000 rpm holds the largest effect size (d = −1.87) in decreasing the passage time, followed by a 1300 N downforce level (d = −1.54). Moreover, rotational speeds of 7000 rpm (d = −1.10) and 1200 N downforce (d = −0.90) were also identified with large effect sizes (d ≥ 0.5).

3.4. Prediction and Evaluation

Adequate validation for linear-regression models is usually conducted by checking the five major assumptions, including independence between errors and residuals, linearity and collinearity between variables, normality of residuals, and homogeneity of variances [21]. The model checking was performed on RStudio software, and the results showed that the multiple linear-regression model in this study meets all the validation requirements.

The predictive capability of a regression model is usually measured by the coefficient of determination, also called $R^2$. $R^2$ follows that 0 ≤ $R^2$ ≤ 1. $R^2$ close to 1 means the model is well-fitted, indicating how much percent of the variability in the dependent variable can be accounted for by the independent variables [21]. The $R^2$ model in this study was 0.8646, indicating that about 86% of the variability in the passage time can be explained by this model.

The model was trained (built) using the 518 training data (80% of the total mentioned in Section 2.2). After checking the model’s validation and goodness of fit, the model was tested by fitting with the rest of the 130 testing data (20% of the total). Figure 6 compares the predicted and actual passage time (logarithmic). The y-axis represents the actual values, and the y-axis represents the predicted value; the diagonal line is the estimated regression where the x equals y. As shown in Figure 6a, both the predicted and actual values show a similar pattern and are close to the regression line, which suggests that the regression model was reasonably well-fitted. Figure 6b shows the QQ plot comparing the quantile distribution of predicted values to the quantile of the standard normal distribution. The pattern suggests that the data followed the reference distribution properly.

4. Discussion

Since all three process parameters—rotational speed, downforce, and switch point— play roles in the passage time and the stability of the passage process, the significance of each parameter needs to be investigated. For instance, rotational speed is mainly treated as the main factor to control the heat during the hole-forming and thread-forming processes. However, the role of downforce on the relatively thicker joint cannot be ignored in heat generation and process stability. Based on a series of joining data from the physical experiments in the lab, the data analysis in the study is substantial for understanding the relationship among each parameter. With data visualization (Figure 4) and a multiple linear-regression model analysis (as presented in Table 4), the results found that both rotational speed and downforce significantly impacted the passage time. When holding downforce and switch point constant, passage time decreased significantly as the rotational speed changed from 6000 rpm to 8000 rpm. When keeping rotational speed and switch points at the same level, passage time also shortened as the downforce increased from 1100 N to 1300 N. In addition, both model results and the interaction plot indicated that the interaction between rotational speed and downforce significantly affected passage time.

From the modeling results, the linear-regression model examines whether there are relationships/associations between independent variables ($x_1,x_2\dots .x_n$) and dependent variables (y). The null hypothesis of the model is that no significant differences (relationships/associations) were observed between the variables (levels) [30]. One of the key test statistics of the linear-regression model is the p-value. A p-value is the measurement of the “statistical significance,” which describes a probability that the null hypothesis is true, suggesting that the tested variables do not have significant effects on the dependent variables. Variables with p-value < 0.05 were commonly considered significant variables [31]. According to the model results in Table 4, the p-values of different levels of rotational speed and downforce were all less than 0.01 (<0.05), while the p-values of the different levels of switch point (0.09, 0.52, respectively) were greater than 0.05. Based on the test statistics, both rotational speed and downforce have a significant impact on the passage time, while the switch point was not a significant factor in determining passage time. A final equation with parameters estimated for each variable is listed in Section 3.2, which may assist engineers with the screwing processing time analysis and the machine parameter setting (RQ1).

The model and interaction analyses found that there were significant relationships between variables. To better understand the stability of the screwing process, an analysis of effect size was conducted to measure the magnitude of the effect on processing variability from each variable. Passage time was found to be more sensitive to some levels of machine setting than others. The findings suggest, compared to the base level of rotational speed (6000 rpm) and downforce (1100 N), that the effect size from larger to smaller is generated from rotational speed 8000 rpm, 1300 N downforce, to rotational speed 7000 rpm, and 1200 N downforce, respectively. The findings may help to detect and investigate the causes of processing instability.

An adequate validation check suggested that the model meets the requirements for validation, and the R squared indicated that 86% of the variability in the passage time can be explained by this model. The model was trained using 80% of the data collected, and then the model was tested by the remaining 20% of the data. Figure 6a compares the predicted and actual passage time, and the data pattern suggests the regression model was fitted reasonably well. The modeling results can help predict passage time and processing variability by given conditions, which may assist in processing design, refining, and cost-saving (RQ2).

5. Conclusions

For the FDS process, thousands of data points are generated globally in everyday production. Data analysis is significant not only to optimize the process stability; it is also substantial to understand the joining challenges from the material-variation and product-design perspectives. Statistical modeling from this study becomes an innovative method to control the process based on the data analysis. Therefore, this study aims to apply linear-regression modeling to analyze and model the manufacturing screwing process. Six hundred and forty-eight data points were collected, splitting into an 80% training set for model building and a 20% test set for model validation. A multiple linear-regression model was built to analyze significant relationships between variables. The results indicate that, compared to the base levels of independent variables (6000 rpm rotational speed and 1100 N downforce), higher rotational speed and greater downforce are significantly associated with shorter passage time; their interaction was also a significant predictor for passage time. Switch point was not a significant factor in determining passage time. The interaction plot and effect size were adopted to analyze the processing stability, and measurements of the effect magnitude on processing time variation were provided. After the validation check, the model was fitted by testing data to make predictions. The predicted and actual passage time values were plotted, and the data pattern suggests the regression model was fitted reasonably well. The coefficient of determination indicated that 86% of the variability in the passage time can be explained by this model.

The findings of this study show that statistical analysis, like data visualization, statistical modeling, and other data-driven analysis methods, could be used to enhance the stability and efficiency of the screwing process. Multi-disciplinary collaboration algorithms can help to detect the underlying relationships among variables, investigate variations, and make predictions.

The coefficient of determination (R squared) is crucial to understanding the model’s effectiveness. The results showed that 86% of the variability in the passage time can be explained with the model, suggesting the model fits well with the data. However, there are three main limitations to this model. First, the sample size is not large enough. To generate a robust result for manufacturing analysis, more data are needed. The relationships and trends generated/found from the model are mainly based on the input data used to train the model. Therefore, the more data covered in the training set, the more accuracy the analysis result might generate. Second, only three independent variables were involved in the model; more variables should be considered to better understand the relationships and variations. Other variables not involved and analyzed in the model would be considered to deal with the remaining 14% deviation, which this model cannot explain. Third, since there were only three different levels (distinct values) in each variable, the independent variables in this model were all considered categorical data (discrete variables) consisting of the data distribution. To extend the variety and range of the value of the variables, using more data or continuous variables would increase the accuracy of the model results. Based on the above limitations, the coefficient generated from this model can guide other relevant regression models but cannot be used directly. Using data visualization plots or mean squared error calculation, a local (not global) minimal passage time can be identified based on the current training set. Modeling with more data, variables, and data from various experiments on different materials and conditions would be needed to get a global minimal passage time. Since the model results were generated from the variables from the training set, they cannot be transferred/used directly to analyze other variables/factors/materials that were not covered in the current training set. More tests and extended analyses are needed to increase the generality of the model results. However, the model results can reveal the current dataset’s relationship and trend within the variables or provide insight/guidance for future analysis.