## 2 Materials and Methods
### Statistical Analysis  
The experimental investigation followed a systematic methodological framework to optimize the ball burnishing process for Ti grade 2 titanium alloy (Figure 4). The methodology commenced with selecting Ti grade 2 as the workpiece material, chosen for its widespread industrial applications and specific mechanical properties. The ball burnishing  
Figure 3: Schematic representation of surface roughness [37].  
process was the primary surface enhancement technique following material selection. A mixed-level Taguchi L18 experimental design was implemented to investigate the process parameters efficiently while minimizing the number of experimental trials.  
The process parameters were carefully defined based on the preliminary studies and literature review. Three primary variable input parameters were identified: burnishing force, feed rate, and number of passes. To comprehensively understand their effects on surface quality, these parameters were investigated under two distinct processing environments--dry conditions and MQL. The output parameter was established as surface roughness, the primary indicator of surface quality enhancement.  
The statistical analysis framework employed ANOVA using statistical software to evaluate the experimental results [38; 39]. The analysis was conducted with a significance level (\(\alpha\)) of 0.05 and a 95% confidence interval. This statistical approach enabled the quantification of each parameter's influence on surface roughness through the following:  
* Calculation of sum of squares (SS) and degrees of freedom (DOF).
* Determination of F-ratio and \(p\)-values.
* Assessment of percentage contribution ratio (PCR).
* Evaluation of parameter interactions.
* Validation of model adequacy through residual analysis.  
The culmination of this systematic approach led to the determination of optimal design parameters for the ball burnishing process. The optimization considered statistical significance and practical applicability, ensuring that the recommended parameter combinations would be feasible for industrial implementation while achieving superior surface quality outcomes.  
To analyze the effects of the ball burnishing parameters on surface roughness (Ra), a regression model was developed using the general linear model (GLM). The GLM approach enables the evaluation of the relationship between the response variable (Ra) and the independent variables (burnishing environment, force, feed rate, and number of passes). The coded values of the parameters were used to normalize the data for statistical analysis. Ra was chosen as the main focus for surface roughness optimization and analysis,  
Figure 4: The experimental methodology.  
as it is the most widely accepted parameter for evaluating surface quality. While Rz is included in Table 2 for additional insight, no statistical analysis was performed for Rz. The general form of the regression model is expressed as follows:  
\[\text{Ra}=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\beta_{3}X_{3}+\beta_{4}X_{4} \tag{4}\]  
The response variable, Ra, represents the arithmetic average roughness. The intercept of the regression model is denoted as \(\beta_{0}\), while \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\) are the regression coefficients corresponding to the independent variables. These independent variables, represented as X\({}_{1}\), X\({}_{2}\), X\({}_{3}\), and X\({}_{4}\), include the coded values for the burnishing environment, force, feed rate, and number of passes. These coded values, as presented in Table 1, were assigned specific codes: E1 and E2 for the burnishing environment (dry and MQL); F1, F2, and F3 for burnishing force (50 N, 100 N, and 200 N); Fr1, Fr2, and Fr3 for feed rate (0.5 mm/min, 1 mm/min, and 2 mm/min); and P1, P2, and P3 for the number of passes (one, two, and four).