Experimental Characterization and Regression Modeling of Arc Height and Surface Roughness in Shot Peening

2.1 Shot peening procedure

SAE J442 Type-A Almen strips were used as specimens. All strips were cleaned and surface-prepared prior to peening to minimize external variability. Shot peening was performed using an automated system, as shown in Figure 1(a), equipped with hard-cast steel shot (ASH230, 55–62 HRC). Figure 1 illustrates (b) the shot peening schematic and (c) the spatial orientation of Almen strip positioning. Strip placement followed a symmetric layout, including upper (Z+), lower (Z–), left (Y+), right (Y–), inlet (X–), outlet (X+), and inclined 45° regions to simulate realistic impact conditions on curved aerospace components.

Fig. 1

Shot penning experimental methods

Processing conditions were kept consistent throughout the experiments by maintaining the wheel rotational speed between 1200 and 1300 rpm, regulating the shot flow rate in pounds per minute, adjusting the impact angle to either horizontal, 30^o, or 45^o, and operating at feed rates of 0.3, 0.6, 1.2, and 2.4 m/min, while the shot peening duration was varied as the primary experimental parameter.

Table 1 summarizes the process conditions used in the wheel-blast shot peening experiments. All parameters except feed rate, strip orientation, and position were kept constant to isolate the effects of exposure time and geometric factors on arc height and surface roughness.

Table 1

Process conditions for wheel-blast shot peening experiments

2.2 Measurement and analysis method

In the current wheel-blast configuration, the effective peening duration at a given surface location is inversely proportional to the feed rate. Accordingly, the inverse feed rate (1/F) was adopted as a representative parameter for local exposure time throughout this study.

Arc height was measured using a certified Almen gauge, as shown in Figure 2(a). The saturation curve was constructed by plotting Arc height versus the inverse feed rate to determine the saturation point. For each experimental condition, five repeated measurements were conducted. The reported values represent the arithmetic mean of the repeated measurements.

Fig. 2

Measurement methods

Surface roughness (Ra, Rz) was measured using a stylus-type profilometer at three representative surface locations per strip, as shown in Figure 2(b). Surface roughness Ra obtained from nine measurements per condition, corresponding to three repeated measurements at three different surface locations on each Almen strip.

A regression analysis workflow was applied, beginning with data collection and followed by the selection of key variables, including |nx|, |ny|, pz², and feed-rate–related terms. The variables nx and ny represent the directional cosine components of the Almen strip normal vector relative to the wheel coordinate system, reflecting strip orientation. The parameter pz denotes the normalized vertical position of the strip relative to the wheel centerline, and pz² was used to capture symmetric positional effects. Tables 2 and 3 are the strip orientation and position used for modeling and verification. Transformed feed-rate terms such as 1/F^0.25 and 1/F² were introduced to account for the nonlinear accumulation of impact density and exposure time, as confirmed by regression residual analysis. Multivariate regression was then performed, after which significance was evaluated using p-values and ANOVA. The resulting model was validated to confirm its reliability. These procedures enabled the development of predictive models for arc height and surface roughness.

Table 2

Strip orientation and position for modeling

3.1 Arc height

Figure 3(a) illustrates the variation in arc height as a function of shot peening duration. The arc height increased rapidly during the early stage and approached saturation, consistent with typical saturation curve behavior. Arc height saturation was defined as the condition under which further reductions in feed rate resulted in less than a 5% increase in measured arc height. Based on this criterion, saturation was observed at feed rates below approximately 1.0 m/min.

Fig. 3

Measurement results

This trend arises because compressive residual stress accumulates quickly at the beginning of peening but stabilizes once the surface layer reaches its plastic deformation capacity^[1]. Additional peening beyond this point contributes minimally to arc height improvement, thus offering diminishing returns for fatigue enhancement.

3.2 Surface roughness

Surface roughness showed weaker correlations with feed rate and arc height than the saturation trend observed for arc height. Figure 3(b) presents the variation of surface roughness Ra as a function of the inverse feed rate (1/F), plotted separately for different absolute position levels |pz|. Overall, Ra increases with increasing 1/F, indicating that lower feed rates (i.e., longer exposure time per unit length) produce rougher surfaces due to higher cumulative impact density. This trend is most pronounced for |pz| = 1, where Ra rises sharply at the highest 1/F values, reaching more than 0.18 μm. The |pz| = 0.5 and |pz| = 0.3 cases show moderate roughness growth with increasing 1/F, although the slope is noticeably smaller than that of |pz| = 1. In contrast, the |pz| = 0 condition exhibits nearly constant Ra with minimal sensitivity to feed rate, indicating that peening exposure at the central position results in a more uniform impact distribution.

The separation between the curves demonstrates that surface roughness depends not only on feed rate but also strongly on spatial position relative to the wheel centerline. Positions with larger |pz|, corresponding to more inclined or peripheral impact conditions, experience greater surface deformation and thus higher roughness. This positional effect becomes more significant at low feed rates, where the cumulative impact intensity amplifies geometric sensitivity.

The average Ra was approximately 0.07 μm, and Rz was approximately 10 × Ra. Beyond the arc height saturation point, roughness continued to increase, reaching 0.72 μm for extended periods.

Figure 3(c) illustrates the standard deviation of surface roughness Ra obtained from nine measurements per condition, corresponding to three repeated measurements at three different surface locations on each Almen strip. The results show that the Ra standard deviation increases gradually with increasing inverse feed rate (1/F), indicating that surface roughness variability becomes more pronounced at lower feed rates. At higher 1/F values, prolonged peening exposure intensifies local plastic deformation and stochastic shot–surface interactions, resulting in greater spatial heterogeneity in surface topography.

At relatively high feed rates (low 1/F), the Ra standard deviation remains below approximately 0.03 μm, suggesting uniform surface modification with limited scatter. In contrast, at low feed rates (high 1/F), the standard deviation increases to values exceeding 0.08 μm in some cases, reflecting increased sensitivity to local impact conditions and cumulative shot overlap effects. This trend suggests that excessive peening duration not only increases the average surface roughness but also amplifies measurement dispersion, potentially affecting the consistency of surface quality.

These results demonstrate that although lower feed rates enhance peening intensity, they also introduce greater roughness variability. Therefore, from a process control perspective, both the mean Ra value and its standard deviation should be considered when defining optimal shot peening conditions, particularly for components requiring strict surface quality uniformity.

4.1 Arc height

Arc height was modeled using |nx|, |ny|, pz², and 1/F^0.25 as independent variables. Figure 5(a) shows the predicted and measured arc height. The regression analysis yielded a strong correlation coefficient of 0.952, with R-squared values of 0.906 and 0.900, respectively, indicating excellent explanatory power. The standard error of 0.0142 confirmed the model’s precision, and the ANOVA results, with an F-value of 142.62 and a p-value of 1.3 × 10^-29, demonstrated that the model was highly statistically significant. All independent variables were statistically significant (p < 0.01). pz² and 1/F^0.25 exhibited the strongest influence. The final regression equation is equation (1).

Fig. 5

Multiple regression model arc height

Figure 5(b) compares standardized coefficients, confirming the dominant effects of feed rate (F) and strip height (pz).

4.2 Surface roughness

Surface roughness was modeled using |nx|, pz², and 1/F², yielding a correlation coefficient of 0.675, an R-squared of 0.455, and an adjusted R-squared of 0.428. Figure 6(a) shows the predicted and measured surface roughness. The standard error was 0.0191, and the ANOVA results (F-value = 16.7, p-value = 5.20 × 10^-8) confirmed that the model was statistically significant. Surface roughness shows greater variability than arc height, which explains the lower R² value. The regression model is equation (2)

Fig. 6

Multiple regression model of surface roughness

Figure 6(b) shows the standardized effect of each variable. Among the variables, pz² and |n_x| were identified as highly influential predictors of Ra.

Table 2 Strip orientation and position for modeling

5.1 Arc height

Figure 8(a) presents the relationship between the predicted and measured arc heights for the validation dataset. The data points follow a clear upward trend, and the fitted line exhibits an R-square value of 0.615, indicating that the model captures a substantial portion of the physical variation in arc height, even under unseen processing conditions. Although the scatter increases at higher arc heights, the slope of the regression line approaches unity, suggesting that the model maintains consistent proportionality between predictions and measurements. This level of agreement confirms that the regression formulation, based on feed rate, impact momentum, and surface loading terms, can reliably approximate arc height generation in practical operational windows.

Fig. 8

Relationship between predicted and measured arc heights

To further examine model behavior under changing feed rate conditions, Figure 8(b) compares the predicted and experimental arc heights as a function of 1/F. Both curves exhibit the expected trend of increasing arc height at lower feed rates, reflecting longer exposure times and greater cumulative energy imparted to the surface. The predicted values remain slightly higher than the experimental measurements across the full range, implying a mild systematic overestimation. Despite this offset, the parallel nature of the two curves demonstrates that the model correctly reproduces the functional dependence on feed rate and is suitable for parametric process optimization.

5.2 Surface roughness

Figure 9(a) illustrates the predictive performance for surface roughness Ra, using the model constructed from |nx|, pz², and 1/F². The distribution of data points exhibits a moderate positive correlation, consistent with the reported quantified R² value of 0.455. The surface roughness regression model’s predictive capability is limited compared with that of the arc height model. Therefore, the roughness model should be interpreted as a trend-level estimator rather than a high-precision predictive tool. In practical applications, the model is suitable for identifying dominant process tendencies and relative parameter sensitivity, while local surface effects and stochastic shot–surface interactions may cause deviations in absolute Ra values.

Fig. 9

Relationship between predicted and measured surface roughness

To assess feed-rate dependency in surface roughness more explicitly, Figure 9(b) compares predicted and measured roughness values across the same 1/F range. The experimental roughness varies only slightly with feed rate, whereas the model predicts a more pronounced upward trend at lower feed values. This difference suggests that the influence of feed-related exposure time is weaker in practice than the model assumes.

Overall, the comparative results demonstrate that the arc height model offers robust predictive capability across both training and validation datasets, with clear physical consistency and strong correlation. The roughness model, while statistically significant, shows greater variability and could be refined. These observations confirm that the developed modeling framework is effective for predicting arc height behavior; however, it requires further enhancement for applications where surface finish is the dominant quality requirement.