## 2 Theoretical Background and Mathematical Formulation
### Analytical Impact Force Model for Impact Ball  
As with the case of the bang machine, the analytical impact force model was developed with respect to the impact ball. The mean value and data obtained from 10 field measurements are shown in Figure 5. The maximum value of the measurements ranged from 1498.48 to 1544.31 N, and the mean value was 1517.91 N. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Korean industrial Standards**} \\  & \multicolumn{4}{c}{**(KS) Level (dB)**} & \multicolumn{2}{c}{**Mean Case (dB)**} & \multicolumn{1}{c}{**Proposed (dB)**} \\ \cline{2-6}  & **Low** & **Middle** & **Upper** & & \\ \hline
31.5 & 46.0 & 47.0 & 48.0 & 46.306 & 46.305 \\
63.0 & 38.5 & 40.0 & 41.5 & 39.355 & 39.360 \\
125.0 & 20.5 & 22.0 & 23.5 & 21.787 & 21.722 \\
250.0 & 9.5 & 11.5 & 13.5 & **17.564\({}^{1}\)** & 13.330 \\
500.0 & 3.5 & 5.5 & 7.5 & **11.288\({}^{1}\)** & 4.174 \\ \hline \hline \end{tabular} \({}^{1}\) Violation of KS.  
\end{table} 
Table 4: Impact exposure level for the bang machine case. Figure 4: Analytical impact force model for the bang machine: (**a**) time series data; and (**b**) computed impact exposure level. Figure 5: Ten experimental test data of the impact ball: (**a**) time series data; and (**b**) computed impact exposure level. According to the study of [PERSON] _et al._[23], as well as KS, the maximum load was 1500 N for the impact ball. The impact time subjected to the impact load was measured at 19.294 ms, and is deemed to be appropriate when compared with the impact time presented in KS (20 \(\pm\) 2 ms). With respect to the mean value of data obtained from the 10 measurements, the impact exposure level was calculated and depicted in Figure 5b and summarized in Table 5.  
As shown in Table 5, there was an excess of 1.25 dB in comparison with the KS with respect to 250 Hz. With the use of the same method as in the bang machine, the analytical impact force model on the impact ball was developed. As in Equation (5), the analytical impact force model for the impact ball is an equation model, and the coefficients are summarized in Table 6.  
Figure 6 shows the impact load of the impact ball which was artificially generated using the proposed analytical impact force model. The maximum value of the generated impact source was 1484.73 N, and the impact time was 19.41 ms (\(\Delta t\) = 1/8192), which is deemed to be appropriate when compared with the impact time presented in KS (20 \(\pm\) 2 ms). Figure 6b shows the impact exposure level of the analytical impact force model and the comparison results showed that the impact exposure level was 2.7 dB lower than KS at 500 Hz. \begin{table}
\begin{tabular}{c c} \hline
**Coefficient** & **Value** \\ \hline C\({}_{0}\) & \(-\)0.002691 \\ C\({}_{1}\) & 0.164272 \\ C\({}_{2}\) & \(-\)3.850763 \\ C\({}_{3}\) & 43.537537 \\ C\({}_{4}\) & \(-\)258.213331 \\ C\({}_{5}\) & 862.686073 \\ C\({}_{6}\) & \(-\)105.837420 \\ \hline \end{tabular}
\end{table} 
Table 6: The coefficients for the proposed impact ball impact force model. Figure 6: Analytical impact force model for the impact ball: (**a**) time series data; and (**b**) computed impact exposure level. A comparison between the results calculated based on the mean value, as in Figure 5, and that calculated using the analytical impact force model, as in Figure 6, revealed that the impact force level was low at 500 Hz. The reason is that the curved sections, indicated in red as shown in Figure 6, were not taken into account when comparing the analytical impact force model consisting of only the actual measurement data and a polynomial expression. Accordingly, the accuracy was improved by adding a sine function to compensate for the analytical impact force model. The improved analytical impact force model is as follows.