## 3 Results
### Effects of Liner Thickness and Head-Helmet Friction  
The modified helmet was used to study the effect of the liner thickness and head-helmet friction. The work employed the POD and Kriging method to obtain a reduced-order model using the training data from finite element simulations. Nine training data were required for each impact locations. The peak linear acceleration was extracted from the DOT FMVSS 218 drop test, while the rotational acceleration was extracted from the inclined flat  
Figure 12: Comparisons of the peak rotational acceleration obtained from the inclined anvil drop test for five combinations of helmet material. anvil drop test. Table 4 illustrates the training data obtained from finite element simulation for the establishment of two ROMs. Two ROMs were established to predict the linear and rotational accelerations and validated against the finite element simulations. Figure 13 shows comparisons of finite element calculation and the prediction from the ROMs for various impact locations, liner thickness, and the friction coefficient as stated in Table 3. The prediction errors for both linear and rotational accelerations were within \(\pm\)5% corridor. The ROMs for the helmet drop test were employed to predict the peak linear acceleration and the rotational acceleration for a helmet with the inner liner thickness ranging from 10 to 20 mm and the head-helmet friction coefficient ranging from 0.05 to 0.65. The prediction results are shown in Figures 14 and 15. The peak linear acceleration decreased with the increasing of the liner thickness for all impact locations. However, at the range of 17-20 mm thickness, the peak linear acceleration changed insignificantly with thickness for all friction coefficients. The peak linear acceleration hardly increased with the friction coefficient. The effect of the friction coefficient on the peak linear acceleration was more obvious at the front and the rear locations than the others. All predicted cases gave the peak linear acceleration lower than the 400 g criteria as stated in the DOT FMVS 218. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Inner Liner Thickness** & \multicolumn{2}{c}{**Head–Helmet**} & \multicolumn{2}{c}{**Peak Linear Acceleration(g)**} & \multicolumn{4}{c}{**Rotational Acceleration (rad/s\({}^{2}\))**} \\ \hline  & & **Crown** & **Front** & **Rear** & **Side** & **Incl-Crown**  
The peak rotational acceleration increased with the friction coefficient for all impact locations, as shown in Figure 15. It slightly decreased with increasing of the liner thickness. However, for the friction coefficient below 0.2, the liner thickness hardly affected the peak rotational acceleration for all impact locations. Similar trend was observed apart from the one at the front impact location. With the rotational acceleration criteria of 8000 rad/s\({}^{2}\)[31], the suggested ranges of both helmet parameters are presented in Table 5.  
Figure 14: Relationship between helmet parameters and the peak linear acceleration predicted by the ROM with the DOT drop test at (**a**) crown, (**b**) front, (**c**) rear, and (**d**) side impact locations. Figure 15: Relationship between helmet parameters and the rotational acceleration predicted by the ROM with the inclined anvil drop test at (**a**) crown, (**b**) front, (**c**) rear, and (**d**) side impact locations.