## 2 Materials and Methods
### Finite Element Analysis  
The finite element models of the bats were built using HyperMesh (Altair Engineering Inc., Troy, MI, USA) and were analyzed using LS-DYNA R10.0 (Ansys Inc., Canonsbury, PA, USA). All post-processing procedures were completed using LS-PrePost V4.7.0. The profile points in Figure 2 were inputted to HyperMesh as temporary nodes, and a spline was fitted to these points to generate the profile. Using the splines, individual surfaces were defined, and each surface was meshed with quad elements. The two-dimensional  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Profile** & **Volume** & **Density** & \begin{tabular}{c} **Min Handle** \\ **Diameter** \\ \end{tabular}  & \begin{tabular}{c} **Max Barrel** \\ **Diameter** \\ \end{tabular}  & \ \begin{tabular}{c} **6-Inch Sweet** \\ **Spot Diameter** \\ \end{tabular}  & \begin{tabular}{c} **11-Inch Transition** \\ **Diameter** \\ \end{tabular}  \\ \hline \hline \multicolumn{3}{c}{**in\({}^{3}\) (cm\({}^{3}\))**} & **1b/in\({}^{3}\) (g/cm\({}^{3}\))** & **in. (cm)** & **in. (cm)** & **in. (cm)** & **in. (cm)** \\ \hline A & 79.7 & (1306) & 0.0243 & (0.673) & 0.47  
mesh was revolved around a center dowel of the bat, resulting in a three-dimensional mesh of brick elements (8-noded brick elements with single Gauss-point integration). To capture an acceptable resolution of bat failure during post processing, the nominal height dimension of the elements was 0.1 in (0.254 cm), and the model had 36 elements around the center line. The bat-ball impacts were simulated to occur in the Bat Durability System (Automated Design Corporation, Romeoville, IL, USA) at the Baseball Research Center at the University of Massachusetts Lowell. This system approximates an on-field bat/ball collision in a controlled lab environment. It uses two pairs of rollers to simulate a player's grip and to keep the bat stationary [17]. The top set of rollers firmly holds the knob area of the bat, while the bottom ones loosely touch the bat's handle. A comparison of a bat tested in the durability system and one of the bat models made using the finite element method is shown in Figure 3. The full model consists of the baseball bat, the roller grips, a rotating back plate, and the baseball. Once the finite element meshes for each of the components were assembled in HyperMesh, the baseball bat model was exported to an LS-DYNA keyword model file, and all analyses were completed. Further details of this modeling framework for baseball bats is presented in [PERSON] et al. [18]. The finite element models were analyzed over a span of bat/ball impact velocities ranging from 90 to 200 mph (145 to 322 kph) in 5 mph (8 kph) increments. All models represented maple wood over a SoG range of \(-3^{\circ}\) to \(+3^{\circ}\) in \(1^{\circ}\) increments and the SoG in each case is limited to being uniform over the entire length of the bat. A LS-DYNA Material Model #143 (\"MAT_WOOD) was employed to represent the maple wood species. Previous studies of this material model demonstrated that it can be used as an effective model to simulate wood structures subjected to dynamic loading [19; 20]. As shown in Table 1, the bat profiles vary in volume. As a result, the density of the maple wood varies among the profiles to maintain the bats such that they all have the same mass of 31 oz (879 g). Previous empirical testing has allowed for the mechanical and failure properties of the maple wood to be related to density [11; 16]. The properties used in the finite element keyword files for the respective bat profiles are shared in Appendix A, in both US customary units (Table A1) and SI units (Table A2). The AOPT option in  
Figure 3: Comparison of (**a**) bat in ADC Bat Durability System and (**b**) finite element model of the bat in the ADC system. Note: rectangular box on the bat surface indicates the contact area for impact. the \"MAT_WOOD material model accommodated the changing of the SoG among the models [21].