# Experimental Modal Analysis of Hand-Arm Vibration in Golf: Influence of Grip Strength
## 4 Finite Element Numerical Model  
### Shaft Modeling  
The first step in the development of the numerical model consisted in the scan of the head, while using the DAVID 3D Scanner 4 system (HP, Palo Alto, CA, USA) comprised of a projector, a camera, and a rotating plate. The resulting geometry, as depicted in Figure 4a, was then imported in general purpose CAD software SolidWorks (Dassault Systems, Paris, France). The dimensions of the shaft and grip were too large to allow for scanning, so the decision was made to directly model them using the CAD software, with geometry measurements being carried out using a measuring tape and a sliding caliper. The whole shaft model, as presented in Figure 4b, was then imported in FEA software Abaqus 6.14 (Simulia, Paris, France). For computational cost purposes, the shell elements were selected due to the low thickness of all three components in the assembly: the carbon-epoxy composite shaft is 1.5 mm thick, while the aluminium head and rubber grip are both 3 mm thick. Ultimately, the overall mesh contains 27061 four-node shell elements with reduced integration (S4R elements). Based on the works by [36; 37], the shaft material is assumed to be an orthotropic, carbon-epoxy composite, while the grip and head can reasonably be considered isotropic. Table 1 summarizes all of the required mechanical properties.  
Figure 4: (**a**) Three-dimensional (3D) scan of the head, imported in SolidWorks. (**b**) Whole numerical model of the golf shaft.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**Free-Free**} & \multirow{2}{*}{**Fixed-Free**} & \multicolumn{2}{c}{**Strong**} & \multirow{2}{*}{**Medium**} & \multirow{2}{*}{**Weak**} & \multirow{2}{*}{**Grip-Free**} & \multirow{2}{*}{**Plane**} \\  & & & & & & & \\ \hline
1 & 45.25 (1.0) & 3.79 (1.7) & 2.03 (24.2) & 2.03 (12.3) & 2.06 (10.5) & xy \\
2 & 51.08 (0.9) & 4.44 (1.8) & 3.02 (32.2) & 3.05 (14.5) & 3.05 (15.0) & xz \\
3 & 124.43 (0.50) & 72.25 (1.4) & 83.00 (15.5) & 84.06 (9.0) & 84.38 (8.9) & xy \\
4 & 146.86 (0.8) & 92.45 (1.5) & 95.45 (-) & 96.33 (13.4) & 97.99 (12.2) & xz \\
5 & 286.75 (0.4) & 187.94 (0.8) & 188.51 (10.2) & 189.99 (5.3) & 193.19(1.0) & xy \\
6 & 298.92 (1.6) & 239.50 (0.5) & 276.68 (19.4) & 277.59 (12.1) & 278.79 (14.2) & xz \\
7 & 361.88 (-) & 320.20 (1.3) & 357.48 (-) & 377.97 (12.5) & 388.55 (-) & xy \\
8 & 477.88 (2.7) & 478.74 (1.6) & 444.60 (35.2) & 456.22 (-) & 472.29 (-) & xz \\ \hline \hline \end{tabular}
\end{table}
Table 3: Identification of modal data. The deformed shapes correspond to bending. The results are expressed in Frequency-Hz (Damping-%). (-) corresponds to an undefined value.  
A three-step model updating was performed. First, two three-point bending tests, one numerical and the other experimental, were compared, so as to update the Young's modulus \(E_{1}\) of the shaft. The updated value is 92,550 MPa. After updating and considering the fixed-free boundary conditions, the MAC (Modal Assurance Criterion) matrices were used to determine the similarity of the identified and numerical eigenmodes (Figure 5). The coefficients are bounded between 0 and 1, with 1 indicating fully consistent eigenmodes. A value near 0 indicates that the modes are not consistent [38]. Prior to updating, the discrepancies between model and experiment ranged from 4.82 to 36.93% but decreased to a 2.04-4.21% interval afterwards, Table 4. Only the first two modes retain significant post-updating differences, 18.24 and 33.00%, respectively. The Modal Assurance Criterion (MAC) matrices highlight good identification of modal shapes, with diagonal values above 0.80, except for mode 2 (0.56). However, it should be noted that modes 4 and 2 are only slightly decoupled, which may be attributable to a lack of observability during experimental modal analysis.