# Experimental Modal Analysis of Hand-Arm Vibration in Golf: Influence of Grip Strength
## 3 Experimental Model  
The studied shaft, as presented in Figure 1, is a Hybrid 500 MEN'S RH that is made by INESIS (Decathlon, Lille, France), displaying a 22\({}^{\circ}\) lott. The structure is divided in three parts, each made of a different material: an aluminium head, a carbon-epoxy composite shaft, and an ethylene propylene diene monomer (EPDM) rubber grip. The corresponding mechanical parameters are extracted from the literature [34] and are presented in Table 1.  
A discretized model is designed using OROS Modal software (OROS, Grenoble, France) in order to rebuild the modal shapes, and it comprises four segments three connected by five nodes. One node represents the three-axis observabilities recorded by a piezoelectric accelerometer (B&K4525, Bruel and Kjaer, Duluth, USA). The other four nodes stand for the three-axis controllabilities, i.e., twelve controllabilities. Excitation is performed using a shock hammer equipped with a force sensor (208C02, PCB Piezotronics, Buffalo, USA), as seen in Figure 2. Synchronous data collection is supplied by an OROS acquisition system (OR36, OROS, France). Five different sets of boundary conditions are tested. The first two are classical free-free and fixed-free boundary conditions. In the free-free case, the shaft is hung while using elastic cords in order to isolate it from external perturbations. In the fixed-free case, the grip of the shaft measuring 21.5 cm is clamped in a bench vice that is mounted on a slotted plate. With regard to the last three boundary conditions, called \"grip-free\", a male participant (37 years old, 173 cm tall, 70 kg) was involved in the study. The local ethic committee approved the experiment and the participant signed a consent form. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Shafit (Composite)** & **Grip (EDPM)** & **Head (Aluminium)** \\ \hline \(E_{1}=138,585\) MPa & \(E=13\) MPa & \(E=190,000\) MPa \\ \(E_{2}=1066\) MPa & \(\
u=0.5\) & \(\
u=0.29\) \\ \(\
u_{12}=0.26\) & Bulk density \(=1200\) kg/m\({}^{3}\) & Bulk density \(=7900\) kg/m\({}^{3}\) \\ \(G_{12}=5130\) MPa & & \\ \(G_{13}=2950\) MPa & & \\ \(G_{23}=5130\) MPa & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical characteristics of the golf shaft before updating, [34]. The parameters are as follows: \(E\) for Young’s modulus, \(\
u\) for Poisson’s ratio, and \(G\) for shear modulus. Figure 2: (**a**) Discretized model in OROS Modal. The sensor is a piezoelectric accelerometer, while 1, 2, 3, 4 are the impact locations spaced 12.5 cm apart. (**b**) Example of experimental modal analysis. Modal analyses are performed while having the participant exert hand pressure on the grip, at three different levels. The contact pressure is recorded and visualized in real time using two pre-calibrated pressure mats (Grip System, Tekscan, Boston, MA, USA) (see Figure 3). The participant is able to achieve near constant pressure intensity by visualizing the value of mean pressure on both hands, as is depicted in Table 2.  
Modal characteristics, frequencies, and damping are extracted using OROS Modal software (OROS, France) and Table 3 summarizes them. The Broband algorithm is used for identification in the 0-500 Hz range, which is characterized by a vibratory risk probability between 0.025 and 1, according to ISO 5349 standard, Equation (2). This algorithm implements the LSCF (Least Squares Complex Frequency) identification method, which offers higher accuracy when compared to SIMO (Single Input Multiple Outputs) methods on coupled modes, thus leading to improved modal estimations [35]. The modal density over the studied frequency range is low, as only eight modes are identified. Four modes are located in the x-y plane and four others in the x-z plane. It is worth mentioning that the identified modal frequencies are consistent with the works by [PERSON] [29]. These modes are the first four bending modes in the x-y and x-z planes. No torsion mode is highlighted, with the sensor and impact locations being aligned on the longitudinal axis of the golf club.