# A Novel Method for the Mechanical Testing of Sports Mouthguards
## 3 New Method for Testing Sports Mouthguards
### Methods
#### 3.1.3 Data Acquisition Method  
High-speed images are the basis of the data analysis. A black point was placed on each mouthguard and the hockey ball before the impact test. The location of the marks was tracked throughout the duration of the impact to calculate the displacement of the ball and mouthguard in each frame. The mouthguard was placed on the aluminium fixture and the marks on the ball and mouthguard that were used to track the deformation are shown in Figure 3b. Displacement-time data were obtained for each test by tracking the mark on mouthguard and hockey ball. Tracking was performed automatically with a MATLAB script which took each frame of the impact video and determined the position of the mark of the mouthguard and ball. The position of the mark at the beginning of each impact was set to zero so that the distance became the displacement during the impact. The pixel displacement was then translated to a distance in metres by using the small piece of tape on the aluminium mount as a reference distance of 0.018 m. Considering this, the pixel length amounts to 0.000046 m. The script took the impact videos and outputs a graph of displacement against time for the duration of the impact. Using a spring-damper system, acceleration, force, impulse and energy can be analytically derived for the hockey ball and the mouthguard [21]. The impact can be thought of as a two degrees of freedom spring-mass-damper system. The system is illustrated in Figure 3c, and shows the ball modelled as a spring and damper in parallel attached to the mouthguard which was also modelled as a spring and damper in parallel, attached to a rigid base, representing the aluminium teeth model. Isolating the masses, the force between the ball and the mouthguard during the impact can be obtained using equations of motion for each mass. For simplicity, the force from gravity on the masses was disregarded as it is unlikely to have a substantial effect over such a small distance. The force \(F\) acting on the mouthguard can be calculated considering the equation of motion in the two degree of freedom mass-spring-damper system as  
\[F=F_{m}=M_{m}\dot{x}_{m}=k_{b}(x_{m}-x_{b})+c_{b}(\dot{x}_{m}-\dot{x}_{b})-k_{ b}x_{m}-c_{m}\dot{x}_{m} \tag{1}\]  
where \(k_{i}\) is the stiffness of the spring component and is calculated with  
\[k_{i}=M_{i}\Big{(}\frac{\pi}{\Delta T}\Big{)}^{2}\Bigg{[}1+\Big{(}\frac{\ln( CR_{i})}{\pi}\Big{)}^{2}\Bigg{]} \tag{2}\]and \(c_{i}\) is the damping coefficient, which is obtained through  
\[c_{i}=-\frac{2M_{i}}{(\Delta T)}\ln(CR_{i}) \tag{3}\]  
where the indices \(i=b,m\) stand for the hockey ball and the mouthguard, respectively, and \(M_{i}\) is the mass of the relevant object. The equations utilise the coefficient of restitution, \(CR_{i}\), which was found using the velocity at the end and at the beginning of the impact with  
\[CR_{i}=\frac{|\dot{x}_{i}(\Delta T)|}{|\dot{x}_{i}(0)|} \tag{4}\]  
where \(\Delta T\) is the total contact time of the impact. The values of the constants \(k\) and \(c\) are not the same for each impact test, thus, they are recalculated using the velocity-time data to obtain the coefficient of restitution [22]. To support studying the effect on whether an orodental injury occurs during an impact, different variables were investigated. The first of these variables was the force (\(F\)) that the ball exerts on the mouthguard, calculated using Equation (1). From the force, the impulse \(I\) during the impact time \(t\) can be obtained through  
\[I=\int Fdt \tag{5}\]  
Finally, using the force and displacement (\(x\)), the energy experienced by the mouthguard (\(W\)) is found using  
\[W=\int Fdx \tag{6}\]  
To verify the method used and the results obtained are correct, the concept of energy conservation was used. It is known that, at the start of the impact, all the energy is contained in the kinetic energy of the drop mass.