# Modeling a Wave Energy Harvester for Supplying Data Buoys
## 2 Materials and Methods
### Harvester Structure  
The considered energy harvester is located inside a floating buoy and moves together with the buoy due to the pitching effect of the sea waves. The harvester is based on a cylinder-shaped permanent magnet with the north-south magnetic axis coinciding with the rotation axis. As a result of the buoy's oscillations, the cylinder rolls back and forth on a track equipped with coils on both sides, as shown in Figure 1a. As an example, only one pair of coils is shown in Figure 1a, but the track can be equipped with multiple pairs of coils. The movement of the cylinder near the coils causes a time-varying magnetic field through the coils. This variation over time generates an electromotive force that induces a current flowing through a load connected at the coil terminals. In this way, the energy of the sea waves is harvested and converted to electric energy delivered to the load. In order to ensure that the cylinder does not come off the track, the cylinder has flanges near the ends. In particular, the magnetic cylinder is composed of five elementary cylinders alignedon the same axis and glued together, as shown in Figure 1b. Two of the five cylinders have the base radius \(R\) slightly smaller than the base radius \(R_{e}\) of the remaining cylinders in order to create the flanges. The length of the external cylinders is equal to \(L_{1}\), that of the cylinders with reduced radius is \(L_{2}\) and that of the central cylinder is \(L_{3}\). The considered energy harvester can be modeled by the system in Figure 2, where the cylinder rolls on the track of length \(L=x_{1}-x_{2}\), inclines with an angle \(\alpha\), and is subject to the force of gravity and friction. Moreover, due to the effect of the electromagnetic coupling with the coils placed on the sides of the track, the cylinder is also subject to a dissipative force. The [PERSON]'s law for the center of mass in the \(x\) direction can be written as  
\[m\ \ddot{x}=-mg\text{sin}\,\alpha+f_{s}-\mu(x)\ \dot{x}-\theta(x)\ i \tag{1}\]  
where \(m\) is the mass of the rolling cylinder, \(\alpha\) the tilt angle of the inclined plane, and \(f_{s}\) the static friction force, which is sufficient to keep the cylinder moving without slipping. The single and double dots over a quantity denote the first and second time derivative, respectively. \(\mu(x)\ \dot{x}\) models the braking force of the track ends and is different from zero only when the cylinder reaches one of the track ends. \(\theta(x)\ i\) describes the electromagnetic force due to the interaction between the magnetic field of the cylinder and the current, \(i\), flowing into the coils. \(\theta(x)\) is the electromechanical coupling coefficient, which is different from zero only near the coils. Moreover, according to the [PERSON]'s second law for rotation, it results  
\[I_{cm}\ \ddot{\theta}=-f_{s}\ R \tag{2}\]  
where \(I_{cm}\) is the moment of inertia of the cylinder, \(R\) the radius of the cylinder and \(\theta\) the rotation angle of the cylinder. By taking into account that \(\ddot{x}=R\ddot{\theta}\), Equation (2) leads to  
\[f_{s}=-\frac{I_{cm}}{R^{2}}\ \ddot{x} \tag{3}\]  
and thus Equation (1) reduces to  
\[k\ m\ \ddot{x}\ =-m\ g\text{sin}\,\alpha-\mu(x)\ \dot{x}-\theta(x)\ i \tag{4}\]  
with \(k=1+I_{cm}/\left(m\ R^{2}\right)\). For a cylinder with flanges for the track, whose structure is shown in Figure 1b, the moment of inertia is  
\[I_{cm}=\frac{1}{2}m\ R_{e}^{2}\left(1-\frac{2\ L_{2}\ \pi\ R^{2}}{2\ L_{1}\ \pi\ R_{e} ^{2}+2\ L_{2}\ \pi\ R^{2}+L_{3}\ \pi\ R_{e}^{2}}.\frac{R_{e}^{2}-R^{2}}{R_{e}^{2}}\right) \tag{5}\]  
Figure 1: Scheme of the energy harvester based on a cylinder rolling on an inclined track. **(a)** The magnetic cylinder rolling on a track with a pair of coils on the sides. **(b)** The structure of the magnetic cylinder with flanges. In order to fully describe the harvester behavior, electrical equations should be added to Equation (4).