# Design, Modeling, and Simulation of a Wing Sail Land Yacht
## 2 Materials and Methods
### Wing Sail Design  
The method adopted to define the physical characteristics of the wing sail was suggested by Reference [29]. The angle of attack \(\alpha\) of a wing sail corresponds to the angle between the wing chord and the apparent wind direction. Non-null \(\alpha\) values generate perpendicular and parallel forces relative to the apparent wind, known as lift (\(L\)) and drag (\(D\)), and their magnitude depends on the corresponding lift (\(C_{L}\)) and drag (\(C_{D}\)) coefficients [30]. The selected wing profile was a NACA63(3)-018 symmetrical airfoil as it provides a high lift to drag ratio (\(73\)) for an \(\alpha=6.75^{\circ}\), as well as keeps a low drag coefficient (0.02) at \(\alpha=10^{\circ}\), with a Reynolds number of 500,000 [31; 32]. The 2D profile of the NACA63(3)-018 airfoil is presented in Figure 1b. The lift and drag forces produced in a sail depend on its shape and dimensions. In this case, the area of the wing sail surface needs to generate sufficient lifting force to move the land yacht. The lift force is presented in Equation (1), where \(\rho\) is the air density, \(V\) is the apparent wind velocity, \(A\) is the airfoil lateral area (chord \(\times\) wingspan), and \(C_{L}\) is the airfoil lift coefficient. \[L=\frac{1}{2}\rho V^{2}AC_{L}. \tag{1}\]  
The drag force is given by Equation (2), where \(\rho\) is the air density, \(V\) is the apparent wind velocity, \(A\) is the airfoil lateral area (chord \(\times\) wingspan), and \(C_{D}\) is the airfoil drag coefficient. \[D=\frac{1}{2}\rho V^{2}AC_{D}. \tag{2}\]  
Figure 2 displays the lift \(L\) and drag \(D\) components of the generated aerodynamic force \(R\), the angle between the vehicle's heading and the apparent wind \(\theta\), the wing sail angle of attack \(\alpha\), and the driving or thrust \(T\) force and lateral or heel \(H\) force applied to the vehicle. Figure 1: Land yacht design. The force required to put the land yacht in motion is given by [PERSON]'s second law presented in Equation (3), where \(m\) and \(a\) are the land yacht mass and acceleration, respectively. In this case, the mass of the vehicle, comprising the chassis, wing set and maximum payload, is assumed to be \(20.0\,\mathrm{kg}\). \[F=ma. \tag{3}\]  
Replacing the force with the lift expression presented in Equation (1) and adding the friction force, results in Equation (4), where \(\mu\) is the friction coefficient, and \(g\) is the gravitation acceleration. \[\frac{1}{2}\rho V^{2}AC_{L}=ma+\mu mg. \tag{4}\]  
Given that the lift force is perpendicular to the apparent wind direction and the drag force is parallel to the apparent wind direction, the thrust force required to move the land yacht forward \(T\), which is parallel to the vehicle's \(x\) axis, is related to the lift and drag through \(\theta\), the angle of the apparent wind. This thrust force can, then, be expressed by Equation (5). \[T=L\sin\theta-D\cos\theta. \tag{5}\]  
From the combination of Equations (4) and (5) results in Equation (6), which represents the equilibrium of forces involved in the forward motion of the land yacht. \[\frac{1}{2}\rho V^{2}AC_{L}(L\sin\theta-D\cos\theta)=ma+\mu mg. \tag{6}\]  
Identically, the lateral or heel force applied to the vehicle's \(y\) axis is expressed by Equation (7), where \(\theta\) represents the apparent wind angle.