## 3 Analytical Model Conceptual Development and Discussion
At any given time, the vehicle's trajectory places it at a point on the circle described in the \(x-z\) plane of Figure 2. Then, it is vital to notice that the trajectory plane is not parallel to the inclined plane of the pavement in Figure 2. A visual scheme of precisely that \"trajectory\" is provided in Figure 3. On the other hand, the validity of the instantaneous dynamic situation portrayed in Figure 2 means that \(v\) in Equation (6) corresponds to the case in which there is no drift between the vehicle's tires and pavement. This means that when solving for \(v\), the system of scalar equations provided by vector Equation (6) provides us with the maximum speed value permitted by a given highway at which the vehicle is in dynamic equilibrium along it with no drift. That is the maximum velocity for safe highway operation. The solution for \(v\) is as follows:  
\[v=\sqrt{\frac{gR(sin\theta+\mu cos\theta)}{cos\theta-\mu sin\theta}}=\sqrt{ \frac{gR(\mu+tan\theta)}{1-\mu tan\theta}} \tag{7}\]  
Figure 3: General layout of the problem and the geometrophysical scheme applied to account for it. In this figure, \(R\) is the parameter of the curvature considered (in this case, a radius of curvature), and \(\vec{v}\) is the vehicle’s velocity. In order for this model application to be meaningful regarding highway design (particularly on the physical and engineering meaningfulness of the curvature radius parameter \(R\)), one has to take into account the following:  
1. A 2D model is used to ascertain a 3D situation. The motion of a vehicle on the highway can not be reduced to normal and tangential components. For the model to make sense, another coordinate along \(\hat{y}\) is used. 2. A radius of the curvature (\(R\)) notion only makes sense when considering sequences of precise segments of circular/curvilinear sections; that is, definite arc segments of the vehicle's trajectory, as shown in Figure 3.
3. Equations (1) and (7) (right) are functionally equivalent since they both are the results of point mass modeling considerations. It can be noticed that Equation (1) is a reparametrization of Equation (7) (right), performed by AASHTO in [25]. Highway Vehicle Dynamics Studied by Cylindrical-Polar Coordinates: Plane of Vehicle Motion Rotated with Respect to Horizontal Plane  
The situation is reviewed with cylindrical-polar coordinates, as shown in Figure 4. This way, one would have two coordinates normal to the motion's trajectory, \(r\) and \(z\), and one tangential (\(\phi\)). As is noticeable in the figure, the \(\phi\) coordinate's unit vector goes into (i.e., enters) the page's plane. This way, \(r,\phi\), and \(z\) set up an orthonormal coordinate system. On the other hand, the main difference between the models in Figures 2 and 4 is that the plane of the circular trajectory in the former is unrotated with respect to the \(x\) axis, while in the latter, it is rotated by an amount of \(\theta\), the highway super-elevation parameter. In other words, the vehicle's trajectory is the same as shown in Figure 3 but with its plane parallel to the pavement's plane, as portrayed in Figure 4. Both possibilities are reasonable since, as said, the dynamic situation is considered to be instantaneous without a critical loss of generality. In practice, the vehicle is neither expected to fulfill the trajectories envisioned by Figures 2 or 4. In this system, the position vector is as follows in terms of Cartesian unit vectors:  
\[\overrightarrow{r}=\textit{rcos}\phi\hat{i}+\textit{rsin}\phi\hat{j}+z\hat{k} \tag{8}\]  
Figure 4: Dynamic situation description in terms of cylindrical-polar coordinates: “normal”, “tangential”, and \(z\) coordinates; that is, (\(r\);\(\phi\);\(z\)). Plane of motion considered rotated with respect to horizontal plane. In this figure, \(\tilde{N}\), \(\tilde{F}\), and \(\tilde{W}\) are the forces acting on the vehicle: normal, friction, and weight, respectively. The angle \(\theta\) is the highway’s considered super-elevation; \(R\) is the radius of curvature (curvature parameter considered).