## 3 Analytical Model Conceptual Development and Discussion  
In this section, details on developing the analytical models to assess the simplified dynamics of a four-rubber-wheeled vehicle on the pavement are provided. Most of the geometrophysical models to be discussed are common knowledge in the literature. However, our interest centers on their conceptual setting up, discussion, and interpretation from a geometrophysical perspective, which has been not carried out to the extent of our knowledge. This way, we can search for insights regarding their application in highway design, construction, and operation. As mentioned, the problem consists of analyzing forces acting on a vehicle supported by rubber wheels while in motion over the pavement. For simplicity, highways are considered single slope cambered with a super-elevation angle (Cf. Figure 1, and then Figure 2). The majority of curves anticipated in the design of the layout of a highway are of this type. Furthermore, this is the strategy adopted by AASHTO when proposing the foundations of the analysis of safe speeds in curves in [25]. Furthermore, the problem is considered to be a single particle dynamics one, i.e., the vehicle is represented entirely by its center of gravity. However, that does not imply critical loss of generality, since particular interest centers on the role of super-elevation rate, curvature radius, and other parameters similar to the latter intervening in vehicle behavior and highway operation. Instantaneous Radius of Curvature Model (IRCM) for Highway Vehicle Dynamics: Normal and Tangential Coordinates  
Motion is considered on a plane with two coordinates, one normal or perpendicular to the trajectory and another tangential or parallel. Mathematically, this model could be assimilated to the 2D motion studied via polar coordinates (not cylindrical-polar 3D ones). The radial coordinate would be the one normal to the trajectory, and the angular coordinate the one tangential to it, as depicted in Figure 1. It should be noticed that conceptually speaking, this model, or slight variations on it, is the one most governmental-associated regulators use to provide highway design criteria and is based, as mentioned, on a point-mass dynamics consideration of the problem combined with the criterion of favoring driver comfort [8; 9]. Particularly, authorities such as FRA, AREMA, Amtrak, and OSHA set \(0.1\ g[m/s^{2}]\) and \(0.03\ g[s^{-1}]-[m/s^{3}]\) as maximum acceptable values for radial/normal acceleration and jerk, respectively [13; 25; 26]. The model gives the following result, called the basic curve equation Cf. [25]:  
\[v=\sqrt{\frac{127R(f_{d}+0.01e)}{1.0-0.01 ef_{d}}} \tag{1}\]  
In it, \(v\) is the design speed in \([km/h]\), \(f_{d}\) is the estimated side friction demand factor, \(R\) is the horizontal curve radius in \([m]\), and \(e\) is the super-elevation in \(\%\) ratio. In order to compare this model to others, friction must be taken into account as a free parameter (subjected to applicable physical restrictions) instead of as a parameter according to the driver's comfortable acceleration friction demand term (\(f_{d}\)). On the other hand, in highway design, it is customary to consider the product \(ef_{d}\) invariably as very small, so the term \(1.0-0.01 ef_{d}\simeq 1.0\) (Cf. [25]). This way, the basic curve equation turns into  
\[v=\sqrt{127R(f_{d}+0.01e)} \tag{2}\]  
In the case of this manuscript, the model is discussed from principles in the following lines. The kinematics/dynamics analysis begins by considering the following expressions:  
\[ \begin{split}\overrightarrow{v}&=v\hat{\theta}\\ \overrightarrow{a}&=\overrightarrow{v}=\hat{v} \hat{\theta}+v\hat{\theta}\end{split}  \tag{3}\]  
Resulting in  
\[\overrightarrow{a}=a_{r}\hat{r}+a_{\theta}\hat{\theta}=-\frac{v^{2}}{r}\hat{ r}+\hat{v}\hat{\theta} \tag{4}\]  
Therefore, the velocity direction is tangent to the motion's trajectory and has only that component.