# Feasibility of a Quasi-Static Approach in Assessing Side-Wind Hazards for Running Vehicles
## 2 Quasi-Static Approach for the Evaluation of Tire Forces
### Equations for Vehicles with Two Axles  
In order to establish a two-axle, quasi-static vehicle model, the basic concept proposed by [PERSON] [3] and [PERSON] and [PERSON] [10] was adopted. Six equilibrium equations for x, y, and z axes were derived to consider the longitudinal, lateral, and vertical tire forces, as shown in Figure 1. After adding the weight of the wheel, we obtained six equilibrium equations as follows.  
\[i_{1}q(F_{z1}+F_{z3})+i_{2}q(F_{z2}+F_{z4})-f_{R}\sum_{i\;=\;1}^{4}F_{zi}\;=\; F_{D} \tag{1}\]  
\[\sum_{i\;=\;1}^{4}F_{zi}\;=\;\Big{(}m_{s}+m_{f}+m_{r}\Big{)}g-F_{L} \tag{2}\]  
\[F_{yf}+F_{yr}\;\;=\;\;F_{S} \tag{3}\]  
\[\frac{c}{2}(F_{z3}+F_{z4}-F_{z1}-F_{z2})\;=\;M_{R}+h_{c}\cdot F_{S} \tag{4}\]\[aF_{yf}-bF_{yr}+\frac{c}{2}(i_{1}q-f_{R})(F_{z3}-F_{z1})+\frac{c}{2}(i_{2}q-f_{R}) (F_{z4}-F_{z2})\ =\ M_{Y} \tag{5}\] \[b(F_{z2}+F_{z4})-a(F_{z1}+F_{z3})\ \ =\ \ \Big{(}-m_{f}\cdot a+m_{r}\cdot b \Big{)}g+M_{P}+h_{c}\cdot F_{D} \tag{6}\]  
Equations (1)-(6) use the following designations: \(a\) and \(b\) are the distance of the front and rear axles from the center of gravity; \(c\) is the track width; \(h_{c}\) is the vertical distance between the ground and the center of gravity; \(g\) is the acceleration of gravity (=9.81 m/s\({}^{2}\)); and, \(m_{s}\), \(m_{f}\), and \(m_{r}\) are the masses of the sprung vehicle body, the front axle, and the rear axle, respectively. There are six wind-induced forces and moments: Drag force, \(F_{D}\); lift force, \(F_{L}\); side force, \(F_{S}\); rolling moment, \(M_{R}\); yawing moment, \(M_{Y}\); and, pitching moment, \(M_{P}\). \(F_{zi}\) (\(i\) = 1, 2, 3, 4) are the vertical reaction forces of tires, and \(F_{yi}\) (\(i\) = \(f\) and \(r\)) are the lateral friction force of front and rear axles. The symbols \(i_{1}\) and \(i_{2}\) are variables that can only be 0 or 1 according to whether the corresponding axle is driven or not. Traction parameter \(q\) and rolling resistance coefficient \(f_{R}\) are the ratio of vertical reaction forces to tire traction forces \(T_{i}\) (\(i\) = 1, 2, 3, 4) and to rolling resistance forces \(F_{xi}\) (\(i\) = 1, 2, 3, 4), respectively [3]. \(q\) is an unknown value which is estimated from Equations (1)-(6) whereas \(f_{R}\) is a constant value which is determined by vehicle speed \(v\) (unit: km/h) according to Equation (7) [6].  
\[f_{R}\ =\ 0.0041-0.0000256\times v \tag{7}\]  
A compatibility condition whereby all wheel centers constantly lie on a plane was also introduced to solve the above seven unknowns. When the stiffness of all suspension was assumed to be equal, the compatibility condition was represented as shown by Equation (8) [3].  
\[F_{z1}-F_{z3}+F_{z4}-F_{z2}\ =\ 0 \tag{8}\]  
Since the number of unknowns in the equations were equal, the tire forces of a vehicle could be determined for given levels of wind-induced forces and moments. The quasi-static approach suggested by [PERSON] [3] and [PERSON] and [PERSON] [10], however, provided a negative contact force for the tires when one of the wheels had lost contact with the ground, which is unrealistic. Therefore, we established two modifications in order to calculate the tire forces under situations where any wheel loses contact with the ground. First, the vertical tire force, \(F_{zi}\), of the wheel being lifted up was fixed at 0 during the calculation. Second, the compatibility condition illustrated by Equation (8) was no longer effective. The tire forces could then be evaluated by considering the remaining three tires.