# Numerical Investigation on Handling Stability of a Heavy Tractor Semi-Trailer under Crosswind
## 3 Computation of the Aerodynamic Characteristics
### Results and Discussion  
Assuming the vehicle drives at a steady velocity, relative inflow angular orientation increases with the growth of crosswind. The crosswind intensities change from zero level to the seventh level of the Beaufort scale, which means the relative inflow angle varies from 0 to 21 degrees. Aerodynamic forces and moments on a road vehicle include drag \(F_{D}\), lift \(F_{L}\), lateral force \(F_{S}\), rolling moment \(M_{R}\), pitching moment \(M_{P}\) and yawing moment \(M_{Y}\). The definitions and positive directions of the forces and moments are shown in Figure 10. These forces and moments refer to the gravity center of the vehicle in the present work. The expressions for aerodynamic coefficients are as follows:  
\[C_{D}=\frac{F_{D}}{\frac{1}{2}\rho V^{2}A} \tag{8}\]  
\[C_{L}=\frac{F_{L}}{\frac{1}{2}\  
By changing the direction and value of the relative speed, the aerodynamic forces and moments can be simulated under different speeds and crosswind strength. The origin of coordinate in this numerical simulation is located at the projection of the center point between the third axis and the fourth axis of the container truck on the ground. The variation of the aerodynamic coefficients with relative inflow angles plotted in Figure 11 reveals that all aerodynamic coefficients trend to increase with these relative inflow angles, and the lateral force, and rolling and yawing moment, are the most sensitive to the relative inflow angle. In addition, the lateral force, yawing moment and rolling moment are the main causes of side-slip and rollover accidents. The lift and pitching moment may reduce the grip of tyres and thus are harmful for driving safety. It is obvious that the yawing moment and rolling moment increased notably with the increase of the crosswind speed. Previous studies [24] have shown that yawing moment made a greater impact on the handling stability, but the specific effects on the performance need to be investigated further in the dynamic simulation.  
To identify the mathematical relationship of aerodynamic characteristics under crosswind, and couple with crosswind stability analysis, a polynomial was used to fit the variation of the aerodynamic coefficient with relative inflow angles. The polynomial models for the curves were given by:  
\[y=\sum_{i=1}^{n+1}p_{i}x^{n+1-i} \tag{14}\]  
in which \(n+1\) is the order of the polynomial, \(n\) is the degree of the polynomial. The order gives the number of coefficients to be fit, and the degree gives the highest power of the predictor variable. In the present work, a two-order polynomial was adopted to get the polynomial model of the aerodynamic coefficients. To evaluate the fitting precision, _R-square_ was introduced. _R-square_ is defined as the ratio of the sum of squares of the regression (SSR) and the total sum of squares (SST); the value of _R-square_ closer to one indicates that the model accounts for a greater proportion of variance. \[SSR=\sum_{i=1}^{n}\omega_{i}(\hat{y_{i}}-\overline{y})^{2} \tag{15}\]  
\[SSR=\sum_{i=1}^{n}\omega_{i}(y_{i}-\overline{y})^{2} \tag{16}\]  
\[R-square=\frac{SSR}{SST} \tag{17}\]  
Figure 10: Aerodynamic forces and moments on the vehicle. The error analysis showed that the _R-square_ for all fitting curves was larger than 0.95, which indicated good fitting accuracy. The fitting curves of aerodynamic coefficients compared with the simulation data were also shown in Figure 11, and the fitting formulas are as follows:  
Aerodynamic drag coefficient:  
\[C_{D} =-0.00051632\beta^{2}+0.03461\beta+0.70049 \tag{18}\]  
Figure 11: The variation curve of the aerodynamic coefficients to the relative inflow angles: (**a**) drag coefficient; (**b**) lateral force coefficient; (**c**) lift coefficient; (**d**) rolling moment coefficient; (**e**) pitching moment coefficient; (**f**) yawing moment coefficient.