## 3 Results
### The Effects of Time Delay in Drivers' Steering Inputs on the Vehicle's Reaction to the Crosswind  
In this part of the study, the researchers explored the impacts of delay times in the drivers' steering inputs on the dynamics of vehicles exposed to crosswind. The lateral displacement, \(y\), yaw angle, \(\psi\), and roll angle, \(\phi\), results of three drivers with different driving skills are presented in Figure 10 at three different delay times, i.e., \(t_{\delta,delay}\) = 0 s, 0.5 s, and 1.0 s. An ideal steering response time of \(t_{\delta,delay}\) = 0 s was employed to assess the effects of different delay times in the steering inputs. As the delay time in the drivers' steering inputs increased, the largest deviations in the yaw angles, roll angles, and lateral displacements of the vehicle due to the crosswind increased. For example, at their initial peaks, the maximum values of the lateral displacements for Driver 1 and Driver 3 at the delay time of \(t_{\delta,delay}\) = 1.0 s were 1.39 and 1.56 times higher than their respective maximum lateral displacement results at \(t_{\delta,delay}\) = 0 s. On the other hand, the results obtained at \(t_{\delta,delay}\) = 0 s and 0.5 s were close to each other. The yaw angle experienced the most significant differences from the delays in drivers' steering inputs because it had the largest percentage variations between the results obtained at \(t_{\delta,delay}\) = 0 s and \(t_{\delta,delay}\) = 1.0 s, as presented in Figure 11. The maximum absolute relative percent difference in the results of the drivers between \(t_{\delta,delay}\) = 0 s and 1.0 s was at least more than 29%, whereas the highest relative percentage variances in the results of the drivers between \(t_{\delta,delay}\) = 0 s and 0.5 s were less than 30%. Consequently, the impact of delay time on the vehicle's response to crosswind disturbances became crucial as the delay time increased from \(t_{\delta,delay}\) = 0.5 s to \(t_{\delta,delay}\) = 1.0 s.  
Figure 9: (**a**) Comparisons of lateral accelerations, \(a_{y}\), and yaw velocity, \(\dot{\psi}\), of the vehicle (solid lines) with drivers’ steering inputs, \(\delta\), (dotted lines). (**b**) Cross-correlations between the lateral acceleration and the steering input by drivers, \(R_{a_{y},\delta}\), and yaw velocity and steering inputs by drivers, \(R_{\dot{\psi}\delta}\). Figure 12 depicts the driver's steering inputs in the instances of three delay times. The magnitudes of the steering inputs of drivers at their initial peaks heightened with both the delay in the steering inputs of drivers and the level of driving skills. For example, the  
Figure 11: The highest percentage variations in the (**a**) lateral displacement, (**b**) yaw angle, and (**c**) roll angle results of different drivers between delay times of \(t_{S,delay}=0\) s and \(t_{S,delay}=0.5\) s and \(1.0\) s at their first peaks. Figure 10: Lateral displacement, \(y\), yaw angle, \(\psi\), and roll angle, \(\phi\), results of the vehicle obtained in the cases of three delay times, i.e., \(t_{S,delay}=0\) s, \(0.5\) s, and \(1.0\) s, in the steering inputs of three drivers with different driving skills.