# Evaluation of Ride Comfort under Vortex-Induced Vibration of Long-Span Bridge
## 2 Modelling of Vehicle and Excitation under VIV
### Input Model of Wheel Excitation under Vortex Vibration of Main Beam  
During the early exploration of vortex-induced vibration, researchers noticed that the observed vibrational behavior closely resembled that of simple harmonic motion. Consequently, they postulated that the forces exerted by vortex-induced phenomena on the structure exhibited characteristics akin to simple harmonic forces [29]. Therefore, for the purpose of this study, simple harmonic vibration will be employed to simulate the vibrational response at any given point along the bridge. Compared with the assumption of simple harmonic vibration, there are some enhanced methods [30] to calculate the VIV response with improved accuracy, including [PERSON]'s model, the polynomial model, the  
Figure 3: Four-wheel road excitation curve under A-class road surface. describing function-based model, [PERSON]'s model and a newly developed aerodynamic envelope model. While these methods can enhance VIV amplitude calculation accuracy, they also decrease efficiency. Simultaneously, this study assumes constant amplitude vortex vibration to investigate how frequency, amplitude, and other factors of vortex-induced vibration affect driving comfort. For constant amplitude and stable vortex-induced vibrations, aerodynamic forces exhibit harmonic stability. Therefore, simplifying constant amplitude vortex-induced vibrations into harmonic vibrations is feasible. This forms the basis for simplifying vortex-induced vibration as simple harmonic vibration in this article. \[f(t)=A\sin(2\pi f_{v}\cdot t+\theta), \tag{29}\]  
where \(A\) is the amplitude of vortex-induced vibration, \(f_{v}\) is the frequency of vortex vibration, and \(\theta\) is the phase angle. The vehicle travels in a straight-line motion at a constant speed along the centerline of the bridge, so the influence of torsional vortex-induced vibrations is minimal. In addition, based on the on-site monitoring data we measured, among the vortex-induced vibration events that have occurred, vertical bending vortex-induced vibrations are predominant, with very few instances of torsional vortex-induced vibrations. Therefore, the primary focus is on vertical bending vortex-induced vibrations, while neglecting torsional vortex-induced vibrations. When the vertical bending vortex-induced vibration with frequency A occurs on the main girder, the corresponding unit mode is S, and the vibration state at position \(x\) at time \(t\) on the bridge is:  
\[y(x,t)=\psi(x)\cdot f(t), \tag{30}\]  
In this study, if the vehicle moves in a straight line at a uniform speed, then the wheel excitation input is:  
\[y(x,t)=\psi(vt)\cdot f(t), \tag{31}\]  
Through utilizing Equation (31) as the input for the vortex vibration excitation of the front wheel of the vehicle, we can determine the input for the vortex vibration excitation of the rear wheel as follows:  
\[y_{B}(x,t)=\psi\bigg{(}v(t-\frac{P}{v})\bigg{)}\cdot f\bigg{(}(t-\frac{P}{v}) \bigg{)}, \tag{32}\]  
The bridge model's effect on the vehicle is transmitted through the simplified mass-spring-damper model representing the vehicle's wheels in this study. Consequently, this can be translated into inputs associated with wheel displacement and velocity excitations. We simplify the vortex-induced vibrations on the bridge as Equation (30). The bridge's impact on the vehicle is described by Equations (31) and (32). Furthermore, in the presence of constant amplitude and stable vortex-induced vibrations, the vortex causes little change the vibration characteristics of the bridge, including frequency and amplitude. The vortex's impact on the bridge is minimal; therefore, this study has neglected the effect of vortices. Taking a certain bridge in China as a case study, we consider the presence of vortex-induced vibrations on the bridge with a frequency of 0.328 Hz and an amplitude of 0.1 m. The mode in Figure 4 is obtained through finite element calculation and verified with reference [28] to ensure the accuracy of the finite element calculation result.