# Evaluation of Ride Comfort under Vortex-Induced Vibration of Long-Span Bridge
## 2 Modelling of Vehicle and Excitation under VIV
### Four-Wheeled Road Excitation Model
\[G_{q}(\omega)=\frac{G_{q}(f)}{2\pi}=\frac{G_{q}(n_{0})n_{0}^{2}v}{2\pi(\frac{ \omega}{2\pi})^{2}}=\frac{2\pi G_{q}(n_{0})n_{0}^{2}v}{\omega^{2}}, \tag{9}\]  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]0  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]1  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]2  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]3  
The ring space frequency corresponding to the cutoff frequency \(f_{0}\) is \(n_{00}\). Since the road surface spectrum is approximately horizontal in the low frequency range, the cutoff frequency \(f_{0}\) can be introduced into the model:  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]4  
Then, the frequency-response function is as follows:  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]5Finally, the road surface roughness model is obtained, which can be expressed as:  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]6  
where \(w(t)\) denotes the white noise signal with a unilateral power spectral density of 1. In order to verify the correctness of the model, the power spectrum of the generated road excitation curve of class A is compared with the power spectrum in the ISO standard. In the generated road excitation curve, the vehicle speed is 10 m/s, and the simulation time is 165 s, as shown in \[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]7  
In Figure 2 and Table 1, the spectral density of the road random excitation generated through the simulation is basically consistent with the spectral density in the ISO 2631 standard [14], and the corresponding root mean square value is close. Thus, the road random excitation model obtained via simulation is reliable. In order to ensure that the time domain signal is not distorted, the sampling frequency should not be less than twice the maximum frequency of the road spectrum. The maximum spatial frequency of the standard spectrum is 10 m\({}^{-1}\), so the maximum spatial frequency of the simulated road spectrum is 5 m\({}^{-1}\). To emulate genuine driving scenarios on the road and account for the spatiotemporal correlation amongst all four wheels, having previously established the single-wheel road random excitation model, it becomes essential to formulate a quad-wheel road random excitation model. \[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]8  
where \(G_{12}(\omega)\) and \(G_{21}(\omega)\) represent the cross-spectrum between the left and right wheel tracks of the vehicle's front axle; \(coh(\omega)\) is the square root of the coherence function between the left and right wheel tracks. Equation (17) is predicated on the assumptions that the auto-power spectra of the car's left and right wheel tracks are identical, the phase difference is zero, and the statistical properties are consistent. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Road Class** & \(G_{q}(n_{0})/(10^{-6}\text{m}^{3})\) & \(\sigma_{qsim}/(10^{-3}\text{m})^{1}\) & \(\sigma_{qiso}/(10^{-3}\text{m})^{2}\) \\ \hline A & 16 & 3.60 & 3.81 \\ B & 64 & 7.10 & 7.61 \\ C & 256 & 14.30 & 15.23 \\ \hline \hline \end{tabular} \({}^{1}\sigma_{qsim}\) is the geometric mean of simulated road model. \({}^{2}\sigma_{qiso}\) is geometric mean of road model in ISO 8608 [28].  
\end{table} 
Table 1: Comparison between simulated road model and standard road model. Figure 2: Single-wheel road excitation curve. (**a**) Time domain diagram of road excitation curve. (**b**) Frequency domain diagram of road excitation curve.