# 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  
Typically, road roughness is defined as the variation in height of the road surface with respect to a reference plane along the road's length. It exhibits characteristics such as randomness, stability, ergodicity, and a zero mean value. This roughness is often modeled as a random process following a Gaussian probability distribution. In spectral analysis, the power spectrum with statistical characteristics is commonly employed. Currently, there are two methods for assessing road surface roughness. The first involves utilizing specialized instruments and continuous measurements to acquire the power spectral density of the road surface. However, this approach is labor-intensive and time-consuming. The second method relies on pre-existing power spectral density data of road surface roughness, which are commonly employed to simulate road surface unevenness. According to ISO 8608 [28], the road surface power spectral density is fitted with the following formula:  
\[G_{q}(n)=G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-W}, \tag{1}\]  
where \(G_{q}(n_{0})\) is unevenness index, which is classified into eight levels, from A to H; \(W\) is waviness; \(n\) is the spatial frequency, indicating the number of wavelengths contained in each meter of length; \(n_{0}\) is the reference spatial frequency with a value of 0.1. Waviness \(W=2\) denotes the case when the change in the road vertical displacement per unit distance travelled is the frequency index, which indicates the frequency structure of the road surface power spectral density. While the frequency index \(W=2\), the velocity power spectrum can be expressed as:  
\[G_{\hat{q}}(n)=(2\pi n)^{2}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2}=(2 \pi n_{0})^{2}G_{q}(n_{0}), \tag{2}\]  
Equation (2) eliminates the spatial frequency; then, the speed power spectral density is only related to the road roughness coefficient but has nothing to do with frequency, which is the same as the white noise power spectrum characteristic. Therefore, the space velocity power spectrum can be simulated with white noise, or the displacement power spectrum can be simulated with integrated white noise, which is also the method used in this paper to construct the road roughness model. Figure 1: Model of the two-axle vehicle. (**a**) Elevation view. (**b**) Side view. The relationship between time frequency \(f\), space frequency \(n\), and vehicle speed \(v\) is as follows:  
\[f=nv, \tag{3}\]  
and, then, the relationship between time frequency bandwidth and corresponding space frequency bandwidth is:  
\[\Delta f=\Delta n\cdot v, \tag{4}\]  
and if the power spectral density is the \"power\" (mean square value) in the unit frequency band, then the spatial power spectral density is:  
\[G_{q}(n)=\lim_{\Delta n\to 0}\frac{\sigma_{q\sim\Delta n}^{2}}{\Delta n}. \tag{5}\]  
\(\sigma_{q\sim\Delta n}^{2}\) is the power contained in the frequency band a of the road surface power spectral density. No matter how the speed changes, the power is always the same. So, the time power spectrum and velocity power spectrum can be expressed as:  
\[G_{q}(f)=\lim_{\Delta f\to 0}\frac{\sigma_{q\sim\Delta n}^{2}}{\Delta f}=\lim_{ \Delta f\to 0}\frac{\sigma_{q\sim\Delta n}^{2}}{v\Delta n}==\frac{1}{v}G_{q}(n_{0}) \bigg{(}\frac{n}{n_{0}}\bigg{)}^{-2}=G_{q}(n_{0})n_{0}^{2}\frac{v}{f^{2}}, \tag{6}\]  
\[G_{q}(f)=(2\pi f)^{2}G_{q}(f)=4\pi^{2}G_{q}(n_{0})n_{0}^{2}v, \tag{7}\]  
According to the random vibration theory, the road surface roughness model based on the filtered white noise method can be derived as follows:  
\[G_{q}(\omega)=\left|H(j\omega)\right|^{2}G_{w}(\omega), \tag{8}\]  
where \(G_{q}(\omega)\) is the displacement power spectrum of road surface roughness; \(G_{w}(\omega)=G_{w}(f)/2\pi\) is the power spectrum of white noise; \(G_{w}(f)=1\).