# 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
The transfer function between the left and right wheel road surface white noise excitation inputs can be expressed as follows:  
\[G_{21}(s)=\frac{W_{2}(s)}{W_{1}(s)}=\frac{a_{0}+a_{1}s+a_{2}s^{2}+\cdots+a_{n}s^{ n}}{b_{0}+b_{1}s+b_{2}s^{2}+\cdots+b_{n}s^{n}}, \tag{18}\]  
\[|H(w)|=|G(jw)|=\left|\frac{a_{0}+a_{1}(jw)+a_{2}(jw)^{2}+\cdots+a_{n}(jw)^{n}}{b _{0}+b_{1}(jw)+b_{2}(jw)^{2}+\cdots+b_{n}(jw)^{n}}\right|, \tag{19}\]  
where \(W_{1}\) and \(W_{2}\) denote the left and right wheel road excitation inputs. As for the coherence function \(coh(\omega)\), it is the result of processing the driving signal collected by the CA141 vehicle on a certain asphalt road on the MTS road simulator of the Changchun Research Institute. The fitting expression of the coherence function is:  
\[coh^{2}(f)= \begin{cases}1-0.45f&f\leq 2 Hz\\ 0.1&f>2 Hz\end{cases} , \tag{20}\]  
Using optimization theory to solve the coefficients of the numerator and denominator of the transfer function, then Equation (18) can be simplified as follows:  
\[G(s)=\frac{W_{2}(s)}{W_{1}(s)}=\frac{a_{0}+a_{1}s+a_{2}s^{2}}{b_{0}+b_{1}s+b_{2 }s^{2}}=\frac{a_{0}s^{-2}+a_{1}s^{-1}+a_{2}}{b_{0}s^{-2}+b_{1}s^{-1}+b_{2}}, \tag{21}\]  
\[ \begin{array}{ll}N(s)&=\frac{1}{b_{2}}W_{1}(s)-\frac{b_{0}}{b_{2}}s^{-2}N(s) -\frac{b_{1}}{b_{2}}s^{-1}N(s)\\ &=\frac{1}{a_{2}}W_{1}(s)-\frac{a_{0}}{a_{2}}s^{-2}N(s)-\frac{a_{1}}{a_{2}}s^{ -1}N(s)\end{array} , \tag{22}\]  
Then, the state equation and output equation of left and right wheel road excitation can be obtained through the Laplace transform and inverse transform. \[ \begin{array}{ll}\dot{X}_{1}=\left[\begin{array}{cc}-\frac{b_{1}}{b_{2}}& -\frac{b_{0}}{b_{2}}\\ 1&0\end{array}\right]X_{1}+\left[\begin{array}{c}\frac{1}{b_{2}}\\ 0\end{array}\right]W_{1}\\ W_{2}=\left[\begin{array}{cc}(a_{1}-a_{2}\frac{b_{1}}{b_{2}})&(a_{0}-a_{2} \frac{b_{0}}{b_{2}})\end{array}\right]X_{1}+\frac{a_{2}}{b_{2}}W_{1}\end{array} , \tag{23}\]  
Next, it is time to consider the correlation between front and rear wheel road excitation. The paper assumes that if the front and rear wheelbases of the vehicle are equal, and the vehicle is moving in a straight line at a constant speed, then the rear wheel input is the lag of the front wheel input for a period of time A, and then the road surface excitation of the front and rear wheels has the following relationship:  
\[W_{3}(t)=W_{1}(t-t_{d}), \tag{24}\]  
where \(W_{1}(t)\) and \(W_{3}(t)\) is the input of front and rear wheel road excitation, \(t_{d}=L/v\), \(L\) is the wheelbase, and \(v\) is the speed of vehicle. For pure time delay systems, we can use the Pade algorithm to approximate the calculation:  
\[G_{31}(s)=\frac{W_{3}(s)}{W_{1}(s)}=\frac{1-\frac{t_{d}s}{2}+P_{2}(t_{d}s)^{2} -P_{3}(t_{d}s)^{3}+\cdots+(-1)^{N}P_{N}(t_{d}s)^{N}}{1+\frac{t_{d}}{2}s+P_{2}(t  
Drawing parallels to Equation (23), we obtain the state and output equations representing the correlation between the vehicle's front and rear wheel tracks as follows:  
\[ \begin{array}{l}\dot{X}_{2}=\left[\begin{array}{cc}-\frac{6}{l_{d}}&-\frac{1 2}{l_{d}^{2}}\\ 1&0\end{array}\right]X_{2}+\left[\begin{array}{c}\frac{12}{l_{d}^{2}}\\ 0\end{array}\right]W_{1}\\ W_{3}=\left[\begin{array}{cc}(1-t_{d}/2)&0\end{array}\right]X_{2}+W_{1}\\ \end{array} \, \tag{27}\]  
According to Equations (16), (23), and (27), the road surface excitation of the right front wheel, left front wheel, and right rear wheel can be calculated; then, the excitation of the left rear wheel can be expressed as follows:  
\[