# Wind Speed Spectrum of a Moving Vehicle under Turbulent Crosswinds
## 2 Wind Speed Spectrum Models
### Wind Speed Spectrum of a Moving Vehicle  
Figure 2 shows two points (Points \(P\) and \(P^{\prime}\)) fixed on a vehicle and the corresponding equivalent points. The wind speed is \(\overline{V}\). The moving vehicle travels at a constant speed \(V_{\mathrm{T}}\) along the straight line ON, which passes through the origin of the overall coordinate system O-XYZ. The angle between the line ON and the \(x\)-axis is \(\phi\), which is the wind yaw angle. The overall coordinate system is fixed to the ground and the moving coordinate system N-\(\xi\eta\zeta\) is fixed to the moving vehicle. The \(\eta\)-axis is the moving direction of the vehicle, and the \(\zeta\)-axis is parallel to the \(z\) axis. Points \(P\) and \(P^{\prime}\) are the fixed points on the moving vehicle, whose coordinates in the local coordinate system N-\(\xi\eta\zeta\) are \((0,n,\,\zeta)\) and \((0,n^{\prime},\,\zeta^{\prime})\), respectively. In the overall coordinate system O-XYZ, the position vector of point \(P\) at t and \(t+\tau\) moments could be accessible. The time delay \(\tau\) can be transformed into an equivalent spatial interval \(\tau\overline{V}\) according to [PERSON]'s freezing hypothesis. Thus, for the physical point \(P^{\prime}\) (at time \(t+\tau\)), there are corresponding equivalent points \(P^{\prime}_{e}\) in the frozen turbulent flow field (at time \(t\)). The cross-correlation characteristics between point \(P\) and \(P^{\prime}\) that move with the vehicles could be equivalent to the cross-correlation between point \(P\) and points \(P^{\prime}_{e}\) in the frozen turbulent flow field. In the frozen turbulent flow field, the equivalent interval between point \(P\) and points \(P^{\prime}_{e}\) could be the projected distances on the \(x\) axis, \(y\) axis and \(z\) axis, respectively:  
\[ \begin{array}{l}\Delta x_{e}=(\eta^{\prime}-\eta-V_{T}\tau)\cos\phi- \overline{V}\tau\\ \Delta y_{e}=(\eta^{\prime}-\eta-V_{T}\tau)\sin\phi\\ \Delta z_{e}=\zeta^{\prime}-\zeta\end{array}  \tag{13}\]  
where \(\overline{V}\) is the mean wind speed of the incoming flow. The cross-correlation function between the two points on a moving vehicle could be expressed as follows:  
\[\rho_{uu}(r,r^{\prime},\tau)=c_{u}f(\widetilde{r})+(1-c_{u})g(\widetilde{r}) \tag{14}\]  
Figure 2: Two points and their equivalent points on a moving vehicle. where \(c_{u}=\left(\Delta x_{e}/\Delta r_{e}\right)^{2}\), \(\Delta r_{e}=\left(\Delta x_{e}^{2}+\Delta y_{e}^{2}+\Delta z_{e}^{2}\right)^{1/2}\); \(\widetilde{r}=\alpha\frac{\Delta r_{e}}{L_{u}}\), \(L_{u}/\alpha=a\), \(a=\Gamma(1/3)/\Gamma\), \(\left[\sqrt{\pi}\cdot\Gamma(5/6)\right]\approx 1.339L\), \(\alpha=0.747\) and \(L_{u}=\left[ \begin{array}{c}\left({}^{x}L_{u}\Delta x_{e}/\Delta r_{e} \right)^{2}+\left({}^{y}L_{u}\Delta y_{e}/\Delta r_{e}\right)^{2}\\ +\left({}^{z}L_{u}\Delta z_{e}/\Delta r_{e}\right)^{2}\end{array} \right]^{1/2}\). \({}^{x}L_{u}\). \({}^{y}L_{u}\) and \({}^{z}L_{u}\) is the longitudinal integral scale in \(x\), \(y\) and \(z\) direction. When point \(P\) and point \(P^{\prime}\) coincide, Equation (13) becomes as follows:  
\[ \begin{array}{l}\Delta x_{e}=\left(-V_{T}\tau\right)\cos\phi-\overline{V} \tau\\ \Delta y_{e}=\left(-V_{T}\tau\right)\sin\phi\\ \Delta z_{e}=0\end{array}  \tag{15}\]  
where \(\Delta r_{e}=\tau V_{R}\), \(V_{R}=\left(\overline{V}^{2}+V_{T}^{2}+2\overline{V}V_{T}\cos\phi\right)^{1/2}\). In the turbulent flow field, the turbulent power spectral density function of a single moving point can be obtained from the autocorrelation function by Fourier transform. \[ \begin{array}{l}S_{uu}(f)=4\sigma_{u}^{2}\int\limits_{0}^{\infty}f(\widetilde {r})\cos 2\pi f\tau d\tau\\ =4\sigma_{u}^{2}c_{u}\int\limits_{0}^{\infty}f(\widetilde{r})\cos(2\pi f\tau) d\tau+4\sigma_{u}^{2}(1-c_{u})\int\limits_{0}^{\infty}g(\widetilde{r})\cos(2\pi f \tau)d\tau\end{array}  \tag{16}\]  
where \(\sigma_{u}\) is the root mean square of the fluctuating wind speed in the longitudinal direction.