## 5 Unsteady Aerodynamic Forces Prediction with a Complete Turbulence
### Prediction Model
}\theta \tag{22}\]  
The following simplification is utilized for the sake of the easy writing:  
\[C_{F}(\beta,\theta)\approx C_{F}\left(\overline{\beta}\right)+C_{F}{}^{\prime }\left(\overline{\beta}\right)\left(\beta^{\prime}+\beta^{\prime\prime} \right)+C_{F}{}^{\prime}(0)\theta \tag{23}\]  
Due to the small longitudinal and lateral turbulence, the fluctuation of a resultant yaw angle can be approximated:  
\[\beta^{\prime}+\beta^{\prime\prime}\approx\sin\left(\beta-\overline{\beta} \right)=\sin\beta\cos\overline{\beta}-\cos\beta\sin\overline{\beta} \tag{24}\]  
Referring to Figure 18, it is easy to be found:  
\[\left\{ \begin{array}{l}\sin\overline{\beta}=\frac{U\sin\alpha}{V_{R}}\\ \cos\overline{\beta}=\frac{V\pm U\cos\alpha}{V_{R}}\\ \sin\beta=\frac{(U+U)\sin\alpha-v\cos\alpha}{V_{R}+V_{R}}\\ \cos\beta=\frac{V+(U+U)\cos\alpha+v\sin\alpha}{V_{R}+V_{R}}\end{array} \right. \tag{25}\]  
Equations (24) and (25) lead to  
\[\beta^{\prime}+\beta^{\prime\prime}\approx\frac{u\sin(\alpha-\overline{\beta} )-v\cos(\alpha-\overline{\beta})}{V_{R}} \tag{26}\]  
It is clear that the lateral turbulence contributes to a negative effect on the resultant fluctuating yaw angle. The small attack angle \(\theta\) is probably calculated by  
\[\theta\approx\sin\theta\approx\frac{w}{V_{R}} \tag{27}\]  
Substituting Equations (18), (23), (26) and (27) into Equation (21), the unsteady aerodynamic force of a moving train in a complete turbulence field can be written as:  
\[ \begin{array}{l}F=\overline{F}+F^{\prime}=\overline{F}+{F_{u}}^{\prime}+{F_ {v}}^{\prime}+{F_{w}}^{\prime}\\ \approx\frac{1}{2}\rho AC_{F}\left(\overline{\beta}\right)\overline{V_{R}}^{ 2}+\frac{1}{2}\rho AC_{u}\overline{V_{R}}u+\frac{1}{2}\rho AC_{v}\overline{V_ {R}}v+\frac{1}{2}\rho AC_{u}\overline{V_{R}}w\end{array}  \tag{28}\]where  
\[\left\{ \begin{array}{l}C_{u}=2C_{F}\big{(}\overline{\beta}\big{)}\cos\big{(} \alpha-\overline{\beta}\big{)}+C_{F}{}^{\prime}\big{(}\overline{\beta}\big{)} \sin\big{(}\alpha-\overline{\beta}\big{)}\\ C_{v}=2C_{F}\big{(}\overline{\beta}\big{)}\sin\big{(}\alpha-\overline{\beta} \big{)}-C_{F}{}^{\prime}\big{(}\overline{\beta}\big{)}\cos\big{(}\alpha- \overline{\beta}\big{)}\\ C_{w}=C_{F}{}^{\prime}(0)\end{array} \right. \tag{29}\]  
\(F_{u}{}^{\prime}\), \(F_{v}{}^{\prime}\), and \(F_{w}{}^{\prime}\) are the fluctuating value caused by the longitudinal, lateral, and vertical turbulence, respectively. When only the longitudinal turbulence \(u(t)\) is included and crosswind is perpendicular to the traveling line (\(\alpha=90^{\circ}\)), the fluctuating value with respect to unsteady force can be simplified as follows:  
\[F^{\prime}=\frac{1}{2}\rho A\big{[}2C_{F}\big{(}\overline{\beta}\big{)}\sin \big{(}\overline{\beta}\big{)}+C_{F}{}^{\prime}\big{(}\overline{\beta}\big{)} \cos\big{(}\overline{\beta}\big{)}\big{]}\overline{V_{R}}u \tag{30}\]  
which is an equivalent form proposed by [PERSON] [34]. For a case where the longitudinal turbulence \(u(t)\) and vertical turbulence \(w(t)\) are concerned, and assume \(\beta^{\prime}=0\) and \(\alpha=90^{\circ}\), it can lead to:  
\[F^{\prime}=\frac{1}{2}\rho A\overline{V_{R}}\big{[}2C_{F}\big{(}\overline{ \beta}\big{)}\sin\big{(}\overline{\beta}\big{)}u+C_{F}{}^{\prime}(0)w\big{]} \tag{31}\]  
which is similar to the results by [PERSON] et al. [40]. While the fluctuating turbulence \(u(t)\) and \(v(t)\) are taken into account, we can obtain the same results by [PERSON] et al. [36]. It should be mentioned that force fluctuation through the use of the quasi-steady theory fully follows the velocity fluctuation. In reality, this assertion does not hold completely since the shape and suspension characteristics of vehicles are able to filter out some small scales and high frequency turbulences [34; 48; 49]. To allow for this the aerodynamic weighting function \(h_{F}(\tau)\) approach is used to relate the instantaneous variations from force to wind velocity in the time domain.