# The Refined Simulation Methods of Fluctuating Wind and Aerodynamic Forces Relative to Moving Trains
## 5 The Aerodynamic Force Simulation of Moving Trains  
### The Admittance Function of Trains  
To consider the phase characteristic of the admittance function, [PERSON] proposes the following complex admittance function for trains [25]:  
\[\chi(n)=|\chi(n)|\Big{[}cos\phi(n)+\sqrt{-1}sin\phi(n)\Big{]} \tag{44}\]  
\[|\chi(n)|=\frac{1/\kappa}{\Bigg{\{}\left[1-\left(\frac{\overline{n}}{\overline{ n}^{\prime}}\right)^{2}\right]^{2}+\left[2\varsigma\frac{\overline{n}}{ \overline{n}^{\prime}}\right]^{2}\Bigg{\}}^{\frac{1}{2}}} \tag{45}\]  
\[\phi(n)=tan^{-1}\Bigg{\{}\left(2\lambda\frac{\overline{n}}{\overline{n}^{ \prime}}\right)/\Bigg{[}1-\left(\frac{\overline{n}}{\overline{n}^{\prime}} \right)^{2}\Bigg{]}\Bigg{\}} \tag{46}\]  
where \(\chi(n)\) is the complex admittance function; \(|\chi(n)|\) is the amplitude; \(\phi(n)\) denotes the phase; \(\overline{n}=nL/\overline{V}_{R}\) represents the dimensionless frequency; and \(\kappa\), \(\lambda\), and \(\overline{n}^{\prime}\) are the fitting coefficients. Notably, the phase angle in the above equations could be regarded as the phase of the unsteady aerodynamic forces lagging the fluctuating winds. Therefore, a negative sign should be added to the phase angle to indicate the lagging effect.  
[PERSON] summarizes the admittance functions of different types of trains and proposes the following simplified admittance function of trains only considering the amplitude [15]:  
\[|\chi(n)|=\frac{1}{1+\left(\frac{\overline{n}}{\overline{n}^{\prime}}\right)^{ 2}} \tag{47}\]  
where \(\overline{n}^{\prime}=\gamma sin\varphi\) is a dimensionless parameter; \(\gamma=2.0\) for the lateral force; and \(\gamma=2.5\) for the lift force. Comparing Equation (47) with Equation (45), it can be observed that the simplified admittance function of the train is taken as \(\kappa=1.0\) and \(\lambda=2.0\) by default.  
In consideration of only the along-wind fluctuating wind and application of the Fourier transform to the admittance function of Equation (47), [PERSON] provides the time domain expression of the unsteady lateral force of moving trains in the case of \(\varphi_{0}=90\) degrees [15] as follows:  
\[F_{S}(t)=\rho UHLC_{Su}(\varphi)\Bigg{[}1+\frac{1}{2C_{Su}(\varphi)}\frac{dC_ {Su}(\varphi)}{d\varphi}cot(\varphi)\Bigg{]}\int_{0}^{\infty}h_{S}(\tau)u(t- \tau)d\tau \tag{48}\]  
\[h_{S}(\tau)=\overline{h}_{S}(\overline{\tau})\overline{V}_{R}/L \tag{49}\]  
\[\overline{h}_{S}(\overline{\tau})=\left(2\pi\overline{n}^{\prime}\right)^{2} \overline{\tau}e^{-2\pi\overline{n}^{\prime}\overline{\tau}} \tag{50}\]  
where \(h_{S}(\tau)\) is the weighting function of the lateral force. It is obtained by the Fourier transform of Equation (47); \(\overline{\tau}=\tau\overline{V}_{R}/L\) is dimensionless time. The expression of the unsteady lift force of moving trains is similar to Equation (48). The solution is to replace the subscript '\(S\)' in the expression with '\(L\)'. In other words, all the aerodynamic parameters related to the lateral force should be replaced by that of the lift force.  
Setting the train speed as 250 km/h and the wind speed as 35 m/s, the amplitude and the phase of the lateral force admittance function are calculated by Equations (46) and (47), respectively. Then, the complex admittance function of the train expressed by the rational function (see Equation (31)) is obtained by using the least squares method to fit the coefficients. The target and fitting values of the admittance function are shown in Figure 6. The complex admittance function fitted by the rational function is entirely consistent with the target value.