# The Refined Simulation Methods of Fluctuating Wind and Aerodynamic Forces Relative to Moving Trains
## 3 The Refined Aerodynamic Model of Moving Trains
### Time-Frequency Transform of Admittance Function  
Equation (23) is a mixed expression in the time and frequency domains, as the admittance function is the frequency domain function. For simplification, it usually either sets the admittance function as 1.0 or takes the amplitude of the admittance function. To accurately simulate the unsteady aerodynamic force acting on the moving train in the time domain, it is necessary to convert the complex admittance function in the frequency domain into a time domain function.  
In the time domain, the unsteady lateral force of the moving train can be expressed by the convolution integral form as follows [21]:  
\[F_{S}(t)=\frac{1}{2}\rho\bar{V}_{R}^{2}HL\int_{-\infty}^{t}\bigg{[}I_{Su}(t- \tau)\frac{u(t)}{\bar{V}_{R}}+I_{S_{0}}(t-\tau)\frac{v(t)}{\bar{V}_{R}}+I_{S_{ 0}}(t-\tau)\frac{w(t)}{\bar{V}_{R}}\bigg{]}\,d\tau \tag{29}\]The Fourier transform is applied to Equation (29), which compares with Equation (23). Thus, the following is obtained:  
\[I_{Su}(i\omega)=C_{Su}(\varphi)\chi_{Su}(i\omega);\ I_{S_{0}}(i\omega)=C_{S_{0}}( \varphi)\chi_{S_{0}}(i\omega);\ I_{Sw}(i\omega)=C_{Sw}(\varphi)\chi_{Sw}(i\omega) \tag{30}\]  
where \(i\omega\) is the complex frequency; \(\omega=2\pi n\) is the circular frequency; \(I_{S_{0}}(i\omega)\) and \(I_{S_{0}}(t-\tau)\) are Fourier transform pairs; and \(\chi_{S_{0}}(i\omega)\) is a complex admittance function.  
Taking the admittance function \(\chi_{Su}(i\omega)\) as an example, it can be rewritten using the rational function approximation method. Thus, the following is obtained:  
\[\chi_{Su}(i\omega)=A_{1}+\sum_{n=1}^{k}\frac{A_{n+1}i\omega}{i\omega+B_{n} \overline{V}_{R}/L} \tag{31}\]  
where \(A_{1}\), \(A_{n}\), and \(B_{n}\) are the frequency-independent fitting coefficients and \(B_{n}>0\); \(k\) is the order of the rational function terms. The greater the \(k\), the higher the fitting accuracy. Equation (31) is a complex function. To determine the coefficients \(A_{1}\), \(A_{n}\), and \(B_{n}\), the least squares method is employed to fit the real and imaginary parts simultaneously.  
Substituting Equation (31) into Equation (30) and applying the inverse Laplace transform to the Equation, the time domain expression of the pulse function is obtained as follows:  
\[I_{Su}(t)=C_{Su}(\varphi)\Bigg{[}A_{1}+\sum_{n=1}^{k}\int_{-\infty}^{t}A_{n+1 }exp\bigg{(}-\frac{B_{n}\overline{V}_{R}}{L}(t-\tau)\bigg{)}\dot{\delta}(t) \Bigg{]} \tag{32}\]  
where \(\dot{\delta}(t)\) is the Dirac function.  
It is easy to obtain the other time domain pulse functions of Equation (30) by the same method. For simplification, only considering the along-wind fluctuating wind, the unsteady lateral force is gained by substituting Equation (32) into Equation (29), which is expressed as follows:  
\[F_{S}(t)=\frac{1}{2}\rho\overline{V}_{R}^{2}HLC_{Su}(\varphi)\Bigg{[}\Bigg{(} A_{1}+\sum_{n=1}^{k}A_{n+1}\Bigg{)}\frac{u(t)}{\overline{V}_{R}}-\sum_{n=1}^{k} \frac{B_{n}\overline{V}_{R}}{L}\psi_{n}(t)\Bigg{]} \tag{33}\]  
where \(\psi_{n}(t)\) is a variable describing the time difference of the aerodynamic force, corresponding to the phase difference of the admittance function. It is expressed by the state equation as follows:  
\[\dot{\psi}_{n}(t)=-\frac{B_{n}\bar{V}_{R}}{L}\psi_{n}(t)+A_{n+1}\frac{u(t)}{ \overline{V}_{R}} \tag{34}\]  
The solution to the above state equation is as follows:  
\[\psi_{n}(t)=exp\bigg{(}-\frac{B_{n}\overline{V}_{R}}{L}\Delta t\bigg{)}\phi_{ n}(t-\Delta t)+\bigg{[}exp\bigg{(}-\frac{B_{n}\overline{V}_{R}}{L}\Delta t \bigg{)}-1\bigg{]}-\frac{L}{B_{n}\overline{V}_{R}}A_{n+1}\frac{u(t)}{\overline {V}_{R}} \tag{35}\]  
The time domain expressions of \(\chi_{S_{0}}(i\omega)\), \(\chi_{Sw}(i\omega)\) and the unsteady lift complex admittance functions are achieved by a similar derivation method. As a result, the unsteady lift expressions of the moving train are obtained. The expression form is consistent with the above results.