# The Refined Simulation Methods of Fluctuating Wind and Aerodynamic Forces Relative to Moving Trains
## 4 The Fluctuating Wind Simulation of Moving Trains  
### The Wind Environment Parameters  
The Kaimal spectrum, recommended by the International Electrotechnical Commission [22], describes the wind spectrums of longitudinal, lateral, and vertical by the same functional form as follows:  
\[\frac{fS_{\xi}(f)}{\sigma_{\xi}^{2}}=\frac{4f_{L\xi}}{\left(1+6f_{L\xi}\right)^ {5/3}} \tag{36}\]  
where \(f_{L\xi}=fL_{\xi}/U\) is the reduced frequency; \(u\), \(v\), \(w\) correspond to the longitudinal, lateral, and vertical fluctuating wind, respectively; and \(\sigma_{\xi}\) and \(L_{\xi}\) are the standard deviation and the turbulence integral scale of the fluctuating wind, respectively.  
The inhomogeneously distributed mean wind and mean square deviation of the fluctuating wind are described by the following equations:  
\[U(x)=5+30 exp\left[-(0.0025x-2.5)^{2}\right] \tag{37}\]  
\[\sigma_{\xi}(x)=I_{\xi}U(x) \tag{38}\]  
where \(U(x)\) is the mean wind in location \(x\); \(\sigma_{\xi}(x)\) and \(I_{\xi}\) are the mean square error and turbulence intensity of the \(\xi\)-component fluctuating wind in location \(x\), respectively; \(x\) is spatial position; and unit is meter.  
Assuming the wind environment is inhomogeneously in the range of \(0<x\leq 2000\) m with the parameters defined in Reference [23]. For simplification, taking the along-wind fluctuating wind as an example, the turbulence integral scale and turbulence intensity in the inhomogeneous region are set as constants, i.e., \(L_{u}=151\) m and \(I_{u}=0.17\). The inhomogeneous wind field parameters are shown in Figure 2.  
A simple modulation function is used to characterize the non-stationary wind field in this study by the following Equation [24]:  
\[A(t)=\alpha_{0}t^{\beta_{0}}e^{-\lambda t} \tag{39}\]  
\[\alpha_{0}=\frac{\lambda\beta_{0}}{\beta_{0}^{\beta_{0}}}e^{\beta_{0}};\lambda =\frac{\beta_{0}}{t_{max}}(\alpha_{0}>0,\beta_{0}\geq 0,\lambda\geq 0) \tag{40}\]  
where \(A(t)\) is the modulation function, which is time-varying; \(t\) is time, and the unit is second; \(t_{max}\) is the time as the modulation function value reaches 1.0; \(\beta_{0}\) is a non-negative integer; and \(\lambda\) and \(\alpha_{0}\) are non-negative integers calculated by \(\beta_{0}\) and \(t_{max}\).  
Figure 2: Characteristic parameters of inhomogeneous wind environments: (**a**) mean wind speed; (**b**) mean square deviation of the fluctuating wind.  
The modulation function is independent of the frequency, assuming that the modulation function of the mean wind speed is the same as that of the fluctuating wind. In this study, the parameters of the modulation function are taken as \(t_{max}=30\) s and \(\beta_{0}=2.0\). Thus, the modulation function is calculated and depicted in Figure 3.  
The above wind environment parameters are substituted into the derived equations to calculate the fluctuating wind spectrum of the moving train under the four kinds of wind environments. The parameter values for different wind environments are listed in Table 1. It notes that the mean wind speed in the coherence function under the non-stationary wind environments should regard the influence of the modulation function. For simplicity purposes, only the along-wind fluctuating wind is analyzed in this paper.