## 2 Characterization of Rail Unevenness Profile
### The Generation of Synthetic Unevenness Profiles  
As stated before, railway operators routinely collect rail unevenness data for maintenance purposes. Although these datasets are large and site-dependent, statistical analysis has shown that they can be approximated numerically using PSD functions. Therefore,  
Figure 2: Power spectral density function of vertical unevenness profile: (**a**) rail 1; (**b**) rail 2.  
using spectral representation theory [10], sinusoidal functions can be superimposed to generate a rail profile:  
\[\Delta u(x)=\sum_{j=1}^{N}\delta u_{j}e^{i\frac{2\pi}{\gamma_{j}}x} \tag{8}\]  
where \(\lambda_{j}\) is the wavelength within the range of interest and \(N\) is the total number of harmonics considered. \(\delta u_{j}\) is a complex number representing the amplitude of the corresponding harmonic, which is given by the following:  
\[\delta u_{j}=A_{j}e^{-i\theta_{j}} \tag{9}\]  
In Equation (9), \(\theta_{j}\) is the phase angle, taken as a random variable with uniform distribution in the interval [0, 2\(\pi\)], and \(A_{j}\) is the amplitude of the corresponding harmonic (\(\Delta k_{1}\) is the wavenumber spacing):  
\[A_{j}=\sqrt{2S(k_{1j})\Delta k_{1}} \tag{10}\]  
The experience accumulated by the various railway administrations has enabled the systematization of different power spectral density functions, typically defined as a function of the track quality. Particular emphasis is placed on the proposals of the Federal Railroad Administration (FRA) [11], those proposed by the French Railway Administration (SNCF), and those resulting from studies endorsed by [PERSON] [12] or [PERSON] [13]. The FRA's proposal [11] defines the following power spectral density function (PSD):  
\[S(k_{1})=\frac{10^{-7}Ak_{3}^{2}(k_{1}^{2}+k_{2}^{2})}{k_{1}^{4}(k_{1}^{2}+k_ {3}^{2})} \tag{11}\]  
where \(k_{1}\) is the wavenumber, and \(k_{2}\) and \(k_{3}\) are constants which take the values of 0.1464 rad/m and 0.8246 rad/m, respectively. Parameter \(A\) corresponds to a roughness constant and it is a function of the track geometric quality, with the entity proposing six distinct classes, as shown in Table 1.  
Alternatively, studies elaborated by [PERSON] and [PERSON] [12] (referenced by [PERSON] [14]) suggest the use of the analytical expression defined by ISO 8608 [15] to define the power spectral density function of road pavement irregularity profiles, suitably adapted to meet railway conditions. The expression proposed by these authors is formulated as:  
\[S(k_{1})=B\bigg{(}\frac{k_{1}}{k_{B}}\bigg{)}^{-w} \tag{12}\]  
here, k\({}_{B}\) takes the value of 1 rad/m and the exponent \(w\) assumes values in the range of 3 to 4 (a value of 3.5 is assumed along the present paper). The parameter \(B\) varies between 5 \(\times\) 10\({}^{-7}\) m\({}^{3}\)/rad for poor quality tracks to 1 \(\times\) 10\({}^{-9}\) m\({}^{3}\)/rad for very good quality tracks. According to studies conducted by SNCF, the PSD function defined by Equation (13) offers a suitable approximation to measured results for wavelengths ranging from 2 m to 40 m:  
\[S(k_{1})=\frac{10^{-6}C}{\left(1+\frac{k_{1}}{k_{C}}\right)^{3}} \tag{13}\]  
Regarding the parameters involved, \(C\) reflects the quality of the track, taking the value 160 for tracks of good quality and 550 otherwise, while \(k_{C}\) is equal to 0.307 rad/m. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Class** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(A\) [m\({}^{-1}\)] & 16.72 & 9.53 & 5.29 & 2.96 & 1.69 & 1.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Track geometry quality: roughness constant A [11].