## 3 Frequency Domain Methods of Calculation
### Superposition of Axle Pulses from Passing Trains
#### 3.5.1 Regular Track Support  
The passage of the static train loads is represented by a number of impulses at a line of discrete excitation points \(y_{j}\). The passage of a train yields impulses \(F_{0}dt\) on the track due to the static axle load \(F_{0}\). The length \(dt\) of the impulse follows from the distance \(dy\) of the discrete excitation points and the train speed \(v_{\mathsf{T}}\) as \(dt=dy/v_{\mathsf{T}}\). The impulse on the rail is a Dirac function \(\delta(t)\). The bending stiffness of the track filters out the high frequency content of this infinitely sharp impulse. This is expressed by the filter function (Figure 3a):  
\[H_{F}(\omega=\xi v_{\mathsf{T}})=\frac{F_{\mathsf{S}}}{F_{\mathsf{R}}}= \mathbf{e_{1}}^{\mathsf{T}}\Big{(}\mathbf{EI}\xi^{4}+\mathbf{K_{TS0}}\Big{)} ^{-1}\mathbf{e_{n}}k_{s}=\frac{s_{R}}{s_{S}}=H_{sT}(\omega=\xi v_{\mathsf{T}}) \tag{16}\]  
Figure 3: Spectra for the calculation of a standard ballast track with varying track support stiffness, (**a**) filter function of the track force, (**b**) axle-sequence spectrum of two axles of a bogie, train speed \(\times\) 63, + 80, \(\Delta\) 100, \(\Circle\) 125, and \(\square\) 160 km/h. The filter function for the force is the same as the filter function (12) for the geometric track support irregularities due to reciprocity. In the case of a standard ballast track, the filter function is reduced to:  
\[H_{F}(\omega=\xi v_{T})=\frac{k_{S}}{EI\xi^{4}+k_{S}}=\frac{1}{1+\left(\omega/ \omega_{0}\right)^{4}} \tag{17}\]  
where the cut-off frequency \(\omega_{0}\) is ruled by the bending stiffness \(EI\) of the track, the support stiffness \(k_{S}\) of the ballast and soil, and the train speed \(v_{T}\) according to:  
\[\omega_{0}\sim\sqrt[4]{\frac{k_{S}}{EI}}v_{T} \tag{18}\]  
For a ballast track and a train speed of 160 km/h, a frequency of \(f_{0}=13\) Hz is used. The impulses at different places \(y_{j}\) are applied at different times \(t_{j}=y_{j}/v_{T}\). The delay time \(t_{j}\) is included in the frequency domain as the factor:  
\[\exp\bigl{(}-\mathrm{i}\omega t_{j}\bigr{)} \tag{19}\]  
Then the response at a distance \(x\) from the track to the sequence of impulses due to a single axle is given as the spectral density:  
\[u(x,\omega)=H_{F}(\omega)\ F_{0}\mathrm{dt}\sum_{j=-n/2}^{+n/2}H_{S}\bigl{(}x, y_{j},\omega\bigr{)}\exp\bigl{(}-\mathrm{i}\omega t_{j}\bigr{)} \tag{20}\]  
for the displacements and:  
\[v(x,\omega)=\mathrm{i}\omega u(x,f) \tag{21}\]  
for the particle velocities. \(H_{S}(x,y_{j},\omega)\) is the transfer function of the soil and is calculated by an infinite integration in the frequency-wavenumber domain (see [35] and Appendix A) and summed for the \(n\)+1 excitation points \(y_{j}\). The response of a whole train follows by the multiplication of (21) with the axle-sequence spectrum (Figure 3b):  
\[X(f)=\left|\sum_{k=1}^{2n_{B}}{F_{k}}^{*}\exp(-i\omega T_{k})\right|\approx \sqrt{n_{B}}X_{A}(f)=\sqrt{n_{B}}2\biggl{|}\cos\biggl{(}\pi f\frac{2I_{B}}{v_ {T}}\biggr{)}\biggr{|} \tag{22}\]  
where \(n_{B}\) (=20) is the number of bogies and \(2I_{b}\) (=2.5 m) is the axle distance in the bogie. The spectrum \(X\)(\(f\)) of all axles with arrival times \(T_{k}\) and relative static loads \(F_{k}\)* (=1) can be approximated with the axle-distance spectrum \(X_{A}\)(\(f\)) of a single bogie in the frequency range of interest (8-30 Hz). The axle-sequence spectrum \(X_{A}\)(\(f\)) has two characteristic minima, which are shifted in frequency with train speed. The results are presented as one-third of octave band spectra that follow from the spectral densities (20), (21) as:  
\[v(x,f_{T})=2\sqrt{\sum_{l=l_{u}}^{l_{u}}\frac{v^{2}(x,f_{l})}{n_{T}}\sqrt{ dfdf_{T}}} \tag{23}\]  
where \(f_{T}\) is the centre frequency of the one-third of octave band, and \(f_{l}\) are the \(n_{T}\) frequencies of the spectral density within one-third of the octave band (\(l_{u}<l<l_{0}\)).