## 5 Calculated and Measured Results at a Specific Site
### Wheelset Accelerations from Axle-Box Measurements and Corresponding Irregularities and Forces  
First, the measured axle-box accelerations are shown in Figure 8a for five train speeds. The spectra are almost constant with frequency. The only exception is for the lowest speed of 63 km/h where an increase at 32 Hz can be related to the sleeper passage excitation. The accelerations are strongly increasing with train speed, and an amplitude factor of 10 is observed for train speeds between 63 and 160 km/h. Four different axle boxes were  
Figure 7: Train passage with \(v_{T}=160\) km/h over an irregular track and a soil with \(v_{S}=200\) m/s, irregularity \(q=\) (**a**) 3%, (**b**) 10%, (**c**) 30%, (**d**–f**) 50%, track stiffness (**a**–d**) \(EI\) ballast track, (**e**) 10 \(EI\), (**f**) 100 \(EI\) slab track, one-third of the octave band spectrum at distances \(x=\square\) 3, \(\Circle\) 5, \(\Delta\) 10 m.  
measured and all four measurement points give very similar results. Only the out of roundness at 8, 10, 12, or 16 Hz is specific for each wheel.  
The measured wheelset accelerations \(a_{W}\) are divided by \(\omega^{2}\) to yield displacements (Figure 8b). These wheel displacements are the same as the effective irregularities \(s\) in this frequency range; see [31]. The irregularities are strongly decreasing with frequency. The accelerations \(a_{W}\) multiplied by the wheelset mass of \(m_{W}=1500\) kg are almost identical to the excitation forces \(F\) acting on the track (Figure 8c), with small deviations at low frequencies due to bogie contributions [31]). Therefore, the forces are almost constant with frequency and strongly increasing with train speed in the same way as the accelerations.  
Corresponding results are calculated with the transfer functions of Section 3.2 and the following irregularities \(s_{S}\) of the subgroup and \(s_{W}\) the combined unevenness of wheel and rail:  
\[\begin{split} s_{S}=s_{S1}(\lambda/\lambda_{1})^{2}\text{ with }s_{S1}=0.02\text{ mm at }\lambda_{1}=1.2\text{ m}\\ s_{W}=s_{W1}(\lambda/\lambda_{1})\text{ with }s_{W1}=0.01\text{ mm at } \lambda_{1}=1.2\text{ m}\end{split} \tag{26}\]  
The stiffness variation is assumed to be constant at \(k_{1}/k_{0}=20\%\), and the stiffness variation by the sleeper passage is included as an effective irregularity of \(s_{D}=0.012\text{ mm}\). The resulting effective irregularity at rail level is shown in Figure 8e. For the train speed of 100 km/h (curve \(\Delta\)), the following components can be observed. At low frequencies, the geometric track support irregularities dominate, but also strongly decrease with \(s_{R}\sim\)\(f^{-2}\). The transfer function \(H_{ST}\) for the geometrical track support irregularity has a cut-off at about 8 Hz and, therefore, the effective irregularity \(s^{*}\) from the support stiffness variation dominates at about 12 Hz. This component is initially constant with frequency and the cut-off of the corresponding transfer function \(H_{KT}\) is at about 20 Hz, from where the stiffness component also decreases until the wheel and track unevenness \(s_{W}\) is dominant at 25 Hz.  
In the next step, the decreasing irregularities have to be multiplied by the increasing vehicle-track transfer functions (6) and (7). The resulting axle-load spectra in Figure 8f are almost constant with frequency. The amplitudes increase with increasing train speed as \(F\sim\)\(\
u\)\(\tau^{2}\). The dynamic load is about \(F\approx\) 0.4 kN for 100 km/h and reaches \(F\approx\) 1.0 kN for 160 km/h.  
The measurements and calculations of the train-track interaction are in good agreement in all details: the irregularities are strongly decreasing with frequency, and the accelerations and forces are constant in frequency and strongly increasing with train speed \(A\sim\)\(\
u\)\(\tau^{2}\), and have the same amplitude of 1.0 kN for 160 km/h.