## 4 Detailed Analysis of Tunnel and Bridge Characteristics
### Time Histories and Narrow-Band Spectra of The Bridge Vibrations  
Figure 12 compiles the detailed results for the bridge for five train speeds, the time histories in the left column (Figure 12a,d,g,j,m), the corresponding Fourier spectra in the second column (Figure 12b,e,h,k,n) and the axle sequence spectra in the third column (Figure 12c,f,i,l,o). The axle sequence of the seven-unit test train is calculated according to the formula:  
\[X(\omega)=\left|\sum_{k=1}^{n}A_{k}\ \exp(-i\omega T_{k})\right|\]  
where \(A_{k}\) is a non-dimensional load factor (\(A_{k}=1\) for the passenger cars and \(A_{k}=2\) for the locomotives) and \(T_{k}=x_{k}/v_{T}\) is the delay time of the \(k\)-th axle at position \(x_{k}\) along the train. The lowest pair of peak frequencies can be found for all train speeds, for example at 5 and 6 Hz for 160 km/h (Figure 12n). The highest pair of peak frequencies are clearly visible for example at 13 and 14 Hz for 125 km/h (Figure 12h). The bridge response is strongly influenced by this axle sequence and by the different natural frequencies which are at 3.5 Hz (first bending mode), 7 Hz (first torsional mode), 11 Hz (second bending mode) and 13.5 Hz (second torsional mode), see [2]. The spectrum of the bridge response \(v_{j}(\omega)\) for mode \(j\) can be represented as a product of three spectra:  
\[v_{j}(\omega)=F_{S}W_{j}(\omega)H_{j}(\omega)X(\omega)\]  
where \(X(\omega)\) is the axle sequence spectrum of the train, which is independent of the specific bridge, and \(H_{j}(\omega)\) is the bridge transfer function and \(W_{j}(\omega)\) is the modal force spectrum, which are independent of the specific train, and \(F_{S}=100\) kN. If the resonance frequency \(\omega_{j}\) in \(H_{j}(\omega)\) meets the maximum (minimum) region of the axle sequence spectrum \(X(\omega)\), the corresponding resonance \(v_{j}(\omega_{j})\) is amplified (reduced). The first torsional mode at 7 Hz is clearly raised for the train speed of 100 km/h (Figure 12d) whereas the second bending mode at 11 Hz is dominant for 140 km/h (Figure 12j). At the same train speeds, the other bridge modes are cancelled as they lie in a minimum region of the axle sequence spectrum. As a result, the time histories on the left look almost mono-frequent with 7 Hz for 100 km/h and with 11 Hz for 140 km/h. For the other train speeds, two modes are excited at the same time, but not so strongly: the first and second mode for 80 km/h, the first and fourth mode for 125 km/h and the third and fourth mode for 160 km/h. As indicated in [1], the bridge resonances are also present and probably dominant in the ground vibration. The Amplitudes and The Amplitude Ratio of The Ground Vibrations from the Tunnel and the Surface Line for Normal and High Train Speeds  
In this section, the amplitudes of the tunnel and the surface line are compared directly where the amplitudes for all train speeds are shown in Figure 13a,b for the tunnel line and in Figure 13c,d for the surface line. In addition to the passages of the normal passenger train with low speeds (Figure 13a,c), the high-speed passages of the Inat the same sites are presented in Figure 13b,d. For all four measurement series, two main frequency ranges can be observed. A high-frequency range from 30 to 80 Hz (Figure 13a,c) or 64 to 128 Hz (Figure 13b,d) includes the sleeper-passage component and the vehicle-track resonance which is dominant for the tunnel measurements. A mid-frequency component is strongest for the high-speed train on the surface line which ranges from 20 to 32 Hz, 16 to 25 Hz, 12 to 20 Hz and 10 to 16 Hz depending on the train speed (Figure 13d). This mid-frequency component can also be found for the passenger train with 125 and 160 km/h in Figure 13a,b. It is clearly reduced for the high-speed train in the tunnel (Figure 13c).