## 2 Admittance-Based Immission Prediction
### How to Determine the Transfer Admittance  
In order to obtain the admittance \(Y\), the ground is subject to controlled, measurable forces at several points along an existing or planned railway corridor. These tests are ideally performed on the subgrade or inside tunnel structures, using a vibration generator or an impact tester. The resulting vibrations are measured at the immission points of interest (e.g., inside nearby houses), as shown in Figure 1. By dividing the vibration velocity, \(v\), by the force, \(F\), as shown in Equation (1), the point transfer spectra are obtained as a function of frequency, \(\omega\), usually in one-third octave bands (where \(\omega\) presents the center frequency of the bands). \[Y(\omega)=\frac{v(\omega)}{F(\omega)} \tag{1}\]  
Figure 1: Schematic explanation of the admittance-based immission prediction procedure. Transfer admittances characterize the entire propagation path between the emission and impression points, including the site's geology, the interaction between structure and foundation, and the dynamics due to the supporting structures. Previous experiments by the authors have shown that artificial vibration sources provide a sufficiently accurate simulation of an object's structural dynamics, such as the floor amplification, as observed during train passages even when very short excitation signals such as those from an impulse generator are used. The steady state of a floor slab excited by an artificial source is normally reached very quickly, as shown by Equation (2) [13]. Given the floor's typical damping, \(\zeta\), the degree of transient response (amplitude \(u_{k}\) relative to the maximum amplitude \(u_{max}\) during steady state excitation) after a certain number of load cycles, \(k\), can be calculated as  
\[\frac{|u_{k}|}{u_{max}}=1-e^{-\zeta(\hat{\alpha}k-1)\cdot\frac{\pi}{2}} \tag{2}\]  
A concrete floor with 5% damping and a resonance frequency of 30 Hz would thus reach 69% of its maximum vibration amplitude after only 4 cycles of excitation in its eigenfrequency (0.13 s). This effect can also be demonstrated through in situ measurements. Figure 2 shows the spectral amplification function measured in one-third octave bands of a timber ceiling in a two-story building. The peak represents the ceiling's low eigenfrequency in the 10 Hz band. It was determined by dividing the measured vibration velocity spectra on the top floor by the measured vibration velocity spectra on the foundation. During the tests, a vibration generator near the building was used to emit a 60 s signal into the ground that corresponded to a sweep from 4-80 Hz as well as white noise in the range 1-80 Hz. Using the same sensor setup, a tram and an impact tester (with an impact duration of 0.1 s) were also used as sources at the foundation. Figure 2 furthermore shows that the amplification factors obtained through artificial impacts reach around 70% of the vibration amplitudes obtained during longer excitations using sweeps or rail-bound vehicles that resemble more of a steady state excitation. Since rail vehicles, particularly in the proximity of tracks, correspond to a line source rather than a point source, vibrations resulting from artificial point excitation are subject to significantly stronger damping with increasing distance from the source compared to passing trains. Figure 3 illustrates the decay of the measured vibration velocities due to train traffic and artificial sources as a function of distance from the source for two exemplary one-third octave bands, (a) 16 Hz and (b) 32 Hz. The time weighting applied to the raw vibration signals per octave band in this article is Slow (S, time constant \(\tau=1\) s) unless otherwise stated. The technical specifications of both vibration sources, which were used during the on-site measurements described herein, are given in Table 2.  
Figure 2: Floor amplification spectra using different means of vibration excitation.