Mitigation of Railway-Induced Ground Vibration by Soft Support Elements and a Higher Bending Stiffness of the Track

Abstract

The mitigation of train-induced ground vibrations by track solutions is investigated by calculations and measurements. The calculation by a wavenumber domain method includes the correct vehicle–track interaction and the correct track–soil interaction. Some theoretical results for elastic elements and an increased bending stiffness of the track are presented where the force transfer of the track and the vehicle–track interaction are calculated for the high-frequency dynamic mitigation, and the force distribution along the track is calculated for the low-frequency mitigation which is due to the smoother impulses from the passing static loads. Measurement results for the ground vibration near isolated and un-isolated tracks are given for several under-sleeper pads, for under-ballast mats, and for several under-ballast plates and ballast troughs. The elastic elements yield a resonance frequency of the vehicle–track–soil system and a high-frequency reduction of the dynamic axle loads which depends mainly on the softness of the pads or mats and which can be improved by a higher sleeper mass. In addition, all troughs and most of the soft elements show a low-frequency reduction which is attributed to the scattered impulses of the static axle loads. Besides this main contribution of the article, the problem of a soft reference section on a different soil is discussed and recommendations for better ground vibration measurements of mitigation effects are given.

1. Introduction

Passing trains generate ground vibration which propagate through the soil and excite neighbouring buildings. To avoid annoyance of the inhabitants or to protect sensitive equipment it is sometimes necessary to provide mitigation measures against these vibrations. Mitigation measures can be built in the building, in the transmission path through the soil or at the emission. The present contribution focusses on the latter possibility where mainly measures at the railway track are considered. The usual concept of mitigation is to add elastic elements to the track and to increase the mass and the bending stiffness of the track. These measures should reduce the forces that are acting on the soil.

This topic can be followed back to the 1980s [1] when the Federal Institute of Material Research and Testing (BAM) participated in a German research programme [2] about noise and vibration abutment for urban railways where many track elements had been tested in situ. Standards exist for how to verify the characteristics and the effects of track elements. For example [3], the European research project “Railway Induced Vibration Abutment Solutions” (RIVAS) had one special focus on under-sleeper pads under heavy sleepers [4]. Besides laboratory and in situ tests, detailed calculations were performed. The track–soil interaction has been modelled by finite-element boundary-elements, by wavenumber domain methods, and by combined so-called two-and-a-half-dimensional methods [5]. Similar methods have also been used for soft rail pads [6] and under-ballast mats [7,8]. An overview of the numerical methods can be found in [9]. Many contributions to track solutions have been published in the series of International Workshops on Railway Noise and Vibration (IWRN), for example [10].

Several state-of-the-art reports summarize knowledge on mitigation of railway vibration [11,12,13,14,15,16]. Some early references already mention a low-frequency reduction that can sometimes be found in measurements [12,13,17] and the possible mitigation by a higher bending stiffness [11]. The present article does not aim at another state-of-the art report. The well-known principle of reducing the dynamic axle loads at high frequencies will be demonstrated, but in addition to the state of the art, the main focus is now laid on the low-frequency reduction of the impulses from the passage of static axle loads. A theoretical explanation of this low-frequency mitigation and a model for its evaluation is given, and existing measurements are further studied to find out the rules, for example, for the influence of the speed and the axle sequence of the train which have been overseen in previous works on mitigation measures.

The frequency–wavenumber method is explained in Section 2. The method takes into account the correct vehicle–track interaction and the correct track–soil interaction. Some theoretical results for elastic elements and an increased bending stiffness of the track are presented in Section 3 where the force transfer of the track and of the vehicle–track interaction are calculated for the high-frequency (dynamic) mitigation, and the force distribution along the track is calculated for the low-frequency (static) mitigation. Measurement results for the ground vibration near isolated and un-isolated tracks are presented in Section 4 for several under-sleeper pads, for under-ballast mats, and for several under-ballast plates and ballast troughs. The interpretation of some experimental observations is given in Section 5 and some recommendations for measurements are derived. The conclusions in Section 6 give the main mitigation effects that were found in the measurements and demonstrate the originality of this article as the new low-frequency reduction of scattered axle impulses and the new method of its calculation by the force distribution of the isolated track.

2. Methods of Calculation

The track model is built of one-dimensional support chains, consisting of rail pads, sleepers, under-sleeper pads, the ballast, and an under-ballast mat, which are homogenized along the track and are coupled by the two rails (and the under-ballast plate/the trough) to a two-dimensional track model. The track model is coupled with a three-dimensional continuous soil and with a multi-body vehicle model where a rigid wheelset mass is usually sufficient. The small deformations across the track and the periodic rail support on sleepers are neglected. This model has been validated with measurements in [18] where the continuous soil results in a load distribution along the track which is more realistic than the Winkler soil model. The behaviour of this model is described in frequency–wavenumber domain as a chain of transfer matrices T_i which relate the state z = (F,u)^T (F force, u displacement) of the bottom and the top of each support element as

z₁ = T₁ z₂

(1)

or

$$\left[\begin{array}{c}{F}_1\\ u_1\end{array}\right]=\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]\left[\begin{array}{c}{F}_2\\ u_2\end{array}\right],$$

(2)

(see the list of symbols in

Table 1. Symbols.

EI	bending stiffness of the track beams
f	frequency
F	force
FS	force on the soil
FT	force on the track
FV	force on the vehicle
FS’	force on the soil per track length
hB	height of the ballast
HTS	force transfer of the track–soil system
HVT	force transfer of the vehicle–track interaction
HVS	total force transfer of the vehicle–track–soil system
k,K	stiffness
K	stiffness matrix
kB	static ballast stiffness
kM″	stiffness per area of a ballast mat
kR	rail pad stiffness
kU	stiffness of the sleeper pad
KT	dynamic stiffness of the track
KV	dynamic stiffness of the vehicle
KTS	stiffness matrix of the track–soil system
mW	mass of the wheelset
NTS	inverse stiffness matrix of the track–soil system
T	transfer matrices
u	displacement
uT	track displacement
u	displacement vector
uT	displacement vector of the track
v	velocity, particle velocity of the soil
vB	wave velocity of the ballast
vI	ground vibration velocity for the isolated track
vU	ground vibration velocity for the un-isolated track
x	coordinate across the track, distance from the track
y	coordinate along the track
z	state vector (F,u)
ω	circular frequency, 2πf
ξ	wavenumber

). The forces point to the element and F₁ and u₁ have the same direction. A spring element (stiffness k, for example, a sleeper pad) yields

$${\mathbf{T}}_{\mathbf{F}}=\left[\begin{array}{cc}1& 0\\ 1/k& 1\end{array}\right]$$

(3)

a mass element (mass m, for example, the sleeper mass)

$${\mathbf{T}}_{\mathbf{M}}=\left[\begin{array}{cc}1& -m{\omega }^2\\ 0& 1\end{array}\right]$$

(4)

and a (rail or slab) beam with bending stiffness EI

$${\mathbf{T}}_{\mathbf{R}}=\left[\begin{array}{cc}1& EI{\xi }^4-m{\omega }^2\\ 0& 1\end{array}\right]$$

(5)

The transfer matrix of the ballast as a one-dimensional continuum reads

$${\mathbf{T}}_{\mathbf{B}}=\left[\begin{array}{cc}\cos ({\xi }_B{h}_B)& -\sin ({\xi }_B{h}_B)k_B{\xi }_B{h}_B\\ \sin ({\xi }_B{h}_B)/k_B{\xi }_B{h}_B& \cos ({\xi }_B{h}_B)\end{array}\right]$$

(6)

with the static stiffness k_B, the height h_B, and the wavenumber ξ_B = ω/v_B of the longitudinal wave velocity v_B of the ballast. The soil is represented as

$${\mathbf{T}}_{\mathbf{S}}=\left[\begin{array}{cc}1& K_S(\omega ,\xi )\\ 0& 1\end{array}\right]$$

(7)

where the dynamic stiffness K_S(ω,ξ) of the layered soil is calculated from an integration in wavenumber domain; see the details in [18].

The transfer function of a support section is achieved by multiplying the transfer functions of all support elements (for example, sleeper pad, ballast, and ballast mat; see Figure 1) as

z₁ = T₁ T₂ T₃ z₃ = T z₃.

(8)

The transfer matrix T is transformed into the stiffness matrix K_TS of the track–soil system as

$$\left[\begin{array}{c}{F}_1\\ -F_2\end{array}\right]=\frac{1}{T_{21}}\left[\begin{array}{cc}{T}_{11}& -1\\ -1& T_{22}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]={\mathbf{K}}_{\mathbf{T}\mathbf{S}}\mathbf{u}$$

(9)

(note that the sign definition is different for F₂, namely, F₂ is in the same direction as u₂ for the stiffness matrix).

Finally, the behaviour of this track–soil system is described by the following equation in the frequency–wavenumber domain

$${\mathbf{K}}_{\mathrm{T}\mathrm{S}}\left(\xi ,\omega \right){\mathbf{u}}_{\mathrm{T}}\left(\xi ,\omega \right)={\mathbf{F}}_{\mathrm{T}}\left(\omega \right)=\left[\begin{array}{c}{F}_T\left(\omega \right)\\ 0\end{array}\right]$$

(10)

where ξ is the wavenumber along the track axis. This equation is solved as

$${\mathbf{u}}_{\mathrm{T}}\left(\xi ,\omega \right)={{\mathbf{K}}_{\mathrm{T}\mathrm{S}}}^{-1}\left(\xi ,\omega \right){\mathbf{F}}_{\mathrm{T}}\left(\omega \right)={\mathbf{N}}_{\mathrm{T}\mathrm{S}}\left(\xi ,\omega \right){\mathbf{F}}_{\mathrm{T}}\left(\omega \right)$$

(11)

for the displacements u_T(ξ,ω), and the force on the soil F_S(ξ,ω) can be calculated as

$$F_S(\xi ,\omega )=K_S(\xi ,\omega )u_1(\xi ,\omega )$$

(12)

The soil force distribution along the track (as well as any displacement distribution) can then be achieved by the Fourier integral

$${F_S}^{\prime }(y,\omega )=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{F}_S(\xi ,\omega )\exp (i\xi y)d\xi$$

(13)

The total force that acts on the track–soil interface is calculated as the integral over the infinite track length

$$F_S(\omega )={\int }_{-\infty }^{+\infty }{F_S}^{\prime }(y,\omega )dy={F_S}^{\prime }(\xi =0,\omega )=F_T(\omega )H_{TS}(\omega )$$

(14)

This soil force can easily be obtained as the transformed integrand at ξ = 0 without any integration. The total force transfer H_TS(ω) = F_S/F_T of the track is one part of the dynamic mitigation effect.

The displacements of the track under the axle load can be calculated by the Fourier transformation of the first element of the inverse stiffness matrix N_TS as

$$u_T(\omega )=\frac{F_T(\omega )}{2\pi }{\int }_{-\infty }^{+\infty }{N}_{\mathrm{T}\mathrm{S},11}\left(\xi ,\omega \right)d\xi =\frac{F_T(\omega )}{K_T(\omega )}$$

(15)

where the dynamic stiffness K_T(ω) of the track has been introduced. The track model is combined with a vehicle model. A single rigid wheel mass m_W is used throughout this paper. The dynamic stiffness K_V(ω) = −m_Wω² of the vehicle is introduced into the vehicle–track interaction analysis with the unsprung or wheelset mass m_W. A force F_V acting on the vehicle yields a force F_T acting on the track giving the vehicle–track force transfer

$$H_{VT}(\omega )=\frac{F_T}{F_V}(\omega )=\frac{K_T(\omega )}{K_T(\omega )+K_V(\omega )}$$

(16)

which is the second part of the dynamic mitigation effect. The total force transfer H_VS(ω) is the product of the force transfer H_TS(ω) of the track–soil system and the force transfer H_VT(ω) of the vehicle–track interaction

$$H_{VS}(\omega )=\frac{F_S}{F_V}(\omega )=\frac{F_T}{F_V}(\omega )\frac{F_S}{F_T}(\omega )=\frac{K_T(\omega )}{K_T(\omega )+K_V(\omega )}\frac{F_S}{F_T}(\omega )=H_{VT}(\omega )H_{TS}(\omega ).$$

(17)

The “static” mitigation effect is calculated with the force distribution on the soil in Equation (13) by transforming it in a force distribution in time and, finally, in the corresponding Fourier spectrum where the latter are dependent on the train speed.

3. Calculated Results for Mitigation Measures at the Track

Two different effects of the mitigation of train-induced ground vibrations are presented in the following sections. The most common effect is the reduction of the dynamic axle loads which is presented at first. The second and newly presented effect is the reduction of the axle impulses from the passage of the static axle loads.

3.1. The Reduction of the Dynamic Axle Loads

Usually, the mitigation due to a special track is analysed for the dynamic effects. The dynamic loads are generated by the passage over irregularities of the train or the track. The dynamic interaction between the track and the vehicle yield a wheelset-track eigenfrequency and a force reduction for higher frequencies. Elastic track elements such as rail pads, sleeper pads and ballast mats shift this vehicle–track eigenfrequency to lower frequencies so that the frequency range of reduction is widened to lower frequencies. The effects for the force acting on the ground are presented for typical stiffness values of the elastic track elements in Figure 2; see the standard parameters in

Table 2. Standard parameters of the vehicle–track–soil systems (parameters with * varied).

Wheelset
Mass	1500 kg
Rail
Bending stiffness	6.4 × 106 Nm2
Mass per length	60 kg/m
Sleeper
Width	2.6 m
Distance	0.6 m
Mass	338 kg
Ballast
Longitudinal wave velocity	400 m/s
Static stiffness	5 × 108 N/m
Height	0.35 m
Damping ratio	10%
Ballast plate
Young’s modulus	3 × 1010 N/m2
Mass densitiy	2500 kg/m3
Width	3 m
Height	0.3 m *
Soil
Shear wave velocity	200 m/s *
Mass density	2000 kg/m3
Poisson’s ratio	0.33
Damping ratio	5%

. The minimum stiffness per sleeper is quite similar for all elastic elements due to safety regulations. The lower the elastic element placed in the track, the more mass is activated. Therefore, the lowest eigenfrequencies and the best reduction effects can be found for the under-ballast mats where eigenfrequencies of 20 Hz can be established (Figure 2c). The second-best reduction can be achieved by under-sleeper pads with eigenfrequencies starting from 25 Hz and maximum reductions below 0.1 at 100 Hz (Figure 2b). The eigenfrequencies that can be established with elastic rail pads are normally quite high (between 40 and 80 Hz) so that train-induced ground vibrations up to 100 Hz are hardly affected (Figure 2a). A reduction of one-third is the best that can be reached by the softest rail pads with a stiffness of k_R = 20 kN/mm. The dynamic reduction effects of elastic track elements lie generally at high frequencies, mainly above 50 Hz, maybe above 30 Hz.

The bending stiffness of an under-ballast plate has no influence on the reduction of the dynamic axle loads, as can be seen by Equation (14): the bending stiffness EIξ⁴ for ξ = 0 is zero. The influence on the dynamic track stiffness (15) and the force transfer of the vehicle–track interaction (16) is weak, as well as the influence of the mass of the under-ballast plate.

3.2. The Reduction of the Axle Impulses from Static Axle Loads

The passage of the static loads is shown in Figure 3 as one-third octave band spectra for two wheelsets of a bogie with an axle distance of 2.5 m. The typical shape of this spectrum, which can be often found in measurements, has three to four-thirds of octaves of increased amplitudes (here between 6 and 12 Hz for 100 km/h train speed) and characteristic minima can be found for 5 and 16 Hz. Soft track elements have an influence on the load distribution under the axle load and, therefore, can reduce the axle impulse spectra (Figure 3a–c). The reduction depends on the stiffness of the elastic elements which are nearly the same for rail pads, sleeper pads, and ballast mats, whereas the different masses that are supported by the elastic element have no influence on the static distribution of the load. The reduction of the impulse spectra is even more pronounced for the under-ballast plates. It can be seen that in the most important frequency range up to 16 Hz, the effect of the plates is already present (Figure 3d). At 12 Hz, the plate of h_P = 0.3 m reduces the force amplitude to one-fourth compared to the standard ballast track. The effect of ballast plates is even stronger if the sub-soil is softer (Figure 3e). The ballast plate is often combined with a ballast mat. Figure 3f shows the mitigation effect of the mat and the plate alone and finally the mitigation effect of the combination of mat and plate. The reduction of the maximum amplitudes reaches down to one-tenth.

4. Measurement Results

The measurements of BAM were performed with calibrated geophones (velocity transducers, Geospace, Houston, TX, USA) and a measurement system (SCADAS, DIFA, Breda, The Netherlands) with 72 channels of sample and hold amplifiers and adjustable analogue filters. Similar measurements have been performed with similar results at the Regensburg site (Heiland&Mistler, Bochum, Germany), at the Lengnau site (Ziegler Consultants, Zürich, Switzerland), and at the Altheim site (Müller BBM, Planegg, Germany).

4.1. Measurements Related to Under-Sleeper Pads

Three measurement campaigns about under-sleeper pads were performed during the RIVAS project [4,14,19]. At first, a measurement site near Regensburg with six track sections equipped with three different under-sleeper pads were investigated where the nominal static stiffnesses were 0.10, 0.15 and 0.22 N/mm³. The ground vibration at the reference section without under-sleeper pad are shown in Figure 4a for an ICE train. For all distances to the track, the dominant frequencies can be found around 16 Hz and at 64 Hz. Similar results are presented in Figure 4b for one of the sections with under-sleeper pads. The maximum around 16 Hz is still present (somewhat reduced at the far field), but the peak at 64 Hz (the sleeper-passage component) is almost completely missing. The ground vibration reduction v_I/v_U between the isolated and the un-isolated track (Figure 4c) clearly presents this effect and some smaller effects at high frequencies.

The second measurements were performed near the villages Lengnau and Pieterlen in Switzerland where three different under-sleeper pads with a nominal stiffness of 0.12 N/mm³ have been installed at a straight and a curved track. The ground vibration near the un-isolated reference track and one of the isolated tracks are shown in Figure 5. The reference track (Figure 5a) shows a dominating low-frequency part at 12 to 16 Hz which is clearly reduced by the under-sleeper pads (Figure 5b). The high frequency maximum at 50 to 60 Hz, therefore, is more dominant for the isolated track, and the vehicle–track resonance may be expected here. The higher frequencies are then also reduced by the under-sleeper pads (Figure 5c).

The measurements of all sections at the Lengnau site have been evaluated for an average reduction in [19] (Figure 6, triangle markers). These results are presented together with three more measurements of under-sleeper pads at Timelkam in Austria from [20] and at Sempach in Switzerland [21]. The high-frequency reduction is between 10 and 15 dB in fair agreement with the softness of the under-sleeper pads which have static stiffnesses of 0.1, 0.15, 0.17, and 0.3 N/mm³. All these measurements show also a low-frequency reduction of 5 to 15 dB.

Finally, newly designed tracks have been installed in Herne at Eiffage Rail. Wide and heavy sleepers from Rail One have been equipped with very soft under-sleeper pads with a static stiffness of 0.03, 0.06, and 0.10 N/mm³. The ten different tracks were excited by a special fixed (non-moving) shaker and an additional static load. The results for the reduction of the ground vibration are shown in Figure 7a. Figure 7b shows calculations from BAM where the nominal track parameters were adjusted to the measurements. The high-frequency reduction was clearly found for all isolated solutions. The reduction starts at a frequency which depends on the stiffness of the under-sleeper pad and lies between 32 and 64 Hz. As the shaker simulates the mass and load distribution of a bogie, this resonance frequency is also expected for a running train. The softest under-sleeper pads yield the lowest resonance frequency and the strongest reduction of up to 20 dB. Because of the fixed (non-moving) excitation, there are no impulses from the passing static loads, and no low-frequency reduction can be found. The shaker measurements are better for the high-frequency resonance and reduction effects than the train measurements in Figure 4c, Figure 5c and Figure 6 which show weaker resonance amplifications and weaker reductions that start at frequencies between 50 and 100 Hz in agreement with the calculations in Figure 2b with similar pad stiffnesses. Under-sleeper pads yield a high-frequency reduction which depends primarily on the stiffness of the rail pad and secondarily on the mass of the sleeper.

4.2. Measurements Related to under Ballast Mats

Measurements at Raron in Switzerland [23] with and without under-ballast mats (static stiffness 0.06 N/mm³) are here analysed for the reduction effects at different train speeds. Figure 8a shows the ground vibration of the reference track without under-ballast mat. For a distance of 8 m, a maximum can be found at 20 Hz for 175 km/h, at 16 Hz for 160 km/h, at 12 Hz for 120 km/h and at 8 Hz for 80 km/h. This maximum is reduced by the under-ballast mat (Figure 8b) and a reduction of up to 10 dB is evaluated in Figure 8c. An amplification around 40 Hz can be observed in Figure 8b,c, and the frequencies above 64 Hz are reduced by about 10 dB. Other measurements in Switzerland (Raron 2, Nüziders and Rothrist) with softer under-ballast mats (static stiffness between 0.025 and 0.06 N/mm³) show a lower resonance frequency and a stronger ground vibration reduction; see Annex A2 in [14].

4.3. Measurements Related to Under-Ballast Plates and Troughs

The idea of a mitigation of ground vibration by a concrete plate under the ballast was first realised near Altheim in Germany. BAM measured the ground vibration along a measuring axis at the reference track (Figure 9a) and at the track with a 0.3 m thick concrete plate (Figure 9b). Some irregularities can be seen at distances between 10 and 25 m on the embankment. The regular measuring points on the original soil at longer distances show a maximum region around 12 Hz which is clearly reduced by the under-ballast plate; see also Figure 9c.

The same mitigation principle has been used for several solutions with concrete troughs which are often combined with under-ballast mats. The measurement results from [24] are further analysed for the reduction effect at different sites, for different trains and different train speeds. Figure 10 shows the ground vibrations at 8 m distance with and without concrete trough. Generally, the high-frequency components between 32 and 64 Hz dominate. At low frequencies, however, a speed-dependent maximum can be observed, most clearly at 10 Hz for the freight train. The slow regional (“Doppelstock”) train in Baden has also a maximum at 10 Hz. The faster passenger trains at the Ulm site show all the same results, a maximum at 12 to 16 Hz, and the fastest train ICE at Sinzheim has the maximum at 25 Hz. Each of these low-frequency maxima is clearly reduced by the trough (Figure 10b), whereas the high-frequency maximum is more pronounced for the trough solutions due to the under-ballast mat which is used together with the trough. The low-frequency component has a strong reduction of up to 15 dB and a wider frequency range up to 20 Hz for the slowest and up to 32 Hz for the fastest train (Figure 10c). In addition, a reduction of about 10 dB can be found at higher frequencies for the Sinzheim and Baden measurement sites where soft under ballast mats (k_M″ ≈ 0.06 N/mm³) have been used. The average reduction effect for all troughs, trains and sites, which is shown in Figure 11 (square markers [24]), is less strong and less specific for the low-frequency reduction. On the other hand, the first measurements at the Sinzheim trough (Figure 11, circle markers [25]) resulted in a stronger and wider reduction, as well as measurements at an under-ballast plate in Graz (Figure 11, triangle markers [26]).

All under-ballast plates and troughs yield a considerable low-frequency reduction, namely at the speed-dependent maximum of the ground vibration.

4.4. Summary of the Measurement Results

The soft mitigation measures, the under-sleeper pads and the under-ballast mat show a reduction of the ground vibration at high frequencies because of the reduction in the dynamic axle loads. The tracks with an increased bending stiffness, the under-ballast plate and the troughs have their characteristic reduction at low frequencies. A low-frequency reduction can also be observed for most of the presented soft mitigation measures, and it could be related to a specific speed-dependent ground vibration component.

5. Interpretation and Recommendations for the Measurement of Mitigation Effects

The train-induced ground vibration is usually strong in the frequency range of 10 to 100 Hz, whereas the high frequencies above 100 Hz and the very low frequencies below 10 Hz are small because of the material damping and the layering of the soil. Therefore, the reduction effects in these extreme frequency ranges are uncertain, namely the intended mitigation effects of soft elements are difficult to assess. The vehicle–track resonances usually fall in the range of high amplitudes and help to check the properties of the elastic elements. The low-frequency mitigation is mainly between 10 and 30 Hz at a typical maximum in the high-amplitude region. This low-frequency maximum can be attributed to the static axle loads and their impulses on the railway track. This has been checked by axle-box measurements and narrow-band spectra of the ground vibration [21]. The regular response to the axle impulses is limited to the close near field of the track and to frequencies below 10 Hz. Only the irregular response due to the scattering by a randomly varying ballast and soil can yield far-field amplitudes at higher frequencies of 10 to 30 Hz [27]. Reductions at frequencies below 10 Hz are ratios from small numbers and should be carefully checked. They can probably stem from different layering of the isolated and un-isolated track section, but also reduced irregularities of a newly installed isolated track should be considered.

Besides the problem of amplitude ratios for small amplitudes, the presentation as a relative reduction makes the mitigation effect strongly dependent on the reference system. This can be avoided by the calculation of absolute force reductions. In measurements, however, only relative mitigation effects can be established which are usually better for a stiff reference system (for example, a tunnel line compared to a surface line).

The following recommendations for measuring mitigation effects can be given:

• The original spectra for the isolated and un-isolated track should be reported for to check if the established reduction is for the main amplitudes or for smaller and more random amplitudes (see Figure 4, Figure 5, Figure 8, Figure 9 and Figure 10).
• The reduction should not be measured in the near field of the track as the quasi-static response could lead to a too positive mitigation effect. A measurement point at the tunnel wall, for example, is generally too close to the track.
• It is better to measure a line of measurement points to check the regularity of the soil (see Figure 9).
• It is always useful to measure the properties of the soil at each track section, that is, the wave velocity (stiffness), the damping (the attenuation with distance), and the transfer function of the soil. Even nearby track sections can have different soil properties as was found for the Regensburg, the Lengnau site, and the Ulm site.
• If the soil is different at the isolated and un-isolated track section, a correction should be made based on the measured transfer functions of the soil.
• The reduction depends strongly on the reference system. A stiff reference would give the “best”-mitigation effect. (The disappointing results at the Regensburg site (Figure 4) are probably due to the soft rail pads of k_R = 40 kN/mm that have been used for the un-isolated reference track and for all isolated track sections).
• As the low-frequency reduction depends on the train speed, it is recommended to analyse the reduction separately for different train types and train speeds, as has been demonstrated by the measurements for the under-ballast mat (Figure 8) and the trough solutions (Figure 10).

6. Conclusions

The mitigation of train-induced ground vibrations by track solutions was investigated by calculations and measurements. The calculation by a wavenumber domain method includes the correct vehicle–track interaction and the correct track–soil interaction. Calculation and measurement results were given for several under-sleeper pads, for under-ballast mats, and for several under-ballast plates and ballast troughs. The elastic elements yield high-frequency reductions in the dynamic axle loads of up to 10 or 15 dB for frequencies above 50 to 100 Hz depending on the softness of the pads or mats. All troughs, the under-ballast mat, and some under-sleeper pads show an additional low-frequency reduction due to smoother axle impulses. The sharp impulses from the passage of static-axle loads over a stiff un-isolated track can give a considerable ground vibration component (“scattered-axle impulses”) if the soil and ballast have a random spatial variation of the stiffness. Elastic elements and especially a higher bending stiffness of the track smoothen the axle impulses and, therefore, strongly reduce the axle-impulse and the ground vibration spectra by 7 to 15 dB at frequencies typically between 10 and 30 Hz (and up to 50 Hz for high-speed trains). According to the experience with the measurement results, recommendations for better ground vibration measurements of mitigation effects are given.