Modeling the Influence of Under-Sleeper Pads on the Vibration Emis- sions from Railway Traffic Nils Mahlert 1 DB Systemtechnik GmbH Völckerstraße 5, D - 80939 Munich, Germany Sascha Hermann 2 DB Systemtechnik GmbH Völckerstraße 5, D - 80939 Munich, Germany Dr. Haike Brick 3 German Centre for Rail Traffic Research August Bebel-Straße 10, D - 01219 Dresden, Germany Rüdiger Garburg 4 Deutsche Bahn AG Europaplatz 1, D - 10557 Berlin, Germany

ABSTRACT One of the aims of the European research project FINE-2 is to contribute to reducing noise and vibrations caused by railway traffic. In this context, the development of models describing the influence of relevant parameters on the excitation of train-induced vibration emissions were investigated in the “Ground Vibration” work package (WP 8), which is led by DB Systemtechnik. Here, resilient track components and especially under-sleeper pads were con- sidered by a new modelling approach. The model is based on the German standard DIN 45673- 4, which describes an approach to compute the insertion loss of under-ballast mats by using an impedance model. It is highly dependent on the dynamic stiffness of the relevant superstructure components. The paper shows a general description of the model, the steps taken to adapt the existing model to consider under-sleeper pads instead of under-ballast mats as well as results validated by using existing measurements. 1. INTRODUCTION

For planning new lines or upgrading existing lines, ground vibrations caused by railway traffic must be taken into account. The European research project FINE-2 contributes to reducing the annoyance caused by vibrations by developing calculation methods and software solutions for predicting vibra- tion emissions from railway traffic. The FINE-2 project is divided in different work packages (WP). Within the “Ground Vibrations” work package (WP 8), mathematical models describing the influence of relevant parameters on the excitation and transmission of vibrations were developed [6]. This in- cluded investigating the influence of elastic elements as under-ballast mats and under-sleeper pads

1 Nils.Mahlert@deutschebahn.com 2 Sascha.Hermann@deutschebahn.com 3 BrickH@dzsf.bund.de 4 Ruediger.Garburg@deutschebahn.com

worm 2022

on railway-induced vibrations. For this purpose, an impedance model based on the dynamic stiffness characteristics of the individual superstructure components according to DIN 45673-4 [1] was used. The procedure described in the standard was originally proposed for under-ballast mats but can be used also for the computation of the efficiency of under-sleeper pads.

This paper explains the general model description of the standard DIN 45673-4 [1] for under- ballast mats and the steps taken to adapt the existing model to consider under-sleeper pads. In addi- tion, the results of the developed model are presented and compared with existing measurements. 2. EXISTING MODEL FOR UNDER-BALLAST MATS

Under-ballast mats are elastic elements situated below the ballast. In order to model their influence on the railway-induced vibration, the modelling approach of the standard DIN 45673-4 can be used [1]. The approach calculates the insertion loss by considering an equivalent mass-spring-system with one degree of freedom and subtracting the transfer function of the system with the elastic element 𝐻 𝑚𝑖𝑡 from the transfer function of the system without the elastic element 𝐻 𝑅𝑒𝑓 (Figure 1).

j ferheostans — TF agincostant keg ‘TL Atlifcostanen [Fara costanen hey ‘without elastic element (reference system) - Hey {| Fe=Fecos(2ntey 4 FF eineostonen Fi Mae pincstetn Paimeesanrh Ln with elastic element (ced frame) ~ Hoe

Figure 1: Equivalent mass-spring-system with and without the elastic element [1].

The insertion loss 𝐷 𝐹 ሺ𝑓ሻ is defined as the quotient of both transfer functions in a logarithmic representation:

(1)

𝐷 𝐹 ሺ 𝑓 ሻ = 20 log 10 ቤ 𝐻 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ

𝐻 𝑚𝑖𝑡 ሺ𝑖𝑓ሻ ቤ

The formula for the transfer function 𝐻 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ of the reference system without the elastic element is defined as follows:

(2)

𝐻 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ= 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ−ሺ2𝜋𝑓ሻ 2 𝑚 𝐸𝑟𝑠

with 𝑓 frequency 𝑚 𝐸𝑟𝑠 dynamic mass of the vehicle and the track

(3)

𝑘 𝑅𝑒𝑓 ሺ 𝑖𝑓 ሻ = ሺ 1 𝑘 𝑘,𝑜 ሺ𝑖𝑓ሻ + 1 𝑘 𝑘,𝑢 ሺ𝑖𝑓ሻ ሻ −1 = 𝑘 𝑘,𝑜 ሺ𝑖𝑓ሻ∙ 𝑘 𝑘,𝑢 ሺ𝑖𝑓ሻ

𝑘 𝑘,𝑜 ሺ𝑖𝑓ሻ+ 𝑘 𝑘,𝑢 ሺ𝑖𝑓ሻ

worm 2022

𝑘 𝑘,𝑜 ሺ𝑖𝑓ሻ complex dynamic stiffness of the upper part system (includes the superstructure com- ponents above the elastic element) 𝑘 𝑘,𝑢 ሺ𝑖𝑓ሻ complex dynamic stiffness of the bottom part system (includes the superstructure com- ponents under the elastic element) The formula for the transfer function 𝐻 𝑚𝑖𝑡 ሺ𝑖𝑓ሻ of the system with the installed elastic element is defined as follows:

𝐻 𝑚𝑖𝑡 ሺ𝑖𝑓ሻ= 𝐹 ෠ 𝑢 ሺ𝑖𝑓ሻ

(4)

= 𝑘 𝑚𝑖𝑡 ሺ𝑖𝑓ሻ 𝑘 𝑚𝑖𝑡 ሺ 𝑖𝑓 ሻ − ሺ 2𝜋𝑓 ሻ 2 𝑚 𝐸𝑟𝑠

𝐹 ෠ 𝐸

with

𝑘 𝑚𝑖𝑡 ሺ𝑖𝑓ሻ= ሺ 1 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ + 1 𝑘 𝑘,𝑒𝑙 ሺ𝑖𝑓ሻ ሻ −1 = 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ∙ 𝑘 𝑘,𝑒𝑙 ሺ𝑖𝑓ሻ

𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ+ 𝑘 𝑘,𝑒𝑙 ሺ𝑖𝑓ሻ (5)

𝑘 𝑘,𝑒𝑙 ሺ𝑖𝑓ሻ complex dynamic stiffness of the elastic element 3. ADAPTED MODEL FOR UNDER-SLEEPER PADS

An under-sleeper pad is an elastic intermediate layer installed underneath a concrete sleeper, i.e. above the ballast For the pads, various materials as polyurethane, rubber or cork with different mate- rial properties were used in the past. Their field of application lies primarily in the reduction of vi- brations and structure-borne noise from rail traffic as well as the protection of ballast from destruc- tion. The material characteristics including the relevant acoustical and dynamical parameters of un- der-sleeper pads are determined according to the European standard EN 16730 [4].

The model to compute the insertion loss of under-sleeper pads contains the upper part system 𝑘 𝑘,𝑜 , the bottom part system 𝑘 𝑘,𝑢 and the pad as an elastic element 𝑘 𝑘,𝑒𝑙 (Figure 2). The complex dynamic stiffness of the upper part system describes the rail and the sleeper. The complex dynamic stiffness of the bottom part system describes the ballast and the planum. To calculate the influence of under- sleeper pads, the existing model for under-ballast mats was adapted. The described model within the DIN 45673-4 for under-ballast mats contains values for 𝑘 𝑘,𝑜 and 𝑘 𝑘,𝑢 based on linearization of the stiffness- and damping characteristics which were determined from experimental investigations [1]:

𝑚 𝐸𝑟𝑠 = 2000 𝑘𝑔

𝑁

𝑘 𝑘,𝑜 = 2.2 ∙10 8

𝑚 ,𝜂 𝑜 = 0.35 (rail, sleeper, ballast)

𝑁

𝑘 𝑘,𝑒𝑙 = 1.4 ∙10 8

𝑚 , 𝜂 𝑜 = 0.05 (under-ballast mat)

𝑁

𝑚 , 𝑑 𝑢 = 1.2 ∙ 10 6 𝑁𝑠

𝑘 𝑘,𝑢 = 15 ∙10 8

𝑚 (planum)

worm 2022

Figure 2: Railway components of the upper part and bottom part system [3]. In this figure, the ballast is already separated from the dynamic stiffness of the upper part system 𝑘 𝑘,𝑜 and shifted to the bottom part system 𝑘 𝑘,𝑢 to add the elastic element 𝑘 𝑘,𝑒𝑙 between the two systems.

Thus, the dynamic stiffness from the upper part 𝑘 𝑘,𝑜 includes the rail, sleeper and ballast. The dynamic stiffness from the bottom part 𝑘 𝑘,𝑢 includes the planum. The complex dynamic stiffness of the upper part system 𝑘 𝑘,𝑜 and the elastic element 𝑘 𝑘,𝑒𝑙 can be described with a hysteretic description:

𝑘 𝑘𝑖𝑛,𝑆 ሺ𝑖𝑓ሻ= 𝑘 𝑘𝑖𝑛 ∙ሺ1 + 𝑖𝜂ሻ (6) with: 𝑘 𝑘𝑖𝑛 kinetic stiffness 𝜂 loss factor The complex dynamic stiffness of the bottom part system 𝑘 𝑘,𝑢 follows the approach of a viscous system:

ns ng plan

𝑘 𝑘 𝑣 ሺ𝑖𝑓ሻ= 𝑘 𝑘𝑖𝑛 + 𝑖2𝜋𝑓𝑑 𝑢 (7)

with: 𝑘 𝑘𝑖𝑛 kinetic stiffness

𝑑 𝑢 damping coefficient of the bottom part system The existing model can be changed by shifting the complex stiffness of the ballast from the upper part system to the bottom part system, because the ballast is below the elastic element for the appli- cation of under-sleeper pads. The complex stiffness of the ballast can be described with a hysteretic description using formula (6).

Based on measured data, the complex stiffness from the ballast can be assumed as follows [2]:

𝑚 ∙ሺ1 + 𝑖0.5ሻ (8)

𝑘 𝑏𝑎𝑙 = 𝑘 𝑘𝑖𝑛,𝑆 ሺ𝑖𝑓ሻ= 𝑘 𝑘𝑖𝑛,𝑏𝑎𝑙 ∙ሺ1 + 𝑖𝜂 𝑏𝑎𝑙 ሻ= 5 ∙ 10 8 𝑁

It should be noted that the complex stiffness of the ballast variates depending on the type and the height of ballast as well as on the loading (dynamic or static). Typical values for the stiffness of the ballast are between ሺ2 𝑡𝑜 5ሻ∙ 10 8

𝑁

𝑚 . The loss factor variates between 0.35 and 0.5 [1]. To shift the stiffness of the ballast to the bottom part system 𝑘 𝑘,𝑢 (Figure 2), the complex stiffness of the ballast (red frame in eq. 9) and the complex stiffness of the planum (blue frame in eq. 9) are connected in series according to the rules of an electrical serial connection:

worm 2022

−1

𝑘 𝑘,𝑢 = ቌ 1

+ 1

(9)

ቍ

5 ∙10 8 𝑁

15 ∙10 8 𝑁

𝑚 + 𝑖2𝜋𝑓∙1.2 ∙10 6 𝑁𝑠

𝑚 ∙ሺ1 + 𝑖0.5ሻ

𝑚

For the under-sleeper pads, the complex stiffness of the upper part system 𝑘 𝑘,𝑜 includes the rail and the sleeper. It can be assumed that these two components are much stiffer than the bottom part system 𝑘 𝑘,𝑢 so that the forces resulting from the unsprung mass act directly on the under-sleeper pad and the bottom part. On that basis, the complex stiffness for the reference system 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ can be defined as follows:

𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ= ሺ 1

ሻ −1 = 𝑘 𝑘,𝑢 (10)

𝑘 𝑘,𝑢

𝑁

The complex stiffness of the under-sleeper pads variates between 𝑘 𝑘,𝑒𝑙 = 3 ∙10 7

𝑚 and

𝑁

𝑘 𝑘,𝑒𝑙 = 8 ∙ 10 8

𝑚 depending on the material properties and the frequency range. The loss factor has typically values between 0.1 and 0.5. As the ballast, the under-sleeper pads can be described with a hysteretic approach using formula (6).

In this approach the under-sleeper pads are described with a dynamic and frequency-dependent stiffness. Table 1 shows three different types of under-sleeper pads and their frequency-dependent stiffness (high frequency dynamic stiffness) measured in a test bench with a preload of 0.12 N/mm². The stiffness of an under-sleeper pad depends also on the sleeper type used. In the following example, the pads are used in combination with a B-70-sleeper. It should be noted that with changing sleeper properties, the pad stiffness variates due to the different geometrical dimensions of the sleeper.

Table 1: High frequency dynamic stiffness and loss factor of three different types of under-sleeper pads (USP 1, USP 2, USP 3) as a function of frequencies (preload 0.12 N/mm²; B-70-sleeper) [5].

Frequency USP 1 USP 2 USP 3 1 Hz 7.62∙10 7 𝑁

𝑚 5.39∙10 7 𝑁

𝑚 1.14∙10 8 𝑁

𝑚

10 Hz 8.24∙10 7 𝑁

𝑚 5.74∙10 7 𝑁

𝑚 1.19∙10 8 𝑁

𝑚

100 Hz 1.43∙10 8 𝑁

𝑚 8.02∙10 7 𝑁

𝑚 1.55∙10 8 𝑁

𝑚

1000 Hz 7.58∙10 8 𝑁

𝑚 2.24∙10 8 𝑁

𝑚 4.51∙10 8 𝑁

𝑚

loss factor 0.2 0.2 0.5

4. RESULTS OF THE PARAMETER STUDY

To calculate the insertion loss 𝐷 𝐹 ሺ𝑓ሻ resulting from the under-sleeper pads, the complex stiffness of the reference system 𝑘 𝑅𝑒𝑓 ሺ𝑖𝑓ሻ as well as the complex stiffness of the under-sleeper pads 𝑘 𝑘,𝑒𝑙 have to be filled in formula (2) and (4).

In our approach, the under-sleeper pads are described by a dynamic and frequency-dependent stiff- ness. Figure 3 shows the different insertion loss values dependent on the complex stiffness of the under-sleeper pads. From the calculations we conclude that with rising stiffness the insertion loss

worm 2022

resulting from the under-sleeper pads deteriorates. The resonance frequency also shifts to higher fre- quencies with an increasing stiffness of the under-sleeper pads. At the resonance frequency, which is located between 25 Hz to 35 Hz the insertion loss is minimal, there may even be an amplification of the vibration transmission.

Insertion loss USP ey

Figure 3: Calculated insertion loss of three different under-sleeper pads.

It should be noted that the level of impact in Figure 3 should be used with caution. The calculated values can overestimate the insertion loss. Depending on a change of the complex stiffness of the under-sleeper pads, Figure 3 shows that the insertion loss variates up to 5 dB.

In addition, the ballast stiffness can have an influence on the insertion loss of the under-sleeper pad. Typically, the stiffness of the ballast variates between ሺ2 𝑡𝑜 5ሻ∙ 10 8

𝑁

𝑚 [1]. Figure 4 shows calculations varying the ballast stiffness by considering USP 1.

Insertion loss of USP dependent on the stitfes of the ballast (USP 2) ea Pa ey

Figure 4: Insertion loss of USP 1 dependent on the stiffness of the ballast.

In conclusion, the insertion loss of under-sleeper pads variates depending on the specific stiffness of the product. The resonance frequency is located between 25 Hz to 32 Hz. A positive effect of

worm 2022

under-sleeper pads results above the resonance frequency. In measurements, the resonance frequency is often not as pronounced as in the mathematical model.

Furthermore, the insertion loss of under-sleeper pads is not only dependent on the dynamic stiff- ness of the used elastic element, but also the dynamic properties of the superstructure components like the ballast, the sleepers, or the unsprung mass of the vehicle. For an accurate simulation these parameters must be determined as accurately as possible. 5. VALIDATION OF THE MODEL

For the validation of the model approach, measured data for the efficiency of under-sleeper pads were used [7]. As input data, the frequency-dependent dynamic stiffness of the under-sleeper pads were used. For the dynamic stiffness of the ballast a value of 𝑘 𝑏𝑎𝑙 = 5 ∙ 10 8

𝑁

𝑚 ∙ሺ1 + 𝑖0.5ሻ was consid- ered. Figure 5 shows the comparison of the calculated and the measured insertion loss for the pass- by of a double-deck train with a speed of 160 km/h.

The validation of the model with measurements shows that considering the prognosis and meas- urement uncertainties, the developed model can describe the insertion loss of the under-sleeper pads. The comparison of the calculated results to measured data shows also that there are more effects, especially within the lower frequency range up to 25 Hz which are not described by the model.

Insertion loss of USP - double deck tran v ey

Figure 5: Comparison between measured and calculated insertion loss of the under-sleeper pads.

6. CONCLUSION

In WP 8 “Ground vibrations” of the project FINE-2, models describing the efficiency of elastic ele- ments to reduce train-induced vibrations were proposed. For the use of under-sleeper pads, a model originally proposed for under-ballast mats was adapted. The validation of the model by using meas- ured data show that the calculated insertion loss of under-sleeper pads is qualitatively comparable with measurement results in consideration of the prediction and measurement uncertainties. However, the comparison between measurements and calculations shows discrepancies opening room for future research. On the one hand, there are characteristics in the low-frequency range below 25 Hz which are not considered by the model. On the other hand, the dynamic stiffness characteristics of the indi- vidual superstructure parameters especially those of the ballast are highly variable. During the devel- opment of the model, these values were set based on the existing literature [1], [2]. In practice, it is

worm 2022

recommended to measure them as accurately as possible preferably in the installed condition. The parameter analysis presented in Chapter 3 shows that varying the dynamic stiffness of the ballast can result in partial differences in the effect of under sleeper pads of up to 5 dB. 7. ACKNOWLEDGEMENTS

The authors thank the Shift2Rail Joint Undertaking for founding the project FINE-2 under Grant Agreement No 881791 in the European Union’s Horizon 2020 research and innovation programme. 8. REFERENCES

1. Standards Committee Acoustics, Noise Control and Vibration Engineering, DIN 45673-4, Me-

chanical vibration – Resilient elements used in railway tracks – Part 4: Analytical evaluation of insertion loss of mounted track systems, 2008. 2. R.G. Wettschureck, U.J. Kurze, ACUSTICA 1985, Einfügungsdämm-Maß von Unterschottermat-

ten, Page 177-182. 3. Stefan Lutzenberger, Dorothee Stiebel, Christian Gerbig, Rüdiger Wettschureck, Taschenbuch

der technischen Akustik, Kapitel: Erschütterungen und sekundärer Luftschall aus dem Schienen- verkehr, Springer-Verlag, 2015. 4. Standards Committee Railway (FSF), EN 16730:2016, Railway applications – Track – Concrete

sleepers and bearers with under sleeper pads, German version, 2016. 5. Collection of data based on the data sheets of different under-sleeper pads. 6. Shift2Rail, Project FINE-2, Deliverable 8.2 Specification how to define and characterize typical

vehicles, track and soil conditions , June 2021. 7. DB Netz AG, Abschlussbericht Initiative Lärmschutz-Erprobung neu und anwendungsorientiert

(I-LENA) , February 2022.

worm 2022