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Static and dynamic behaviours of isolated or unisolated ballast tracks using a fast wavenumber domain method

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Abstract

The dynamics of unisolated and isolated ballast tracks have been analysed by multi-beam models for the track and by a layered half-space model for the soil. The solution is calculated in frequency–wavenumber domain and transformed back to space domain by a wavenumber integral. This is a faster method compared to other detailed track–soil interaction methods and almost as fast as the widely used Winkler soil method, especially if the compliances of the soil have been stored for repeated use. Frequency-dependent compliances and force transfer functions have been calculated for a variety of track and soil parameters. The ballast has a clear influence on the high-frequency behaviour, whereas the soil is dominating the low-frequency behaviour of the track. A layering of the soil may cause a moderate track–soil resonance, whereas more pronounced vehicle–track resonances occur with elastic track elements like rail pads, sleeper pads and ballast mats. Above these resonant frequencies, a reduction in the excitation forces follows as a consequence. The track deformation along the track has been analysed for the most interesting track systems. The track deformation is strongly influenced by the resonances due to layering or elastic elements. The attenuation of amplitudes and the velocity of the track–soil waves change considerably around the resonant frequencies. The track deformation due to complete trains have been calculated for different continuous and Winkler soils and compared with the measurement of a train passage showing a good agreement for the continuous soil and clear deviations for the Winkler soil model.

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Abbreviations

a :

Constant amplitude vector

b :

Width of the track

\(C_{T}\) :

Compliance of the track (rail)

\(C_{T\textit{0}}\) :

Static compliance of the track

\(C_{T\mathrm{max}}\) :

Maximum compliance of the rail

d :

Sleeper distance

D :

Material damping as \(k^*=k\) (\(1+2D\)i)

\({\mathbf{e}}_\mathbf{1 }\), \({\mathbf{e}}_\mathbf{3 }\) :

Base vector (first and last track beam)

\(EI_{R}\) :

Bending stiffness of the rail

\({EI}_{i}\) :

Bending stiffness of the jth track beam

EI :

Matrix of the bending stiffnesses of the multi-beam track model

f :

Frequency

\(f_{0}\) :

Resonant frequency

\(f_{T}\) :

Frequency of the track resonance

\(f_{VS}\) :

Frequency of the vehicle–track resonance

F :

Force

\({F}^\prime \) :

Force per length

\(F_{i}\) :

Axle load of the train

\(F_{T}, {F_{T}}^\prime \) :

Dynamic force (per length) on the track

\(F_{S}, {F_{S}}^\prime \) :

Dynamic force (per length) on the soil

\(F_{V}\) :

Dynamic force on the vehicle

\({\mathbf{F}_{T}}^{\varvec{\prime }}\) :

Track load vector

G :

Shear modulus of the soil

\(G_{\textit{0}}\) :

Real part of the shear modulus of the soil

\(h_{B}\) :

Height of the ballast

\(h_{L}\) :

Height of the soil layer

\(H_{S}\) :

Compliance of the soil (in wavenumber domain)

\(H_{SS}\) :

Strip load compliance of the soil

\(H_{VT}\) :

Vehicle–track force transfer function

\(H_{TS}\) :

Track–soil force transfer function

\(H_{VS}\) :

Total force transfer function

i:

Imaginary unit

\(k_{R}, {k_{R}}^\prime \) :

Rail pad stiffness (per length)

\({k_{B}}^\prime , {k_{ij}}^\prime \) :

Stiffness per length of the ballast

\(k_{S}, {k_{S}}^\prime \) :

Stiffness (per length) of the under-sleeper pad

\({K_{B}}^\prime \) :

Matrix of the ballast stiffness

\(K_{T}\) :

Dynamic stiffness of the track under a wheelset load

\(K_{V}\) :

Dynamic stiffness of the vehicle

\({K_{S}}^\prime \) :

Dynamic stiffness (per length) of the soil

\({\mathbf{K}_{\mathbf{T}}}^{\varvec{\prime }}\) :

Dynamic track stiffness of the multi-beam track model

\({\mathbf{K}_{\mathbf{S}}}^{\varvec{\prime }}\) :

Dynamic soil stiffness of the multi-beam track model

\({\mathbf{K}_{\mathbf{TS}}}^{\varvec{\prime }}\) :

Dynamic support stiffness of the multi-beam track model

\(l_{A}\) :

Axle distance

\(l_{B }\) :

Bogie distance

\(l_{C}\) :

Carriage distance

\({m_{j}}^\prime \) :

Mass (per length) of the jth track beam

\(m_{S}\) :

Mass of the sleeper

\(m_{W}\) :

Mass of the wheelset

\({\mathbf{m}}^{\varvec{\prime }}\) :

Mass matrix of the multi-beam track model

\(p_{\textit{1}}\) :

Load distribution across the track (wavenumber transform)

t :

Thickness of the under-ballast plate

T :

Transfer matrix of a support element

\(u_{R}, u_{\textit{1}}\) :

Displacement of the rail under the wheelset load

\(u_{S}, u_{\textit{3}}\) :

Displacement of the soil under the track

\(\mathbf{u}_{S}\) :

Displacement vector of the continuous soil

\({ {\ddot{\mathbf{u}}}}_{S}\) :

Acceleration vector of the continuous soil

\(\mathbf{u}_{T}\) :

Displacement vector for all track beams

\(v_{S}\) :

Shear wave velocity of the soil

\(v_{S\textit{1}}\) :

Shear wave velocity of the soil layer

\(v_{S\textit{2}}\) :

Shear wave velocity of the underlying half-space

\(v_{P}\) :

Compression wave velocity of the soil

\(v_{R}\) :

Rayleigh wave velocity of the soil

\(v_{B}\) :

Longitudinal compression wave velocity of the ballast

x :

Coordinate vector, position vector

x :

Coordinate across the track

y :

Coordinate along the track

z :

State vector for a transfer matrix

\(\delta \) :

Dirac delta function

\(\lambda \) :

Wavelength

\(\nu \) :

Poisson’s ratio of the soil

\(\rho \) :

Mass density of the soil

\(\xi \) :

\(= 2\pi \)/ \(\lambda \), wavenumber

\({\varvec{\upxi }}\) :

Wavenumber vector

\(\xi _{B}\) :

Longitudinal wavenumber of the ballast

\(\xi _{S}\) :

Shear wavenumber of the soil

\(\xi _{P}\) :

Compression wavenumber of the soil

\(\xi _{x}\) :

Wavenumber across the track

\(\xi _{y }\) :

Wavenumber along the track

\(\omega \) :

Angular frequency

References

  1. Esveld, C.: Modern Railway Track, 2nd edn. MRT-Productions, Zaltbommel (2001)

    Google Scholar 

  2. Auersch, L.: Zur Parametererregung des Rad-Schiene-Systems: Berechnung der Fahrzeug-Fahrweg-Untergrund-Dynamik und experimentelle Verifikation am Hochgeschwindigkeitszug Intercity Experimental. Arch. Appl. Mech. (Ingenieur-Archiv) 60, 141–156 (1990)

    Google Scholar 

  3. Nielsen, J.: Train/Rrack Interaction. Dissertation. Chalmers University of Technology, Götheburg (1993)

  4. Auersch, L.: The excitation of ground vibration by rail traffic: theory of vehicle–track–soil interaction and measurements on high-speed lines. J. Sound Vib. 284, 103–132 (2005)

    Article  Google Scholar 

  5. Maldonado, M.: Vibrations dues au passage d’un tramway - mesures expérimentales et simulations numériques. Dissertation, École Centrale de Nantes (2008)

  6. Auersch, L.: Theoretical and experimental excitation force spectra for railway induced ground vibration - vehicle-track soil interaction, irregularities and soil measurements. Veh. Syst. Dyn. 48, 235–261 (2010)

    Article  Google Scholar 

  7. Krüger, F.: Schall- und Erschütterungsschutz im Schienenverkehr: Grundlagen der Schall- und Schwingungstechnik - praxisorientierte Anwendung von Schall- und Erschütterungsschutzmaßnahmen. Expert Verlag, Renningen-Malmsheim (2001)

    Google Scholar 

  8. Müller-Boruttau, F., Breitsamter, N.: Elastische Elemente verringern die Fahrwegbeanspruchung. Eisenbahntechnische Rundschau 49, 587–596 (2000)

    Google Scholar 

  9. Auersch, L.: Dynamic axle loads on tracks with and without ballast mats—numerical results of three-dimensional vehicle–track–soil models. J. Rapid Transit. 220, 169–183 (2006)

    Article  Google Scholar 

  10. Auersch, L.: Force and ground vibration reduction of railway tracks with elastic elements. J. Vib. Control 21, 2246–2258 (2015)

    Article  Google Scholar 

  11. Zimmermann, H.: Die Berechnung des Eisenbahnoberbaus. Ernst & Korn, Berlin (1888)

    Google Scholar 

  12. Winkler, E.: Die Lehre von der Elastizität und Festigkeit. Dominicus, Prag (1867)

    Google Scholar 

  13. Knothe, K., Wu, Y.: Receptance behaviour of railway track and subgrade. Arch. Appl. Mech. 68, 457–470 (1998)

    Article  MATH  Google Scholar 

  14. Li, D., Selig, E.: Wheel/track dynamic interaction: track substructure perspective. Veh. Syst. Dyn. 24, 183–196 (1994)

    Article  Google Scholar 

  15. Auersch, L.: Zur Dynamik einer unendlichen Platte auf dem Halbraum—Fundamentnachgiebigkeit und Wellenfeld bei harmonischer Punktlast. Arch. Appl. Mech. 64, 346–356 (1994)

    Article  Google Scholar 

  16. Auersch, L.: Dynamic interaction of various beams with the underlying soil—finite and infinite, half-space and Winkler models. Eur. J. Mech. A/Solids 27, 933–958 (2008)

    Article  MATH  Google Scholar 

  17. Jones, C.: Use of numerical models to determine the effectiveness of anti-vibration system for railways. Proc. Inst. Civil Eng.-Transp. 105, 43–51 (1994)

    Google Scholar 

  18. Lieb, M., Sudret, B.: A fast algorithm for soil dynamics calculations by wavelet decomposition. Arch. Appl. Mech. 68, 147–157 (1998)

    Article  MATH  Google Scholar 

  19. Sheng, X.: Ground Vibration Generated from Trains. Dissertation. University of Southampton (2001)

  20. Lombaert, G., Degrande, G., Vanhauwere, B., Vandeborght, B., François, S.: The control of ground-borne vibrations from railway traffic by means of continuous floating slabs. J. Sound Vib. 297, 946–961 (2006)

    Article  Google Scholar 

  21. Auersch, L.: The dynamic behaviour of slab tracks on homogeneous and layered soils and the reduction of ground vibration by floating slab tracks. J. Eng. Mech. 138, 923–933 (2012)

    Article  Google Scholar 

  22. Sheng, X., Jones, C., Thompson, D.: Prediction of ground vibration from trains using discrete wavenumber finite and boundary element methods. J. Sound Vib. 293, 575–586 (2006)

    Article  Google Scholar 

  23. Galvin, P., Francois, S., Schevenels, M., Bongini, E., Degrande, G., Lombaert, G.: A 2.5D coupled FE-BE model for the prediction of railway induced vibrations. Soil Dyn. Earthq. Eng. 30, 1500–1512 (2010)

    Article  MATH  Google Scholar 

  24. Alves Costa, P.: Vibrações do sistema via-maciço induzidas por tráfego ferroviário - modelação numérica e validação experimental. Dissertation. University of Porto (2011)

  25. Auersch, L.: Dynamics of the railway track and the underlying soil—the boundary-element solution, theoretical results and their experimental verification. Veh. Syst. Dyn. 43, 671–695 (2005)

    Article  Google Scholar 

  26. Romero, A.: Predicción, medida experimental y evaluación de las vibraciones producidas por el tráfico ferroviario. Dissertation. University of Sevilla (2012)

  27. Ju, S.H.: Finite element analysis of structure-borne vibration from high-speed train. Soil Dyn. Earthq. Eng. 27, 259–273 (2007)

    Article  Google Scholar 

  28. Kouroussis, G.: Modélisation des effets vibratoires du traffic ferroviaire sur l’environnement. Dissertation. University of Mons (2009)

  29. Conolly, D.: Ground Borne Vibrations from High-Speed Trains. Dissertation. University of Edinburgh (2013)

  30. Auersch, L.: Ground vibration due to railway traffic—the calculation of the effects of moving static loads and their experimental verification. J. Sound Vib. 293, 599–610 (2006)

    Article  Google Scholar 

  31. Auersch, L.: Zur Entstehung und Ausbreitung von Schienenverkehrs-erschütterungen: Theoretische Untersuchungen und Messungen am Hochgeschwindigkeitszug Intercity Experimental. Forschungsbericht 155, BAM Berlin (1988)

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Appendices

Appendix 1: Compliance of a layer-on-half-space system

For harmonic waves, the field equation and the interface condition between the layer and the half-space can be used to establish the stiffness matrix of the layer-on-half-space system which relates the horizontal (x) and vertical (z) displacements and stresses of the surface (1) and the interface between layer and half-space (2)

$$\begin{aligned} \left[ {{\begin{array}{l} {\sigma _{x1} } \\ {\hbox {i}\sigma _{z1} } \\ {\sigma _{x2} } \\ {\hbox {i}\sigma _{z2} } \\ \end{array} }} \right] =\left[ {{\begin{array}{cccc} {K_{11} }&{} {K_{12} }&{} {K_{13} }&{} {K_{14} } \\ {K_{12} }&{} {K_{22} }&{} {-K_{14} }&{} {K_{24} } \\ {K_{13} }&{} {-K_{14} }&{} {K_{11} +K^{H}_{11} }&{} {-K_{12} +K^{H}_{12} } \\ {K_{14} }&{} {K_{24} }&{} {-K_{12} +K^{H}_{12} }&{} {K_{22} +K^{H}_{22} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {u_{x1} } \\ {\hbox {i}u_{z1} } \\ {u_{x2} } \\ {\hbox {i}u_{z2} } \\ \end{array} }} \right] \end{aligned}$$
(25)

The stiffness matrix of the layer is calculated according to

$$\begin{aligned} \begin{array}{l} K_{11} /A=K_{33} /A=\frac{1}{t}C_s S_t +sS_s C_t \\ K_{22} /A=K_{44} /A=tC_s S_t +\frac{1}{s}S_s C_t \\ K_{12} /A=-K_{34} /A=\frac{3-t^{2}}{1+t^{2}}\left( {1-C_s C_t } \right) +\frac{1+2s^{2}t^{2}-t^{2}}{st(1+t^{2})}S_s S_t \\ K_{13} /A=-sS_s -\frac{1}{t}S_t \\ K_{14} /A=-K_{23} /A=C_s -C_t \quad \\ K_{24} /A=-\frac{1}{s}S_s -tS_t \\ \end{array} \end{aligned}$$
(26)

where the following abbreviations are used

$$\begin{aligned} \begin{array}{ll} s=-\hbox {i}\sqrt{1-\left( {\frac{\xi _P }{\xi }} \right) ^{2}}&{} t=-\hbox {i}\sqrt{1-\left( {\frac{\xi _S }{\xi }} \right) ^{2}} \\ C_s =\cos \xi sd &{} \;C_t =\cos \xi td \\ S_s =\sin \xi sd &{} \;\;S_t =\sin \xi td \\ N=2(1-C_s C_t )+\left( {st+\frac{1}{st}} \right) S_s S_t\; &{}\qquad A=\frac{(1+t^{2})\xi G}{N} \\ \end{array}. \end{aligned}$$
(27)

The corresponding stiffness matrix of the underlying half-space is calculated as

$$\begin{aligned} \begin{array}{l} {K^{H}}_{11} =\xi G\frac{\hbox {i}s(1+t^{2})}{1+st} \\ {K^{H}}_{22} =\xi G\frac{\hbox {i}t(1+t^{2})}{1+st}\quad \\ {K^{H}}_{12} =\xi G\left( {2-\frac{(1+t^{2})}{1+st}} \right) \\ \end{array}. \end{aligned}$$
(28)

The dynamic stiffness matrix K (25) is inverted

$$\begin{aligned} \mathbf{K}^{-1} = \mathbf{F} \end{aligned}$$
(29)

to obtain the flexibility matrix F of which the vertical element of the surface is chosen as the transfer function of the soil in the frequency–wavenumber domain

$$\begin{aligned} F_{zz} =H_S (\xi ,\omega ). \end{aligned}$$
(30)

Appendix 2: Transfer matrices for the stiffness of the track support

The stiffness matrix of a support section of the track is calculated by transfer matrices T which relate the state z \(=\) (\(F,u)^{\mathrm{T}}\) (F force, u displacement) of the bottom and the top of each support element as

$$\begin{aligned} \mathbf z _\mathbf{1 } = \mathbf{T z}_\mathbf{2 } \end{aligned}$$
(31)

or

$$\begin{aligned} \left[ {{\begin{array}{l} {F_1 } \\ {u_1 } \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {T_{11} }&{} {T_{12} } \\ {T_{21} }&{} {T_{22} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {F_2 } \\ {u_2 } \\ \end{array} }} \right] . \end{aligned}$$
(32)

The forces point to the element and \(F_{1}\) and \(u_{1}\) have the same direction. A spring element (stiffness k, for example a sleeper pad) yields

$$\begin{aligned} \mathbf{T}_\mathbf{F} =\left[ {{\begin{array}{ll} 1&{} 0 \\ {1/k}&{} 1 \\ \end{array} }} \right] \end{aligned}$$
(33)

as

$$\begin{aligned}&F_{\textit{1}}=F_{\textit{2}} \quad \text {and}\nonumber \\&u_{\textit{1}} - u_{\textit{2}}=F_{\textit{2}}/k . \end{aligned}$$
(34)

A mass element (mass m) would yield

$$\begin{aligned} \mathbf{T}_\mathbf{M} =\left[ {{\begin{array}{cc} 1&{} {-m\omega ^{2}} \\ 0&{} 1 \\ \end{array} }} \right] \end{aligned}$$
(35)

as

$$\begin{aligned}&F_{\textit{1}}- F_{\textit{2}} = - m\omega ^{2}u_{\textit{2}}\nonumber \\&u_{\textit{1}}=u_{\textit{2}} . \end{aligned}$$
(36)

The transfer matrix of a column, which is used for the ballast, reads

$$\begin{aligned} \mathbf{T}_\mathbf{C} =\left[ {{\begin{array}{cc} {\cos (\xi _B h_B )}&{} {-\sin (\xi _B h_B ) k_B \xi _B h_B } \\ {\sin (\xi _B h_B )/k_B \xi _B h_B }&{} {\cos (\xi _B h_B )} \\ \end{array} }} \right] \end{aligned}$$
(37)

with the static stiffness \(k_{B}\), the height \(h_{B}\), and the wavenumber \(\xi _{B}=\omega \)/\(v_{B}\) of the longitudinal wave velocity \(v_{B}\) of the column (the ballast).

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 Fig. 1) as

$$\begin{aligned} \mathbf z _\mathbf{1 } = \mathbf{T}_{1}{} \mathbf{T}_{2}{} \mathbf{T}_{3}\, \hbox {z}_{3} = \mathbf{T z}_{3} . \end{aligned}$$
(38)

The transfer matrix T is transformed to the stiffness matrix K as

$$\begin{aligned} \left[ {{\begin{array}{l} {F_1 } \\ {-F_2 } \\ \end{array} }} \right] =\frac{1}{T_{21} }\left[ {{\begin{array}{cc} {T_{11} }&{} {-\det \mathbf{T}} \\ {-1}&{} {T_{22} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {u_1 } \\ {u_2 } \\ \end{array} }} \right] =\frac{1}{T_{21} }\left[ {{\begin{array}{cc} {T_{11} }&{} {-1} \\ {-1}&{} {T_{22} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {u_1 } \\ {u_2 } \\ \end{array} }} \right] =\mathbf{Ku} \end{aligned}$$
(39)

(note that det T = 1 for passive systems, and that the sign definition is different for \(F_{\textit{2}}\), namely \(F_{\textit{2}}\) is in the same direction as \(u_{\textit{2}}\) for the stiffness matrix).

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Auersch, L. Static and dynamic behaviours of isolated or unisolated ballast tracks using a fast wavenumber domain method. Arch Appl Mech 87, 555–574 (2017). https://doi.org/10.1007/s00419-016-1209-6

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