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
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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)
The stiffness matrix of the layer is calculated according to
where the following abbreviations are used
The corresponding stiffness matrix of the underlying half-space is calculated as
The dynamic stiffness matrix K (25) is inverted
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
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
or
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
as
A mass element (mass m) would yield
as
The transfer matrix of a column, which is used for the ballast, reads
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
The transfer matrix T is transformed to the stiffness matrix K as
(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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DOI: https://doi.org/10.1007/s00419-016-1209-6