Combined approach for analysing evolutionary power spectra of a track-soil system under moving random loads

  • Y. ZhaoEmail author
  • L. T. Si
  • H. Ouyang
Research Paper


The pseudo-excitation method combined with the integral transform method (PEM-ITM) is presented to investigate the ground vibration of a coupled track-soil system induced by moving random loads. Commonly in the track model, the rail, sleepers, rail pads, and ballast are modelled as an infinite Euler beam, discretely distributed masses, discretely distributed vertical springs, and a viscoelastic layer, respectively. The soil is regarded as a homogenous isotropic half-space coupled with the track using the boundary condition at the surface of the ground. By introducing a pseudo-excitation, the random vibration analysis of the coupled system is converted into a harmonic analysis. The analytical form of evolutionary power spectral density responses of the simplified coupled track-soil system under a random moving load is derived in the frequency/wavenumber domain by PEM-ITM. In the numerical examples, the effects of different parameters, such as the moving speed, the soil properties, and the coherence of moving loads, on the ground response are investigated.


Track-soil system Moving random loads Evolutionary power spectrum Pseudo-excitation method Integral transform method Vibration transmission 



This work was supported by the National Basic Research Program of China (Grant 2014CB046803) and the National Natural Science Foundation of China (Grant 11772084).


  1. 1.
    Zhou, S.H., He, C., Di, H.G., et al.: An efficient method for predicting train-induced vibrations from a tunnel in a poroelastic half-space. Eng. Anal. Bound. Elem. 85, 43–56 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Li, G.Q., Wang, Z.L., Chen, S.W., et al.: Field measurements and analyses of environmental vibrations induced by high-speed Maglev. Sci. Total Environ. 568, 1295–1307 (2016)CrossRefGoogle Scholar
  3. 3.
    Fryba, L.: Vibration of Solids and Structures Under Moving Loads. Springer Press, New York (2013)zbMATHGoogle Scholar
  4. 4.
    Alabi, B.: A parametric study on some aspects of ground-borne vibrations due to rail traffic. J. Sound Vibr. 153, 77–87 (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Krylov, V.V.: Generation of ground vibrations by superfast trains. Appl. Acoust. 44, 149–164 (1995)CrossRefGoogle Scholar
  6. 6.
    Takemiya, H., Goda K.: Prediction of ground vibration induced by high-speed train operation. In: Proceedings of the 18th Sino-Japan Technology Seminar, pp. 1–10 (1997)Google Scholar
  7. 7.
    Gunaratne, M., Sanders, O.: Response of a layered elastic medium to a moving strip load. Int. J. Numer. Anal. Methods. 20, 191–208 (1996)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dieterman, H.A., Metrikine, A.: Critical velocities of a harmonic load moving uniformly along an elastic layer. J. Appl. Mech. Trans. ASME 64, 596–600 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lefeuve-Mesgouez, G., Peplow, A.T., Le Houédec, D.: Surface vibration due to a sequence of high speed moving harmonic rectangular loads. Soil Dyn. Earthq. Eng. 22, 459–473 (2002)CrossRefGoogle Scholar
  10. 10.
    Jones, C.J.C., Sheng, X., Petyt, M.: Simulations of ground vibration from a moving harmonic load on a railway track. J. Sound Vibr. 231, 739–751 (2000)CrossRefGoogle Scholar
  11. 11.
    Bierer, T., Bode, C.: A semi-analytical model in time domain for moving loads. Soil Dyn. Earthq. Eng. 27, 1073–1081 (2007)CrossRefGoogle Scholar
  12. 12.
    Sheng, X., Jones, C.J.C., Petyt, M.: Ground vibration generated by a harmonic load acting on a railway track. J. Sound Vibr. 225, 3–28 (1999)CrossRefGoogle Scholar
  13. 13.
    Hung, H.H., Yang, Y.B.: Elastic waves in visco-elastic half-space generated by various vehicle loads. Soil Dyn. Earthq. Eng. 21, 1–17 (2001)CrossRefGoogle Scholar
  14. 14.
    Koziol, P., Mares, C., Esat, I.: Wavelet approach to vibratory analysis of surface due to a load moving in the layer. Int. J. Solids Struct. 45, 2140–2159 (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sheng, X., Jones, C.J.C., Thompson, D.J.: A comparison of a theoretical model for quasi-statically and dynamically induced environmental vibration from trains with measurements. J. Sound Vibr. 267, 621–635 (2003)CrossRefGoogle Scholar
  16. 16.
    Lombaert, G., Degrande, G.: Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains. J. Sound Vibr. 319, 1036–1066 (2009)CrossRefGoogle Scholar
  17. 17.
    Xia, H., Cao, Y.M., De Roeck, G.: Theoretical modeling and characteristic analysis of moving-train induced ground vibrations. J. Sound Vibr. 329, 819–832 (2010)CrossRefGoogle Scholar
  18. 18.
    Hunt, H.: Modelling of road vehicles for calculation of traffic-induced ground vibration as a random process. J. Sound Vibr. 144, 41–51 (1991)CrossRefGoogle Scholar
  19. 19.
    Sun, L., Greenberg, B.S.: Dynamic response of linear systems to moving stochastic sources. J. Sound Vibr. 229, 957–972 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Metrikine, A.V., Vrouwenvelder, A.C.W.M.: Surface ground vibration due to a moving train in a tunnel, two-dimensional model. J. Sound Vibr. 234, 43–66 (2000)CrossRefGoogle Scholar
  21. 21.
    Paolucci, R., Maffeis, A., Scandella, L., et al.: Numerical prediction of low-frequency ground vibrations induced by high-speed trains at Ledsgaard, Sweden. Soil Dyn. Earthq. Eng. 23, 425–433 (2003)CrossRefGoogle Scholar
  22. 22.
    Sheng, X., Jones, C.J.C., Thompson, D.J.: A theoretical model for ground vibration from trains generated by vertical track irregularities. J. Sound Vibr. 272, 937–965 (2004)CrossRefGoogle Scholar
  23. 23.
    Lai, C.G., Callerio, A., Faccioli, E., et al.: Prediction of railway-induced ground vibrations in tunnels. J. Vib. Acoust. Trans. ASME 127, 503–514 (2005)CrossRefGoogle Scholar
  24. 24.
    Lin, J.H.: A fast CQC algorithm of PSD matrices for random seismic responses. Comput. Struct. 44, 683–687 (1992)CrossRefGoogle Scholar
  25. 25.
    Lu, F., Gao, Q., Lin, J.H., et al.: Non-stationary random ground vibration due to loads moving along a railway track. J. Sound Vibr. 298, 30–42 (2006)CrossRefGoogle Scholar
  26. 26.
    Lombaert, G., Degrande, G., Clouteau, D.: Numerical modelling of free field traffic-induced vibrations. Soil Dyn. Earthq. Eng. 19, 473–488 (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Bendat, J.S., Piersol, A.G.: Random Data: Analysis and Measurement Procedures, vol. 729. Wiley, New York (2011)zbMATHGoogle Scholar
  28. 28.
    Peng, Y.B., Chen, J.B., Li, J.: Nonlinear response of structures subjected to stochastic excitations via probability density evolution method. Adv. Struct. Eng. 17, 801–819 (2014)CrossRefGoogle Scholar
  29. 29.
    Zhang, Y.W., Zhao, Y., Zhang, Y.H., et al.: Riding comfort optimization of railway trains based on pseudo-excitation method and symplectic method. J. Sound Vibr. 332, 5255–5270 (2013)CrossRefGoogle Scholar
  30. 30.
    Zhao, G.Z., Chen, G., Kang, Z.: An iterative algorithm for analysis of coupled structural–acoustic systems subject to random excitations. Acta. Mech. Sin. 28, 458–467 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lin, Z.Q., Gea, H.C., Liu, S.T.: Design of piezoelectric energy harvesting devices subjected to broadband random vibrations by applying topology optimization. Acta. Mech. Sin. 27, 730–737 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhao, Y., Si, L.T., Ouyang, H.: Dynamic analysis of an infinitely long beam resting on a kelvin foundation under moving random loads. Shock Vibr. 2017, 1–13 (2017)Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Center for Computational MechanicsDalian University of TechnologyDalianChina
  2. 2.School of EngineeringUniversity of LiverpoolLiverpoolUK

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