Calculation of the Dynamic Responses of Rails Subjected to Moving Loads on Ballasted Railway Track
Main Article Content
Abstract
The dynamical responses of railway track has been carried out by different ways. In this work, by developing an analytical model for the ballasted railway track which includes two rails connected to the railway sleeper, a fast method to calculated the dynamic responses of the two rails is presented. The rail is modelled as the infinite beams posed on the periodically supports which are rested on a viscoelastic foundation. We consider the dynamic equation in the steady-state of rails subjected to the moving loads. By using Fourier transform together the periodically conditions, a relation between the reaction force and the beam displacement in the frequency domain has been demonstrated. Then, by performing this relation into the dynamic equation of sleeper laying on viscoelastic foundation, the dynamic responses of the two rails can be obtained with a help of Green’s function. The numerical example demonstrates the effect of the support on the beam responses.
Keywords:
Dynamic, Structure, Railway track, periodically supported beam, Euler-Bernoulli beam.
References
[1] L. Frýba, Vibration of Solids and Structures Under Moving Load, 3rd edition, Research Institute of Transport, Thomas Telford, 1972.
[2] V. H. Nguyen, D. Duhamel, Finite Element Procedures for Nonlinear Structure in Moving Coordinates – Part 1: Infinite Bar Under Moving Axial Loads, Computer and Structures, Vol. 84, 2006, https://doi.org/10.1016/j.compstruc.2006.02.018.
[3] V. H. Nguyen, D. Duhamel, Finite Element Procedures for Nonlinear Structure in Moving Coordinates – Part 2: Infinite Bar Under Moving Harmonic Loads, Computer and Structures, Vol. 86, 2008, https://doi.org/10.1016/j.compstruc.2008.04.010.
[4] T. Hoang et al., Respone of a Periodically Supported Beam on a Nonlinear Foundation Subjected to Moving Loads, Nonlinear Dynamic, Vol. 86, 2016, https://doi.org/10.1007/s11071-016-2936-5.
[5] D. Mead, Free Wave Propagation in Periodically Supported, Infinite Beam, Journal of Sound and Vibration,
Vol. 11, 1970, https://doi.org/10.1016/S0022-460X(70)80062-1.
[6] D. Mead, Wave Propagation in Continuous Periodic Structures: Research Contributions from Southampton: 1964-1965, Journal of Sound and Vibration, Vol. 190, 1996, https://doi.org/10.1006/jsvi.1996.0076.
[7] A. V. Metrikine, Vibration of a Periodically Supported Beam on an Elastic Half-space, European Journal of Mechanic - A/Solid, Vol. 18, 1999, https://doi.org/10.1016/S0997-7538(99)00141-2.
[8] P. M. Belotserkovskiy, On the Oscillations of Infinite Periodic Beams Subjected to a Moving Concentrated Force, Journal of Sound and Vibration, Vol. 196, 1996, https://doi.org/10.1006/jsvi.1996.0309.
[9] A. Nordborg, Vertical Rail Vibrations: Pointforce Excitation, Acta Acustica Vol. 84, 1998.
[10] A. Nordborg, Vertical Rail Vibrations: Parametric Excitation, Acta Acustica, Vol. 84, 1998.
[11] T. Hoang et al., Dynamical Response of a Timoshenko Beams on a Periodical Nonlinear Support Subjected to Moving Forces, Engineering Structures, Vol. 176, 2018, https://doi.org/10.1016/j.engstruct.2018.09.028.
[12] L. H. Tran et al., Calculation of the Dynamic Responses of a Railway Track on a Non-uniform Foundation, Journal of Vibration and Control, 2022. https://doi.org/10.1177/10775463221099353.
[13] L. H. Tran et al., A Fast Analytic Method to Calculate the Dynamic Response of Railway Sleeper, Journal of Vibiration and. Acoustic, Vol. 141, 2019, https://doi.org/10.1115/1.4040392.
[14] L. H. Tran et al., A Comparison of Beam Models for the Dynamics Responses of Railway Sleepers, International Journal of Rail and Transportation, Vol. 10, 2022, https://doi.org/10.1080/23248378.2022.2034062.
[15] L. H. Tran et al., Identification of Train Loads from the Dynamic Responses of an Integrated Sleeper in
Situ, Journal of Intelligent Material Systems and Structures, Vol. 31, 2020, https://doi.org/10.1177/1045389X20922905.
[16] X. Yang et al., An Explicit Periodic Nonlinear Model for Evaluating Dynamic Responses of Damaged Slab Track Involving Material Nonlinearity of Damage in High-speed Railway, Construction and Building Materials,
Vol. 168, 2018, https://doi.org/10.1016/j.conbuildmat.2018.02.147.
[17] X. Xiao and W. X. Ren, A Versatile 3D Vehicule-track-bridge Element for Dynamics Analysis of Railway Bridges Under Moving Train Loads, International Journal of Structural Stability and Dynamics, Vol. 19, 2019, https://doi.org/10.1142/S0219455419500500.
[18] L. H. Tran et al., Influence of Non-homogeneous Foundation on the Dynamic Responses of Railway Sleepers, International Journal of Structural Stability and Dynamics, Vol. 21, 2020. https://doi.org/10.1142/S0219455421500024.
[19] R. Gustavson and K. Gylltoft, Influence of Cracked Sleepers on the Global Track Responses: Coupling of a Linear Track Model and Nonlinear Finite Element Analyses, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Vol. 216, 2002, https://doi.org/10.1243/0954409021531674.
[20] J. F. Ruiz et al., Study of Ground Vibrations Induced by Railway Traffic in a 3D FEM Model Formulated in the Time Domain: Experimental Validation, Structure and Infrastructure Engineering, Vol. 13, 2017, https://doi.org/10.1080/15732479.2016.1172649.
[21] T. Hoang et al., Calculation of Force Distribution for a Periodically Supported Beam Subjected to Moving Loads, Journal of Sound and Vibration, Vol. 388, 2017, https://doi.org/10.1016/j.jsv.2016.10.031.
[22] Vietnam Building codes on railway TCCS 04:2014/VNRA, http://vnra.gov.vn/Media/AuflaNews/Attachment/ TCCS.04.2014.VNRA.30.12.2014.pdf, 2014 (acces on: March 8th, 2022).
[23] L. H. Tran et al., Analytical Model of the Dynamics of Railway Sleeper, presented at 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 2017, https://doi.org/10.7712/120117.5695.18372.
[2] V. H. Nguyen, D. Duhamel, Finite Element Procedures for Nonlinear Structure in Moving Coordinates – Part 1: Infinite Bar Under Moving Axial Loads, Computer and Structures, Vol. 84, 2006, https://doi.org/10.1016/j.compstruc.2006.02.018.
[3] V. H. Nguyen, D. Duhamel, Finite Element Procedures for Nonlinear Structure in Moving Coordinates – Part 2: Infinite Bar Under Moving Harmonic Loads, Computer and Structures, Vol. 86, 2008, https://doi.org/10.1016/j.compstruc.2008.04.010.
[4] T. Hoang et al., Respone of a Periodically Supported Beam on a Nonlinear Foundation Subjected to Moving Loads, Nonlinear Dynamic, Vol. 86, 2016, https://doi.org/10.1007/s11071-016-2936-5.
[5] D. Mead, Free Wave Propagation in Periodically Supported, Infinite Beam, Journal of Sound and Vibration,
Vol. 11, 1970, https://doi.org/10.1016/S0022-460X(70)80062-1.
[6] D. Mead, Wave Propagation in Continuous Periodic Structures: Research Contributions from Southampton: 1964-1965, Journal of Sound and Vibration, Vol. 190, 1996, https://doi.org/10.1006/jsvi.1996.0076.
[7] A. V. Metrikine, Vibration of a Periodically Supported Beam on an Elastic Half-space, European Journal of Mechanic - A/Solid, Vol. 18, 1999, https://doi.org/10.1016/S0997-7538(99)00141-2.
[8] P. M. Belotserkovskiy, On the Oscillations of Infinite Periodic Beams Subjected to a Moving Concentrated Force, Journal of Sound and Vibration, Vol. 196, 1996, https://doi.org/10.1006/jsvi.1996.0309.
[9] A. Nordborg, Vertical Rail Vibrations: Pointforce Excitation, Acta Acustica Vol. 84, 1998.
[10] A. Nordborg, Vertical Rail Vibrations: Parametric Excitation, Acta Acustica, Vol. 84, 1998.
[11] T. Hoang et al., Dynamical Response of a Timoshenko Beams on a Periodical Nonlinear Support Subjected to Moving Forces, Engineering Structures, Vol. 176, 2018, https://doi.org/10.1016/j.engstruct.2018.09.028.
[12] L. H. Tran et al., Calculation of the Dynamic Responses of a Railway Track on a Non-uniform Foundation, Journal of Vibration and Control, 2022. https://doi.org/10.1177/10775463221099353.
[13] L. H. Tran et al., A Fast Analytic Method to Calculate the Dynamic Response of Railway Sleeper, Journal of Vibiration and. Acoustic, Vol. 141, 2019, https://doi.org/10.1115/1.4040392.
[14] L. H. Tran et al., A Comparison of Beam Models for the Dynamics Responses of Railway Sleepers, International Journal of Rail and Transportation, Vol. 10, 2022, https://doi.org/10.1080/23248378.2022.2034062.
[15] L. H. Tran et al., Identification of Train Loads from the Dynamic Responses of an Integrated Sleeper in
Situ, Journal of Intelligent Material Systems and Structures, Vol. 31, 2020, https://doi.org/10.1177/1045389X20922905.
[16] X. Yang et al., An Explicit Periodic Nonlinear Model for Evaluating Dynamic Responses of Damaged Slab Track Involving Material Nonlinearity of Damage in High-speed Railway, Construction and Building Materials,
Vol. 168, 2018, https://doi.org/10.1016/j.conbuildmat.2018.02.147.
[17] X. Xiao and W. X. Ren, A Versatile 3D Vehicule-track-bridge Element for Dynamics Analysis of Railway Bridges Under Moving Train Loads, International Journal of Structural Stability and Dynamics, Vol. 19, 2019, https://doi.org/10.1142/S0219455419500500.
[18] L. H. Tran et al., Influence of Non-homogeneous Foundation on the Dynamic Responses of Railway Sleepers, International Journal of Structural Stability and Dynamics, Vol. 21, 2020. https://doi.org/10.1142/S0219455421500024.
[19] R. Gustavson and K. Gylltoft, Influence of Cracked Sleepers on the Global Track Responses: Coupling of a Linear Track Model and Nonlinear Finite Element Analyses, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Vol. 216, 2002, https://doi.org/10.1243/0954409021531674.
[20] J. F. Ruiz et al., Study of Ground Vibrations Induced by Railway Traffic in a 3D FEM Model Formulated in the Time Domain: Experimental Validation, Structure and Infrastructure Engineering, Vol. 13, 2017, https://doi.org/10.1080/15732479.2016.1172649.
[21] T. Hoang et al., Calculation of Force Distribution for a Periodically Supported Beam Subjected to Moving Loads, Journal of Sound and Vibration, Vol. 388, 2017, https://doi.org/10.1016/j.jsv.2016.10.031.
[22] Vietnam Building codes on railway TCCS 04:2014/VNRA, http://vnra.gov.vn/Media/AuflaNews/Attachment/ TCCS.04.2014.VNRA.30.12.2014.pdf, 2014 (acces on: March 8th, 2022).
[23] L. H. Tran et al., Analytical Model of the Dynamics of Railway Sleeper, presented at 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 2017, https://doi.org/10.7712/120117.5695.18372.