{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:38:44Z","timestamp":1760150324033,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T00:00:00Z","timestamp":1700006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Portuguese Foundation for Science and Technology","award":["UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/50022\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Ballasted railway tracks can be modeled using reduced\/simplified models composed of several layers of discrete components. This paper deals with the two-layer model, which is very popular due to its computational efficiency. In order to provide some recommendations for track design, it is necessary to identify which set of parameters leads to some irregular\/unexpected behavior. In this paper, irregularities are investigated at three levels, namely, (i) the critical velocity of a moving constant force, (ii) the instability of one moving mass, and (iii) the instability of two moving masses. All results are presented in a dimensionless form to cover a wide range of real parameters. Irregular cases are identified by sets of parameters leading to them, which is the main finding of this paper; then, general conclusions are drawn. Regarding the method, all results are obtained analytically or semi-analytically, where \u201csemi\u201d refers to solving the roots of a given polynomial using predefined numerical procedures in symbolic software. No numerical integration is involved in any of the results presented. This means that the results are highly accurate and refer to exact values, so any kind of parametric or sensitivity analyses is readily possible.<\/jats:p>","DOI":"10.3390\/app132212356","type":"journal-article","created":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T10:51:37Z","timestamp":1700045497000},"page":"12356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Instability of Vibrations of Mass(es) Moving Uniformly on a Two-Layer Track Model: Parameters Leading to Irregular Cases and Associated Implications for Railway Design"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2210-3660","authenticated-orcid":false,"given":"Zuzana","family":"Dimitrovov\u00e1","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, NOVA School of Science and Technology, NOVA University of Lisbon, 2829-516 Caparica, Portugal"},{"name":"Institute of Mechanical Engineering (IDMEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fr\u00fdba, L. (1999). Vibration of Solids and Structures under Moving Loads, Research Institute of Transport, Prague (1972), Thomas Telford. [3rd ed.].","DOI":"10.1680\/vosasuml.35393"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1002\/nag.1135","article-title":"Analytical solutions for Euler-Bernoulli beam on visco-elastic foundation subjected to moving load","volume":"37","author":"Basu","year":"2013","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.ijsolstr.2017.10.005","article-title":"Universal analytical solution of the steady-state response of an infinite beam on a Pasternak elastic foundation under moving load","volume":"132\u2013133","author":"Froio","year":"2018","journal-title":"Int. J. Solids Struct."},{"key":"ref_4","first-page":"295","article-title":"Steady state displacements of a beam on an elastic half-space due to uniformly moving constant load","volume":"16","author":"Dieterman","year":"1997","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/j.jsv.2015.12.023","article-title":"Critical velocity of a uniformly moving load on a beam supported by a finite depth foundation","volume":"366","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_6","first-page":"128","article-title":"Analysis of the critical velocity of a load moving on a beam supported by a finite depth foundation","volume":"122\u2013123","year":"2017","journal-title":"Int. J. Solids Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1007\/s10409-012-0021-y","article-title":"On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method","volume":"28","author":"Kiani","year":"2011","journal-title":"Acta Mech. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1016\/j.jsv.2008.08.010","article-title":"Prediction capabilities of classical and shear deformable beam models excited by a moving mass","volume":"320","author":"Kiani","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s10409-010-0365-0","article-title":"Assessing dynamic response of multispan viscoelastic thin beams under a moving mass via generalized moving least square method","volume":"26","author":"Kiani","year":"2010","journal-title":"Acta Mech. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1115\/1.3408901","article-title":"Dynamic stability of a beam carrying moving masses","volume":"38","author":"Nelson","year":"1971","journal-title":"J. Appl. Mech. Trans. ASME"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1115\/1.3423435","article-title":"Dynamic stability of a beam loaded by a sequence of moving mass particles","volume":"41","author":"Benedetti","year":"1974","journal-title":"J. Appl. Mech. Trans. ASME"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1115\/1.2888325","article-title":"The response of an infinite railroad track to a moving, vibrating mass","volume":"57","author":"Duffy","year":"1990","journal-title":"J. Appl. Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1006\/jsvi.1996.0783","article-title":"Instability of vibrations of a mass moving uniformly along an axially compressed beam on a visco-elastic foundation","volume":"201","author":"Metrikine","year":"1997","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s004190100177","article-title":"Instability of vibrations of a moving two-mass oscillator on a flexibly supported Timoshenko beam","volume":"71","author":"Metrikine","year":"2001","journal-title":"Arch. Appl. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1850125","DOI":"10.1142\/S0219455418501250","article-title":"Vibrations of a Multi-Span Beam Structure Carrying Many Moving Oscillators","volume":"18","author":"Yang","year":"2018","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jsv.2017.10.021","article-title":"Coupled dynamics of a viscoelastically supported infinite string and a number of discrete mechanical systems moving with uniform speed","volume":"415","author":"Roy","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s40997-018-0224-3","article-title":"Dynamic response of Timoshenko beam resting on nonlinear Pasternak foundation carrying sprung masses","volume":"43","author":"Nassef","year":"2019","journal-title":"Iran. J. Sci. Technol. Trans. Mech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"(2011). Mazilu, T.; Dumitriu, M.; Tudorache, C. On the dynamics of interaction between a moving mass and an infinite one-dimensional elastic structure at the stability limit. J. Sound Vib., 330, 3729\u20133743.","DOI":"10.1016\/j.jsv.2011.02.026"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1007\/s00419-009-0350-x","article-title":"Interaction between a moving two-mass oscillator and an infinite homogeneous structure: Green\u2019s functions method","volume":"80","author":"Mazilu","year":"2010","journal-title":"Arch. Appl. Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1007\/s11071-011-0068-5","article-title":"Instability of an oscillator moving along a Timoshenko beam on viscoelastic foundation","volume":"67","author":"Mazilu","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4597","DOI":"10.1016\/j.jsv.2013.03.022","article-title":"Instability of a train of oscillators moving along a beam on a viscoelastic foundation","volume":"332","author":"Mazilu","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.ijsolstr.2016.12.010","article-title":"Dynamic instability and critical velocity of a mass moving uniformly along a stabilized infinity beam","volume":"108","year":"2017","journal-title":"Int. J. Solids Struct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.euromechsol.2019.01.020","article-title":"Stability and vibrations of an overcritical speed moving multiple discrete oscillators along an infinite continuous structure","volume":"75","author":"Deng","year":"2019","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.apm.2021.07.022","article-title":"Dynamic interaction and instability of two moving proximate masses on a beam on a Pasternak viscoelastic foundation","volume":"100","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107042","DOI":"10.1016\/j.ijmecsci.2021.107042","article-title":"Two-layer model of the railway track: Analysis of the critical velocity and instability of two moving proximate masses","volume":"217","year":"2022","journal-title":"Int. J. Mech. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"116788","DOI":"10.1016\/j.engstruct.2023.116788","article-title":"Non-stability of a bogie moving along a specific infinite complex flexibly beam-layer structure","volume":"295","author":"Deng","year":"2023","journal-title":"Eng. Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"118020","DOI":"10.1016\/j.jsv.2023.118020","article-title":"Dynamics of moving coupled objects with stabilizers and unconventional couplings","volume":"570","author":"Deng","year":"2024","journal-title":"J. Sound Vib."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1007\/s11012-020-01258-3","article-title":"Semi-analytical analysis of vibrations induced by a mass traversing a beam supported by a finite depth foundation with simplified shear resistance","volume":"55","year":"2020","journal-title":"Meccanica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1007\/s00419-020-01656-9","article-title":"Nonlinear dynamic response of an Euler\u2013Bernoulli beam under a moving mass\u2013spring with large oscillations","volume":"90","author":"Jahangiri","year":"2020","journal-title":"Arch. Appl. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2435","DOI":"10.1007\/s11071-019-05083-6","article-title":"Transition radiation in a piecewise-linear and infinite one-dimensional structure\u2013a Laplace transform method","volume":"98","author":"Metrikine","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1007\/s11071-020-06117-0","article-title":"Transition radiation in a nonlinear and infinite one-dimensional structure: A comparison of solution methods","volume":"103","author":"Keijdener","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"115804","DOI":"10.1016\/j.jsv.2020.115804","article-title":"Transition radiation in an infinite one-dimensional structure interacting with a moving oscillator\u2014The Green\u2019s function method","volume":"492","author":"Mazilu","year":"2021","journal-title":"J. Sound Vib."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1002\/nme.624","article-title":"Moving element method for train-track dynamics","volume":"56","author":"Koh","year":"2003","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2954","DOI":"10.1016\/j.jsv.2013.01.005","article-title":"Response analysis of high-speed rail system accounting for abrupt change of foundation stiffness","volume":"332","author":"Ang","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5427","DOI":"10.1016\/j.jsv.2014.05.053","article-title":"Vertical dynamic response of non-uniform motion of high-speed rails","volume":"333","author":"Tran","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Dai, J., Lim, J.G.Y., and Ang, K.K. (2023). Dynamic response analysis of high-speed maglev-guideway system. J. Vib. Eng. Technol.","DOI":"10.1007\/s42417-023-00995-5"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"031001","DOI":"10.1115\/1.4045788","article-title":"A new three-dimensional moving Timoshenko beam element for moving load problem analysis","volume":"142","author":"Xu","year":"2020","journal-title":"J. Vib. Acoust. Trans. ASME"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"04021012","DOI":"10.1061\/(ASCE)EM.1943-7889.0001915","article-title":"Novel Nonlinear Dynamic Beam-Foundation Interaction Model","volume":"147","author":"Elhuni","year":"2021","journal-title":"ASCE J. Eng. Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.jsv.2019.05.006","article-title":"Railway track degradation: The contribution of a spatially variant support stiffness-Local variation","volume":"455","author":"Sadri","year":"2019","journal-title":"J. Sound Vib."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"114992","DOI":"10.1016\/j.jsv.2019.114992","article-title":"Railway track degradation: The contribution of a spatially variant support stiffness-Global variation","volume":"464","author":"Sadri","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3390\/vibration4010013","article-title":"Applicability of a Three-Layer Model for the Dynamic Analysis of Ballasted Railway Tracks","volume":"4","author":"Rodrigues","year":"2021","journal-title":"Vibration"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"113","DOI":"10.3390\/vibration6010009","article-title":"On the Critical Velocity of Moving Force and Instability of Moving Mass in Layered Railway Track Models by Semianalytical Approaches","volume":"6","year":"2023","journal-title":"Vibration"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"115363","DOI":"10.1016\/j.jsv.2020.115363","article-title":"A three-dimensional modal theory-based Timoshenko finite length beam model for train-track dynamic analysis","volume":"479","author":"Yang","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"895","DOI":"10.3390\/vibration6040053","article-title":"Predicting Critical Speed of Railway Tracks Using Artificial Intelligence Algorithms","volume":"6","author":"Ramos","year":"2023","journal-title":"Vibration"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1243\/JMES_JOUR_1982_024_016_02","article-title":"The dynamic response of railway track to high frequency vertical excitation","volume":"24","author":"Grassie","year":"1982","journal-title":"J. Mech. Eng. Sci."},{"key":"ref_46","unstructured":"Rodrigues, A.S.F. (2017). Viability and Applicability of Simplified Models for the Dynamic Analysis of Ballasted Railway Tracks. [Ph.D. Thesis, Faculdade de Ci\u00eancias e Tecnologia da Universidade Nova de Lisboa]."},{"key":"ref_47","unstructured":"Van Dalen, K.N. (2006). Ground Vibration Induced by a High-Speed Train Running over Inhomogeneous Subsoil, Transition Radiation in Two-Dimensional Inhomogeneous Elastic Systems. [Master\u2019s Thesis, Department of Structural Engineering, TUDelft]."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1006\/jsvi.2000.3333","article-title":"Response of an Infinite Timoshenko Beam on a Viscoelastic Foundation to a Harmonic Moving Load","volume":"241","author":"Chen","year":"2001","journal-title":"J. Sound Vib."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1016\/S0022-460X(03)00186-X","article-title":"Modelling and experiment of railway ballast vibrations","volume":"270","author":"Zhai","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Sa\u00f1udo, R., Miranda, M., Alonso, B., and Markine, V. (2022). Sleepers Spacing Analysis in Railway Track Infrastructure. Infrastructures, 7.","DOI":"10.3390\/infrastructures7060083"},{"key":"ref_51","unstructured":"IMV-019 (2000). Fabrico e Fornecimento de Travessas Monobloco UIC54 e 60, Rede Ferrovi\u00e1ria Nacional, REFER EPE. (In Portuguese)."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"124326","DOI":"10.1016\/j.conbuildmat.2021.124326","article-title":"The importance of sleepers spacing in railways","volume":"300","author":"Ortega","year":"2021","journal-title":"Constr. Build. Mater."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.advengsoft.2012.02.011","article-title":"Critical Velocity of a Uniformly Moving Load","volume":"50","author":"Rodrigues","year":"2012","journal-title":"Adv. Eng. Softw."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/13\/22\/12356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:23:31Z","timestamp":1760131411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/13\/22\/12356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,15]]},"references-count":53,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["app132212356"],"URL":"https:\/\/doi.org\/10.3390\/app132212356","relation":{},"ISSN":["2076-3417"],"issn-type":[{"type":"electronic","value":"2076-3417"}],"subject":[],"published":{"date-parts":[[2023,11,15]]}}}