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2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 International Journal of Computational Methods Vol 11, No (2014) 1343007 (12 pages) c World Scientific Publishing Company DOI: 10.1142/S021987621343007X ANALYSIS OF HIGH-SPEED RAIL ACCOUNTING FOR JUMPING WHEEL PHENOMENON KOK KENG ANG, , JIAN DAI,Đ , MINH THI TRAN,ả and VAN HAI LUONG†, Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only ∗Department of Civil and Environmental Engineering National University of Singapore, Singapore †Department of Civil Engineering Ho Chi Minh City University of Technology HCM City, Vietnam ceeangkk@nus.edu.sg Đdaijian@nus.edu.sg ảtranminhthi@nus.edu.sg lvhai@hcmut.edu.vn Received March 2012 Accepted July 2012 Published 20 September 2013 In this paper, a computational study using the moving element method (MEM) was carried out to investigate the dynamic response of a high-speed train–track system Results obtained using Hertz contact model and linearized Hertz contact model are compared and discussed The dynamic responses of a train travelling across a uniform foundation and a transition region are also investigated Parametric study is performed to understand the effect of various factors on the occurrence and patterns of the jumping wheel phenomenon such as the variation of foundation stiffness, travelling speed of the train and the severity of railhead roughness Keywords: Moving element method; track transition; wheel–rail interaction; track irregularity Introduction Railway transportation is one of the key modes of travel today The advancement in train technology leading to faster and faster trains is without doubt a positive development, which makes high-speed rail (HSR) system more attractive as an alternative to other modes of transportation for long distance travel Due to the high speed of train moving over the track, the chance of occurrence of the “jumping wheel” phenomenon is high in particular when the railhead roughness or so-called “track irregularity” is significant As the name implies, the phenomenon describes the situation when there is a momentary loss of contact between the wheel and rail It is expected that the response of the train–track system would be significantly 1343007-1 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only K K Ang et al higher when there is occurrence of such phenomenon The deterioration rate of the railway system is also accelerated and the risk of derailment increased Thus, it is important to model correctly the dynamic behavior of train–track system accounting for the possible occurrence of the jumping wheel phenomenon Another issue that is heightened with the increase in speed of train is the traveling condition of the train–track system at railway track transitions Transition regions are places where the stiffness of the foundation experiences an abrupt change They are often located at the entrance and exit points of a train tunnel or a bridge Such transition regions have been known to cause problems [Esveld (2001); Lei and Mao (2004); Dimitrovov´ a and Varandas (2009); Lei (2006)] However, the response of the train–track system at track transition areas has not been extensively studied, i.e., the aforementioned research works assumed smooth railhead surface without any consideration of the initial railhead surface imperfections It would therefore be worthwhile to investigate the response of train–track system at track transitions, in particular, the combined effect of railhead roughness and variation of foundation stiffness on the occurrence of the jump wheel phenomenon as well as its patterns The objective of this paper is to investigate the response of HSR systems with a realistic computational model based on the moving element method (MEM) The train is modeled as a sprung-mass system comprising of car body, bogie and wheelset to account for the effect of moving train load A linearized contact model is used to account for the contact between wheel and track but which does not model correctly the jumping wheel phenomenon A more correct model of the jumping wheel phenomenon is developed using nonlinear Hertz contact theory Difference between the results generated using the two contact models is investigated Parametric studies are carried out to investigate the effect of existence of track transitions, severity of railhead roughness and train speeds on the inducement of the occurrence of the jumping wheel phenomenon Formulation and Methodology 2.1 Modeling of train–track system In this paper, the train load is assumed to traverse the railway track at a constant velocity v The railhead is considered to be not smooth but assumed to have some imperfections resulting in so-called track irregularity The moving sprung-mass model, as shown in Fig 1, is employed to model the train–track system as a coupled system composed of the train, railway track and foundation [Ang and Dai (2013)] The railway track is modeled as an Euler–Bernoulli beam resting on a viscoelastic foundation subject to a moving train load The governing equation of motion of the railway beam can be written as [Ang and Dai (2013)]: EI ∂ 2y ∂y ∂4y + k(x)y = Fc δ(x − vt), +m ¯ + c(x) ∂x ∂t ∂t 1343007-2 (1) 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Analysis of High-Speed Rail Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only Fig Moving sprung-mass model where EI and m ¯ refer to the flexural rigidity and mass per unit length of the track, respectively; k(x) and c(x) denote the variation of the vertical stiffness and damping properties of the foundation along the longitudinal direction of the railway; y denotes the transversal displacement of the track; x is the spatial coordinate along the longitudinal direction whose origin is fixed at the initial location of the train; t the time; δ the Dirac-delta function; and Fc the contact force The Hertz contact theory is employed to account for the interaction between the wheel and rail According to the theory, the contact surface between the wheel and rail is an ellipse The shape of the elliptic contact surface changes according to the location of the contact indentation As it is difficult to trace the instantaneous location of the contact surface, a reasonable assumption can be made such that the contact surface is always circular [Esveld (2001)], which gives rise to the simplified form as: KH ∆y Fc = KH = E2 ∆y ≥ , ∆y < (2a) Rwheel Rrailprof , (1 − υ )2 (2b) where KH denotes the Hertzian spring constant; Rwheel and Rrailprof denote the radii of the wheel and railhead, respectively; v the Poisson’s ratio of the material; ∆y the indentation at the contact surface which can be written as: ∆y = yc + yt − u3 (2c) in which yc and u3 denote the displacements of the track and wheel-set, respectively; and yt the magnitude of the track irregularity at the contact point Track irregularity is a major source of the dynamic excitation According to the recommendation in literature [Yang et al (2004); Nielsen and Igcland (1995)], the track irregularity can be expressed as: yt = −at − exp − x xc 1343007-3 sin 2πx , λt (3a) 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only K K Ang et al where at and λt denote the amplitude (wave depth) and the wavelength of the irregularity, respectively; and xc is a constant associated with the condition of the railhead When the train travels far away from its initial position, the exponential term in Eq (3a) will soon become negligible; thus for simplicity, the expression for the vertical track irregularity profile can be written in terms of a sinusoidal function as: 2πx (3b) yt = −at sin λt As the relationship between the contact force and indentation at contact surface is nonlinear, the computational effort required from adopting such a contact model in the study of train–track dynamics is generally high Thus, to avoid the high computational cost and complexity of the problem, many researchers have adopted a simplified approach by linearizing the contact force model The linearized contact force may be written as: Fc = KL ∆y, (4a) where KL is the linearized Hertzian spring constant evaluated by the relationship between the force and displacement increments around the static loading condition [Esveld (2001)], in which the reaction force at the contact point equals the self-weight of the upper structure of the train–track system Thus, the linearized Hertzian spring constant may be expressed as: KL = 3E W Rwheel Rrailprof 2(1 − υ )2 (4b) It is to be noted that the linearized contact model is inappropriate in accounting for the jumping wheel phenomenon in view that Eq (4a) indicates that an erroneous tensile force exists between the wheel and railhead when the phenomenon occurs In the treatment of the problem involving a railway transition, it is assumed that the entire foundation is composed of two adjacent uniform subdomains [Ang and Dai (2013)] The stiffness and damping properties of the foundation can be written as: k(x) = k1 H(−x + x0 ) + k2 H(x − x0 ), (5a) c(x) = c1 H(−x + x0 ) + c2 H(x − x0 ), (5b) where x0 denotes the location of the transition point where the two uniform subdomains meet; k1 and c1 refer to the stiffness and damping of the foundation before and after the transition point, respectively; while k2 and c2 the stiffness and damping of the after the transition point, respectively; and H the Heaviside function 2.2 Moving element method Standard finite element method (FEM) usually suffers from the difficulty encountered due to the moving load eventually reaching the boundary of the finite domain, 1343007-4 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only Analysis of High-Speed Rail rendering the artificial boundary conditions invalid [Ang and Dai (2013)] In an attempt to overcome the complication, Krenk et al [Krenk et al (1999)] gave a FE solution to the response of an elastic half-space subject to a moving load in convected coordinates Later on, Koh et al [2003, 2006, 2007] solved different kinds of problems involving moving loads by adopting the idea of attaching the origin of the spatial coordinates system to the point of application of the moving load, and named the numerical method as the MEM In view that the method had been limited to applications involving horizontally homogeneous foundation, Ang and Dai [2013] extended the usage of MEM to deal with problems involving horizontally inhomogeneous foundation In the MEM, a traveling r-axis is used The origin of the moving axis is fixed at the same position of the moving load (see Fig 2) and is thus traveling at the same velocity as the load The relationship between the fixed x and moving r coordinates is given by x = r + vt (6) In order to consider the existence of track transitions for train–track dynamic analysis, the formulation of the equations considering the case in which the vertical stiffness and damping of the foundation is variable is presented below In view of Eq (6), the governing equation for the rail beam given in Eq (1) may be rewritten as ∂y ∂ 2y ∂y ∂2y ∂2y ∂4y + + c(r) −v ¯ v 2 − 2v + k(r)y = Fc δ(r), EI + m ∂r ∂r ∂r∂t ∂t ∂t ∂r (7) where c(r) and k(r) are functions of the moving r-coordinate By adopting Galerkin’s approach, the mass, damping, and stiffness matrices of the moving element can be obtained After assemblage, the equations of motion for the traintrack model can be written as: Mă z + Cz + Kz = P, (8) where z is the global displacement vector of the train–track system; M, C, and K are the global mass, damping, and stiffness matrices, respectively; and P the global external load vector Fig Coordinate systems for moving load problem 1343007-5 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 K K Ang et al The computational procedure including multiple phases for treating problems involving a transition region, which is elaborated in [Ang and Dai (2013)], is adopted in this study Numerical Results Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only The effect of various combinations of parameters, including parameters relating to track transition, on inducing the occurrence of the jumping wheel phenomenon is investigated The difference between results obtained using the Hertz contact model and the linearized contact model is analyzed and discussed 3.1 Uniform foundation In this numerical case study, the MEM model comprises of a truncated railway track of 60 m length uniformly discretized into 600 moving FEs Newmark’s constant acceleration method is applied to solve the equations using a time-step of 0.0005 s Note that this configuration of the MEM mesh and time-step size have been decided based on the outcome of a convergence study It should also be noted that the testing speeds adopted in the study are far below the critical speed of the system (subcritical cases) so that the use of transmitting boundary conditions or energy absorbing layers is not essential [Ang and Dai (2013); Nguyen and Duhamel (2008)] The parameters for the train model recommended by Koh et al [2003] are adopted in the study whereas parameters for the track-foundation model are listed in Table Unless noted otherwise, all data presented here will be used throughout this paper Three typical track irregularities with a wavelength of m and ranging from “near smooth” to severe condition are used to investigate its effect on the occurrence of the jumping wheel phenomenon The amplitudes of the track irregularities are given in Table Results obtained from the MEM analyses are presented in Table 3, which shows the occurrence or nonoccurrence of the jumping wheel phenomenon for various Table Parameters for track-foundation model Parameter Flexural rigidity Track section Value Parameter Value 6.12 × 106 Nm2 UIC 60 (60 E1) Stiffness of foundation Damping ratio × 107 N/m2 0.1 Table Track irregularities Severity Amplitude (mm) Near smooth Moderate Severe 1343007-6 0.05 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Analysis of High-Speed Rail Table Occurrence of jumping wheel phenomenon Severity Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only Near smooth Moderate Severe Speed (m/s) 50 70 90 0 S 0 S S S speeds of train and severity of track irregularity A zero value implies that no jumping wheel phenomenon took place and a “S” entry indicates that the phenomenon occurred and is sustained throughout the journey It is found that track irregularity and speed of train are two key factors affecting the occurrence of the jumping wheel phenomenon Jumping wheel is noted to easily occur when the track irregularity is considered severe For less severe condition, there is also a good possibility for the occurrence of the phenomenon when the speed of train is high As to be expected, when the track is nearly smooth, jumping wheel is unlikely to occur even at very high train speed Thus, the simpler linearized contact model without allowing for the possible loss of contact between the wheel and rail is not suitable to account for the wheel–rail interaction when the two factors are not considered to be small enough Figures and show, respectively, the displacement profiles of the rail at contact point for near smooth and severe track irregularities when the speed of train is 90 m/s (324 km/h) In the figures, a nondimensional variable N is introduced as: N= vt λt Fig Displacement profile of rail at contact point (near smooth track irregularity) 1343007-7 (9) 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only K K Ang et al Fig Displacement profile of rail at contact point (severe track irregularity) As can be seen from Fig 3, results obtained using the nonlinear Hertz and linearized contact models are found to be comparable with each other when there is no occurrence of jumping wheel phenomenon However, as can be seen from Fig 4, the difference between the two contact models is large when there is an occurrence of the jumping wheel phenomenon due to the incapability of the linearized contact model in simulating such a phenomenon 3.2 Track transition The effect of track transition on the dynamic response of HSR system is next investigated The parameter measuring the “magnitude” of the transition effect is described by the ratio of the foundation stiffness after and before the transition point A computational study to investigate the combined effects of track irregularity and foundation stiffness ratio on the dynamic behavior of HSR system is carried out The train is assumed to be travelling at a constant velocity of 90 m/s Table shows the parameters of the track irregularities adopted in this study A same track Table Track irregularities Track irregularity Irregularity Irregularity Irregularity Irregularity Amplitude (mm) 0.05 0.10 0.30 0.50 1343007-8 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only Analysis of High-Speed Rail irregularity of wavelength of m is considered for all cases Note that “Irregularity 1” pertains to that of a near smooth track The degree of track irregularity increases from “Irregularity 1” to “Irregularity 4” All these track conditions may be considered to be not as severe than a moderately corrugated track For such track irregularity conditions and same properties of track and foundation listed in Table 4, no jumping wheel phenomenon is expected to occur when the foundation is uniform (n = 1) The aim of this investigation is therefore to determine what degree of track transition will induce the occurrence of the jumping wheel phenomenon Table presents the results showing the occurrence or nonoccurrence of the jumping wheel phenomenon for various track irregularity conditions and foundation stiffness ratios, n The numerical value listed in the table denotes the number of times the wheel jumps in the vicinity of the transition point No jumping wheel phenomenon is found to occur for the near smooth track for all values of n considered There is also no occurrence for all track conditions considered when the degree of track transition is not large (n < 4) However, jumping wheel is found to occur occasionally for certain combinations of track condition and degree of track transition Also, when the degrees of track transition or track irregularity increase, the jumping wheel phenomenon is observed to occur and sustained after the train passes the transition point Figure shows the dynamic amplification factor (DAF) in contact force in the vicinity of the transition point The DAF is computed by taking the ratio of the maximum dynamic contact force to the combined self-weights of car body, bogie, and wheel-set In all cases, it is found that increasing stiffness ratio has the effect of increasing the maximum contact force, which agrees with one of the findings from [Ang and Dai (2013)] It is also observed that the effect of stiffness ratio within the range of 8–16 tends to have smaller effect on the increase in DAF when compared with that within the range of 4–8, which implies that the stiffness ratio smaller than tends to have more impact on the responses of the HSR system Figures and present the contact force distributions along the railhead in the vicinity of the transition point for various track conditions with n = and various foundation stiffness ratios for track “Irregularity 2”, respectively Note that x − x0 = refers to the transition point As can be seen from these figures, the maximum contact Table Occurrence of jumping wheel phenomenon n Irregularity Irregularity Irregularity Irregularity Irregularity 4 16 0 0 0 0 0 S S S S S 1343007-9 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only K K Ang et al Fig Effect of stiffness ratio and track irregularity on DAF Fig Effect of track irregularity on contact force force occurs after the wheel passes the transition point It is also observed from Fig that when n = 4, the contact force attains a zero value momentarily at a location about 2.1 m after passing the transition point indicating that the wheel jumps once at this location For a large stiffness ratio of or 16, the jumping wheel phenomenon is found to occur after the wheel passes the transition point and is observed to be sustained as the train travels over the second foundation, resulting in a sharp increase in the DAF in contact force as shown in Fig It can be seen from Fig that the phenomenon occurred twice due to the existence of a track 1343007-10 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only Analysis of High-Speed Rail Fig Effect of foundation stiffness ratio on contact force transition of stiffness ratio For a larger stiffness ratio of 16, the phenomenon is induced and sustained after the train travelled past the transition point Conclusions In this paper, a computational study on the dynamic response of HSR system using the MEM is carried out Both the situations of a uniform foundation and a transition region are considered The proposed computational model adopts Hertz contact theory to account for the wheel–rail interaction The results obtained using Hertz contact model and linearized contact model are compared and discussed The occurrence of the jumping wheel phenomenon is accounted for and examined In the parametric study on the occurrence of the jumping wheel phenomenon, it was found that the speed of the travelling train and the severity of track irregularity are key factors affecting the occurrence of this phenomenon The jumping wheel phenomenon generally does not occur when either the speed of the train is relatively low or the track surface nearly smooth As to be expected, the dynamic response of the train–track system is found to be significantly higher when there is an occurrence of jumping wheel This has important implication on the track maintenance program It is critical that track maintenance is properly exercised and/or train operational speed be moderated to avoid any occurrence of the jumping wheel phenomenon, especially for old tracks where track corrugation is likely to be severe Also, it is critical that the nonlinear Hertz contact model be adopted to model correctly the wheel–rail interaction, especially when there is strong possibility of the occurrence of the jumping wheel phenomenon In the response study of the train–track system involving railway track transitions, it is found that increasing foundation stiffness ratio has the effect of increasing 1343007-11 2nd Reading September 18, 2013 16:52 WSPC/0219-8762 196-IJCM 1343007 K K Ang et al Int J Comput Methods 2014.11 Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/07/15 For personal use only the maximum contact force In general, it is found that large change in foundation stiffness and higher degree of track irregularity tends to induce the occurrence of the jumping wheel phenomenon Thus, no such phenomenon is observed to occur for near smooth railhead condition for all foundation stiffness ratios considered Similarly, the phenomenon does not occur for smaller values of foundation stiffness ratios even for track irregularity considered moderate The jumping wheel phenomenon is triggered under certain combinations of severity of track irregularity and degree of track transition When both parameters are large enough, the phenomenon is observed to occur sporadically or repeatedly after the train passes the transition point References Ang, K K and Dai, J [2013] “Response analysis of high-speed rail system accounting for abrupt change of foundation stiffness,” J Sound Vib 332, 2954–2970 Dimitrovov´ a, Z and Varandas, J N [2009] “Critical velocity of a load moving on a beam with a sudden change of foundation stiffness: Applications to high-speed trains,” Comput Struct 87, 1224–1232 Esveld, C [2001] Modern Railway Track, 2nd edn (MRT Productions, Duisburg) Koh, C G., Chiew, G H and Lim, C C [2007] “A numerical method for moving load on continuum,” J Sound Vib 300, 126–138 Koh, C G., Ong, J S Y., Chua, D K H and Feng, J [2003] “Moving element for train–track dynamics,” Int J Numer Meth Eng 56, 1549–1567 Koh, C G., Sze, P P and Deng, T T [2006] “Numerical and analytical methods for in-plane dynamic response of annular disk,” Int J Solids Struct 43, 112–131 Krenk, S., Kellezi, L., Nielsen, S R K and Kirkegaard, P H [1999] “Finite elements and transmitting boundary conditions for moving loads,” Proc 4th European Conf Structural Dynamics, Eurodyn’ 99, Praha, June 7–1, Vol 1, pp 447–452 Lei, X Y [2006] “Effects of abrupt changes in track foundation stiffness on track vibration under moving loads,” J Vib Eng 19(2), 195–199 Lei, X Y and Mao, L J [2004] “Dynamic response analysis of vehicle and track coupled system on track transition of conventional high speed railway,” J Sound Vib 271, 1133–1146 Nguyen, V.-H and Duhamel, D [2008] “Finite element procedures for nonlinear structures in moving coordinates Part II: Infinite beam under moving harmonic loads,” Comput Struct 86, 2056–2063 Nielsen, J C O and Igeland, A [1995] “Vertical dynamic interaction between train and track-influence of wheel and track imperfections,” J Sound Vib 187(5), 825–839 Yang, Y B., Yau, J D and Wu, Y S [2004] Vehicle-Bridge Interaction Dynamics: With Applications to High-Speed Railways (World Scientific, Singapore) 1343007-12 ... 05/07/15 For personal use only Analysis of High-Speed Rail Fig Effect of foundation stiffness ratio on contact force transition of stiffness ratio For a larger stiffness ratio of 16, the phenomenon. .. 3E W Rwheel Rrailprof 2(1 − υ )2 (4b) It is to be noted that the linearized contact model is inappropriate in accounting for the jumping wheel phenomenon in view that Eq (4a) indicates that an... 05/07/15 For personal use only Analysis of High-Speed Rail irregularity of wavelength of m is considered for all cases Note that “Irregularity 1” pertains to that of a near smooth track The degree of