Study on the dynamic response of a curved railway track subjected to harmonic loads based on periodic structure theory

Transcription

1 Accepted for publication in Proceedings of the Institution of Mechanical Engineers Part, Journal of Rail and Rapid Transit, doi: / Stud on the dnamic response of a cured railwa track subjected to harmonic loads based on periodic structure theor Weifeng Liu 1*, Linlin Du 1, Weining Liu 1, Daid J. Thompson 2 1. School of Ciil Engineering, Beijing Jiaotong Uniersit, Beijing , China. 2. Institute of Sound and Vibration Research, Uniersit of Southampton, Highfield, Southampton SO17 1BJ, UK Corresponding author Weifeng Liu, School of Ciil Engineering, Beijing Jiaotong Uniersit, No.3 Shanguancun, Haidian District, Beijing , People s Republic of China. wfliu@bjtu.edu.cn Abstract: The dnamic response of a cured railwa track subjected to moing and non-moing harmonic loads is studied in this paper. The track is considered as a cured Timoshenko beam supported b periodicall-spaced discrete fasteners. The displacement and rotation of the cured rail are epressed as the superposition of track modes in the frequenc domain. Periodic structure theor is applied to the equations of motion of a cured track, allowing the dnamic response of the track to be calculated efficientl in a reference cell. The effect of the stiffness and damping of the fasteners, the fastener spacing and the radius of curature on the mobilit and deca rate of the track is analsed for non-moing loads on the rail head. The ibration of the rail under moing loads is also discussed. It is found that the dnamic response of a cured rail with a large radius has the same characteristics as that of a straight track. Howeer, the dnamic response of the track is significantl affected when the radius of curature becomes small. The radius affects the mobilit, the deca rate below 2000 H and the elocit of the rail in the ertical direction when the radius is smaller than about 15 m and for the lateral direction when it is less than about 30 m. Moreoer, the curature has a significant influence on the ertical/lateral cross mobilit, the magnitude of which increases as the radius is reduced. When the radius is larger than 10 m, the lateral ibration amplitude under a moing ertical load and the ertical response to a moing lateral load are inersel proportional to the radius. Kewords: dnamic response; cured track; harmonic load; periodic structure theor; mode superposition 1

2 1 Introduction Vibration induced b railwa or metro trains running in urban areas and transmitted through the ground can cause annoance to residents, and ma also affect historic buildings and sensitie instruments. The ibration leels are often higher in cured sections of line where the dnamic responses in the tunnel and on the ground surface are often greater than those for straight tracks 1. The higher ibration leels are caused b the dnamic interaction of the train and the cured tracks. In the literature there are man models reported for the dnamic behaiour of railwa tracks. Most of these focus on the response of straight tracks, whereas much less attention has been paid to cured tracks because of the higher compleit associated with the modelling of a cured track compared with a straight one. The dnamic behaiour of rails is often studied b representing them as Euler-Bernoulli or Timoshenko beams. An Euler-Bernoulli beam has been found to be acceptable for frequencies below 500 H and a Timoshenko beam for frequencies up to at least 2 kh 2,3. A model based on a cured beam is required for a cured track. The dnamic response of cured beams has been studied for man ears. Both analtical methods and the finite element method hae been emploed in these preious studies. Generall, the finite element method is more applicable to studies of the dnamic response of cured beams with complicated structures than the analtical methods. The finite element method, howeer, remains an approimate approach, and its calculation results are affected b some factors such as the element tpes and the boundar conditions. Moreoer, for a railwa track, an infinite length should generall be considered, which is more difficult to implement in a finite element approach. Although it is sometimes difficult or een impossible to sole some problems using analtical methods, for eample the response of a beam with an irregular cross-section, the analtical methods can proide more theoretical insight into the dnamic behaiour. In the analtical models, some assumptions are made to allow solution of the equations of motion. Proided that the beam cross-section is smmetric, the motion of a cured beam can be decoupled into in-plane (i.e. in the plane of curature) and out-of-plane motions. There hae been a number of analtical studies on cured beams in recent ears. or eample, Yang et al. established a complete theor for treating the ibration of a horiontall cured Euler-Bernoulli beam subjected to a series of moing masses, each of which was simulated as a combination of a graitational force and a centrifugal force 4. Kang et al. proided a concise and efficient method for determining the free ibration of a multi-span circular cured beam with general boundar conditions and supports 5. Yu et al. 6 carried out an analtical stud of the free ibration of a naturall cured and twisted beam with uniform cross-sectional shape using spatial 2

3 cured beam theor based on Washiu s static model 7. Çalım performed the forced ibration analsis of a cured beam on a two-parameter elastic foundation subjected to impulsie loads based on the Timoshenko beam theor 8. Lee analed the in-plane free ibration of circularl cured Timoshenko beams through the pseudospectral method 9. Howson and Jemah calculated the eact out-of-plane natural frequencies of cured Timoshenko beams b the dnamic stiffness method 10. Although the dnamic response of cured beams has been etensiel studied, the application to cured railwa tracks is limited, which is mainl attributed to the higher compleit of cured track models. Kostoasilis et al. established a finite element model of a cured track and compared the dnamic response obtained using straight beam elements and cured beam elements 11. The cured beam model used in their stud added compleit without giing substantial improement for the specific application and therefore the straight element method was preferred. In another paper, Kostoasilis et al. used an analtical model to discuss the ertical/lateral coupling of the rail on a continuous elastic foundation including the effects of initial curature 12. In this model, the track is subjected to a non-moing harmonic load and the solution is obtained in the waenumber domain using the ourier transform method. Ang and Dai gae an analtical solution to the response of a cured railwa track resting on a iscoelastic foundation subjected to a moing load 13,14. In this work, trigonometric functions were emploed as the trial functions to approimate the displacement of the cured rail. Li et al. performed an analtical stud of the dnamic response of a cured track subjected to moing loads, and presented a model of a cured Timoshenko beam periodicall supported b double-laer spring-damping elements 15,16. In this model, the displacement of the rail under moing loads is epressed as the product of the load and the transfer function for the cured track based on the Duhamel integral and the dnamic reciprocit theorem. The transfer function was deried b using the transfer matri method in the frequenc domain. Zhang et al. discussed the dnamic response of a cured rail subjected to a moing train based on Li s research b coupling a two-dimensional ehicle model to the track model 17. Because most railwa tracks are periodicall supported b the sleepers, researchers hae deeloped models of a straight track in which the periodicit of the track is eploited to increase the calculation efficienc. Some papers discuss the dnamic behaiour of track due to non-moing loads. Grassie et al. deeloped a model of a track represented as a periodicall supported Timoshenko beam 18. Gr and Gontier presented a periodic railwa track model including cross-section deformation based on the notion of generalied cross-section displacements of a beam 19. In this model, the deformation of the beam s cross-section is described approimatel in terms of cross-section modes, so the periodic analsis can be efficientl made using a small sied matri equation. Sheng et al. gae a detailed discussion on the propagation and resonance properties 3

4 of the track modelled as an infinitel long periodic Euler-Bernoulli beam 20. Degrande et al. and Clouteau et al. presented a periodic track model as part of a periodic finite element-boundar element model based on the loquet transform 21,22. The dnamic response of track, tunnel and ground were calculated first in a reference cell, and then the response in other cells could be obtained through the inerse loquet transformation. The dnamic response of the track under moing loads has also been calculated through some models based on the periodicit of the track. Based on the work in [21, 22], Gupta et al. discussed the dnamic response of track, tunnel and ground due to moing loads 23,24. Chebli et al. established a periodic track-ground model based on loquet decomposition to predict the ibration of the track and ground 25. Sheng et al. proposed a more general, waenumber-based approach to stud the response of an infinite periodic track under moing harmonic loads 26. In this approach the periodicall supported structure is represented as either a multiple-beam model or a two-and-half-dimensional finite-element model. Ma et al. considered a subwa track as an infinite periodicall supported Euler-Bernoulli beam and set up an analtical model; the ibration of a floating slab track and a general non-ballast track was calculated through this model 27,28. In this model, the displacement of the track in the frequenc domain was epressed as the superposition of track modes. In all the aboe studies, howeer, onl straight tracks were considered. This paper presents an analtical model of a cured, periodicall supported track based on periodic structure theor. The periodicit of the track structure is applied to the equations of motion of a cured track to obtain efficientl the dnamic response in a reference cell of the track. The analtical solutions for a cured track under non-moing and moing harmonic loads are deried b separating the in-plane and out-of-plane motions. The effect of arious parameters on the frequenc response and deca rate of the track due to a non-moing load is analsed, and the ibration of the track under a moing load is also discussed. 2 Dnamic response of a periodic cured track 2.1 Equations of motion of a cured track The cured rail is considered as a cured Timoshenko beam, as shown in ig. 1, in which denotes the subtended angle, and R is the radius of curature. A right-handed coordinate sstem is used, the - and -ais of which coincide with the principal aes of the cross-section, and the -ais is tangential to the centroidal ais of the beam. of each cross-section of the cured beam along the three aes, and u, u and u denote the displacements of the centroid, and are the rotations 4

5 about the three aes. All the deformations are assumed to be small so that linear theor applies. The cured beam is assumed to hae a constant cross-section with negligible warping resistance. ig. 1 Coordinates of cured beam i Let ertical and lateral harmonic forces e t i and l e t, where is the ecitation frequenc, moe along the rail head of the cured track at a speed, as shown in ig. 2. It should be noted that the ertical and lateral forces can be applied at an point on the rail head. The lateral force on the rail head is equialent to a lateral force at the centroid and a moment about the -ais, similarl a ertical force and a moment i e t h acting i 1 le t b for the ertical force (see ig. 3). The rail is supported b a finite number Ns of periodicall-spaced discrete fasteners. The fasteners are considered as springs and dashpots connected to the rail foot in the ertical, lateral and aial directions and torsional springs and dashpots connected at the shear centre of the rail. i l e t i e t L R kr, cr k c h, h k, k, c c ig. 2 Periodic cured track subjected to harmonic loads i l e t B b A i e t h 1 h2 h 3 C S kr, c r k c k c, h h ig. 3 Cross section of a rail (C is the centroid, S the shear centre, A and B the force locations) 5

6 According to the theor of cured Timoshenko beam, the equations of motion can be diided into those for the in-plane (lateral bending and aial) and the out-of-plane (ertical bending and torsional) motions. The equations for the in-plane motion of the cured track can be written as follows 9, Ns u K AG EA u EA u it KAG u ( ) ( ) e ( 2 2 A f 2 hj t sj l 0 t), (1) R R t 2 2 N u s K AG EA u K AG u u EA ( ) ( ) 0 2 A f 2 j t sj R R R, (2) t j1 j1 EI K AG I R 2 2 u u t, (3) where E and G denote the Young s modulus and shear modulus, respectiel, of the rail, A the cross-sectional area, I the second moment of area about the -ais, K the cross-sectional shape factor about the -ais, the densit of the rail, 0 the initial position of the moing load, and sj the position of the j th fastener. The fasteners hae been replaced b forces: f hj and f j are the lateral and aial forces applied on the rail b the j th fastener. Equations (1), (2) and (3) correspond to the lateral displacement, the aial displacement along the -ais and the rotation about the -ais, respectiel. The equations for the out-of-plane motion of the cured track can be written as follows 10,29. u u K AG K AG A f ( t) ( ) e ( t), (4) 2 2 Ns it 2 2 j sj 0 t j1 EI GI u GI EI K AG K AG I R R t 2 2 d d , (5) EI GI EI GI I T ( t) ( ) h e + b e ( t), (6) 2 2 Ns d it it d j sj 1 l 0 R R t j1 where I denotes the second moment of area about the -ais, I0 the polar moment of area, torsional constant, K the cross-sectional shape factor about -ais, Id the fj and T j the ertical force and torsional moment applied on the rail b the j th fastener respectiel. Equations (4), (5) and (6) correspond to the ertical displacement, and the rotations about the -ais and -ais, respectiel. Equations (1)-(3) for the in-plane displacements and rotation of the cured track are independent of equations (4)-(6) for the out-of-plane displacement and rotations apart from the forcing terms, so the in-plane and out-of-plane dnamic responses can be determined separatel. 6

7 2.2 Transformation to the frequenc domain Based on the ourier transformation, the frequenc spectrum of a dnamic response can be analsed. Appling the ourier transformation with respect to time to equations (1)-(3), and introducing the support stiffnesses, the in-plane equations can be obtained in the frequenc domain: 2 * * * N i * s u KAG E A u EA K AG A u ( ) e 2 2 kh u sj l, (7) R R j1 2 * * * * N * u s K AG E A u K AG 2 K AG E A ( ) 0 2 A u 2 k u sj R R R j1, (8) 2 * * u u 2 * E I 2 K AG I K AG 0 R, (9) where ^ is used to indicate epressions in the frequenc domain; damping is introduced b making E * E(1 i ) and * G G(1 i ) comple, in which is the damping loss factor of the rail; the lateral forces applied b the fasteners hae been replaced b introducing the dnamic stiffnesses k k ic and k k ic, in which k h and h h h k are the lateral and aial stiffnesses of the fastener, and ch and c are the corresponding lateral and aial damping coefficients. Similarl the out-of-plane equations in the frequenc domain are obtained as follows. 2 N i * * s u K AG K ( ) e 2 AG A u k u sj, (10) j1 2 * * * * E I G Id u * G Id * 2 E I K 0 2 AG K 2 AG I R R, (11) 2 * * * N i * s 2 0 E I G Id E I h1 l b G Id I 2 0 ( ) e 2 kr sj, (12) R R j1 where the ertical and rotational forces applied b the fasteners hae been replaced b introducing the dnamic stiffnesses k k ic, k k ic, in which k and torsional stiffnesses of the fastener, and of the fastener. r r r k r are the ertical and c and c r are the ertical and torsional damping coefficients If the cured rail is supported b a continuous elastic foundation, the response of the rail can also be calculated. The forces applied on the rail b the fasteners in equations (7)-(12) can be replaced with product of the dnamic stiffness of the foundation and the displacement of the rail, 7

8 for eample f k u in equation (7), where h f k h is the lateral dnamic stiffness of the foundation per unit length. In this calculation, the stiffness and damping of the foundation per unit length are equal to those of the fastener diided b the fastener spacing. 2.3 Periodic structure theor and mode function of the track In this subsection, the periodic structure theor and the mode function of the track are introduced to sole the aboe equations of motion. Based on the periodicit of the track structure along the -ais, periodic structure theor can be used. Assume that the infinitel long track is composed of cells, each with the same properties, and an cell can be chosen as the reference cell. The rail displacements due to a moing harmonic unit load in another cell can be linked to those in the reference cell 30,31, i( ) c / u n L,, e n L u,,, (13) c where is used to indicate the epression in the reference cell, u u,, u,, u, T, L is the length of each cell, and n is the inde of the cells. A function P is defined here: c,, i( / / ),,,, e P u. (14) Substituting equation (13) into equation (14), we can obtain: P n L,, P,,. (15) c Because P is a periodic function, it can be decomposed as a ourier series.,,,,, e n n i P C, (16) n where 2 / n n L, and U,, U,, U, T C is a ector of coefficients. Specificall, the displacements of the rail can be written as (,, ) (, ) C V (,, ) u. (17) n n n (,, ) e n i( / / ) where Vn, which can be called the mode function of the track. In the calculation, 2N+1 modes can be considered, so the displacements are written as N (,, ) (, ) n Vn (,, ) u C. (18) nn N should be enough large to ensure the accurac of the calculation. Based on equation (13), the following boundar conditions of the reference cell should be satisfied. i( ) L / in L e in 0 (19) 8

9 u u (20) i( ) / e L L 0 where in Q, M, N, M, Q, T T represents the internal force ector including the shear force Q, the bending moment M and the aial force N, which are the in-plane terms, and the bending moment M, the shear force Q and the twisting moment T, which are the out-of-plane terms. According to the theor of a cured Timoshenko beam, these internal forces can be written as 9,32,, u u Q, = K AG,, R, M, = EI,, u u, N, = EA, R,, M, = EI, R u, Q, = K AG,,,, T, = GId. R (21) (22) Substituting equation (18) into equations (21) and (22), it can be found that equations (19) and (20) are easil satisfied for each alue of n and hence for the oerall response. 2.4 Solution to equations of motion ig. 4 shows the solution process diagrammaticall. The specific ectors and matrices in ig. 4 will be eplained in the following deriation. 9

10 Propert of periodic structure i( ) c / u n L,, e n L u,, c Define a periodic function i( / / ),,,, e P u ourier series,,, e i P n Cn n u(,, ) (, ) V C (,, ) n n n N u(,, ) (, ) Cn Vn (,, ) nn N ( ) / u(,, ) (, ) Cn Vn (,, ) nn Motion equations in the time domain ourier transform Motion equations in the frequenc domain 1 V m Multipling equations b and integrating the equations oer the reference cell [0,L] K U,,, Motion equations for track modes -i 1 (,, ) e n c L u B(,, ) K(, ) (, ) Inerse ourier transform 1 u(,, ) u(,, ) d t i( +) t e 2π ig. 4 Solution process of equations of motion of a cured track Writing ( ) /, the displacements of the rail from equation (17) can also be written as N (,, ) (, ) n Vn (,, ) u C. (23) nn Multipling both sides of the in-plane equations (7)-(9) b (,, ) 1 e i( m ) m and integrating the equations oer the length 0, L of the reference cell, we can obtain: Ns L 1 i0 im h m ( rj,, ) le e 0 j1 V m N, N EA K AG E A K AG A LU K AG L LU R R * * * * 2 2 * m i 2 m m m i m m 1 k V u d,, (24) 2 2 K * * AG E A K AG A K AG E A LU i LU L R R * * * m m m m m Ns 1 (,, ) m rj 0 j1 k V u, (25) * 2 2 K * * i * AG I K AG E I m L m K AG m LU m Lm 0, (26) R Similarl, the out-of-plane equations become: 10

11 2 2 i K AG A LU K AG L * * m m m m 1 k V u d Ns L 1 i0 im m ( rj,, ) e e 0 j1 G I E I G I R R * * * d * 2 * 2 d K 2 AG I E I m L m i m L m * ik AG m LU m 0, (27), (28) * * * Ns 2 E I * 2 E I G Id i (,, ) I G Id m L m m L m krvm rj R R j1 h L 1l b i0 im e e d. 0 In equations (24), (27) and (29), the integral on the right-hand side can be ealuated as (29) 1 L 0 L 0 m i l m i i l d e 0 0, ( m 0), ( 0) e e =. (30) Because mn, N, in equation (24) Ns 1 (,, ) h m rj j1 k V u V N ( r1) V N ( r 2)... V N ( ) rn ( s V N r1) V N 1( r1)... V N ( 1) U r N V N 1( r1) V N 1( r 2)... V N 1( rn ) k s V N ( r 2) V N 1( r 2)... V N ( 2) U r N1 h V N ( r1) V N ( r 2)... V N ( rn ) V ( ) 1( )... ( ) U N N rn V s s N rn V s N rn s. (31) A similar epression can be obtained for the terms related to the fastener in equations (25), (27) and (29). The in-plane equations (24)-(26) can be written as K, U,,, (32) in in in where, U,, U,,,, U,, U T U = ; K, is the generalied in ( N ) ( N ) ( N ) ( N ) ( N ) ( N ) stiffness matri;, is the force ector, which can also be written as L in in in i 0, e /, (33) in where the j th element of is in in( j) l, ( j N 1) 0, ( j others) =. (34) Similarl, the out-of-plane equations (27)-(29) can be written as K, U,,, (35) out out out where,,,, U,, U,,, T U =. out ( N ) ( N ) ( N ) ( N ) ( N ) ( N ) 11

12 L out out i 0, e /, (36) where the j th element of out is, ( j N 1) out ( j) h1 l b, ( j 5N 3) 0, ( j others) =. (37) The frequenc responses of the rail at an point can be obtained: i 1 (,, ) e n cl uin B(,, ) Kin(, ) in (, ) -, (38) i 1 (,, ) e n cl uout B(,, ) Kout (, ) out (, ) -, (39) where B (,, ) is the mode matri, gien b B i( N- ) i( N- ) e e i( N- ) i( N- ) (,, ) 0 0 e e 0 0 i( N- ) i( N- ) e e. (40) Through the inerse ourier transformation, the dnamic responses of the rail in the time domain due to a moing harmonic force can be obtained as: 1 1 u(, t, ) u(,, ) u(,, ) 1 2π it i( +) t e d e d 2π 2π L - i ( nc L0 ) 1 ' i t it e B(,, )K(, ) e d e, (41) where K(, ) can be K in(, ) or K out (, ), and can be or in out. Equation (41) gies the dnamic response of the rail subjected to the moing harmonic load. If we set 0, the response of the rail due to a non-moing harmonic load applied at =0 can be epressed as u(, t, ) 1 L B(,, )K(, ) i ( nc L0 ) 1 ' it e d e 2π -. (42) rom equation (42), the amplitude of the harmonic response of the rail can be calculated b numerical integration: 1 u(, ) L a B(,, )K(, ) M - i j( ncl 0 ) 1 ' e j j j 2π j1 (43) where a1 0.5, a 0.5 and a 1 ( j (1, M)). should be sufficientl small to ensure the required accurac. M j 12

13 3 Model alidation To erif the alidit of the analtical model described aboe, two comparisons are made between this model and solutions proided in the literature, [11] and [13]. In [11], the ertical displacement of the rail due to a moing load on a cured track was calculated using a finite element model. The parameters of this calculation are shown in Table 1. Table 1 Parameters of calculation from [11] Young s modulus of rail E N/m 2 Shear modulus of rail G N/m 2 Cross-sectional area A m 2 Second moment of area of rail for ertical bending Second moment of area of rail for lateral bending I m 4 I m 4 Polar moment of inertia of rail I m 4 0 Damping loss factor of rail 0.1 Densit of rail 7850 kg/m 3 Vertical stiffness of fastener astener spacing L Radius of curature R Speed of load Load magnitude k 20 MN/m 0.6 m 20 m 20 m/s 1 N The results are compared with those from [11] in ig. 5. This shows the displacement at mid-span between two fasteners due to a harmonic moing load in two ecitation frequenc cases of 0 H and 20 H. It can be seen that the results calculated b the present model show a good agreement with the numerical solutions from [11]. 13

14 (a) Displacement (m/n) Literature [11] Present model Displacement (m/n) Literature [11] Present model Time (s) Time (s) ig. 5 Displacement at mid-span due to a harmonic moing load at (a) 0 H and 20 H A second comparison is made with results from Reference [13]. This gae an analtical solution of response of a cured track continuousl supported b a Winkler foundation under moing loads. Unlike [11], the results include the lateral and torsional response. The calculation parameters from [13] are shown in Table 2. Table 2 Parameters of calculation in [13] Young s modulus of rail E N/m 2 Shear modulus of rail G N/m 2 Cross-sectional area A m 3 Second moment of area of rail for ertical bending Second moment of area of rail for lateral bending Torsional constant of rail I m 4 I m 4 I m 4 d Densit of rail 7800 kg/m 3 Vertical stiffness of foundation Vertical damping coefficient of foundation Lateral and aial stiffnesses of foundation, k c k h and Lateral and aial damping coefficients of foundation, Torsional stiffness of foundation Torsional damping coefficient of foundation Radius of curature R Speed of load k r c r k c h and c 10 MN/m 2.45kNs/m 5.5 MN/m 1.82kNs/m 71kNm/rad 4.84Nms/rad 1000 m 250 km/h 14

15 The lateral, ertical and torsional displacements of the rail under a combination of moing loads of grait and centrifugal forces are compared in ig. 6. As can be seen from ig. 6, ecellent agreement is found for responses of the cured rail calculated b models in [13] and proposed in this paper. (a) Lateral displacement (mm) Literature [13] Present model Distance(m) Vertical displacement (mm) Literature [13] Present model Distance(m) (c) Torsional displacement (rad) Literature [13] Present model Distance(m) ig. 6 (a) Lateral, ertical and (c) torsional displacements of the rail under a moing load calculated b models in [13] and proposed in this paper 4 Mobilit and deca rate of the track due to non-moing harmonic loads In this section, the response of the track subjected to non-moing ertical and lateral harmonic unit forces is discussed. The response at the ecitation point is epressed as the mobilit, that is the elocit due to a unit harmonic force. In addition the deca rates of ibration along the track are 15

16 determined. The ertical response at Position A and the lateral one at Position B on the rail head due to a ertical force applied at Point A or a lateral one at Point B (ig. 3) are calculated. In this calculation, Point A is placed on the centreline of the rail head (b=0). Parameters are chosen to correspond to a general non-ballast metro track. In this track, the GB60 rail and the DTVI2 fastener are used. A fastener spacing of 0.6 m and a track radius of 300 m are considered in the following calculations. Unless otherwise stated the parameters are as listed in Table 3. The fastener is modelled as a single laer of springs and dashpots. The length of the reference cell is chosen as the fastener spacing, so a single fastener is included in the reference cell. Table 3 Parameters of the metro track Young s modulus of rail E N/m 2 Shear modulus of rail G N/m 2 Cross-sectional area A m 3 Second moment of area of rail for ertical bending Second moment of area of rail for lateral bending Torsional constant of rail I m 4 I m 4 I m 4 d Polar moment of inertia of rail I m 4 0 Cross-sectional shear factor for ertical bending Cross-sectional shear factor for lateral bending K K Damping loss factor of rail 0.01 Densit of rail 7850kg/m 3 Vertical stiffness of fastener k Vertical damping coefficient of fastener Lateral and aial stiffnesses of fastener, c k h and Lateral and aial damping coefficients of fastener, Torsional stiffness of fastener Torsional damping coefficient of fastener astener spacing L Radius of curature R k r c r k c h and c 60MN/m 0.04MNs/m 25 MN/m 16.7 kns/m knm/rad knms/rad 0.6 m 300 m If a ertical harmonic force is applied on the centreline of the rail head, onl out-of-plane dnamic responses of the rail are obtained, which include ertical and torsional responses. Howeer, the torsional response around the shear centre of the rail leads to a lateral response at the rail head. On the other hand, a lateral force applied on the rail head can be decomposed into a lateral force at 16

17 the centroid and a torsional moment, so the dnamic response consists of both in-plane and out-of-plane responses according to ig. 3. The ertical and lateral point mobilities and the cross mobilit at the rail head are shown in ig. 7. In each case the mobilit is shown at mid-span between two fasteners and directl aboe a fastener. rom ig. 7a, the ertical mobilit has a peak at about 200 H, which corresponds to the resonance of the rail mass on the ertical fastener stiffness. At about 1100 H there is a sharp peak at mid-span and a corresponding dip aboe a fastener, which is the pinned-pinned resonance 3,18 at which half a bending waelength fits within one sleeper span. Moreoer, the mobilit of the cured rail supported b a continuous elastic foundation is also shown. Compared with the result for discrete fasteners, the results from this model show the mean trend; the pinned-pinned resonance disappears for the continuous foundation, while the first peak has the same frequenc. ig. 7c shows the corresponding lateral mobilit. This has a higher amplitude than the ertical mobilit due to the smaller lateral fastener stiffness, lower bending stiffness and additional fleibilit introduced b the torsion of the rail. The first peak in this case corresponds to the resonance of the rail mass on the lateral stiffness of the fastener. Two peaks are found at mid-span, at about 530 and 640 H, caused b the pinned-pinned resonances of lateral and torsional motions of the rail. Again, corresponding dips are found aboe a fastener. Higher order pinned-pinned modes are seen at higher frequencies. The result for the continuousl supported rail is again shown. ig. 7b shows the ertical/lateral cross mobilit of the rail. The magnitude is much smaller than either ertical or lateral point mobilities. The first peak here occurs at a similar frequenc to those found in the ertical and lateral mobilit. Both the ertical and torsional pinned-pinned modes of the rail appear in the cross mobilit. 17

18 (a) Vertical mobilit (m/sn) mid-span aboe a fastener Continuous foundation Lateral mobilit (m/sn) 10-9 mid-span aboe a fastener Continuous foundation requenc (H) (c) requenc (H) 10-4 Lateral mobilit (m/sn) mid-span aboe a fastener Continuous foundation requenc (H) ig.7 (a) Vertical and lateral mobilit at rail head under a ertical force, and (c) lateral mobilit under a lateral force (0.6 m fastener spacing, 300 m radius) The effects on the frequenc response of changes in the fastener stiffness, the fastener damping, the fastener spacing and the curature of the track are net discussed. The ertical and lateral (cross) mobilit at the rail head, both aboe a fastener and at mid-span, due to a ertical force are shown for different alues of ertical fastener stiffness in ig. 8. rom ig. 8a, the frequenc of the first peak rises with increasing stiffness and the response amplitude at the peak becomes larger. Howeer, the frequenc and amplitude at the pinned-pined resonance are not affected. ig. 8b shows the ertical/lateral cross mobilit of the rail which shows similar trends. The ertical and lateral responses at the rail head under a lateral force are shown for changes in lateral fastener stiffness in ig. 9a and 9b. Similar trends are found to those seen in ig. 8 for the effect of ertical stiffness. Again, the fastener stiffness has little influence on the pinned-pinned resonances of the rail. 18

19 (a) Vertical mobilit (m/sn) k =30 MN/m, mid-span k =60 MN/m, mid-span k =120 MN/m, mid-span k =30 MN/m, aboe a fastener k =60 MN/m, aboe a fastener k =120 MN/m, aboe a fastener Lateral mobilit (m/sn) k =30 MN/m, mid-span k =60 MN/m, mid-span k =120 MN/m, mid-span k =30 MN/m, aboe a fastener k =60 MN/m, aboe a fastener k =120 MN/m, aboe a fastener requenc (H) requenc (H) ig.8 (a) Vertical and lateral mobilit (cross) at rail head under a ertical force, with changes of the ertical fastener stiffness (0.6 m fastener spacing, 300 m radius) (a) 10-4 kh =12.5 MN/m, mid-span k h =25 MN/m, mid-span Vertical mobilit (m/sn) k h =12.5 MN/m, mid-span k h =25 MN/m, mid-span k h =50 MN/m, mid-span k h =12.5 MN/m, aboe a fastener k h =25 MN/m, aboe a fastener k h =50 MN/m, aboe a fastener Lateral mobilit (m/sn) k h =50 MN/m, mid-span k h =12.5 MN/m, aboe a fastener k h =25 MN/m, aboe a fastener k h =50 MN/m, aboe a fastener requenc (H) requenc (H) ig.9 (a) Vertical (cross) and lateral mobilit at rail head under a lateral force, with changes of the lateral fastener stiffness (0.6 m fastener spacing, 300 m radius) ig. 10 shows the effect of changes in the fastener damping coefficient on the ertical and lateral mobilit at the rail head. As the damping is increased, the amplitude of the mobilit at the first peak is reduced. The pinned-pinned resonance frequencies become sharper with increasing damping as the fastener constrains the rail more. or low damping some oscillation can be seen in the mobilit which is caused b the truncation of the number of fasteners. 19

20 (a) 10-4 Vertical mobilit (m/sn) c =0.01 MNs/m, mid-span c =0.04 MNs/m, mid-span c =0.07 MNs/m, mid-span c =0.01 MNs/m, aboe a fastener c =0.04 MNs/m, aboe a fastener c =0.07 MNs/m, aboe a fastener Lateral mobilit (m/sn) c h =8.35 kns/m, mid-span c h =16.7 kns/m, mid-span c h =33.4 kns/m, mid-span c h =8.35 kns/m, aboe a fastener c h =16.7 kns/m, aboe a fastener c h =33.4 kns/m, aboe a fastener requenc (H) requenc (H) ig.10 (a) Vertical mobilit at rail head under a ertical force, and lateral mobilit under a lateral force, with changes of the fastener damping (0.6 m fastener spacing, 300 m radius) The effect of changing the fastener spacing on the ertical and lateral point mobilit of the rail is shown in ig. 11. The fastener spacing affects the pinned-pinned frequencies significantl. These frequencies drop and the corresponding amplitudes rise as the spacing is increased. Moreoer, as the fastener spacing is increased, the effectie stiffness (per unit length) of the support is reduced so that the frequenc of the first peak reduces and the mobilit amplitude at the first peak increases. (a) 10-4 Vertical mobilit (m/sn) L=0.45m, mid-span L=0.6m, mid-span L=0.9m, mid-span L=0.45m, aboe a fastener L=0.6m, aboe a fastener L=0.9m, aboe a fastener requenc (H) Lateral mobilit (m/sn) L=0.45m, mid-span L=0.6m, mid-span L=0.9m, mid-span L=0.45m, aboe a fastener L=0.6m, aboe a fastener L=0.9m, aboe a fastener requenc (H) ig.11 (a) Vertical mobilit at rail head under a ertical force, lateral mobilit under a lateral force, with changes of the fastener spacing (300 m radius) rom the aboe results, which are all for a cured track with a radius 300 m, it can be seen that the effects of changes to the stiffness, the damping and the spacing of the fasteners are similar to the effects seen for a straight track

21 The effect of changing the curature on the mobilit of the rail is shown in igs 12 and 13. The radius of curature has a negligible influence on the ertical and lateral mobilit of the rail when the radius is larger than about 15 m and 30m, respectiel. The result for a straight track is not shown but has almost the same mobilit as the cured track with 300 m radius in igs 12 and 13b. Additionall, the torsional pinned-pinned resonance appears in the ertical mobilit at about 640 H in the case of er small radii (ig. 12a and 12b). The radius of cures on metro or railwa lines is usuall at least 100 m, in which case changes of the radius do not affect the mobilit of the track. On tram tracks smaller radii are found but the are still in the region where the influence is negligible. or er small radii there are some changes, although such small radii are not found in practice. Howeer, the cross mobilit at the rail head is influenced greatl b the curature (ig. 13a). The amplitude of this mobilit increases significantl as the radius is reduced. The cross mobilit for the straight track is also shown in ig. 13a. This is much lower than the results for a cured track. The lateral force at the rail head causes a moment about the -ais (see ig. 3). The ertical displacement and rotation of the rail are produced for the cured track, while onl the rotation of the rail occurs for the straight track. Therefore the cross mobilit has a different trend for the cured and straight tracks. (a) Vertical mobilit (m/sn) R=3m R=6m R=15m R=30m R=300m Vertical mobilit (m/sn) R=3m R=6m R=15m R=30m R=300m requenc (H) requenc (H) ig.12 Vertical mobilit at rail head at (a) mid-span and aboe a fastener under a ertical force with changes of the curature (0.6 m fastener spacing) 21

22 (a) Vertical mobilit (m/sn) Lateral mobilit (m/sn) 10-4 R=3m R=6m R=15m R=30m R=300m R=3m R=6m R=15m R=30m R=300m Straight track requenc (H) requenc (H) ig. 13 (a) Vertical (cross) and lateral mobilit at rail head at mid-span under a lateral force with changes of the curature (0.6 m fastener spacing) ig. 14 shows the track deca rates obtained for different radii of curature. The deca rate (db/m) is ealuated here according to the standard EN b determining the transfer mobilit to positions along the rail due to a point force at mid-span. The deca rate is gien b: DR nma n A A n n where A n is the mobilit at position n along the track, A0 (44) is the mobilit at the ecitation point 0 at mid-span, and n is the distance between n and 0. The curature has an effect on the deca rates below 2000 H to some etent, while the are unaffected at high frequencies. The deca rates appear to increase as the radius reduces. There is a peak at about 640 H in the ertical deca rate due to the torsional pinned-pinned resonance of the rail for small radii. Howeer, when the radius is greater than about 15 m it does not affect the ertical deca rate an more; similarl when the radius is greater than about 30 m it does not affect the lateral deca rate. These effects are mainl determined b the influence of the curature on the point mobilit, which appears in the equation for the deca rate (equation (44)). The track deca rates for a straight track, not shown, are irtuall identical to those for a cured track with 300 m radius in ig

23 (a) Vertical deca rate (db/m) 1 R=3m R=6m R=15m R=30m R=300m Lateral deca rate (db/m) 10 R=3m R=6m R=15m R=30m R=300m requenc (H) requenc (H) ig. 14 (a) Vertical and lateral deca rates with changes of the curature (0.6 m fastener spacing) 5 Velocit of the rail due to moing harmonic loads The ibration elocit of the rail subjected to moing ertical and lateral harmonic unit forces is discussed in this section. The ertical elocit at Position A and the lateral elocit at Position B under a ertical moing load at Point A or a lateral one at Point B (ig. 3) are calculated. The ertical load moes along the centreline of the rail head. The parameters of the track used in these calculations are the same as in the preious section, as listed in Table 3. A fastener spacing of 0.6m and a track radius of 300 m are considered. Unless otherwise stated the load speed is 100 km/h and the ecitation frequenc is 200 H, which is chosen to correspond to the peak in the ertical rail mobilit. igs 15 and 16 show the frequenc content of the elocit at the mid-span point on the rail for different load speeds 50 and 100 km/h. ig 15 shows the results due to a ertical moing load and ig. 16 shows the corresponding results for a lateral load. The frequenc content of the response of a cured rail has similar characteristics to that of straight track under a moing harmonic load 28. The rail has a relatiel large dnamic response at and near the ecitation frequenc of the moing load, and the elocit awa from this frequenc attenuates quickl. There are two peaks, aboe and below the ecitation frequenc, due to the Doppler effect in the rail. or eample, the frequencies of the two peaks are 193 H and 207 H for the load speed 100 km/h in ig. 15. The elocit in the icinit of the ecitation frequenc drops as the load speed increases, while the width of the two peaks increases and the leel rises at other frequencies. Moreoer, some other small peaks can be 23

24 found as well as the aboe-mentioned two large peaks. These small peaks are caused b the discrete fasteners. (a) Vertical elocit (m/s/h) =50km/h =100km/h Lateral elocit (m/s/h) =50km/h =100km/h requenc (H) requenc (H) ig.15 requenc content of (a) ertical and lateral elocit of rail under a ertical moing load at 200 H, with changes of the load speed (0.6 m fastener spacing, 300 m radius) (a) Vertical elocit (m/s/h) =50km/h =100km/h Lateral elocit (m/s/h) =50km/h =100km/h requenc (H) requenc (H) ig.16 requenc content of (a) ertical and lateral elocit of rail under a lateral moing load at 200 H, with changes of the load speed (0.6 m fastener spacing, 300 m radius) The rail elocit obtained in the case of periodicall-spaced discrete fasteners is compared with the result for a continuous foundation in ig. 17. The hae the similar dnamic behaiour at and near the ecitation frequenc. Howeer the small peaks awa from this frequenc are not found for the continuous foundation, which confirms that the are the result of the discrete propert of the fasteners. 24

25 (a) Vertical elocit (m/s/h) Discrete fasteners Continuous foundation Lateral elocit (m/s/h) Discrete fasteners Continuous foundation requenc (H) requenc (H) ig. 17 requenc content of (a) ertical and lateral elocit of rail under a ertical moing load at 200 H in the cases of periodicall discrete fasteners and the continuous foundation (0.6 m fastener spacing, 300 m radius, 100 km/h load speed) To show the effect of different ecitation frequencies, the maimum ertical and lateral elocit amplitudes obtained in the time domain for a ertical moing load are shown in ig. 18 as a function of ecitation frequenc. It can be seen that these elocit amplitudes hae a similar tendenc to the mobilit under the non-moing harmonic load (ig. 7). Howeer, as a result of the moing load, the peak at the pinned-pinned frequencies for the mid-span case is split into two peaks, whereas aboe a fastener the peaks and dips associated with the pinned-pinned frequencies that were found in the mobilit disappear. (a) 10-4 Vertical elocit (m/s) mid-span aboe a fastener Lateral elocit (m/s) mid-span aboe a fastener Ecitation frequenc (H) 10-9 Ecitation frequenc (H) ig. 18 Maimum of (a) ertical and lateral rail elocit amplitude for different ecitation frequencies under a ertical moing load in the time domain (0.6 m fastener spacing, 300 m radius, 100 km/h load speed) 25

26 To see the effect if changing the radius of the curature, the frequenc content of the rail elocit for a moing ertical load at 200 H is shown in ig. 19 for different alues of the radius. Equialent results for a moing lateral load are gien in ig. 20. The curature has little influence on the ertical rail elocit for a ertical load (ig. 19a) or the lateral rail elocit for a lateral load (ig 20b). Howeer, the lateral response to a ertical load (ig. 19b) and ertical response to a lateral load (ig. 20a) are greatl affected b the curature. or a straight track, different from the ertical response to a lateral load, the lateral response to a ertical load is ero as the ertical load at the centreline of the rail head does not cause a moment about the -ais. (a) Vertical elocit (m/s/h) R=3m R=6m R=15m R=30m R=300m requenc (H) Lateral elocit (m/s/h) R=3m R=6m R=15m R=30m R=300m requenc (H) ig. 19 requenc content of (a) ertical and lateral elocit of rail under a ertical moing load at 200 H, with changes of the curature (0.6 m fastener spacing,100 km/h load speed) (a) Vertical elocit (m/s/h) R=3m R=6m R=15m R=30m R=300m Straight track Lateral elocit (m/s/h) R=3m R=6m R=15m R=30m R=300m requenc (H) requenc (H) ig. 20 requenc content of (a) ertical and lateral elocit of rail under a lateral moing load at 200 H, with changes of the curature (0.6 m fastener spacing, 100 km/h load speed) 26

27 ig. 21 shows the maimum elocit amplitudes from the time domain at the mid-span position under a ertical and a lateral moing load, with an ecitation frequenc of 200 H. These results are plotted against the radius of curature. The maimum elocit amplitude becomes larger for small radii of curature apart from the lateral response to a lateral load which reduces. The maimum elocit is greater for a larger fastener spacing due to the reduction in support stiffness per unit length of track. or the ertical elocit under the ertical load, the maimum response is independent of the curature for radii greater than about 15m, while for the lateral elocit under the lateral load it is affected significantl b curature for radii less than about 30 m. The curature has the same effect on the elocit of the rail as on the mobilit due to a non-moing harmonic load (see igs 12 and 13). When the radius of curature is larger than 10 m, the maimum ertical responses to a lateral force (and ice ersa) are inersel proportional to the radius. (a) 6 Vertical elocit (m/s) m fastener spacing 0.6 m fastener spacing 0.9 m fastener spacing Lateral elocit (m/s) 0.45 m fastener spacing 0.6 m fastener spacing 0.9 m fastener spacing 1 Radius (m) 1 Radius (m) (c) (d) 5 Vertical elocit (m/s) 0.45 m fastener spacing 0.6 m fastener spacing 0.9 m fastener spacing Lateral elocit (m/s) m fastener spacing 0.6 m fastener spacing 0.9 m fastener spacing 1 Radius (m) 2 1 Radius (m) ig. 21 Maimum of time histor of (a) ertical and lateral elocit amplitude of rail under a 27

28 ertical moing load at 200 H, and (c) ertical and (d) lateral elocit amplitude under a lateral moing load at 200 H, with changes of the curature (100 km/h load speed) 6 Conclusions An analtical approach has been proposed to determine the response of a cured track subject to a non-moing or a moing harmonic load. The rail is modelled as a cured Timoshenko beam supported b periodicall spaced discrete fasteners, and the dnamic responses in the ertical and lateral directions are taken into account. The displacement of the cured track in the frequenc domain is epressed as the superposition of track modes which are associated with the ourier series representation. The dnamic response of the cured track can be calculated efficientl using the periodic structure theor. The effect of arious parameters on the dnamic behaiour of the track under non-moing and moing harmonic loads is discussed; these include the stiffness and damping of the fasteners, the fastener spacing, the radius of curature, and the ecitation speed. The effects of aring the stiffness, damping and spacing of the fasteners on the dnamic response of the cured track with a large radius are similar to those found b preious authors for a straight track. Howeer, when the radius of curature is er small it has some influence on the dnamic behaiour of the track to some etent. Specificall, the radius significantl affects the ertical mobilit of the cured rail when it is smaller than about 15 m and the lateral mobilit when it is smaller than about 30 m. Coupling between the ertical bending and torsion of the rail affects the ertical mobilit for the cured track when the radius is less than about 15 m. Moreoer, the curature has a significant influence on the ertical/lateral cross mobilit, the magnitude of which increases as the radius becomes small. The curature leads to coupling of ertical and torsional motions and of lateral and longitudinal motions. The curature has an effect on the track deca rate below 2000 H. The deca rates increase as the radius is reduced for both ertical and lateral ecitation. Because the track deca rate (according to the standard [33]) depends on the point mobilit of rail, these effects are mainl determined b the influence of the curature on the point mobilit. The frequenc content of the dnamic response of the cured rail under moing harmonic loads has similar characteristics to those of a straight track. The curature has little influence on the 28

29 ertical or lateral elocities of the rail for forces in the corresponding direction, whereas it has significant effect on the ertical elocit due to a lateral force and ice ersa. When the radius is larger than 10 m, the maimum ertical/lateral cross amplitudes in the time domain are found to be inersel proportional to the radius. urthermore, the amplitude of the ertical rail elocit for a ertical load is affected b the curature when the radius is less than about 15 m, and the lateral ibration under a lateral load is affected when the radius is less than about 30 m. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. unding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is part of a research project supported b National Natural Science oundation of China (grant no ). References [1] Yuan Y, Liu WN, Liu W. Propagation law of ground ibrations in the cure section of metro based on in-situ measurement. China Railwa Science 2012; 33(4): [2] Knothe KL, Grassie SL. Modeling of railwa track and ehicle/track interaction at high frequencies. Vehicle Sstem Dnamics1993; 22(3-4): [3] Thompson DJ. Railwa noise and ibration: mechanisms, modelling and means of control. Oford: Elseier, [4] Yang Y, Wu C and Yau J. Dnamic response of a horiontall cured beam subjected to ertical and horiontal moing loads. Journal of Sound and Vibration 2001; 242 (3): [5] Kang B, Riedel CH, Tan CA. ree ibration analsis of planar cured beams b wae propagation. Journal of Sound and Vibration 2003; 260: [6] Yu A, Yang J and Nie G. Analtical formulation and ealuation for free ibration of naturall cured and twisted beams. Journal of Sound and Vibration 2010; 329: [7]Washiu K. Some considerations on a naturall cured and twisted slender beam. Journal of 29