DSpace at VNU: Vertical dynamic response of non-uniform motion of high-speed rails

DSpace at VNU: Vertical dynamic response of non-uniform motion of high-speed rails tài liệu, giáo án, bài giảng , luận v...

Journal of Sound and Vibration ] (]]]]) ]]]–]]] Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Vertical dynamic response of non-uniform motion of high-speed rails Minh Thi Tran a, Kok Keng Ang a,n, Van Hai Luong b a b Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore Department of Civil Engineering, Ho Chi Minh City University of Technology, Viet Nam a r t i c l e i n f o abstract Article history: Received August 2013 Received in revised form 27 May 2014 Accepted 29 May 2014 Handling Editor: S Ilanko In this paper, a computational study using the moving element method (MEM) is carried out to investigate the dynamic response of a high-speed rail (HSR) traveling at nonuniform speeds A new and exact formulation for calculating the generalized mass, damping and stiffness matrices of the moving element is proposed Two wheel–rail contact models are examined One is linear and the other nonlinear A parametric study is carried out to understand the effects of various factors on the dynamic amplification factor (DAF) in contact force between the wheel and rail such as the amplitude of acceleration/deceleration of the train, the severity of railhead roughness and the wheel load Resonance in the vibration response can possibly occur at various stages of the journey of the HSR when the speed of the train matches the resonance speed As to be expected, the DAF in contact force peaks when resonance occurs The effects of the severity of railhead roughness and the wheel load on the occurrence of the jumping wheel phenomenon, which occurs when there is a momentary loss of contact between the wheel and track, are investigated & 2014 Elsevier Ltd All rights reserved 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 rails (HSRs) more attractive as an alternative to other modes of transportation for long distance travel The HSR has been investigated as a track beam resting on a visco-elastic foundation subject to moving loads varying both in time and space As early as 1926, Timoshenko [1] proposed the use of a moving coordinate system to obtain the quasisteady-state solution of an infinite beam resting on an elastic foundation subject to a constant load moving at a constant velocity The Fourier Transform Method (FTM) is used for solving the differential equation Obtaining analytical solutions however become cumbersome and difficult when dealing with complex HSR modeled as a multi-degree of freedom system with multiple contact points or where there are moving loads that involve acceleration/deceleration The Finite Element Method (FEM) is a well-established numerical method widely used to solve many complicated problems, including problems involving moving loads For example, Frýba et al [2] presented a stochastic finite element analysis of an infinite beam resting on an elastic foundation subject to a constant load traveling at constant speed Another n Corresponding author E-mail address: ceeangkk@nus.edu.sg (K.K Ang) http://dx.doi.org/10.1016/j.jsv.2014.05.053 0022-460X/& 2014 Elsevier Ltd All rights reserved Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] work carried out based on the FEM was made by Thambiratnam and Zhuge [3] They performed a dynamic analysis of a simply supported beam resting on an elastic foundation subjected to moving point loads and extended the study to the analysis of a railway track modeled as an infinitely long beam Various researchers have investigated the problem of loads traveling at non-uniform velocities Suzuki [4] employed the energy method to derive the governing equation of a finite beam subject to traveling loads involving acceleration Involved integrations are carried out using Fresnel integrals and analytical solutions are presented The vibration response of a train– track–foundation system resulting from a vehicle traveling at variable velocities over finite track has been investigated by Yadav [5] Analytical solutions were obtained and the response characteristics of the system examined Karlstrom [6] used the FTM to obtain analytical solutions for the investigation of ground vibrations due to accelerating and decelerating trains traveling over an infinitely long track In dealing with moving load problems, the FEM encounters difficulty when the moving load approaches the boundary of the finite domain and travels beyond the boundary These difficulties can be overcome by employing a large enough domain size but at the expense of significant increase in computational time In an attempt to overcome the complication encountered by FEM, Krenk et al [7] proposed the use of FEM in convected coordinates, similar to the moving coordinate system proposed by Timoshenko [1], to obtain the response of an elastic half-space subject to a moving load The key advantage enjoyed by this approach is its ability to overcome the problem due to the moving load traveling over a finite domain Andersen et al [8] gave an FEM formulation for the problem of a beam on a Kelvin foundation subject to a harmonic moving load Koh et al [9] adopted the idea of convected coordinates for solving train–track problems, and named the numerical algorithm as the moving element method (MEM) The method was subsequently applied to the analysis of inplane dynamic response of annular disk [10] and moving loads on a viscoelastic half-space [11] Ang and Dai [12] and Ang et al [13] applied the MEM to investigate the “jumping wheel” phenomenon in high-speed train motion at constant velocity over a transition region where there is a sudden change of foundation stiffness The phenomenon occurs when there is momentary loss of contact between train wheel and track The effects of various key parameters such as speed of train, degree of track irregularity and degree of change of foundation stiffness at the transition region were examined Safety concerns during the acceleration and deceleration phases of a high-speed train journey have not been adequately addressed in the literature One major concern is the possible occurrence of resonance of the system when the frequency of the external force, in this case the rail corrugation, coincides with the natural frequency of a significant vibration mode of the system When this happens, the response of the system is dynamically amplified and becomes significant large This paper is concerned with a computational study of the dynamic response of HSR systems involving accelerating/decelerating trains using the MEM A new and exact formulation for calculating the generalized structural matrices of the moving element is proposed Parametric study is performed to understand the effects of various factors on the dynamic amplification factor in contact force between the wheel and rail such as the amplitude of acceleration/deceleration of the train, the severity of railhead roughness and the wheel load As the dynamic response of the track depends significantly on the contact between wheel and track, this study is also concerned with examining the suitability of two contact models Formulation and methodology The HSR system comprises of a train traversing over a rail beam in the positive x-direction The origin of the fixed x-axis is arbitrarily located along the beam However, for convenience, its origin is taken such that the train is at x ¼ when t ¼ The velocity and acceleration of the train at any instant are v and a, respectively The railhead is assumed to have some imperfections resulting in the so-called “track irregularity” The moving sprung-mass model, as shown in Fig 1, is employed to model the train The topmost mass m1 represents the car body where the passengers are The car body is supported by the bogie of mass m2 through a secondary suspension system modeled by the spring k1 and dashpot c1 The bogie is in turn supported by the wheel-axle system of mass m3 through a primary suspension system modeled by the spring k2 and dashpot c2 The contact between the wheel and rail beam is modeled by the contact force F c The rail beam rests on a viscoelastic foundation comprising of vertical springs k and dashpots c The vertical displacement of the track is denoted by y, while the vertical displacements of the car body, bogie and wheel-axle are denoted by u1 , u2 and u3 , respectively The governing equation of motion of the rail beam, which is modeled as an infinite Euler–Bernoulli beam resting on a viscoelastic foundation subject to a moving train load, is given by EI ∂4 y y y ỵ m ỵc ỵ ky ¼ À F c δðx À sÞ ∂t ∂x4 ∂t (1) where E, I and m are Young's modulus, second moment of inertia, and mass per unit length of the rail beam, respectively; t denotes time; s the distance traveled by the train at any instant t; and δ the Dirac-delta function The moving element method was first proposed with the idea of attaching the origin of the spatial coordinates system to the applied point of the moving load Fig also shows a traveling r-axis moving at the same speed as the moving load The relationship between the moving coordinate r and the fixed coordinate x is given by r ¼ xÀs (2) Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig HSR model In view of Eq (3), the governing equation in Eq (1) may be rewritten as     ∂4 y ∂ y y y y y y a ỵ ỵc v ỵ ky ẳ F c rị EI ỵ m v2 2v rt r t ∂t ∂r ∂r ∂r (3) By adopting Galerkin's approach and procedure of writing the weak form in terms of the displacement field, the formulation for general mass Me , damping Ce and stiffness Ke matrices of the moving element can be proposed: RL Me ¼ m NT N dr RL RL Ce ¼ À 2mv NT N;r dr þc NT N dr (4) RL RL RL RL Ke ẳ EI NT;rr N;rr dr ỵ mv2 NT N;rr dr ma ỵ cvị NT N;r dr ỵ k NT N dr where ị;r denotes partial derivative with respect to r and ðÞ;rr denotes second partial derivative with respect to r For beam elements, it is common to use the shape function N based on Hermitian cubic polynomials Considering the special case in which the train traverses at a constant velocity V, i.e a ¼ 0; v ¼ V, Eq (4) reduces to RL Me ¼ m NT N dr Z L Z L Ce ẳ 2mV NT N;r dr ỵ c NT N dr Z Ke ¼ EI L NT;rr N;rr dr ỵ mV Z L 0 T N N;rr dr À cV Z L NT N;r dr ỵ k Z L NT N dr (5) It can be seen that the element mass, damping and stiffness matrices derived in Eq (5) are identical to the matrices derived by Koh et al [9] As the dynamic response of the train–track system depends significantly on the accuracy in modeling the contact between the wheel and track, this study will evaluate two contact models In these models, Hertz contact theory [15] is employed to account for the nonlinear contact force F c between the wheel and rail as follows: ( K H Δy3=2 for ΔyZ Fc ¼ (6) for Δyo where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rwheel Rrailprof E KH ẳ ị2 (7) in which K H denotes the Hertzian spring constant; Rwheel and Rrailprof the radii of the wheel and railhead, respectively, υ the Poisson's ratio of the material, and Δy the indentation at the contact surface which can be expressed as y ẳ yr ỵ yt u3 (8) in which yr and u3 denote the displacements of the rail and wheel, respectively, and yt the magnitude of the track Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] irregularity at the contact point Note that track irregularity is a major source of the dynamic excitation According to the recommendation by Nielsen [14], the track irregularity profile can be written in terms of a sinusoidal function as follows: 2π x yt ¼ at sin λt (9) where at and λt denote the amplitude and wavelength of the track irregularity, respectively To avoid high computational cost and complexity of the nonlinear contact problem, many researchers have adopted a simplified approach based on a linearized Hertz contact model in which F c is given by ( K L Δy for ΔyZ Fc ¼ (10) for Δyo where K L is the linearized Hertzian spring constant [1] computed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rwheel Rrailprof 3E W KL ¼ 2ð1 À υ2 Þ2 (11) in which it is assumed that the reaction force at the contact point equals the self-weight of the upper structure W of the train–track system [15] The governing equations for the vehicle model are m1 u€ ỵk1 u1 u2 ị ỵ c1 u_ u_ ị ẳ m1 g m2 u þk2 ðu2 À u3 Þ þ c2 ðu_ À u_ Þ À k1 ðu1 À u2 Þ Àc1 u_ u_ ị ẳ m2 g m3 u€ Àk2 ðu2 À u3 Þ À c2 u_ u_ ị ẳ m3 g ỵ F c (12) where g denotes gravitational acceleration Upon combining Eq (12) with the governing equations for the rail beam given in Eq (3), the equation of motion for the traintrack system may be written as M z ỵC z_ þ Kz ¼ P (13) where z€ , z_ , z denote the global acceleration, velocity and displacement vectors of the train–track system, respectively; M, C and K the global mass, damping and stiffness matrices, respectively; and P the global load vector The above dynamic equation can be solved by any direct integration methods such as Newmark-β method [16] Numerical results To verify the accuracy of the proposed MEM approach in obtaining the dynamic response of a high-speed rail (HSR) considering variable train velocity, the present solutions are compared against solutions obtained by Koh et al [9] using the so-called ‘cut-and-paste’ FEM The latter involves updating the force and displacement vectors in accordance with the position of the vehicle while keeping the structure mass, damping and stiffness matrices intact For the purpose of comparison, the same train speed profile adopted by Koh et al [9] is employed This speed profile is shown in Fig where it can be seen that there are phases of travel The initial phase considers the train to be moving at a constant acceleration of travel and reaching a maximum speed of 20 m s À after s This is followed by the train traveling at the maximum constant speed for another s during the second phase In the final phase, the train decelerates at a constant magnitude to come to a complete halt after another s of travel Values of parameters related to the properties of track and foundation are summarized in Table [9] Results obtained using the proposed method are found to be in excellent agreement with those obtained by the ‘cut-and-paste’ FEM Fig shows the rail displacement profiles at s obtained by the two methods In view there is virtually no visible difference in the plots obtained by both methods In the present study, the stiffness matrix of the moving element depends on the additional term involving the magnitude of the train acceleration/deceleration and mass of rail beam, as can be seen from Eq (4) However, upon close examination of the magnitudes of the various terms contributing to the stiffness, it is found that the contribution from the acceleration component is expected to be small compared to other terms, in particular, the contribution from the foundation stiffness Fig Profile of train speed for comparison purpose Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig General profile of train speed Fig Comparison of rail displacement profiles The additional term becomes significant when the train travels at high acceleration/deceleration on a track of high mass per unit length (such as the slab track investigated by Lei and Wang [19] which was modeled as a beam with a mass distribution of 3675 kg/m) resting on soft foundation In view of this, three values of acceleration/deceleration of train ranging from low to high and subgrade stiffness ranging from soft to stiff [20] will be considered in the study Note that the amplitude and wavelength of all track irregularities are chosen to be 0.5 mm and 0.5 m, respectively Results obtained using the proposed MEM are compared against results obtained via the approach adopted by Koh et al [9] using the MEM formulation based on piecewise constant velocity to account for the non-uniform motion of the train The comparison is presented in Fig 5, which shows the maximum difference of rail displacements for various foundation stiffness and train acceleration It can be seen from Fig that there is virtually no difference in the results for most cases However, the difference becomes significant when the acceleration of the train is high and when the subgrade stiffness is low In the following sections, results from the study of two cases of HSR travel using the proposed MEM approach are presented The first case studies the response of high-speed train moving over a uniform Winkler foundation at constant speed The effects of track irregularity and wheel load on the dynamic response of train–track system and the occurrence of the jumping wheel phenomenon will be investigated using the Hertz nonlinear and linearized contact models In the second case, the response of train–track system moving at varying speed will be investigated The aim of this study is to determine whether the magnitude of acceleration or deceleration affects the dynamic response of the train–track system when the HSR travels at resonant speed The effects of track irregularity and wheel load on the occurrence of the jumping wheel phenomenon and dynamic response of the system during the accelerating or decelerating phases will also be examined 3.1 Case 1: HSR travels at constant speed The MEM model adopted in the study comprises of a truncated railway track of 50 m length discretized non-uniformly with elements ranging from a coarse m to a more refined 0.1 m size Note that refined element sizes are employed in the vicinity of the moving train load in order to capture accurately the maximum response of the train–track system The equations of motion are solved using Newmark's constant acceleration method employing a time step of 0.0005 s This small time step size is necessary in view of the inherent high natural frequency of the train–track system Values of parameters Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig Contribution of additional component due to train acceleration Table Parameters for train model m1 k1 c1 3500 kg 1.41 Â 105 N m À 8.87 Â 103 N s m À m2 k2 c2 250 kg 1.26 Â 106 N m À 7.1 Â 103 N s m À m3 350 kg Table Parameters for track–foundation model Parameter Value Flexural stiffness Track section Stiffness of foundation Damping of foundation 6.12 Â 106 N m2 UIC 60 (60 E1) Â 107 N m À 4900 N s m À related to the properties of train, track and foundation are summarized in Tables and 2, respectively [9] In analyses involving the Hertz nonlinear contact model, Newton–Raphson's method [16] is employed to solve the resulting nonlinear equations of motion Note that the radii of the wheel Rwheel , railhead Rrailprof and the Poisson's ratio of the wheel/rail material υ used in determining the nonlinear and linearized Hertz spring constants are taken to be 460 mm, 300 mm and 0.3, respectively The initial conditions for this analysis are z€ ¼ z_ ¼ 3.1.1 Effect of track irregularity amplitude As the dynamic response of the train–track system depends significantly on the accuracy in modeling the contact between the wheel and track, it would be important to examine the suitability of the aforementioned nonlinear and linearized contact models The effects of train speed and track irregularity amplitude are investigated The wavelength of all track irregularities considered is taken to be 0.5 m [17] Fig shows the variation of dynamic amplification factor (DAF) in wheel–rail contact force against track irregularity amplitude for various train speeds typically associated with today's HSR travels All analyses are carried out twice, each using the nonlinear and linearized contact models Note that DAF is defined as the ratio of the maximum dynamic contact force to the static wheel load which is the sum of the self-weights of car body, bogie and wheel-set For the perfectly smooth (at ¼ mm) track, the DAF is found be as to be expected in view that there is no dynamic load Consequently, the linearized contact model based on spring properties computed in Eq (11) according to the static wheel load condition [15] can be used The results in Fig also show that when the amplitude of track irregularity and/or train speed increase, the DAF is increased Both the linearized and nonlinear contact models were found to produce results, which are in good agreement for low vehicle speeds regardless of the amplitude of the track irregularity Good agreement was also noted to occur at higher speeds provided the amplitudes of track irregularity are smaller than certain critical values, approximately 0.7 mm and 0.4 mm for v ¼ 70 and 90 m s À 1, respectively Beyond these critical values, the difference in the DAF results becomes significant between the two contact models The above results clearly indicate that the simple linearized contact Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig Effects of irregularity amplitude and train speed on DAF in contact force Fig Effects of track irregularity wavelength and train speed on DAF in contact force model may be used only when there is no large dynamic load involved This is to be expected since the spring property used in the linearized contact model is based on the static wheel load Thus, when the train speed is high and/or the track irregularity is considered to be severe, it is necessary to use the more computationally intensive nonlinear contact model in view of the expected high dynamic load 3.1.2 Effect of track irregularity wavelength As the response of high-speed rails system strongly depends on the severity of track irregularity, it is expected that shorter irregularity wavelength would lead to larger vibrations Therefore, it would be useful to investigate the effects of irregularity wavelengths and train speeds on the response of the HSRs The amplitude of all track irregularities considered in this investigation is taken to be mm Fig shows the effects of irregularity wavelengths and train speeds on the DAF of HSRs It can be seen that the DAF is generally close to 1.0 for irregularity wavelengths larger than some critical values This critical value depends on the train speed, being larger when the speed is larger As to be expected, when the wavelength is large enough, the track may be considered to be in a near smooth condition Consequently, there is little dynamic amplification effect Conversely, when the wavelength is small resulting in a more severe track irregularity condition, the DAF is noted to be significantly larger than especially when the wavelength is less than 1.0 m and the train speed is high However, when the train speed is low such as at 50 m s À 1, there is little dynamic effect despite that the track irregularity is considered to be severe Whenever the DAF is Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] large, it can be seen that the difference in results between the linearized and nonlinear contact models is significant As the linearized contact model results are consistently smaller, it may be concluded that it is not conservative to adopt this model especially when the dynamic response of the HSR is expected to be high Fig also shows that there are some localized peaks in the DAF at certain values of the irregularity wavelength for each train speed The frequency of the dynamic excitation f e due to track irregularity depends on the train speed v and irregularity wavelength λt and may be expressed as fe ¼ v (14) λt The natural frequency of the linearized train model fn ¼ ω (15) 2π may be determined by solving the associated characteristic equation detðK À Mị ẳ (16) for the circular natural frequency ω, where M, K are the global mass and stiffness matrices of linearized train–track– foundation system, respectively Resonance occurs when the exciting frequency f e due to track irregularity coincides with the natural frequency f n and this occurs when the train speed matches the resonant speed vr given by vr ¼ λ t f n (17) À1 The frequencies of the dynamic excitation computed from Eq (14) for train speeds of 50, 70 and 90 m s and track irregularity wavelengths ranging from 0.5 m to m are presented in Table Table shows the natural frequencies of various components of the linearized train system determined from Eqs (15) and (16) As can be seen from these two tables, the frequency of the dynamic excitation approaches the natural frequency of the wheel-set component of the train when the track irregularity wavelengths are 1.5, and 2.5 m (values in bold) corresponding to train speeds of 50, 70 and 90 m s À 1, respectively This explains why Fig shows peak dynamic responses occurring at these combinations of train speed and track irregularity wavelength due to the occurrence of near resonance For other wavelength track irregularities, the exciting frequency is noted to be appreciably different in value from the natural frequencies of the various components of the train system 3.1.3 Effect of wheel load It has been shown that the accuracy of the contact force depends on the contact model In order to further establish when it would be important to adopt the more accurate but computationally more intensive nonlinear contact model, it would be critical to investigate the effect of the wheel load parameter In practice, there is a varying range of wheel loads Typical passenger vehicles range from about 40 to 60 kN per wheel load, while goods-carrying vehicles have wheel loads in excess of 100 kN [18] Two magnitudes of wheel loads are thus considered in the study, namely W ¼41 kN to represent the lowest end of a typical passenger vehicle [9] and 81 kN for an medium loaded vehicle Fig 8(a), (b) and (c) illustrate the effect of wheel load on the accuracy of the linearized contact model as compared to the nonlinear contact model in predicting the DAF in contact force of HSRs for train speed equal to 50, 70 and 90 m s À 1, respectively All plots show the variation of DAF in contact force against track irregularity amplitude for the two cases of wheel loads considered using the linearized and nonlinear contact models Table Exciting frequencies f e (Hz) due to track irregularities Train speed (m s À 1) 50 70 90 Track irregularity wavelength (m) 0.5 1.5 2.5 3.5 100 140 180 50 70 90 33.3 46.7 60 25 35 45 20 28 36 16.7 23.3 30 14.3 20 25.7 12.5 17.5 22.5 Table Natural frequencies of the linearized train model Train component Car body Bogie Wheel-set f n (Hz) 0.96 11.64 37.88 Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig Comparison of linearized and nonlinear contact models: (a) v ¼ 50 m s À 1; (b) v ¼ 70 m s À 1; and (c) v ¼ 90 m s À It can be seen from Fig 8(a) that when the train speed is small at 50 m s À 1, there is virtually no difference in results obtained by both contact models for all track irregularities and wheel loads considered This is not surprising in view that the dynamic effect is expected to be small when the train speed is low and hence the linearized contact model is accurate enough to capture the dynamic response of the HSR system Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 10 However, when the train speed is larger as the case in Fig 8(b) and (c), the difference in solutions between the two contact models becomes appreciable especially for larger amplitudes track irregularities For a given track irregularity condition, the difference is also larger when the wheel load is small On the other hand, when the wheel load is large, it appears that the linearized contact model is able to produce results close to the nonlinear model Note that when the wheel load is large, dynamic effect is mitigated as can be seen by lower values of DAF in contact force Under such a condition, it is expected that the linearized contact model is able to give good results, as the contact force magnitude would be largely due to the static wheel load effect From the results presented in Sections 3.1.1–3.1.3, it can be concluded that the computationally cheaper linearized contact model is accurate enough to be used whenever the expected dynamic effect of the HSR system is not large In general, when the train speed is low, track irregularity is near smooth and/or wheel load is large, the DAF in contact wheel force is expected to be low, and hence the use of the linearized contact model would be acceptable On the other hand, it should be emphasized that the computationally more expensive but more accurate nonlinear contact model must be employed whenever the dynamic effect of the HSR system is expected to be significant 3.1.4 Occurrence of jumping wheel phenomenon As presented earlier, the contact force between the wheel and rail strongly depends on the train speed, track irregularity and wheel load When the condition is such that the DAF is relatively large, the possibility of the occurrence of the jumping wheel phenomenon, where there is momentary loss of contact between wheel and rail, becomes high Thus, the aforementioned factors are also critical in affecting the occurrence of the jumping wheel phenomenon Tables and show the occurrence or non-occurrence of the jumping wheel phenomenon for various train speeds, track irregularity and wheel load Note that track irregularity condition is affected by two parameters, namely track irregularity amplitude and wavelength In general, when the wavelength is small and/or amplitude is large, the track irregularity condition may be rated as severe, and vice-versa Table shows the results for a track irregularity wavelength of 0.5 m [17], which is deemed to be small, for cases of track irregularity amplitudes ranging from very small to large Table presents the results for a track irregularity amplitude of mm, which is deemed to be large, for cases of track irregularity wavelength ranging from small to large Note that all results presented are obtained through the use of the nonlinear contact model It can be seen in Tables and that when the track condition is deemed near smooth, there is no occurrence of the jumping wheel phenomenon for all train speeds and wheel loads On the other hand, when the track condition is considered to be severe, the jumping wheel phenomenon occurs for all wheel loads when the train speed is large enough For the case when the train speed is low at 50 m s À 1, the jumping wheel phenomenon is suppressed when the wheel load is large When the track condition is rated as moderate, that is, it is neither near smooth or severe, the jumping wheel phenomenon may or may not occur It tends to occur when the train speed is high enough and when the wheel load is small This observation is consistent with earlier results that a combination of small wheel load and high train speed promote larger dynamic effects and hence the greater chance of occurrence of the jumping wheel phenomenon Table Occurrence of jumping wheel phenomenon (constant train speed, λt ¼ 0:5 m) Train speed (m s À 1) Track irregularity amplitude (mm) 0.01 50 70 90 0.5 1.6 Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN N N N N N N N N Y N N N Y Y Y N Y Y Table Occurrence of jumping wheel phenomenon (constant train speed, at ¼ mm) Train speed (m s À 1) Track irregularity wavelength (m) 0.5 50 70 90 Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Y Y Y N Y Y N Y Y N N N N N N N N N Note that “N” denotes non-occurrence of jumping wheel phenomenon “Y” denotes occurrence of jumping wheel phenomenon Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 11 3.2 Case 2: HSR travels at varying speed In the following sections, the effects of amplitudes of train acceleration/deceleration, track irregularity and wheel load on the dynamic response of the train–track system during the accelerating or decelerating phases using the proposed MEM approach are presented Fig shows the general profile of a train journey The parameters for various train speed profiles considered are presented in Table under Cases 2–4 The proposed MEM model adopted in the study comprises of a truncated railway track of 50 m length uniformly discretized into 250 moving finite elements Values of parameters related to the properties of train, track and foundation are summarized in Tables and 2, respectively [9] The equations of motion are solved using Newmark's constant acceleration method employing a time step of 0.0005 s This small time step size is necessary in view of the inherent high natural frequency of the train–track system In analyses involving the Hertz nonlinear contact model, Newton–Raphson's method [16] is employed to solve the resulting nonlinear equations of motion The radii of the wheel Rwheel , railhead Rrailprof and the Poisson's ratio of the wheel/rail material υ used in determining the nonlinear Hertz spring constant are taken to be 460 mm, 300 mm and 0.3, respectively The initial conditions for this analysis are z€ ¼ z_ ¼ 3.2.1 Effect of amplitudes of train acceleration/deceleration In the moving element method, the stiffness matrix of the moving element depends on the magnitude of the train acceleration/deceleration, as can be seen from Eq (4) In view of this, only two values of acceleration/deceleration of train will be considered in the study These are designated as Cases and in Table corresponding to acceleration/deceleration magnitudes of 2.222 and 0.720 m s À 2, respectively Three track irregularity amplitudes, corresponding to smooth (0.01 mm), moderate (0.5 mm) and severe (2 mm) conditions, are considered Note that the wavelength of all track irregularities is chosen to be m [17] Fig 9(a)–(d) shows the force factor–time history plots for Case during the acceleration phase of the travel Note that the force factor (FF) in contact force is defined as the ratio of dynamic force to static force The vertical lines drawn in Fig 9(a) demarcate the time duration in which the jumping wheel phenomenon occurs Fig 9(b) and (d) show blow-up views of the force factor–time history plot in the vicinity of the onset and ending of the jumping wheel phenomenon, respectively Fig (c) shows a similar blow-up view over a typical period where there is sustained jumping wheel Note that when the FF equals to À 1, there is momentary loss of contact between the wheel and rail, which is what is known as jumping wheel As the plots are similar during the deceleration phase for Case as well as for both phases in Case 3, these are thus not presented It should be noted that when the instantaneous speed of the train is close to the resonant speed of the HSR, high dynamic response is expected to occur leading to the occurrence of the jumping wheel phenomenon Hence, the magnitude of the acceleration/deceleration will only affect the duration in which jumping wheel occurs Thus, it is not surprising that the interval of time in which jumping wheel occurs is found to be longer in Case when the acceleration/deceleration amplitude is smaller as compared to Case Thus, subject to meeting the comfort level of passengers, it is recommended that HSR trains should travel at its highest possible acceleration/deceleration to attain its final speed in order to minimize the duration of the jumping wheel phenomenon However, it is important that such a major recommendation to the operation of HSRs be confirmed experimentally on a real line The maximum FF for both cases are found to be virtually the same thereby confirming the fact that the magnitude of acceleration/deceleration has negligible effect on the stiffness of the system and hence the dynamic response In view of this finding, all other results to be subsequently presented shall pertain to Case 2, considered to be the typical speed profile of today's HSR travels 3.2.2 Effect of track irregularity amplitude The effect of track irregularity amplitude on the DAF is next investigated The results are plotted in Fig 10 The curves drawn correspond to the phases of travel, namely during the acceleration, constant speed and deceleration phases Note that the wavelength λt of all track irregularities is chosen to be m [17] For a near smooth track (at ¼ 0:01 mm), the DAF is found to be approximately 1, as to be expected When the amplitude of track irregularity increases, the DAF is noted to increase gradually and then significantly for the acceleration/deceleration phases For these two phases, when the track irregularity amplitude is large enough for the track condition to be considered as moderate or severe, the DAF increases significantly due to the occurrence of the jumping wheel phenomena for the brief interval in which the train speed is in the vicinity of the resonant speed Note that the frequency of the dynamic excitation due to the track irregularity depends on Table Profiles of train speed Case Maximum train speed V max (m s À 1) 20 70 70 Amplitude of acceleration/ deceleration jaj (m s À 2) 10 2.222 0.720 Time parameters Reference t (s) t (s) t (s) 2.0 31.5 98.0 4.0 33.5 100.0 6.0 65.0 198.0 Koh et al [9] Karlstrom [6] Current high-speed rails Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i 12 M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Fig Force factor–time history: (a) during acceleration phase; (b) in vicinity of onset of jumping wheel phenomenon; (c) over typical period where there is sustained jumping wheel; and (d) before and after end of jumping wheel phenomenon the train speed and irregularity wavelength Thus, when the train speed and wavelength are such that the frequency of the dynamic excitation approaches the natural frequency of the linearized train model, there will be expected resonance effect The DAF is also noted to be slightly larger during the deceleration phase as compared to the acceleration phase For the Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 13 Fig 10 Effect of track irregularity amplitude on DAF in contact force when train travels at varying speed Fig 11 Effect of track irregularity wavelength on DAF in contact force when train travels at varying speed constant speed phase, the DAF is observed to increase gradually as the track irregularity amplitude increases No jumping wheel is noted to occur during this phase 3.2.3 Effect of track irregularity wavelength As the natural frequency of the linearized train model system is fixed, the magnitude of track irregularity wavelength is a significant factor in affecting the occurrence of the resonant phenomenon during the acceleration/deceleration phases As already mentioned, resonance occurs briefly during these two phases when the train speed crosses the magnitude of the resonant speed This occurs only when the maximum train speed is higher than the resonant speed Fig 11 shows the effect of track irregularity wavelength on the DAF for the phases of train travel Note that all track irregularity amplitudes are mm for this study When the wavelength is large, it is noted that the DAF is relatively low, as to be expected As the wavelength decreases, the DAF is found to increase gradually and then abruptly when it approaches m for all phases It should be noted that the frequency of the wheel-set is close to the exciting frequency when the track irregularity wavelength is m Thus, resonance is noted to occur for all phases which accounts why the DAF increases abruptly Also, note that when the track irregularity wavelength is greater than approximately 2.5 m, the resonant phenomenon does not occur, as the theoretical resonant speed is larger than the maximum train speed attained during the travel During the acceleration/deceleration phases, the DAF seems to be initially constant as the track irregularity wavelength decreases below m Note that the DAF tends to increase with decreasing track irregularity wavelength since small values of wavelength are associated with more severe track irregularity conditions However, as the resonant speed decreases with Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i 14 M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] decreasing wavelength, the DAF tends to decrease since the train speed, as it crosses the resonant speed during the acceleration/deceleration phases, are smaller These two opposing effects on the DAF thus explain why the DAF is observed to be approximately constant when the wavelength is between 0.75 m and m As the wavelength decreases further below 0.75 m, the effect of severity of track condition becomes more pronounced and thus the DAF is seen to increase abruptly Note that the trend for the constant speed phase of travel is similar to the acceleration/deceleration phases except that the DAF decreases abruptly as the track irregularity wavelength decreases below m 3.2.4 Effect of the wheel load The effect of wheel load on the dynamic response of the system was earlier investigated and discussed for the case when the train travels at constant speed The conclusion then was that the wheel load effect could be considerable It would thus be important to examine the effect of wheel load when the train travels at variable speed as in Case Fig 12 shows the variation of the DAF against track irregularity amplitude for various wheel loads The track irregularity wavelength is fixed at 1.0 m [17] for this study The results show that the DAF during all phases for small wheel load is always larger as compared to large wheel load Note that when the wheel load is large, it is expected that the dynamic effect is mitigated, which explains why the DAF is found to be larger for the case of the smaller wheel load This finding is the same as that obtained by Herwig [21] Fig 12 shows that there is virtually no difference in DAF for both cases of wheel loads when the track irregularity amplitude is below approximately 0.4 mm Beyond this critical value, the DAF increases significantly during the acceleration/deceleration phases and the difference in DAF for the two wheel loads becomes more pronounced However, during the constant speed phase, it is found that the DAF grows marginally for track irregularity amplitude beyond 0.4 mm and that there is virtually no difference in DAF for both cases of wheel loads Wheel load is thus critical in affecting the response of the system for a brief time interval during the acceleration/deceleration phases as the train speed crosses the resonant speed of the linearized train model A larger wheel load has the advantage of mitigating the dynamic response during the critical period of travel 3.2.5 Occurrence of jumping wheel phenomenon In this section, the occurrence of the jumping wheel phenomenon is investigated when the train travels at varying speed Table shows the occurrence or non-occurrence of the jumping wheel phenomenon for various wheel loads and track irregularity amplitudes The track irregularity wavelength is fixed at 1.0 m [17] in this study It can be seen that when the track irregularity amplitude is less than 0.4 mm corresponding to a near smooth track condition, there is no occurrence of the jumping wheel phenomenon Beyond this critical value, the jumping wheel phenomenon may or may not occur during the acceleration/deceleration phase It tends to occur when the wheel load is small Note that when the wheel load is large, dynamic effect is mitigated as can be seen by lower values of DAF in contact force in Fig 12 When the track condition is considered to be severe, the jumping wheel phenomenon occurs briefly during the critical period of travel of the acceleration/deceleration phase As the exciting frequency is close to the natural frequency of the linearized train model, there is occurrence of resonant phenomenon during these phases With such condition and large amplitude, the jumping wheel phenomenon occurs, as it is anticipated that the dynamic effect is large Table shows the occurrence or non-occurrence of the jumping wheel phenomenon for the two cases of wheel loads considered and various track irregularity wavelengths The jumping wheel phenomenon is noted to occur in all phases of travel when the track irregularity wavelength is small, such as at 0.5 m, since this value corresponds to the case of a more Fig 12 Effects of wheel load and track irregularity amplitude on DAF in contact force when train travels at varying speed Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 15 Table Occurrence of jumping wheel phenomenon (variable train speed, λt ¼ m) Phase Track irregularity amplitude (mm) 0.01 0.4 0.8 1.2 1.6 Wheel load Wheel load Wheel load Wheel load Wheel load Wheel load Wheel load Wheel load Wheel load Wheel load 41 kN 81 kN 41 kN 81 kN 41 kN 81 kN 41 kN 81 kN 41 kN 81 kN Acceleration N Constant N Deceleration N N N N N N N N N N Y N Y N N N Y N Y N N N Y N Y Y N Y Table Occurrence of jumping wheel phenomenon (variable train speed, at ¼ mm) Phase Track irregularity wavelength (m) 0.5 Wheel load 41 kN Acceleration Y Constant Y Deceleration Y 0.75 1.5 2.25 2.5 Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Wheel load 41 kN Wheel load 81 kN Y Y Y Y Y Y Y N Y Y N Y Y N Y Y N Y Y N Y Y Y Y Y Y Y Y Y Y N N N N N N N N N N N N N N N Note that “N” denotes non-occurrence of jumping wheel phenomenon “Y” denotes occurrence of jumping wheel phenomenon severe track condition and hence high dynamic effects At a wavelength of m, the frequency of the linearized train model is close to the exciting frequency and thus there is expected occurrence of resonance resulting in the occurrence of the jumping wheel phenomenon too For wavelengths higher than the critical value of m, the track condition matches closer to a near smooth case Consequently, there is no occurrence of the jumping wheel phenomenon When the wheel load is small, higher DAF is to be expected as compared to the case of larger wheel load Thus, Table shows that the jumping wheel phenomenon does occur under certain conditions for the case of smaller wheel load but not for the larger wheel load case Conclusion In this paper, a numerical study on the dynamic response of non-uniform motion of high-speed rails using the moving element method was carried out A new and generalized formulation for calculating the structural matrices of the moving element was proposed To account for the wheel/rail interaction, two contact models were employed and their accuracy and suitability evaluated The first is an approximate simpler linearized contact model and the second is a computationally more demanding nonlinear Hertz contact model The effects of train speed, amplitudes of train acceleration/deceleration, track irregularity and wheel load on the dynamic amplification factor in contact force between the wheel and rail and the jumping wheel phenomenon are investigated when the train travels at constant and varying speeds In the present study, the stiffness matrix of the moving element includes an additional term relating to the train acceleration The contribution from the acceleration component is expected to be generally small Results obtained using the proposed MEM are thus found to agree well with results obtained using the ‘cut-and-paste’ FEM by other researchers However, when the train travels at high acceleration on a track of high mass per unit length resting on soft foundation, the contribution from the acceleration component becomes significant For the case when the train travels at constant speed, it is found that the computationally cheaper linearized contact model is accurate enough to be used whenever the expected dynamic effect of the HSR system is not large In general, when the train speed is low, track irregularity is near smooth and/or wheel load is large, the DAF in contact wheel force is expected to be small, and hence the use of the linearized contact model would be acceptable On the other hand, it should be emphasized that the computationally more expensive but more accurate nonlinear contact model must be employed whenever the dynamic effect of the HSR system is expected to be significant A combination of small wheel load, high train speed and severe track condition promote larger dynamic effects and hence the greater chance of occurrence of the jumping wheel phenomenon Subject to ensuring that the comfort level of passengers is attained, it is recommended that HSR trains should travel at its highest possible acceleration/deceleration to attain its final speed in order to minimize the duration of the jumping wheel Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i 16 M.T Tran et al / Journal of Sound and Vibration ] (]]]]) ]]]–]]] phenomenon However, it is important that such a major recommendation to the operation of HSRs be confirmed experimentally on a real line As to be expected, the DAF are larger when the HSR travels at a resonant speed during the acceleration/deceleration phases rather than the case in which the train travels at constant speed phase Consequently, there is a greater chance of the jumping wheel phenomenon occurring during the acceleration/deceleration phases Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2013.27 References [1] S.P Timoshenko, Statistical and dynamical stresses in rails, Proceedings of the Second International Congress for Applied Mechanics, Zurich, Switzerland, 12–17 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elements in a moving frame of reference, Journal of Vibration and Control 20 (9) (2013) 1301–1317 [20] Z Dimitrovová, J.N Varandas, Critical velocity of a load moving on a beam with a sudden change of foundation stiffness: applications to high speed trains, Computers and Structures 87 (2009) 1224–1232 [21] A Herwig, Consideration of the dynamic effect of increased train loads for the fatigue examination of concrete bridges, Proceedings of the Sixth International PhD Symposium in Civil Engineering, Zurich, August 23–26, 2006 Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i ... indicate that the simple linearized contact Please cite this article as: M.T Tran, et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration... et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound... et al., Vertical dynamic response of non-uniform motion of high-speed rails, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.05.053i M.T Tran et al / Journal of Sound