Dynamic Responses of Wheel-Rail Systems with Block Dampers

Transcription

1 University of Nebraska - Lincoln DigitalCoons@University of Nebraska - Lincoln Mechanical (and Materials) Engineering -- Dissertations, Theses, and Student Research Mechanical & Materials Engineering, Departent of Dynaic Responses of Wheel-Rail Systes with Block Dapers TzuYu Tseng University of Nebraska-Lincoln, tzuyutseng@gail.co Follow this and additional works at: Part of the Acoustics, Dynaics, and Controls Coons Tseng, TzuYu, "Dynaic Responses of Wheel-Rail Systes with Block Dapers" (2016). Mechanical (and Materials) Engineering -- Dissertations, Theses, and Student Research This Article is brought to you for free and open access by the Mechanical & Materials Engineering, Departent of at DigitalCoons@University of Nebraska - Lincoln. It has been accepted for inclusion in Mechanical (and Materials) Engineering -- Dissertations, Theses, and Student Research by an authorized adinistrator of DigitalCoons@University of Nebraska - Lincoln.

2 DYNAMIC RESPONSES OF WHEEL-RAIL SYSTEMS WITH BLOCK DAMPERS by TzuYu Tseng A THESIS Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillent of Requireents For the Degree of Master of Science Major: Mechanical Engineering and Applied Mechanics Under the Supervision of Professor Cho Wing Soloon To Lincoln, Nebraska Deceber 2016

3 Dynaic Responses of Wheel-Rail Systes with Block Dapers Tzu Yu Tseng, M.S. University of Nebraska, 2016 Advisor: C. W. Soloon To The wheel-rail interaction proble has been widely studied in the past few decades. In this proble, dynaic responses at the contact areas reain the central issue since they induce daage to the rail over tie. In particular, the dynaic responses at the contact areas between the wheels and rails present difficulties in understanding and atheatical odeling. Even with the coputer power one has today, its atheatical odeling eploys the versatile nuerical analysis ethod, the finite eleent ethod (FEM) reains a foridable challenge due to its extreely sall contact areas and in turn the extreely high stress levels. In addition, friction at the contact areas is another challenge in the odeling. These extreely high stress levels and difficulties in odeling friction at the sall contact areas lead to the siplified analytical and finite eleent (FE) odels available in the literature. However, to-date, these siplified atheatical and coputational odels are far fro satisfactory. Therefore, in the investigation reported here, siplified analytical and FE odels are first studied in order to understand the

4 paraeters of the odels and to provide a foundation for ore detailed studies. In these siplified analytical and FE odels, the wheels are treated as traveling point loads. Subsequently, a ore detailed FE odel eploying three-diensional (3D) finite eleents for the wheel and rail is studied while the coputed results are copared with those of the sae FE odel but with the wheel replaced by the traveling point loads. In this odel, frictions at the contact areas are considered. In parallel, the effect of block dapers on the dynaic responses at the contact areas is studied. Various configurations of attaching the block dapers to the rail are considered. Conclusions are then drawn fro the accuracy and efficacy of replacing the wheels by traveling point loads, and the effect on the dynaic responses of added block dapers to the rail.

5 i ACKNOWLEDGMENTS Firstly, I would like to thank y advisor Dr. C.W. Soloon To for his support, guidance, and being patient over these years. He gives e an interesting topic of the study and inspires e with different aspects. I a so glad and honor to do y thesis under Dr. C.W. Soloon To. Secondly, I would also like to express y gratitude to Dr. Mehrdad Negahban and Dr. Benjain Terry for being y exainers and giving e the different point of views for further studying. Thirdly, I would like to thank our departent secretary Kathie Hiatt for helping e to deal with soe paper works for graduating. And I would like to thank y friends who have supported e in these years. Finally, I would like to thank y faily, y father, other, and sister. No atter what decision I ake, they always support e.

6 ii CONTENT ACKNOWLEDGMENTS CONTENT LIST OF FIGURES LIST OF TABLES ⅰ ⅱ ⅵ ⅹ CHAPTER 1 INTROUDCTION 1.1 BACKGROUND AND MOTIVSTION OBJECTIVES OF INVESTIGATION LITERATURE SURVEY Bea Theories Finite eleent analysis ORGANIZATION OF PRESENTS 5 CHAPTER 2 THEORETICAL DEVELOMENT RAIL MODELS BY BEAM THEORIES Euler-Bernoulli Bea Theory Tioshenko Bea Theory FREE VIBRATION OF UNDAMPED SIMPLY SUPPORTED EBB BEAM ON ELASTIC FOUNDATION SIMPLE MODEL FOR WHEEL-RAIL INTERACTION Siply Supported EBB with Traveling Point Load Unifor Bar with Axial Load 17

7 2.5 FINITE ELEMENT MODELS FOR WHEEL-RAIL INTERACTION Bea Finite Eleent Model of Wheel-Rail Interaction Eleent ass and stiffness atrices based on EBB Theory Eleent ass and stiffness atrices based on TB Theory Three-Diensional Finite Eleent Model of Wheel-Rail Syste Dynaic Characteristics of Wheel-Rail Syste Three-Diensional Finite Eleent Model of Wheel-Rail Interaction FINITE ELEMENT MODELS FOR WHEEL-RAIL WITH BLOCK DAMPERS 25 iii CHAPTER 3 WHEEL-RAIL MODEL BY FEM SOFTWARE GEOMETRICAL AND MATERIAL PROPERTIES OF WHEEL AND RAIL PREOCESS OF FE MODEL IN WHEEL-RAIL INTERACTION IMPLICIT AND EXPLICITE TIME-MARCHING SCHEMES BOUNDARY CONDITIONS ELEMENT SIZE AND MESH 31 CHAPTER 4 WHEEL-RAIL INTERACTION MODELS WITHOUT BLOCK DAMPERS EIGENVALUES SOLUTION OF RAIL MODELS Solution by EBB Theory Coputed Results of Bea Eleent and Solid Eleent Model Discussion 45

8 4.2 RESPONSES OF RAIL DUE TO TRAVELING POINT LOAD 46 iv Analytical and FE Models of Rail with Traveling Point Load Coputed Results of 3D FE odel with Traveling Point Load Discussion COMPUTED RESULTS OF 3D FE RAIL MODEL WITH TRAVELING POINT LOAD Coparison of 3D FE Rail with Traveling Point Load Model and 3D FE Wheel-Rail Model Modification of Traveling Point Load By Scaling Aplitude Discussion 107 CHAPTER 5 RAIL MODELS WITH BLOCK DAMPERS MODELING 3D FE RAIL MODELS WITH BLOCK DAMPERS Diension of Block Dapers Locations of Block Dapers on 3D FE Rail Model COMPUTED RESULTS OF 3D FE RAIL MODEL WITH BLOCK DAMPERS The Effects of Block Dapers at Different Locations The Effects of Block Dapers with Different Percentages of Rail Mass Discussion 139 CHAPTER 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 6.1 SUMMARY CONCLUSIONS 142

9 6.2 RECOMMEDATIONS FOR FUTURE WORK 144 APPENDIX APPENDIX 3A Wheel Profile 145 APPENDIX 3B Rail Profile 146 v REFERENCE 147

10 LIST OF FIGURES Figure 2.1 Track structure with foundation. 11 Figure 2.2 Model for Winkler bea. 11 Figure 2.3 Siply-supported EBB subjected to traveling point load. 14 Figure 2.4 Traveling friction force on unifor bar. 17 Figure 2.5 Eight-node brick eleent. 22 Figure 3.1 Geoetrical diensions of rail and wheel: (a) I bea odel for ail, and (b) circular disk for wheel. 27 Figure 3.2 Wheel rail odel with boundary conditions indicated. 30 Figure 3.3 Finite eleent wheel-rail odel with boundary condition. 30 Figure 4.1 First six ode-shapes of SS EBB rail odel. 35 Figure 4.2 First six ode-shapes of bea eleent odels: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, and (f) Mode Figure 4.3 First twelve ode-shapes of solid eleent (C3D8) odel: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, (f) Mode 6, (g) Mode 7, (h) Mode 8, (i) Mode 9, (j) Mode 10, (k) Mode 11, and (l) Mode Figure 4.5 Noralized responses at a quarter-point fro starting end of SS EBB (x =0.15 ): (a) v = 10 /s. (b) v = 20 /s. (c) v = 30 /s. (d) v = 40 /s. (e) v = 50 /s. (f) v = 60 /s. 48 Figure 4.6 Noralized responses at id-point fro starting end of SS EBB (x = 0.30 ): (a) v = 10 /s. (b) v = 20 /s. (c) v = 30 /s. (d) v = 40 /s. (e) v = 50 /s. (f) v = 60 /s. 51 Figure 4.7 Noralized responses at the third quarter-point fro starting end of SS EBB (x = 0.45 ): (a) v = 10 /s. (b) v = 20 /s. (c) v = 30 /s. (d) v = 40 /s. (e) v = 50 /s. (f) v = 60 /s. 54 Figure 4.8 Three bea eleent representation of one SS bea structure. 57 Figure 4.9 Finite eleent 3D odel for traveling point load. 59 Figure 4.10 Signu function f(x). 59 Figure 4.11 Regularization of signu function in discrete tie steps. 60 Figure 4.12(a) Longitudinal responses at quarter-point fro starting node. (b) Vertical responses at quarter-point fro starting node. span. 60 Figure 4.13(a) Longitudinal responses at iddle. (b) Vertical responses at iddle span. 61 vi

11 Figure 4.14(a) Longitudinal responses at quarter-point fro ending node. (b) Vertical responses at quarter-point fro ending node. 62 Figure 4.15 Wheel and rail contact paths: (a) Path 1, (b) Path 2, and (c) Path Figure 4.16 Locations of 9 selected points. 67 Figure 4.17 Vertical deflections of 3D FE wheel-rail odel. 68 Figure 4.18 Vertical deflections 3D FE rail odel with TPL. 68 Figure 4.19 Longitudinal deflection at Point 4:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 69 Figure 4.20 Vertical deflection at Point 4:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 72 Figure 4.21 Longitudinal deflection at Point 5:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 75 Figure 4.22 Vertical deflection at Point 5:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 78 Figure 4.23 Longitudinal deflection at Point 6:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 81 Figure 4.24 Vertical deflection at Point 6:, TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 84 Figure 4.25 Noral and shear (longitudinal) force fro FE wheel-rail odel: Figure 4.26 (a) shear force, and (b) noral force. 88 Longitudinal deflection at Point 4:, Scaled TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 89 Figure 4.27 Vertical deflection at Point 4:, Scaled TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 92 Figure 4.28 (a) v = 10 Longitudinal deflection at Point 5:, Scaled TPL;, WRM;. (b) v = 20 s s. (c) v = 30 s. vii

12 Figure 4.29 (d) v = 40 s. (e) v = 50 s viii.(f) v = 60 s. 95 Vertical deflection at Point 5:, Scaled TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 98 Figure 4.30 Longitudinal deflection at Point 6:, Scaled TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 101 Figure 4.31 Vertical deflection at Point 6:, Scaled TPL;, WRM; (a) v = 10 s s s. (d) v = 40. (e) v = 50 s s s. 104 Figure 4.32 Relation between average aplitude of vertical force and speed 107 Figure 4.33 Relation between average aplitude of shear force and speed (a) Average axiu shear force (b) Average iniu shear force 108 Figure 5.1 Three different locations for placing BD: Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 (a) Location 1, (b) Location 2, and (c) Location Percentage of difference in axiu longitudinal deflection at locations at Point 4 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 120 Percentage of difference in iniu longitudinal deflection at locations at Point 4 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 121 Percentage of difference in iniu vertical deflection at locations at Point 4 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 122 Percentage of difference in axiu longitudinal deflection at locations at Point 5 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 123 Percentage of difference in iniu longitudinal deflection at locations at Point 5 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 124 Percentage of difference in iniu vertical deflection at locations at Point 5 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 125 Figure 5.8 Percentage of difference in axiu longitudinal deflection at locations at Point 6 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 126 Figure 5.9 Percentage of difference in iniu longitudinal deflection at locations at Point 6 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 127 Figure 5.10 Percentage of difference in iniu vertical deflection at locations at Point 6 with block dapers of (a) 22%, (b) 26%, and (c) 30% rail ass. 128 Figure 5.11 Percentage of difference of axiu longitudinal deflection at Point 4.

13 Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.12 Percentage of difference of iniu longitudinal deflection at Point 4. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.13 Percentage of difference of iniu vertical deflection at Point 4. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.14 Percentage of difference of axiu longitudinal deflection at Point 5. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.15 Percentage of difference of iniu longitudinal deflection at Point 5. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.16 Percentage of difference of iniu vertical deflection at Point 5. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.17 Percentage of difference of axiu longitudinal deflection at Point 6. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.18 Percentage of difference of iniu longitudinal deflection at Point 6. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.19 Percentage of difference of iniu vertical deflection at Point 6. Block dapers with 22% of rail ass (a) Location 1, (b) Location 2, and (c) Location Figure 5.20 Vertical deflection of 3D FE rail odel with and without block dapers. 140 ix

14 LIST OF TABLES x Table 2.1 Nodal co-ordinates for 8-node brick eleent. 22 Table 3.1 Evolution of different esh sizes for wheel-rail odel with juping (no contact). 32 Table 3.2 Relationship between eleent size and tie step size. 32 Table 4.1 First six natural frequencies of SS EBB. 35 Table 4.2 First twelve natural frequencies of FE solid eleent. 39 Table 4.3 Tie and aplitude of traveling point load at Node 1 to Node Table 4.4 Tie and aplitude for bea structure represented by N eleents. 58 Table 5.1 Diensions of three types of BD in different locations. 111 Table 5.2 Percentage of difference between odel with and without BD at Point 4: (a) Maxiu longitudinal deflection (b) Miniu longitudinal deflection(c) Miniu vertical deflection. 114 Table 5.3 Percentage of difference between odel with and without BD at Point 5: (a) Maxiu longitudinal deflection. (b) Miniu longitudinal deflection(c) Miniu vertical deflection 116 Table 5.4 Percentage of difference between odel with and without BD at Point 6: (a) Maxiu longitudinal deflection. (b) Miniu longitudinal deflection(c) Miniu vertical deflection. 118 Table 5.5 The first twelve natural frequencies of 3D FE rail odel and results with block dapers. 140

15 CHAPTER 1 INTRODUCTION 1 Railway plays an iportant role during the industrial revolution and, nowadays, it has never stopped changing our lives. The innovation of railways is rapidly developing fro the first stea locootive appeared in 1804 [1.1]. The world s first stea locootive ran on railway syste in England at 8 k/hr. To date, the fastest speed of test railway syste is at 603 k/hr which is based on the application of agnetic levitation or the so-called aglev with passengers in Japan in The fastest speed of coercial railway syste is at k/hr in China in 2003 [1.2]. Also, railway syste is not only used for long distance travel but for short distance in the cities. Railway transportations provide an alternatively safe and clean way of travel in any countries. However, no atter how fast the speeds of trains are the railway copanies have a ajor proble with aintaining their rail tracks. These copanies not only spend a great aount of tie and oney on their aintenance but also create noises during the aintenance work at night. Therefore, how to reduce the ties and, in turn, the cost of aintenance is a ajor challenge to the design and aintenance engineers. One approach to the challenge is by studying the dynaics and reducing the dynaic deflections of the rail syste. This posts a ajor challenge in providing a relatively accurate and efficient dynaic odel. 1.2 OBJECTIVES OF INVESTIGATION One of the ajor goals of the present investigation is to provide a odel wheel-rail syste by using rail and concentrated traveling point loads. This, if proves to be applicable, will circuvent the great difficulty of providing a relatively accurate

16 2 coputational odel, typically applying the finite eleent ethod (FEM). The great difficulty is due to the fact that every wheel-rail syste is under extreely high stresses at the contact areas between the wheels and rails. In the finite eleent (FE) odel, to provide accurate representations of the contact areas, very sall sizes of FE are necessary. In addition, to odel the wheel posts another ajor challenge in the sense that one has to consider sliding and rotating. The second ajor goal of the investigation is to perfor a coparison study between the wheel-rail syste and the rail with traveling point loads. In this part of the investigation, a procedure is established so that the rail with traveling point loads can replace the wheel-rail syste. This, in turn, will reduce the coputational cost drastically and circuvent the difficulty of dealing with extreely sall FE sizes. The final and third ajor goal of this investigation is to understand the dynaic responses of the wheel-rail syste with and without block dapers. Block dapers which are siply ass blocks attached to the sides of the rail. The ain idea of adding block dapers is to change significantly the natural frequencies of rail and reduce the dynaic responses. The effects of different locations and asses of block dapers on the dynaic responses are studied. 1.3 LITERATURE SURVEY The literature survey of wheel-rail systes in this section is separated into two sub-sections. The first sub-section is concerned with bea theories for application to the railway odel. It includes natural frequencies, and dynaics responses with traveling point load and oving ass. The second sub-section deals with applications of the FEM

17 in the wheel-rail odeling and response coputations Bea Theories The ost coon and relatively siple way to odel the rail is by using bea theories. Euler-Bernoulli bea (EBB) and Tioshenko bea (TB) theories are applied. In Chapter 6 of [1.3], two types of vibrations were discussed. One is the low-frequency oscillation due to long wavelength and it usually coes fro structure of rails and sleepers, and soeties rail pad which are odeled as spring set connected between rail and sleeper. The other consists of irregularities due to short wavelength oscillations which occur when the wavelength of span is twice the width of sleeper. This is the so-called pined-pined resonance. In addition, because of the aspect ratio TB theory can give closer solution to easured data. A relatively coprehensive treatent of wheel-rail odeled by EBB with concentrated traveling point load can be found in the book by Fryba [1.4]. Analytical solution for siply-supported (SS) EBB with traveling point load is provided in this reference. In a siilar work Young, et al. [1.5] dealt with dynaic interaction of vehicle-bridge syste with applications to high-speed railways. For high-speed railways proble, reference [1.6], presented by Grassie, et al., is concerned with undergoing high frequency otion. The deficiency by using EBB under certain conditions was entioned. In this reference, corrugation with aplitude and phase of contact force was studied. Bea under a oving load was reported by B. Mehri, et al. [1.7]. The approach of odeling traveling point load was using dynaic Green s function which is different fro the ethod by Fryba. Many studies of rail by applying bea theories are concerned with one span only. In [1.8], Ichikawa, et al. presented a different approach for

18 4 solutions of EBB subjected to traveling point load. They believed their ethod could easily handle any conditions of continuous beas, such as non-unifor, ultiple boundary conditions, and so on. In [1.9] the issue of contact forces in wheel-rail syste was addressed by Ayasse and Chollet. Because of their coplexities frictional forces at the contact areas are frequently disregarded, such that the siple Hertzian contact theory can be eployed. Owing to the fact that the distance between two consecutive sleepers does not satisfy the aspect ratio requireent of SS EBB theory, TB theory is widely used instead. Lin [1.10] pointed out the advantages of using TB theory and inaccuracy of applying EBB theory. In [1.11] Ruge and Brik exained results using EBB and TB on Winkler foundation. In [1.12] Kargarnovin and Younesian included the effects of Pasternak foundation with TB subjected to a oving load. Fourier transforation was eployed. Uzzal, et al. [1.13] also used Pasternak foundation with oving load and oving ass. It was found that the oving ass had a larger effect than the oving load. In [1.14] Wu and Thopson studied the noise and the effect of nonlinearity Finite Eleent Method for Wheel-Rail Interaction The approaches to the analysis of the wheel-rail proble, while can provide soe useful results and insight, they cannot provide detail inforation at the contact areas and three-diensional (3D) responses. In order to overcoe these shortcoings, the versatile nuerical technique, the Finite Eleent Method (FEM) has been widely eployed. However, because of coputational cost, soe researchers applied sipler FE odels for the wheel-rail interaction proble. For exaple, Wu [1.15] applied FE based on TB

19 5 theory for the rail study. In [1.16] Wu and Thopson applied TB finite eleent to study high-frequency lateral response. In this reference, ultiple spans were considered. Bea FE odel for the railway was studied in [1.17]. In [1.18] Ekevid found that when the speed of train or load exceeds the Rayleigh wave velocity larger vibration otion occurred. Many detailed FE odels for wheel-rail interaction proble have been investigated in the last three decades or so. For exaple, the wear proble of the lateral surface of wheel-rail syste was investigated by Vyas and Gupta [1.190]. It was found that the ipact force resulted fro the flat zones of the rail wheel was twice as uch as the noral rolling force. In [1.20] Sladkowski and Sitarz also studied the wear proble of the wheel-rail interaction in which the quasi-hertz contact was assued. In [1.21] Pieringer, et al. found that the contact force level of a 3D FE odel is lower than that of a 2D FE odel. In [1.22] K Nguyen, et al. used a 2D FE odel since they have found that a 3D FE odel was of large coputational cost. Dinh [1.23] considered a two span 3D FE odel. Esen [1.24] presented results of a study using the 3D FE odel in which only the noral contact force was considered. Corrugation was exained. 1.4 ORGANIZATION OF PRESENTATIONS In this thesis, there are six chapters. Chapter 1 includes background and otivation, objectives of the investigation, literature survey, and organization of presentation of this thesis. Chapter 2 deals with theoretical developent. Analytical approaches applying bea theories and traveling point load are introduced. Finite eleent odels eploying bea theories and 3D solid eleents are presented. Frictions at the contact areas are

20 6 included. Chapter 3 shows the details of wheel-rail odeling by the coercially available FE package, Abaqus. In Chapter 4, coputed results and their verifications by those of analytical approaches are given. Coputed results for the 3D FE wheel-rail syste are copared with those of 3D FE rail and traveling point load odel. In addition, it provides an approach to adjust the traveling point loads in order to provide coputed results coparable to those of the wheel-rail syste. Chapter 5 presents the applications of scaled traveling point load on the 3D FE rail odel. Here in particular block dapers are included in the wheel-rail syste. Effects of block dapers, and their locations are exained. Finally, in Chapter 6, it includes conclusions of the investigation and recoendations for future work.

21 CHAPTER 2 THEORETICAL DEVELOPMENT 7 In this chapter, odels of wheel-rail interaction based on bea theories and finite eleents (FE) are presented. The wheel-rail contact forces are odeled as traveling loads. In the FE odels, the wheel-rail syste without and with block dapers are considered. Siilar to the bea odels contact forces of the three-diensional (3D) FE odels are considered as traveling loads. The organization of this chapter is the following. Section 2.1 is concerned with the rail odeled as beas. The bea theories applied are those due to Euler-Bernoulli and Tioshenko. Section 2.2 deals with the free vibration of undaped siply supported Euler-Bernoulli bea (EBB). Wrinkler s bea (WB) theory for a bea with the foundation is briefly introduced in Section 2.3. A siple odel for wheel-rail interaction is presented in Section 2.4. For ore realistic wheel-rail interaction investigation the FE odels are considered in Section 2.5 in which the FE bea odels and 3D finite eleent odels using the 8-node brick eleent are included. Finally, the wheel-rail odel with block dapers is considered in Section RAIL MODELS BY BEAM THEORIES Siple analytical odels of rail have been eployed in the studies of wheel-rail interaction. In this section, vibration of bea structures is considered. The Euler-Bernoulli bea theory and the Tioshenko bea theory are considered in the following sub-sections.

22 2.1.1 Euler-Bernoulli Bea Theory 8 The basic assuptions of the Euler-Bernoulli bea (EBB) theory are that: (a) the aspect ratio of length to transversal diension such as diaeter of a circular cross-section or width a rectangular cross-section is greater than 10, (b) noral to the cross-section before deforation and after deforation reains unchanged, (c) the aterial is hoogeneous and isotropic, and (d) it obeys Hooke s law. The governing partial differential equation of otion for transversal or lateral bea deflection w(x, t) for unifor cross-section is given by [2.1] where, EI 4 w(x, t) x 4 EI = flexural rigidity of the bea, + ρa 2 w(x, t) t 2 = q(x, t), (2.1) ρ = density, A = area of cross section of the bea, ρa = the ass of the bea per unit length, t = tie, q(x, t) = distributed load or load intensity on the bea. Note that for siplicity daping in the bea is not included in this equation Tioshenko Bea Theory In Tioshenko bea (TB) theory, it assues the rotation of the area of cross-section of the bea and shear deforation of the bea. Its aspect ratio can be saller than 10 and generally it is applied when accurate high natural frequencies are required. One general for of the equations of otion consists of two variables, naely, the lateral deflection w(x, t) and shear deforation φ(x, t). These equations are [2.2]

23 ρa 2 w(x, t) t 2 q(x, t) = x t) [κag ( w(x, ) φ(x, t)], (2.2a) x 9 ρi 2 φ(x, t) t 2 = x φ(x, t) (EI ) + κag ( x w(x, t) φ(x, t) ). x (2.2b) If the bea is linear, isotropic, hoogeneous and of unifor cross-section, the last two equations can be cobined to for a single one as [2.2] EI 4 w(x, t) x 4 + ρa 2 w(x, t) t 2 ρi (1 + E κg ) 4 w(x, t) x 2 t 2 + ρ2 I 4 w(x, t) κg t 4 = q(x, t) + ρi 2 q(x, t) κga t 2 EI 2 q(x, t) κga x 2, (2.3) where EI,, A, t, and q(x, t) have already been defined in the foregoing, G = shear odulus, and κ = shear factor. 2.2 FREE VIBRATION OF UNDAMPED SIMPLY SUPPORTED EBB In 2.1.1, the equation of otion for EBB was given. For free vibration excluding daping, it reduces to EI 4 w(x, t) x 4 + ρa 2 w(x, t) t 2 = 0, (2.4) or the if the area of cross-section is not unifor it can be rewritten as 2 x 2 [EI 2 w(x, t) x 2 ] = ρa 2 w(x, t) t 2. (2.5) After using separation of variables, the solution can be shown to be w(x, t) = W(x)[Asin ωt +Bcos ωt], (2.6) in which W(x) is given by W(x) = C 1 sin βx +C 2 cos βx + C 3 sinh βx + C 4 cosh βx where A, B, C 1, C 2, C 3, and C 4 are constant and β 4 = ω2 ρa EI.

24 The boundary conditions for siply supported (SS) bea are W(x) x=0 = 0, d2 W(x) dx 2 x=0 with L being the length of the bea. = 0, W(x) x=l = 0, and d 2 W(x) dx 2 x=l = 0, 10 After applying the above boundary conditions, one has the solution for free vibration as [2.3]. w(x, t) = C n sin ( nπ L x) {A n sin[ω n t] + B n cos[ω n t]}, (2.7) n=1 where C n is constant, ω n = ( nπ L )2 EI ρa, and n = 1, 2, 3,. The integer n is known as the odal nuber. 2.3 BEAM ON ELASTIC FOUNDATION In practice, the rails rest on the track foundation which contains rail pads, fastening, sleepers, ballast, subballast, and subgrade, as shown in Figure 2.1[2.4]. The subgrade is usually used to represent the soil bed when it is under forces. For siplicity, it is coonly assued that the relationship between forces and deforations is linear. When the displaceent only appears in the loading zone, otherwise the displaceent is equal to zero, the beas on elastic foundation can be odeled as series of springs connected between the beas and solid ground, as shown in Figure 2.2. This is the basis of Winkler bea (WB) theory [2.5].

25 11 Figure 2.1 [2.3] Track structure with foundation. The foundation odulus can be represented as k f = A f k s = bk s where A f = section area of the copressed foundation, k s = soil stiffness coefficient, and b = width of the bea. Figure 2.2 Model for Winkler bea.

26 The equation for free vibration of WB theory is therefore defined as [2.5] 12 EI 4 w(x, t) x 4 + k f w(x, t) = ρa 2 w(x, t) t 2. (2.8) This equation has the sae for as Equation (2.5) except the additional ter on the left-hand side (LHS). This ter can siply be incorporated in the stiffness ter of Equation (2.5) to give the new solution for the lateral deflection as w i (x, t) = W i (x)(a i sin ω i t +B i cos ω i t), (2.9) where now W i (x) is governed by the fourth order differential equation d 4 W i (x) dx 4 k i 4 W i (x) = 0 (2.10) in which k i 4 = (ρaω i 2 k f ) EI, and W i (x) = C 1i sin k i x +C 2i cos k i x + C 3i sinh k i x + C 4i cosh k i x (2.11) where C 1i, C 2i, C 3i, and C 4i are constant and can be deterined by applying the boundary conditions. 2.4 SIMPLE MODEL FOR WHEEL-RAIL INTERACTION As pointed out in Chapter 1, to provide a realistic three-diensional (3D) odel for the wheel-rail interaction the issue of contact stresses is paraount and it incurs several ajor coputational probles. For exaple, at the contact points, the stress levels are very high to the extent that the sizes of the finite eleents are extreely sall if one wishes to obtain reasonable nuerical values of the stresses. This akes the coputation of stresses very tie consuing, if not coputationally infeasible. On the other hand, any analytical odels consider this contact proble by representing the rails as an EBB

27 13 or a TB and the contact point as Hertzian contact [2.6]. The latter does not include friction at the contact point and therefore is very different fro reality. In order to provide a eans of including friction at the contact point between the wheel and rail, an analytical odel for the wheel-rail interaction is to consider the SS EBB with a traveling load along with axial oving loads at the contact point. For siplicity, it is assued that lateral and axial deforations are linear so that the principle of superposition can be applied, and the coupling effect between the axial and lateral deflections are disregarded such that this proble consists of two separate equations of otion. The first is the lateral deflection based on the EBB theory with a oving point load, and the second is the axial deflection governed by the second order partial differential equation with a oving load acting longitudinally. This latter longitudinal oving load represents the frictional force at the contact point Siply Supported EBB with Traveling Point Load The case of lateral bea deflection with a oving load has been coonly applied in truck/car-bridge interaction proble [2.7], for exaple. The equation of otion for this case is given by [2.7] EI 4 w(x, t) x 4 + ρa 2 w(x, t) t 2 = Pδ(x u), (2.12) where δ is the Dirac-delta function, P is the external traveling force, u(t) = vt describes the position where the applied external force, and s is the constant velocity of the traveling load. A scheatic diagra for this case is included in Figure 2.3. In this odel the ends of the bea structure are assued to be SS. That is, the boundary conditions at both ends of the bea structures are:

28 W(x) x=0 = 0, d 2 W(x) dx 2 x=0 = 0, W(x) x=l = 0, and 14 d 2 W(x) dx 2 x=l = 0, And the initial conditions are: w(x, 0) = 0, and w(x,t) = 0. t t=0 Figure 2.3 Siply-supported EBB subjected to a traveling point load. The Dirac-delta function is given in the following: δ(x vt) = dh(x) dx, (2.13) where H(x) is called Heaviside function. Equation (2.13) siply eans that the Dirac-delta function is equal to the distributional derivative of H(x). The Dirac-delta function has the following relations with continuous function f(x) in the intervals < x 1, x 2 >: 0, vt < x 1 < x 2 x 2 δ(x vt)f(x)dx = { f(vt), x x 1 < vt < x 2 1 0, x 1 < x 2 < vt (2.14)

29 To solve Equation (2.12) with boundary and initial conditions, one uses the ethod of 15 integral transforation. It starts with Equation (2.12) ultiplied by sin nπx L and integrated with respect to x fro 0 to L. Then, fro fundaental relations of Laplace-Carson integral transforation [2.7], it one can obtain the following relations: L W(n, t) = w(x, t)sin nπx dx, n = 1,2,3, L 0 v(x, t) = 2 L n=1 where W(n, t) is the sine transfor of w(x, t). W(n, t) Taking the sine integral transforing, Equation (2.12) becoes n 4 π 4 L 4 sin nπx L, (2.15) EI W(n, t) + ρa W (n, t) = Psin nπvt L, (2.16) and the angular frequency of n-th ode of vibration of SS EBB is the sae as that in Section 2.2. That is, ω n = ( nπ L )2 EI ρa. The n-th natural frequency is described as f n = ω n 2π, (2.17) and also the angular velocity is related to the constant velocity of the traveling load by Now, Equation (2.16) can be rewritten as ω = πv L. (2.18) 2 W (n, t) + ω n W(n, t) = P sin nωt, (2.19) ρa and Equation (2.19) can be solved by using the ethod of Laplace-Carson integral transforation [2.7], W (n, p) = p W(n, t)e pt dt, 0

30 W(n, t) = 1 2πi a 0 +i etp a 0 i 16 W (n, p) dp, (2.20) p where the paraeter p is a variable in the coplex plane and a 0 in the second relation signifies that the integration is carried out along a straight line parallel to the iaginary axis lying to the right of all the singularities of the function of the coplex variable etp p. Therefore, after applying the Laplace-Carson integral transforation [2.7], Equation (2.19) becoes p 2 W (n, p) + ω n 2 W (n, p) = ( Pnω ρa ) 1 p 2 + ω n 2. (2.21) The transfored solution is W (n, p) = Pnω ρa ( 1 p 2 + n 2 ω 2) ( 1 p 2 + ω n 2 ). (2.22) By using the relation fro Equation (2.20) and Equation (2.15), the original solution of SS EBB subjected with traveling point load with constant velocity is [2.8] w(x, t) = w 0 sin ( nπx L ) 1 n 2 (n 2 α 2 ) [sin(nωt) α n sin(ω nt)], (2.23) n=1 where w 0 is the deflection at id-span of the bea with static load P at x = L 2. The approxiated value given by [2.7] is w 0 = PL3 48EI, (2.24) and α is the characteristic relating to the effect of velocity as α = v v critical = vl π ρa EI. (2.25)

31 2.4.2 Unifor Bar with Traveling Axial Load 17 The vibration of a unifor bar with a oving horizontal load is applied to account for the oving friction force at the contact point. The case is shown in Figure 2.4. The equation of otion is given E 2 U(x, t) x 2 ρ 2 U(x, t) t 2 = μpδ(x u), (2.26) where U(x, t) or siply U is displaceent (longitudinal displaceent) at x, μ is the kineatic coefficient of friction, and the reaining sybols have already been defined in the foregoing. The boundary conditions of a siply supported bar undergoing axial deforation are the sae as a fixed-fixed cable or string. They are: and the initial conditions are: U(x, t) x=0 = 0, and U(x, t) x=l = 0, U(x, 0) = 0, and U(x,0) = 0. t t=0 Also, the steps in the solution process are siilar to those presented in Sub-section Figure 2.4 Traveling friction force on unifor bar.

32 2.5 FINITE ELEMENT MODELS FOR WHEEL-RAIL INTERACTION 18 The analytical odels considered in the foregoing are relatively siple and are confined to siple loadings. However, when odels with ore realistic loadings or odels involved with practical geoetrical configurations are required one usually resorts to the versatile nuerical approach, the finite eleent ethod (FEM). As reviewed in Chapter 1, finite eleent (FE) odels for wheel-rail interaction can be loosely divided into two classes. The first class is concerned with the relatively siple cases of representing the rail as EBB or TB finite eleents and the wheel as a traveling point load. The second class consists of the detailed odels of wheel-rail interaction using three-diensional (3D) finite eleents. In this section, an exaple of the first class of odels is introduced in Sub-section while the 3D FE odel of the second class is outlined in Sub-section Bea Finite Eleent Model of Wheel-Rail Interaction To liit the scope of the present investigation, sall deforations are considered such that linear odels are adequate. In the FE analysis (FEA) the linear equation of otion for a siple wheel-rail interaction odel is given by [M]{q} + [C]{q} + [K]{q} = {F}, (2.27) where [M] is the assebled ass atrix, [C] is the assebled daping atrix, [K] is the assebled stiffness atrix, and {F} is the assebled generalized forcing vector. For exaple, [M] is assebled by aking use of the EBB eleent atrices whereas [K] is assebled by applying the EBB eleent stiffness atrices [2.8]. For direct reference and copleteness, the bea eleent ass and stiffness based on the EBB theory and TB theory are included in the following sub-sections.

33 Eleent ass and stiffness atrices based on EBB Theory 19 For siplicity, the bea finite eleent considered here has 2 nodes with 2 degrees-of-freedo (dof) per node. The nodal dof are the displaceent and angular displaceent or rotation. Based on the EBB theory, its stiffness and ass atrices are, respectively given by [2.9] [k e ] = EI l 3 [ 12 6l 12 6l 6l 4l 2 6l 2l 2 ], (2.28) 12 6l 12 6l 6l 2l 2 6l 4l 2 where l l 54 13l [ e ] = ρal [ 22l 4l 2 13l 3l 2 ], (2.29) l l 13l 3l 2 22l 4l 2 is the eleent length, and the subscript e denotes the eleent nuber Eleent ass and stiffness atrices based on TB Theory As pointed out in Section 2.1 when the aspect ratio is less than 10 the EBB theory is no longer applicable and therefore the TB theory is applied instead. In this case, an additional nodal dof is introduced such that the eleent stiffness and ass atrices [2.10] for a unifor TB are, respectively defined as [k e ] = [k B ] + [k S ] = EI l k 11B k 12B k 13B k 14B k 15B k 16B k 22B k 23B k 24B k 25B k 26B k 33B k 34B k 35B k 36B k 44B k 45B k 46B syetrical k 55B k 56B [ k 66B ] + κ sgal 6 [ syetrical 0 0 2] (2.30)

34 20 k 11B = 12 l 2 = k 14B = k 44B ; k 12B = 6 l = k 13B = k 15B = k 16B = k 24B = k 34B = k 45B = k 46B ; k 22B = 4 = k 52B ; k 23B = 3 = k 26B = k 33B = k 35B = k 36B = k 56B = k 66B ; k 25B = 2, where I is the second oent of area of the cross-section of the bea, G is shear odulus, κ s is shear correction factor given by [2.11], [k B ] is associated with bending, and [k S ] is due to shear. [ e ] = [ B ] + [ R ] = ρal B 12B 13B 14B 15B 16B 22B 23B 24B 25B 26B 33B 34B 35B 36B 44B 45B 46B syetrical 55B 56B 66B ] [ + ρil R 12R 13R 14R 15R 16R 22R 23R 24R 25R 26R 33R 34R 35R 36R 44R 45RR 46R, 2.31) syetrical 55R 56R [ 66R ] 11B = 156 = l 44B, 2 12B = 22 = l 13B = 45B = 46B, 14B = 54, l 2 22B = 4 = 23B = 33B = 55B = 56B = 66B, 15B = 13 l = 16B = 24B = 34B, 25B = 3 26B = 35B = 36B ; 11R = 504 = l 2 14R = 44R, 12R = 42 = l 15R = 24R = 45RR, 13R = 252 l = 16R = 34R = 46R, 22R = 56 = 55R, 23R = 21 = 26R = 35R = 56R, 25R = 14, 33R = 126 = 36R = 66R, where [ B ] is associated with bending and [ R ] is due to rotary inertia Three-Diensional Finite Eleent Model of Wheel-Rail Syste A general 3D FE odel of wheel-rail interaction is challenging in the sense that it possesses two ajor issues. The first issue is to do with the coputation effort required, whereas the second issue is concerned with the physical proble of odeling the contact

35 between the oving wheel and rail. While there are any approaches available in the literature [2.12, 2.13] a particularly relevant and iportant review was presented by Wriggers and Zavarise [2.12]. To liit the scope of the present investigation the atrix equation of otion for a general wheel-rail interaction syste is defined by [2.12] 21 [M]{q} + {R(q)} + {R c (q)} = {F} (2.32) where {R(q)} defines the so-called stress divergence ter that includes nonlinearities due to large deforations, {R c (q)} is the residual due to contact, and the reaining sybols have already been defined in Equation (2.27). The contact ter in Equation (2.32) is further divided into two sub-classes, naely, the sooth contact case and the non-sooth contact case. Coputationally, Equation (2.32) requires a great aount of effort as reviewed in Chapter 1 and in any cases it is infeasible. In order to overcoe this ajor difficulty the present investigation has confined to cases with linear deforations and the ter {R c (q)} is replaced by a forcing vector that contains ultiple oving point loads. In turn, Equation (2.32) reduces to the for given by Equation (2.27). Thus, the 3D finite eleent odel for wheel-rail interaction in the present investigation applies a 3D finite eleent that is available in any coercial FEA coputer packages. This 3D finite eleent is included in the following for siplicity, copleteness, and direct reference. This siple 3D or solid finite eleent has 8 nodes with 3 dof per node and known as the linear brick eleent [2.14]. The scheatic diagra of this eleent is shown in Figure 2.5 [2.14]. The shape functions are given by N i = 1 8 (1 + ξξ i)(1 + ηη i )(1 + ζζ i ), i = 1 8, (2.33)

36 where ξ, η, and ζ are the natural co-ordinates while the subscript i denotes the nodal nuber. The values of ξ i, i, and ζ i are given in Table Figure 2.5 Eight-node brick eleent. Table 2.1 Nodal co-ordinates for 8-node brick eleent. i ξ i i ζ i

37 Therefore, the atrix of shape functions can be deterined to be 23 and the strain atrix is given by [N] = [ 0 1 0] N i, i = 1, 2,, 8 (2.34) [B] = L [N], (2.35) where L is the linear differential operator and not to be confused with the length of the bea structure. The ass atrix of this brick eleent is given as [2.14] [ e ] = ρ[n] T [N]dV V where J is the deterinant of the Jacobian atrix, 1 = ρ[n] T [N] J d ξ d η d ζ, (2.36) 1 [ J ] = [ x ξ x η x ζ y ξ y η y ζ z ξ z η z ζ ], (2.37) where x, y, and z are the Cartesian co-ordinates. The corresponding eleent stiffness atrix for a brick of orthotropic aterial has first been explicitly given by Melosh [2.15] as [k e ] = [B] T [D][B]dV V 1 = [B] T [D][B] J 1 d ξ d η d ζ, (2.38) where [D] is the elastic property atrix. The details of the explicit eleent stiffness atrix are not included here for brevity. However, it ay be appropriate to point that this eleent exhibits locking behavior. Of course, to circuvent this behavior higher order

38 3D finite eleents are preferred. But the latter is relatively expensive to eploy since considerably ore coputer eory is required in the coputation. Therefore, in the present investigation, the eleent derived by Melosh is adopted Dynaic Characteristics of Wheel-Rail Syste Before the coputation responses fro atrix Equation (2.27), the eigenvalue solution has to be addressed first. This is iportant in that the coputed eigenvalues or natural frequencies and eigenvectors or ode-shapes of the bea or 3D FE representations of the rail syste can be copared with those fro the analytical solution. For siplicity, the effect of the rotation of the wheel is disregarded and only the dynaic characteristics of the rail are obtained in the present studies. The eigenvalue solution is obtained by considering the free vibration analysis of the syste. That is, the free vibration of the wheel-rail syste is obtained by the following atrix equation By applying {q} = {Q}e i t, the last equation reduces to 24 [M]{q} + [K]{q} = {0}. (2.39) ( ω 2 [M] + [K] ){Q} = {0} (2.40) in which ω is the angular frequency and {Q} is the vector of aplitudes of displaceent. By writing = ω 2 such that Equation (2.40) becoes ( [M] + [K] ){Q} = {0}. (2.41) The characteristic values or eigenvalues of Equation (2.41) is deterined by solving the characteristic or frequency equation [M] + [K] = 0. (2.42) For the present investigation, [M] and [K] are positive definite and therefore, there are n eigenvalues with n corresponding eigenvectors, where n is the order of the atrix [M].

39 Three-Diensional Finite Eleent Model of Wheel-Rail Interaction In the present investigation, the atrix equation of otion of the 3D FE odel of wheel-rail interaction is given by [M]{q} + [C]{q } + [K]{q} = {F o δ(x vt)} (2.43) in which F o is the vector of applied loads where friction forces at the contact points are included. The daping atrix [C], in general, consists of two parts. One is the daping due to the aterial and the other is associated with the rotation of the wheels. This latter atrix is skew syetric and is called the gyroscopic atrix [2.16]. The reaining sybols have already been defined in Equation (2.39). 2.6 FINITE ELEMENT MODELS FOR WHEEL-RAIL WITH BLOCK DAMPERS As stated previously one of the ain objectives of the present investigation is to investigate the responses in the wheel-rail interaction without or with block dapers. The effects of the block dapers on the responses of the wheel-rail interaction is to be studied, in addition to the treatent of contact forces as traveling point loads. Owing to the relatively large ass and stiffness of every block daper, the attached block dapers essentially odify the distributed nature of the rail. This, in effect, changes the assebled ass and stiffness atrices of the rail. However, the for of the atrix equation of otion for the wheel-rail syste with block dapers is the sae as that given by Equation (2.39). For siplicity and in order to reduce the aount of coputational effort the block dapers are represented by the 8-node brick finite eleent introduced in Section 2.5.

40 CHAPTER 3 WHEEL-RAIL MODEL BY FEM SOFTWARE 26 In order to provide a ore realistic odel for the analysis of wheel-rail interaction, coercially available FEM software, such as Abaqus, ANSYS, COMSOL Multiphysics, and uch ore can be eployed. However, in the present investigation Abaqus [3.1] is eployed since it is relatively easy to ipleent in a high-end laptop or engineering workstation. To analyze the coplicated syste of wheel and rail, there are several siplifications that are coonly applied. First, the wheel is treated as a rotation disk (see, Appendix 3A). Second, the standard UIC60 (this is the coonly used European standard where 60 denotes the ass of the rail in kg/) rail is assued to be an I-bea (see, Appendix 3B). Third, owing to the last two assuptions, the angle of wheel-rail interaction is not included in the studies. To further liit the scope of the present investigation in which one of the ain objectives was to study and explore the replaceent of the wheel/rail interaction or wheel/rail contact action by a traveling load, only vertical and transversal deflections are considered in the first phase of the investigation. In addition, friction at the contact point is included. This is believed to be ore realistic than the Hertzian contact odel adopted by any researchers in the past. During the second phase of the investigation, siple block dapers have been attached to the rail so that their effect on the deflections of the rail can be exained. Of course, ore refined FE odels for the wheel and rail can be investigated but it is outside the scope of the present investigation.

41 GEOMETRICAL AND MATERIAL PROPERTIES OF WHEEL AND RAIL The siplified cross-sectional geoetrical properties of the UIC60 rail are indicated in Figure 3.1(a) in which the diensions are in. The total length of this experiental rail is 4.2. The wheel is siplified as a disk with a center hole and depth is The sketch of this wheel odel is presented in Figure 3.1(b). (a) (b) Figure 3.1 Geoetrical diensions of rail and wheel: (a) I-bea odel for rail, and (b) circular disk for wheel (in eter). The aterial properties for the rail, wheel, and block daper are: Young s odulus E = Pa, Poisson ratio = 0.3, and density = 7850 kg PROCESS OF FE MODEL IN WHEEL-RAIL INTERACTION In wheel-rail interaction studies, the typical process of constructing the finite eleent (FE) odel is to separate into two steps. The first step is called pre-loading. It coputes the deforations and stresses which are caused by applying the loading before the wheel

42 traveling and it is used to ake sure that the wheel reaches the steady-state of rolling. The second step of the process is the initialization of deforation at the first tie step and solving the oving loading proble by the explicit tie-arching schee [3.2]. The tie step size is different fro step to step in the tie-arching schee. For the first step, it is set to 1 s. For the second tie step the following equation is used 28 t 2 = 4 () speed. (3.1) Initially, the wheel is located at 0.1 fro the departing end of the rail. Thus, the total distance of traveling in the second step is 4 since the total length of the rail is IMPLICIT AND EXPLICIT TIME-MARCHING SCHEMES Siilar to any coercially available FEA packages Abaqus have several dynaic analysis procedures. In order to analyze the proble efficiently, the finite eleent dynaics explicit tie schee has been adopted for the present investigation. This is because there are several iportant features in this particular schee. For exaple, when the syste has a short dynaics response tie and large discontinuity, it is efficient to eploy the explicit tie nuerical integrating schee since it is known to be unconditionally stable. Further, various choices for contact odeling are available. For the explicit tie nuerical integration schee, one has to select a sall tie step size in order to provide a stable solution. In this regard, the critical tie step size is governed by the following relation [3.3] where t is the tie step size (s), t = 2 ω ax, ω ax = 2 l e E ρ, (3.2, 3.3)

43 l e is the length of the eleent (), 29 ρ is the density (kg/ 3 ), and ω ax is the highest natural frequency of the syste. As long as the tie step is chosen slightly saller than the critical tie step size defined in Equation (3.2) the solution will be stable. A typical value has been suggested to be 98 or 95% of the critical [3.4]. In the FE odel that adopts the explicit tie schee, the critical tie step size, t = s since for the above wheel-rail syste Young s odulus is Pa, the density of the aterial 7850 kg/ 3, and sallest length of the eleent Clearly, this tie step size is very sall and therefore for response coputation the explicit tie schee is generally preferred for efficient coputation. 3.4 BOUNDARY CONDITIONS In the present investigation, the boundary conditions for the rail between consecutive sleepers are assued to be siply-supported (SS) or hinged. More specifically, the present wheel rail odel has the following features. (1) With reference to Figure 3.2, the width of every sleeper is 0.2 (6 sleepers are indicated in Figure 3.2). (2) The length between two siply supported sleepers is 0.6 (there are 5 spans with 6 SS sleepers as shown in Figure 3.3). (3) The wheel is placed at 0.1 fro the starting end of the rail for the pre-loading step. (4) The external force is 100kN. It is calculated fro Superliner [3.5]. Each car is 74

44 tons with two four-wheel sets plus the weight of the capacity of passengers. 30 (5) The kineatic friction coefficient between wheel and rail is 0.3. (6) The FE contact odel assues the penalty contact ethod [3.6]. (7) The relation between velocity and angular velocity is v = r. (8) There are 8 different velocities of the train/wheel in every case considered in the coputation. That is, v = 10, = 1, 2, 3, 6, (/s). Figure 3.2 Wheel-rail odel with boundary conditions indicated. Figure 3.3 Finite eleent wheel-rail odel with boundary conditions.

45 3.5 ELEMENT SIZE AND MESH 31 As entioned in Section 3.3 in the foregoing, the critical tie step size for explicit tie schee was obtained as t = s which iplies that in order to provide sufficiently accurate coputational results the size of the eleents at the contact point between the wheel and rail would be extreely sall. The size of the eleent also leads to other probles when the contact point is rotating. The proble appears when the size of the rolling part is not sall enough. For exaple, when the rail (or disk) is odel as I shape bea and after a particular esh is chosen, using eleent type C3D8 which is the 8 node lower order 3D eleent available in Abaqus [3.7], such that the eleent size is insufficiently sall in accordance with Equation (3.3), then the coputed displaceent histories exhibit juping (no contacts) phenoenon. Clearly, this phenoenon is not present in reality. To provide a visual appreciation of such a phenoenon three different sizes of the 3D eleent are presented in Table 3.1 in which eleent size eans its largest diension of that eleent. Recall that the 3D solid finite eleent eployed in the present investigation is identified as type C3D8. As can be observed fro this table the two coarser eshes start to experience no contact at 2 s. The effect of eleent size on Juping (no contacts) phenoenon. In Table 3.2 the eleent size and corresponding critical tie step size are given so as to provide an appreciation of the sallness of the critical tie step size for the coputation applying the explicit tie arching schee. Eleent C3D4, which is a 4-node tetrahedral eleent, is applied on wheel odel due to partition. Rail odel is still by using C3D8 eleent.

46 Tie (s) Table 3.1 Evolution of different esh sizes for wheel-rail odel with juping (no contact). 32 Eleent size (c) Table 3.2 Relationship between eleent size and tie step size. Eleent size () Critical t (s)

47 33 CHAPTER 4 WHEEL-RAIL INTERACTION MODELS WITHOUT BLOCK DAMPERS This chapter is divided into two ain parts. The first part is concerned with the deterination of the iportant dynaic characteristics, such as the natural frequencies and ode-shapes of the rails. The second part deals with the evaluation of the responses of wheel-rail interaction odels. No block dapers attached to the rails are included in this chapter. The first part is therefore concerned with the eigenvalue solution and is included in Section 4.1 in which the rail is represented as a SS EBB, TB, and 3D FE odel applying the 8 node brick eleent introduced Chapter 2. The second part that includes the odeling of the wheel-rail contact proble as a traveling point load, is presented in Section 4.2. It is further been sub-divided into three coponents. The first coponent deals with the analytical solution of the responses of the SS EBB under a traveling point load. The second coponent presents the responses of the 3D FE odel under siilar traveling point load and in this coponent friction forces are included in the analysis. The third coponent shows the unscaled traveling point load and the odification of vertical and friction forces by scaling aplitude of each node. In addition, the results of 3D FE rail odel with unscaled and scaled traveling point load are copared with 3D FE wheel-rail odel, are presented in Section 4.3.

48 4.1 EIGENVALUE SOLUTION OF RAIL MODELS 34 The eigenvalue solution is concerned with the deterination of the natural frequencies and ode-shapes of the dynaic syste. In this section, the rail is odeled as a SS EBB, TB, and 3D FE representation. Analytical solution for the SS EBB is included in Sub-section Coputed results applying the FEM are presented in Sub-section in which the EBB finite eleent, TB finite eleent, and 3D finite eleent odels are studied and copared. The discussion is included in Sub-section Solution by EBB Theory As presented in Chapter 2, the natural frequencies of the SS EBB are given by ω n = ( nπ L )2 EI ρa. The geoetrical properties of this EBB are: length L = 0.6, cross-section area of the bea = , and the second oent of area of the cross-section I = The aterial properties are: Young s odulus of elasticity E = Pa, Poisson ratio = 0.3, and density = 7850 kg/ 3. While any natural frequencies and ode-shapes of this rail odel can be calculated, for brevity, only the first six ode-shapes and natural frequencies are presented in Figure 4.1 and Table 4.1, respectively.

49 35 Figure 4.1 First six ode-shapes of SS EBB rail odel. Table 4.1 First six natural frequencies of SS EBB. Mode nuber EBB (Exact) (Hz) EBB eleent (B23) (Hz) TB eleent (B21) (Hz) Coputed Results of Bea Eleent and Solid Eleent Models In the FE odels, bea eleents based on the EBB theory and TB theory, and 3D solid eleent are eployed. The ass and stiffness atrices of the bea eleents have

50 36 already been presented in Sub-sections and Table 4.1 and Figure 4.2 (a) through (f) show the first six natural frequencies and ode shapes by using EBB eleent (which is identified as B23 in Abaqus; henceforth, the notation inside the parentheses is referred to the identification in Abaqus) and TB eleent (B21). Table 4.2 and Figure 4.3 (a) through (l) show the first twelve natural frequencies and ode shapes by using the solid eleent (C3D8). Figure 4.2 (a)

51 37 Figure 4.2 (b) Figure 4.2 (c) Figure 4.2 (d)

52 38 (e) (f) Figure 4.2 First six ode-shapes of bea eleent odels: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, and (f) Mode 6.

53 Table 4.2 First twelve natural frequencies of FE solid eleent. 39 Mode nuber (Hz) Mode nuber (Hz) Figure 4.3 (a)

54 40 Figure 4.3 (b) Figure 4.3 (c)

55 41 Figure 4.3 (d) Figure 4.3 (e)

56 42 Figure 4.3 (f) Figure 4.3 (g)

57 43 Figure 4.3 (h) Figure 4.3 (i)

58 44 Figure 4.3 (j) Figure 4.3 (k)

59 45 (l) Figure 4.3 First twelve ode-shapes of solid eleent (C3D8) odel: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, (f) Mode 6, (g) Mode 7, (h) Mode 8, (i) Mode 9, (j) Mode 10, (k) Mode 11, and (l) Mode Discussion With reference to the results presented in the foregoing, the EBB exact natural frequencies are close to the corresponding FE odel as expected. The differences of first six natural frequencies between those using the EBB eleent and exact solutions are, respectively, 2.5%, 2.4%, 2.4%, 2.4%, 2.4%, and 2.4%, with respect to the exact EBB solution. However, these natural frequencies are significantly different fro those using the TB FE odel, and the 3D brick eleent odel. This indicates that the TB and 3D FE odels are close approxiations to the actual rail whose aspect ratio is L/D = 3.48, where D is the largest diension of the cross-section area of the rail.

60 4.2 RESPONSES OF RAIL DUE TO TRAVELING POINT LOAD 46 As presented in Chapter 2, the contact proble of the wheel-rail interaction odel is treated as a SS structure with a traveling point load. The SS structure is represented by the EBB bea eleent and the 3D brick eleent. The ain objectives are (a) to study analytically the responses of the EBB under a oving point load, and (b) to copute the responses of the EBB bea eleent and the 3D FE odel for a siilar traveling point load. The analytical and nuerical responses of the siple EBB odel are included in Sub-section while the responses of the 3D FE odel are coputed and presented in Sub-section Analytical and FE Models of Rail with a Traveling Point Load The essential objectives of the studies reported in this sub-section are (a) to introduce the detailed steps in ipleenting the traveling point load for the coputation of responses of the SS EBB structure, and (b) to provide siilar results using the FE EBB eleent. To satisfy the aspect ratio criterion for EBB theory, the following geoetrical properties of the rail odel are: length L = 0.6, width of cross-section b = 0.03, height of area of cross-section h = 0.03, the aspect ratio is 20, and the second oent of area of the cross-section = The aterial properties are: Young s odulus of elasticity E = Pa, Poisson ratio = 0.3, and density = 7850 kg/ 3. Before the responses along the length of the FE SS EBB structure are coputed the responses obtained analytically are considered first. Fro the results in Sub-section 2.4.1, the transversal deflection at id-span of the bea, w 0 = 2MgL3 π 4 EI, where Mg is the

61 point load applied at id-span or at x = L In the case of a traveling load, the steps in the response coputations are presented in the following. Firstly, Mg is the vertical point load applied on the bea. To express into the diensionless quantities, it is convenient to noralize the deflection by the id-point static deflection which, in this case, the point load is 125 kn, is w 0 = , and the diensionless tie, τ = t ( L v ). Secondly, Equation (2.17) is a suation series. It can be chosen as any odes as required. However, Equation (2.17) shows that when n is large, the value of α n 2 (n 2 α 2 ) will be sall. Therefore, in this case, the largest nuber n is chosen as 5. The diensionless responses coputed by using the analytical solution are plotted in Figures 4.5 through 4.7. Note that in these figures the coputed responses for n = 1 and n = 5 are very close. Now, the responses of the SS EBB structure approxiated by the FEM are considered. The traveling point load acting on the discretized SS EBB structure is dealt with first. As in the analytical odel presented in the foregoing, for the point load to ove along the length of the bea structure it is required to assign different aplitudes of the load at different stations along the length of the bea structure. For the present FE odel, the traveling load is only considered when it passes the specific node. To illustrate this, Figure 4.8 ay be helpful. For exaple, if the point load F is traveling with a constant speed v through Node i = 1, 2, 3, and 4 (in this particular exaple) the length of the 2 node EBB eleent is L 3, which coes fro the length of span L being divided by three 2 node EBB eleents. Therefore, the ties for the traveling point load to reach

62 48 Nodes i = 1, 2, 3, and 4 are given by t i = L 3 ( i 1 v ). (4.1) (a) w( L 4, t) w 0 (b) w( L 4, t) w 0 Figure 4.5 Noralized responses at a quarter-point fro starting end of SS EBB (x =0.15 ): (a) v = 10 /s, and (b) v = 20 /s.

63 (c) 49 w( L 4, t) w 0 (d) w( L 4, t) w 0 Figure 4.5 Noralized responses at a quarter-point fro starting end of SS EBB (x =0.15 ): (c) v = 30 /s, and (d) v = 40 /s.

64 (e) 50 w( L 4, t) w 0 (f) w( L 4, t) w 0 Figure 4.5 Noralized responses at a quarter-point fro starting end of SS EBB (x =0.15 ): (e) v = 50 /s, and (f) v = 60 /s.

65 (a) 51 w( L 2, t) w 0 (b) w( L 2, t) w 0 Figure 4.6 Noralized responses at id-point fro starting end of SS EBB (x = 0.30 ): (a) v = 10 /s, and (b) v = 20 /s.

66 (c) 52 w( L 2, t) w 0 (d) w( L 2, t) w 0 Figure 4.6 Noralized responses at id-point fro starting end of SS EBB (x = 0.30 ): (c) v = 30 /s, and (d) v = 40 /s.

67 (e) 53 w( L 2, t) w 0 (f) w( L 2, t) w 0 Figure 4.6 Noralized responses at id-point fro starting end of SS EBB (x = 0.30 ): (e) v = 50 /s, and (f) v = 60 /s.

68 (a) 54 w( 3L 4, t) w 0 (b) w( 3L 4, t) w 0 Figure 4.7 Noralized responses at the third quarter-point fro starting end of SS EBB (x = 0.45 ): (a) v = 10 /s, and (b) v = 20 /s.

69 (c) 55 w( 3L 4, t) w 0 (d) w( 3L 4, t) w 0 Figure 4.7 Noralized responses at the third quarter-point fro starting end of SS EBB (x = 0.45 ): (a) v = 30 /s, and (b) v = 40 /s.

70 (e) 56 w( 3L 4, t) w 0 (f) w( 3L 4, t) w 0 Figure 4.7 Noralized responses at the third quarter-point fro starting end of SS EBB (x = 0.45 ): (a) v = 50 /s, and (b) v = 60 /s.

71 For the SS bea structure approxiated by N EBB eleents, the tie is given by 57 t i = L N ( i 1 v ), (4.2), where i = 1, 2, 3,, N. Figure 4.8 Three bea eleent representation of one SS bea structure. The tie and aplitude of the traveling point load for the 3 eleent case are included in Table 4.3. It includes the odifications of the starting node and ending node of the SS EBB FE odel. In this table, the tie and aplitude values under the nodal nubers are the tie and force aplitudes applied at the nodes. Siilarly, Table 4.4 provides the tie and force aplitude values for the case of bea structure represented by N eleents. Table 4.3 Tie and aplitude of traveling point load at Node 1 to Node 4. Node 1 Node 2 Node 3 Node 4 Tie 0 L 3v 0 L 3v 2L 3v L 3v 2L 3v L v 2L 3v L v Aplitude

72 Table 4.4 Tie and aplitude for bea structure represented by N eleents. 58 Node 1 Node 2 Node n Tie t 1 t 2 t i 1 t i t i+1 t n 1 t n Aplitude Coputed responses by the SS EBB FE odel are included in Figures 4.5 through 4.7 for direct coparison to those of the analytical EBB solutions. It should be pointed out that the FE EBB results and those by the analytical EBB solutions are very close to each other Coputed Results of 3D FE odel with Traveling Point Load In this sub-section, the tie and force aplitude values considered in the last sub-section are applied to the 3D FE odel. Thus, Table 4.4 is applied to the path of traveling vertical force along the (X1, Y1, Z1) to (XN, YN, ZN) straight line. In the present case, it is (0, 0, 0) to (0.6, 0, 0), where the unit is in. That is, (0, 0, 0) is the starting node and (0.6, 0, 0) is the ending node (See, Figure 4.9). To provide a ore realistic representation of the wheel-rail interaction proble, friction is included in the present FE odel. The friction force is considered to be equal to the product of the noral or vertical point load and the kineatic coefficient of friction, = 0.3 in the present investigation. In the ost general atheatical odel for friction force it is considered as Coulob friction and therefore it is represented by the signu function such that

73 sgn(x ) = { 1, x < 0 0, x = 0 +1, 0 < x 59 (4.3) The signu function is illustrated in Figure Coputationally it is difficult to operate on this function as it has a discontinuity at x = 0. Therefore, it is generally approxiated as that shown in Figure This approxiation is known as regularization. The coputed responses applying the 3D solid eleent odel are plotted in Figures 4.12 through Figure 4.9 Finite eleent 3D odel for traveling point load. Figure 4.10 Signu function f(x).

74 60 Figure 4.11 Regularization of signu function in discrete tie steps. Figure 4.12(a) Longitudinal responses at quarter-point fro starting node.

75 61 Figure 4.12(b) Vertical responses at quarter-point fro starting node. Figure 4.13(a) Longitudinal responses at iddle span..

76 62 Figure 4.13(b) Vertical responses at iddle span. Figure 4.14(a) Longitudinal responses at quarter-point fro ending node

77 63 Figure 4.14(b) Vertical responses at quarter-point fro ending node Discussion Coputed results of deflection on FE SS EBB shows quite closed to analytical results. In addition, when the analytical results include ore natural frequency into Equation 2.23, the results are ore closed to FE SS EBB. It gives strong verification of the ethod of odeling traveling point load is correct. In 3D FE odel, it does not show the different shapes due to various velocities obviously. The proble of this case is owing to input velocities which are uch saller than critical velocity. The relation between input velocity and critical velocity is defined as α which is presented in Equation By using the analytical solution of SS EBB, the critical velocity is up to 235 /s in this case. α in these cases are very sall which is 0.04, 0.085, 0.12, 0.17, 0.21 and 0.25 with respect to 10 /s to 60 /s.

78 4.3 COMPUTED RESULTS OF 3D FE RAIL MODEL WITH TRAVELING 64 POINT LOAD With the traveling point load in foregoing, it presents the traveling point load on 3D FE rail odel. In order understanding the responses of rail wheel-rail odel. Here, the traveling point load is based on a real case which is 100kN for vertical force and with the friction force 30kN Coparison of 3D FE Rail with Traveling Point Load Model and 3D FE Wheel-Rail Model In the traveling point load (TPL) case considered in Sub-section above the TPL path is along the central line, as indicated in Figure 4.9. Since in practice, the contact is not at a point but rather a sall area and therefore ore TPL paths are required in the coputation. For siplicity and syetry, therefore three TPL paths along the global X-axis are chosen in the present studies. These paths with the nodal DOF indicated in sall red integers and partial wheel and rail are shown in Figures 4.15 in which the first, second, and third paths are shown in Figures 4.15(a), 4.15(b), and 4.15(c), respectively.

79 (a) 65 (b)

80 (c) 66 Figure 4.15 Wheel and rail contact paths: (a) Path 1, (b) Path 2, and (c) Path 3. As entioned in Chapter 3, in the 3D FE wheel-rail odel, there are two ain steps in the coputation process. The first step is for pre-loading the wheel and inputting the forces. The second step is for the wheel to start rotating and oving along the rail. The tie for the first step is 1.0 s and the tie for the second step is based on the speed of wheel such that the tie in the present case is 4.0 v ( s ) since the entire length of the present wheel-rail odel is 4.2. Representative coputed results are selected fro the 9 points shown in Figure The first 3 points (Points 1, 2, and 3) correspond to the quarter, iddle, and third quarter distances fro the starting end of the second span. Points 4, 5, and 6 correspond to the quarter, iddle, and third quarter distances fro the starting end of the third span while Points 7, 8, and 9 correspond to those of the fourth

81 67 span. Note that in the present odel the rail was divided into 5 equal spans as illustrated in Figures 3.2 and 3.3. Since the corresponding results along Paths 1 and 3 are close to those of Path 2, as shown in Figures 4.17 and 4.18, and to reduce the aount of coputed data only those along Path 2. The velocity of 3D FE wheel-rail ode and 3D FE rail odel shown in Figure 4.17 and 4.18 is 10 /s. Figure 4.16 Locations of 9 selected points. The coputed results for the 3D FE wheel-rail odel, denoted by WRM, and those of the 3D FE rail with TPL odel, denoted by TPL, are plotted in Figures 4.19 through For brevity, only coputed results at Points 4, 5, and 6 are included.

82 68 Figure 4.17 Vertical deflections of 3D FE wheel-rail odel. Figure 4.18 Vertical deflections 3D FE rail odel with TPL.

83 69 (a) (b) Figure 4.19 Longitudinal deflection at Point 4:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

84 70 (c) (d) Figure 4.19 Longitudinal deflection at Point 4:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

85 71 (e) (f) Figure 4.19 Longitudinal deflection at Point 4:, TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

86 72 (a) (b) Figure 4.20 Vertical deflection at Point 4:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

87 73 (c) (d) Figure 4.20 Vertical deflection at Point 4:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

88 74 (e) (f) Figure 4.20 Vertical deflection at Point 4:, TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

89 75 (a) (b) Figure 4.21 Longitudinal deflection at Point 5:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

90 76 (c) (d) Figure 4.21 Longitudinal deflection at Point 5:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

91 77 (e) (f) Figure 4.21 Longitudinal deflection at Point 5:, TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

92 (a) 78 (b) Figure 4.22 Vertical deflection at Point 5:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

93 79 (c) (d) Figure 4.22 Vertical deflection at Point 5:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

94 80 (e) (f) Figure 4.22 Vertical deflection at Point 5:, TPL;, WRM; (c) v = 50 s, and (d) v = 60 s.

95 81 (a) (b) Figure 4.23 Longitudinal deflection at Point 6:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

96 82 (c) (d) Figure 4.23 Longitudinal deflection at Point 6:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

97 (e) 83 (f) Figure 4.23 Longitudinal deflection at Point 6:, TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

98 84 (a) (b) Figure 4.24 Vertical deflection at Point 6:, TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

99 85 (c) (d) Figure 4.24 Vertical deflection at Point 6:, TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

100 86 (e) (f) Figure 4.24 Vertical deflection at Point 6:, TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

101 4.3.2 Modification of Traveling Point Load By Scaling Aplitude 87 With the observed significant differences between the coputed results of 3D FE rail with TPL odel and those of the 3D FE wheel-rail odel, it shows a large difference between these two at soe points and speeds. In order understanding the reason why TPL cannot odel FE wheel-rail well, it needs to consider the different contact force which is due to wheel-rail interaction. Therefore, the input TPL needs to follow the results of contact noral force and contact shear force on the surface of rail fro 3D FE wheel-rail odel. In 3D FE wheel-rail odel, every contact point has different contact noral and shear force. Therefore, the scaling aplitudes should be collected fro Path 1, Path2 and Path3 by searching the axiu and iniu shear force and the iniu noral force of each node on the paths. Then, the shear force and noral force collected fro the nodes on contact paths are divided by 30 kn and -100kN, respectively. The 30kN is calculated by noral force ultiplied by friction coefficient 0.3, and the reason why keeping it positive is due to the aplitude of friction force in the foregoing is +1 and -1. Therefore, input friction force follows the aplitude only. Also, keeping noral force negative is due to the contact noral force which always shows negative. Thus, the aplitude of input noral force is only positive. The Figure 4.25 illustrates the siple way to get the aplitudes. The response of scaled TPL on 3D FE rail odel are plotted and copared with FE wheel-rail odel through Figure 4.26 to 4.31.(a)

102 88 (b) Figure 4.25 Noral and shear (longitudinal) forces fro FE wheel-rail odel: (a) shear force, and (b) noral force.

103 89 (a) (b) Figure 4.26 Longitudinal deflection at Point 4:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

104 90 (c) (d) Figure 4.26 Longitudinal deflection at Point 4:, Scaled TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

105 91 (e) (f) Figure 4.26 Longitudinal deflection at Point 4:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

106 92 (a) (b) Figure 4.27 Vertical deflection at Point 4:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

107 93 (c) (d) Figure 4.27 Vertical deflection at Point 4:, Scaled TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

108 94 (e) (f) Figure 4.27 Vertical deflection at Point 4:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

109 95 (a) (b) Figure 4.28 Longitudinal deflection at Point 5:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

110 96 (c) (d) Figure 4.28 Longitudinal deflection at Point 5:, Scaled TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

111 97 (e) (f) Figure 4.28 Longitudinal deflection at Point 5:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

112 98 (a) (b) Figure 4.29 Vertical deflection at Point 5:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

113 99 (c) (d) Figure 4.29 Vertical deflection at Point 5:, Scaled TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

114 100 (e) (f) Figure 4.29 Vertical deflection at Point 5:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

115 101 (a) (b) Figure 4.30 Longitudinal deflection at Point 6:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

116 102 (c) (d) Figure 4.30 Longitudinal deflection at Point 6:, Scaled TPL;, WRM; (c) v = 30 s, and (d) v = 40 s.

117 103 (e) (f) Figure 4.30 Longitudinal deflection at Point 6:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

118 104 (a) (b) Figure 4.31 Vertical deflection at Point 6:, Scaled TPL;, WRM; (a) v = 10 s, and (b) v = 20 s.

119 105 (c) (d) Figure 4.31 Vertical deflection at Point 6:, Scaled TPL;, WRM; (c) v = 40 s, and (d) v = 30 s.

120 (e) 106 (f) Figure 4.31 Vertical deflection at Point 6:, Scaled TPL;, WRM; (e) v = 50 s, and (f) v = 60 s.

121 4.3.3 Discussion 107 Although the difference still appears in longitudinal deflection at soe specific speeds, such as 10 /s, 20 /s, and 30 /s, the vertical deflection fro the scaled TPL is uch closer to the 3D FE WRM than the non-scaled TPL results. However, the vertical deflection fro non-scaled TPL is closed to the FE WRM at 10 /s as indicated in Figure 4.24(a). On the other hand, the results presented in Figure 4.32 indicated that the average aplitudes of the vertical forces are independent of the speed starting around 23 /s. In this figure, the results for Paths 1 and 3 are close to each other because they are obtained fro the equidistant nodal points on both sides of Path 2. Figure 4.32 Relation between average aplitudes of vertical forces and speed for 3D FE wheel-rail odel.

122 108 The coputed average axiu and iniu shear forces are shown in Figure In the latter, the results for Paths 1 and 3 are close to each other for the sae reason as for the results in Figure (a) (b) Figure 4.33 Relation between average aplitudes of shear forces and speed: (a) average iniu shear forces, and (b) average axiu shear forces.

123 109 CHAPTER 5 RAIL MODELS WITH BLOCK DAMPERS Periodically, the train copanies need to aintain rails and the ain part of the aintenance involves with achining soe sections of the rail. Such a aintenance is usually perfored at night when the train syste is closed. However, it takes tie and the process is expensive. If vibration aplitude of the rail is large, aintenance becoes ore frequent. Therefore, how to reduce the vibration of rail is iportant fro the econoical and safety points of view. Aong various approaches the ethod of adding ass block dapers or siply called block dapers has been suggested recently. The philosophy behind this approach sees to be that of the so-called tuned ass daper (TMD) in earthquake-resistant design in which a single degree-of-freedo (dof) ass-dashpot-and-spring syste is attached to the priary building structure so that the response of the building or syste ay be reduced. In the case of railway vibration reduction by attaching ass block dapers (BD), there are very liited nuber of studies reported in the literature. Therefore, there is a need for further understanding the effect of adding BD on the dynaic responses of the rail. To reduce the aount of coputation and therefore cost the approach of replacing the 3D FE wheel-rail syste by the 3D FE rail with scaled TPL is adopted in this chapter. The focus of the studies in this chapter is therefore in the coparison of coputed results between the scaled TPL odel without BD to that with BD. Another ain objective in this chapter is the study of the effect of the locations of BD on dynaic responses. The total ass of the BD is also of interest.

124 5.1 MODELING 3D FE RAIL MODEL WITH BLOCK DAMPERS 110 In Sub-section 4.3.2, it was observed that the scaled TPL can approxiately represent the 3D FE wheel-rail odel. This is particular true of the results of the vertical deflection. In this section, the effects of BD on the responses of the scaled TPL odel is investigated. For adding BD to the 3D FE rail odel, the BD are attached to the rail or the so-called constrained by tie in the context of Abaqus. Fro Abaqus docuentation [5.1], tie constraint is by foring two surfaces tied together. In addition, each nodal value such as displaceent, teperature, pore pressure or electrical potential on slave surface has the sae values of the aster surface. Also, tie constraint allows rapid transitions in eleent esh density within the odel. The discretization ethod of tie constraint is surface-to-surface forulation. The reason to choose this ethod is avoiding stress noise at the tied interfaces. It is iportant for adding BD without bringing ore irrelevant results into the odified odel Diension of Block Dapers Diensions of BD are related to the length of rail span which is 0.6 in the rail syste chosen. By adding the BD the total ass of the rail syste changes without changing the rail itself. Therefore, the diensions of BD are restricted by the length of rail span and the height of rail. In addition, owing to considering the effects of the total ass of BD with respect to the rail ass, the sae location needs to be added with different asses of BD. To study the effects of different total ass at the sae location on the 3D FE rail odel, the only variable is the length. Table 5.1 shows the diensions of three types of BD.

125 111 Table 5.1 Diensions of three types of BD in different locations. Type Width (c) 4.45 Length (c) Height (c) % rail ass Location Location 1 Location 2 Location 3 Nuber of BD Locations of Block Dapers on 3D FE Rail Model To study the effects of different locations with sae total ass of BD, the total ass of BD needs to be kept the sae at each location. Figure 5.1 illustrates the different locations. In Figure 5.1(a) one Type 1 BD on both sides of the single span is identified as Location 1. The different total percentages of ass of BD with respect to that of the rail are 22%, 26%, and 30% rail ass. In Figure 5.1(b) two Type 2 of BD on both sides of the rail are called Location 2. The different total percentages of the ass of BD with respect to that of the rail are 22%, 26%, and 30% rail ass. In Figure 5.1(c) three Type 3 of BD on both sides are referred to as Location 3. The different total percentages of the ass of BD with respect to that of the rail are also 22%, 26%, and 30% rail ass.

126 (a) 112 (b) (c) Figure 5.1 Three different locations for placing BD: (a) Location 1, (b) Location 2, and (c) Location 3.

127 5.2 RESULTS OF 3D FE RAIL MODEL WITH BLOCK DAMPERS 113 In the coputational process and by aking use of the scaled TPL to represent the 3D FE wheel-rail odel, BD is added to the 3D FE rail odel. It should be entioned there are any coputed deflections and only representative ones are presented in this section. The chosen points for coputed results are Points 4, 5, and 6 (see, Chapter 4). In every case the speeds of the train or TPL are 10, 20, 30, 40, 50, and 60 /s. The following sub-sections are concerned with the effects of every specific case. For the vertical deflection, because coputed results of the scaled TPL odel showing the largest agnitude of deflection being negative, the results of the sae rail odel with BD should be negative. For the longitudinal deflection, it involves with axiu and iniu deflections due to friction, and therefore, the 3D FE results also show corresponding axiu and iniu values of deflections. This is consistent with the fact that friction force between the wheel and rail behaves in a atter best described by the signu function. The coputed results of the effects of BD are presented in Sub-suctions and Effects of Block Dapers at Different Locations In the forging section, it was entioned that the sae total ass of BD at different locations was aintained. This helps to understand the effects of BD at different locations. Therefore, the following table shows the coparison of results of the sae total ass of BD at different locations for the case with BD and that without BD. Tables 5.2 through 5.4, and Figures 5.2 through 5.10 show the percentages of difference with respect to the 3D FE rail odel with scaled TPL.

128 Table 5.2 Percentages of difference between odel with and without BD at Point 4: (a) Maxiu longitudinal deflection, and (b) Miniu longitudinal deflection. 114

129 Table 5.2 Percentages of difference between odel with and without BD at Point 4: (c) Miniu vertical deflection. 115

130 Table 5.3 Percentages of difference between odel with and without BD at Point 5: (a) Maxiu longitudinal deflection (b) Miniu longitudinal deflection. 116

131 Table 5.3 Percentage of difference between odel with and without BD at Point 5: (c) Miniu vertical deflection. 117

132 Table 5.4 Percentage of difference between odel with and without BD at Point 6: (a) Maxiu longitudinal deflection (b) Miniu longitudinal deflection. 118

133 Table 5.4 Percentage of difference between odel with and without BD at Point 6: (c) Miniu vertical deflection. 119