Investigation on the Wheel/Rail Contact and Longitudinal Train/Track Interaction Forces

Transcription

1 Investgaton on the Wheel/Ral Contact and Longtudnal Tran/Track Interacton Forces BY ALI AFSHARI B.S., Iran Unversty of Scence and Technology, 2004 M.S., Sharf Unversty of Technology, 2006 THESIS Submtted as partal fulflment of the requrements for the degree of Doctor of Phlosophy n Mechancal Engneerng n the Graduate College of the Unversty of Illnos at Chcago, 2011 Chcago, Illnos Defense Commttee: Ahmed A. Shabana, Char and Advsor Laxman Saggere Thomas J. Royston Mchael A. Brown Behrooz Fallah, Northern Illnos Unversty

2 Ths thess s dedcated to my parents, wthout whose support and encouragement t would never have been accomplshed.

3 ACKNOWLEDGEMENTS I would lke to express my grattude and apprecaton to my thess advsor, Professor Ahmed A. Shabana for hs valuable gudance and constructve suggestons durng my doctorate studes and research at the Dynamc Smulaton Laboratory of the Unversty of Illnos at Chcago. Also, I would lke to apprecate my thess commttee members, Dr. Saggere, Dr. Royston, Dr. Brown, and Dr. Fallah, for ther helpful advce regardng ths thess. I would also lke to thank Mr. Caldwell, Dr. Zaazaa, and Mr. Marqus for ther suggestons that greatly helped me n my research. Last but not least, I would lke to apprecate my parents who were very supportve of me durng my studes at the Unversty of Illnos at Chcago. AA

4 TABLE OF CONTENTS CHAPTER PAGE 1. INTRODUCTION Motvaton Background and Lterature Survey Choce of the Contact Frame Wheel/ral Contact Formulatons and Algorthms Ar Brake Modelng Scope, Objectves and Organzaton of the Thess TANGENTIAL CONTACT FORCE Wheel and Ral Geometry Surface Geometry Ral Geometry Wheel Geometry Rollng Drecton (RD) Frame Contact Ellpse (CE) Frame Longtudnal Tangent (LT) Frame Multbody Contact Formulatons Numercal Results Track wth Lateral Devaton Track wth Varable Profle Extreme Case Concludng Remarks MULTIPOINT CONTACT SEARCH ALGORITHM Contact Formulatons Search Algorthm Numercal Results Sngle Truck Model Subjected to Lateral Forces Suspended Wheelset Model on a Curve Track Concludng Remarks AIR BRAKE FORCE MODEL Ar Flow Equatons Contnuty Equaton Momentum Equaton v

5 TABLE OF CONTENTS (Contnued) Flud Consttutve Equaton Naver-Stokes Equatons One-dmensonal Model Fnte Element Formulaton Alternate Formulaton Wall Frcton Forces Tran Nonlnear Dynamc Equatons Poston, Velocty, and Acceleraton Trajectory Coordnates Equatons of Moton Locomotve Valve Model Regulatng Valve Operaton Brake Applcaton and Release Pressure Rate Equlbrum of the Regulatng Valve Daphragm Relay Valve Operaton Brake Release and Applcaton Mathematcal Model Smplfed Model Brake Ppe Cut-off Valve Operaton Integraton of Locomotve Valve, Brake Ppe and Tran Dynamc Models Brake Release Brake Applcaton Concludng Remarks AIR BRAKE COMPUTER IMPLEMENTATION Car Control Unt (CCU) Control Valve Auxlary and Emergency Reservors Brake Cylnder Brake Rggng Brake Shoe CCU Operaton Brake Applcaton Poston Release Poston Lap Poston v

6 TABLE OF CONTENTS (Contnued) 5.3. CCU Mathematcal Model Mass Flow Rates Pressure Rates The Car Brake Parameters n Dfferent Brake Modes Brake Force Integraton of the Ar Brake and Tran Dynamcs Brake Release Brake Applcaton Numercal Results Brake Applcaton n a 4-Car Model Brake Applcaton and Recharge n a 75-Car Model Servce and Emergency Brake n a 100-Car Model Valdaton Comparson wth Expermental Results Analytcal Valdaton of Arbrake Dry Charge Concludng Remarks SUMMARY AND CONCLUSIONS REFERENCES APPENDIX VITA v

7 LIST OF TABLES TABLE PAGE I. The average devaton of the RD and LT models from the CE model 40 II. The average devaton of the RD and LT models from the CE model 44 III. The varaton range of the parameters assocated wth car 127 IV. Frcton coeffcent for dfferent flow regmes 140 V. Brake ppe and automatc brake valve propertes 140 VI. Car control unt propertes 141 v

8 LIST OF FIGURES FIGURE PAGE 1. The RD, CE, and LT contact frames Surface Geometry The coordnate systems used to defne track geometry The surface parameters at the contact pont of the rght ral Space curve and the horzontal plane The wheel surface parameters Lateral dsplacement of the wheelset Angle of attack at the rght wheel Normal force at the rght wheel Longtudnal force at the rght wheel Lateral force at the rght wheel The angle between the RD and the CE frames The angle between the LT and the CE frames X component of the rght contact force n the global coordnate system Y component of the rght contact force n the global coordnate system Z component of the rght contact force n the global coordnate system Longtudnal creepage at the rght wheel Lateral creepage at the rght wheel Rght ral profles Lateral force at the rght wheel The angle between the RD and the CE frames The angle between the LT and the CE frames Sngle truck model subjected to two lateral forces The wheel profle The ral profle The approxmate locatons of contact ponts Y-coordnate of the left front contact v

9 LIST OF FIGURES (Contnued) 28. Y-coordnate of the left rear contact Ral profle surface parameters of the left front contact Ral profle surface parameters of the left rear contact Normal force of the left front contact Normal force of the left rear contact Suspended wheelset model The top vew of the track used for the suspended wheelset model The wheel profle The ral profle Ral profle surface parameters of the rght contact Normal force of the rght contact Normal force of the rght contact (constrant contact method) Man ar brake components Coordnate systems Trajectory coordnates C valve scheme C regulatng valve Relay valve Brake ppe cut-off valve Car control unt components The slde valve n the brake applcaton poston The slde valve n the brake release poston The slde valve n the lap poston The parameters assocated wth the control unt of car The mass flow rate between components e and f The mass flow rate between the brake ppe, atmosphere, and CCU components The brake force appled on a car wheel Trajectory coordnates x

10 LIST OF FIGURES (Contnued) 56. The relatonshp between the dfferent models developed n ths research Tran mass center acceleraton Velocty of car The brake force of car Pressure dfference between the two sdes of the relay valve daphragm Pressure (gage) nsde the equalzng reservor and the brake ppe Auxlary reservor and brake cylnder pressures (gage) of car Pressure (gage) nsde the equalzng reservor and the brake ppe Auxlary reservor and brake cylnder pressures (gage) of cars 1 and The brake force of car Tran mass center acceleraton Tran mass center velocty Tran mass center acceleraton Pressure (gage) nsde the equalzng reservor and the brake ppe Auxlary reservor and brake cylnder pressures (gage) of cars 1 and Dstance travelled by car Equalzng reservor pressure (gage) comparson Head-end pressure (gage) comparson Rear-end pressure (gage) comparson Equalzng reservor pressure (gage) comparson Head-end pressure (gage) comparson Rear-end pressure (gage) comparson Pressure (gage) at the brake ppe ends durng dry charge Mass transfer rate durng dry charge Valve states and confguratons, Appendx Supply state regons, Appendx x

11 LIST OF ABBREVIATIONS CE Contact Ellpse CCU Car Control Unt LT RD Longtudnal Tangent Rollng Drecton x

12 SUMMARY Prevous analytcal and expermental nvestgatons have shown that the wheel/ral contact forces have a sgnfcant effect on the nonlnear dynamcs, rde comfort, and stablty of ralroad vehcle systems. The wheel/ral contact force can be parttoned nto two man components; the normal and tangental components. Whle the drecton of the normal contact force s well defned, dfferent drectons for the tangental contact forces have been proposed n the lterature. In some of the wheel/ral creep theores used n ralroad vehcle smulatons, the drecton of the tangental creep forces s assumed to be the wheel rollng drecton (RD). When Hertz theory s used, an assumpton s made that the rollng drecton s the drecton of one of the axes of the contact ellpse. In prncple, the rollng drecton depends on the wheel moton, whle the drecton of the axes of the contact ellpse (CE) s determned usng the prncpal drectons whch depend only on the geometry of the wheel and ral surfaces and do not depend on the moton of the wheel. The RD and CE drectons can also be dfferent from the drecton of the ral longtudnal tangent (LT) at the contact pont. In ths nvestgaton, the dfferences between the contact frames that are based on the RD, LT, and CE drectons, that enter nto the calculaton of the wheel/ral creep forces and moments, are dscussed. The choce of the frame, n whch the contact forces are defned, can be determned usng one longtudnal vector and the normal to the ral at the contact pont. Whle the normal vector s unquely defned, dfferent choces can be made for the longtudnal vector ncludng the RD, LT, and CE drectons. In the case of pure rollng or when the slppng s small, the RD drecton can be defned usng the cross product of the angular velocty vector and the vector that defnes the locaton of the contact pont. Therefore, ths drecton does not depend explctly on the geometry of the wheel and ral x

13 SUMMARY (Contnued) surfaces at the contact pont. The LT drecton s defned as the drecton of the longtudnal tangent obtaned by dfferentaton of the ral surface equaton wth respect to the ral longtudnal parameter (arc length). Such a tangent does not depend explctly on the drecton of the wheel angular velocty nor does t depend on the wheel geometry. The CE drecton s defned usng the drecton of the axes of the contact ellpse used n Hertz theory. In the Hertzan contact theory, the contact ellpse axes are determned usng the prncpal drectons assocated wth the prncpal curvatures. Therefore, the CE drecton dffers from the RD and LT drectons n the sense that t s functon of the geometry of the wheel and ral surfaces. In order to better understand the role of geometry n the formulaton of the creep forces, the relatonshp between the prncpal curvatures of the ral surface and the curvatures of the ral profle and the ral space curve s dscussed n ths nvestgaton. Numercal examples are presented n order to examne the dfferences n the results obtaned usng the RD, LT and CE contact frames. The effect of the tangental forces becomes more sgnfcant n the case of multple contact ponts between the wheel and the ral. It s mportant to predct accurately these forces n order to evaluate ther effect on the dynamc behavor of the ralroad vehcle systems. These contact forces can be determned usng parameters such as the wheel/ral geometry, dmensons, materal propertes, contact area, and relatve veloctes, etc. However, n order to correctly determne these contact forces, ncludng the normal forces, t s mportant to determne frst all possble wheel/ral contact ponts. Nonetheless, most exstng wheel/ral contact formulatons can be used to predct only one or two ponts of contact between the wheel and ral. There are, however, mportant scenaros n whch there are more than two wheel/ral contact ponts. The x

14 SUMMARY (Contnued) lack of accurate three-dmensonal multple pont wheel/ral contact formulaton represents a serous lmtaton when deralments and accdents are nvestgated. In ths thess, a new multpont contact search algorthm that can be appled to general three-dmensonal wheel/ral contact problems s developed. The algorthm s capable of fndng any number of wheel/ral contact ponts. Furthermore, the algorthm can be used wth both constrant and elastc contact approaches. The results obtaned usng the constrant and elastc contact approaches wll be compared n the case of multpont wheel/ral contact. The man steps used n the proposed search algorthm to fnd all possble contact ponts are summarzed. Longtudnal tran forces resultng from coupler and brakng forces are other types of forces that must be consdered n deralment and accdent nvestgatons. These forces also have a sgnfcant effect on the wheel/ral contact forces. Nonetheless, the ntegraton of an accurate ar brake model wth a nonlnear tran dynamc model remans a challengng problem. One of the goals of ths nvestgaton s to ntegrate an ar brake model wth effcent tran longtudnal force algorthms based on the trajectory coordnate formulatons. The ar brake model, developed n ths nvestgaton conssts of the locomotve automatc brake valve, ar brake ppe, and car control unt (CCU). The proposed ar brake force model accounts for the effect of the ar flow n long tran ppes as well as the effect of leakage and branch ppe flows. Ths model can be used to study the dynamc behavor of the ar flow n the tran ppe and ts effect on the longtudnal tran forces durng brake applcaton and release. The governng equatons of the ar pressure flow are developed usng the general flud contnuty and momentum equatons, smplfed usng the assumptons of one dmensonal sothermal flow. Usng these assumptons, one obtans two xv

15 SUMMARY (Contnued) coupled ar velocty/pressure partal dfferental equatons that depend on tme and the longtudnal coordnate of the brake ppes. The partal dfferental equatons are converted to a set of frst order ordnary dfferental equatons usng the fnte element method. The resultng ar brake ordnary dfferental equatons are solved smultaneously wth the tran second order nonlnear dynamc dfferental equatons of moton that are based on the trajectory coordnates. The tran car nonlnear dynamcs s defned usng a body track coordnate system that follows the car moton. The body track coordnate system translaton and orentaton are defned n terms of one parameter that descrbes the dstance traveled by the car. The confguraton of the car wth respect to ts track coordnate system s descrbed usng two translaton coordnates and three Euler angles. The operaton modes of the brake system consdered n ths nvestgaton are the brake release mode and the brake applcaton mode that ncludes servce and emergency brakes. A detaled model of the locomotve automatc brake valve s presented n ths nvestgaton and used to defne the nputs to the ar brake ppe durng the smulaton. A smplfed model of ths valve s also proposed n order to reduce the computatonal tme of the smulaton. Then, the detaled CCU formulaton s presented. Furthermore, the relatonshp between the man components of the ar brake system and the tran dynamcs s dscussed, and the fnal set of dfferental equatons that ncludes the two models s presented. Dfferent computer smulaton scenaros are consdered n order to nvestgate the effect of the ar brake forces on the tran longtudnal dynamcs n the case of dfferent brakng modes. The numercal results, obtaned n ths study, are compared wth expermental results publshed n the lterature. xv

16 1. INTRODUCTION Ths chapter dscusses the motvaton for the research presented n ths thess, presents a lterature survey for the topcs addressed n ths nvestgaton, and summarzes the thess objectves Motvaton Ralroad vehcle systems are among the most commonly used methods of transportaton, both for passengers and goods. Ther wdespread use has sparked, over the years, contnuous technologcal developments, wth the objectve of achevng hgher operatng speeds n order to mnmze cost and transportaton tme. Hgher operatng speeds, however, requre a better and more sophstcated approach for the desgn of the ral vehcle system n order to avod deralments and reduce the vbraton and nose levels. Therefore, the development and use of accurate computer models for smulaton of ralroad vehcle systems subjected to dfferent loadng condtons, operatng speeds, track geometres, brakng and tracton scenaros are necessary. By usng these accurate computer models, t s possble to buld vrtual prototypes for the smulaton and nonlnear dynamc analyss of long trans or for the smulaton of detaled sngle or multple car models (Amercan Assocaton of Ralroads, 2002a and b; Amercan Assocaton of Ralroads, 1992; Andrews, 1986; Berzer et al., 2000; Garg, and Dukkpat, 1984; Ralway Techncal Web Pages, 2010; Sanborn et al., 2007; Sanborn et al., 2007a and b; Shabana, 2008). Such studes wll contrbute sgnfcantly to better understandng of the causes of deralments and accdents, and to better understandng of the vbraton, stablty, dynamc characterstcs, and longtudnal shock loads of ralroad vehcle systems. 1

17 2 Prevous analytcal and expermental nvestgatons have shown that the wheel/ral contact forces have a sgnfcant effect on the nonlnear dynamcs, rde comfort, and stablty of ralroad vehcle systems. Investgatons of ralroad vehcle accdents have revealed that many of these accdents are due to deralments caused by sgnfcantly hgh forces at the wheel/ral contact ponts. Usng expermental and analytcal tools, these contact forces can be measured or evaluated numercally n order to develop gudelnes that can be used to avod deralments. For ths reason, the wheel/ral contact analyss has been extensvely nvestgated n the area of ralroad vehcle dynamcs. The wheel/ral contact force can be parttoned nto two man components; the normal and tangental components. The normal force component s n the drecton of the normal to the contact surface, whle the tangental component les n the tangent plane to the ral at the contact pont. The tangental force component, whch s due to the wheel/ral creep phenomenon, has a sgnfcant effect on the dynamcs and stablty of ralroad vehcle systems. In the case of saturaton, where the relatve velocty of the wheel wth respect to the ral s predomnantly sldng as n the case of brakng scenaros, Coulomb frcton law can be used. If the wheel/ral relatve moton s domnated by rollng, the creep effect must be taken nto consderaton. For ths reason, the wheel/ral creep forces must be used n the formulaton of the nonlnear dynamc equatons of ralroad vehcle systems. In most of the computer algorthms used n the analyss of the wheel/ral nteracton, the normal force s frst calculated, and s then used wth geometrc and materal propertes of the wheel and ral surfaces to determne the tangental creep forces that enter nto the formulaton of the nonlnear dynamc equatons of the ralroad vehcle system.

18 3 Whle the drecton of the normal contact force s well defned, dfferent drectons for the tangental contact forces have been proposed n the lterature. That s, several choces can be made for the drecton of the tangental contact forces dependng on the formulaton used. Although these forces can have a sgnfcant effect on the vehcle dynamcs, the choce of the drecton of these forces has not been nvestgated n the lterature. It s, therefore, one of the man objectves of ths thess to shed lght on ths mportant and fundamental ssue by examnng the effect of the drecton of the tangental forces on the wheel/ral dynamc nteracton. The effect of the tangental forces becomes more sgnfcant n the case of multple contact ponts between the wheel and the ral. It s mportant to predct accurately these forces n order to evaluate ther effect on the dynamc behavor of the ralroad vehcle systems. These contact forces can be determned usng parameters such as the wheel/ral geometry, dmensons, materal propertes, contact area, and relatve veloctes, etc. However, n order to correctly determne these contact forces, ncludng the normal forces, t s mportant to determne frst all possble wheel/ral contact ponts. Nonetheless, most exstng wheel/ral contact formulatons can be used to predct only one or two ponts of contact between the wheel and ral. There are, however, mportant scenaros n whch there are more than two wheel/ral contact ponts. One mportant example of these scenaros s when the ral vehcles negotate swtches and turn outs. The lack of accurate three-dmensonal multple pont wheel/ral contact formulaton represents a serous lmtaton when deralments and accdents are nvestgated. It s, therefore, the second objectve of ths thess to develop a new procedure that allows for predctng multple ponts of contact between the wheel and ral.

19 4 Longtudnal tran forces resultng from coupler and brakng forces are other types of forces that must be consdered n deralment and accdent nvestgatons. These forces also have a sgnfcant effect on the wheel/ral contact forces. Brakng forces, for example, lead to sgnfcant tangental frcton forces at the wheel/contact ponts. Nonetheless, the ntegraton of an accurate ar brake model wth a nonlnear tran dynamc model remans a challengng problem. It s, therefore, the thrd objectve of ths thess to address ths fundamental problem by developng a new procedure that leads to a successful ntegraton of pneumatc ar brake model wth a fully nonlnear dynamc tran model Background and Lterature Survey In ths secton, a lterature survey of the topcs consdered n ths nvestgaton s provded Choce of the Contact Frame The wheel/ral contact forces are defned along the axes of a coordnate system called the contact frame whose orgn s located at the contact pont. One of the axes s n the drecton of the normal to the contact surface at the contact pont, whle the other two axes le n the tangent plane. There are several choces for the axes that le n the tangent plane. In many of the wheel/ral creep force theores as the ones developed by Kalker (1990), the wheel rollng drecton (RD) s used wth the normal to defne the contact frame. Many authors assume that the rollng drecton s the same as the drecton of one of the axes of the contact ellpse when Hertz contact theory s used (Guglotta and Soma, 1996; Malvezz et al., 2008; Hoffman, 2006). In general, the rollng drecton depends on the moton of the wheel wth respect to the ral. The axes of the contact ellpse, on the other hand, are determned usng the prncpal drectons whch depend only on the wheel and ral surface geometry and are not

20 5 functon of any dynamc varables. The normal vector can be defned usng the cross product of two tangent vectors defned usng the geometry of the ral surface (Kreyszg, 1991; Shabana et al., 2008). The rollng drecton can then be defned as the drecton of the vector obtaned by the cross product of the angular velocty vector and the vector that defnes the locaton of the contact pont wth respect to the wheel body coordnate system. Ths drecton, therefore, does not explctly depend on the wheel and ral geometry at the contact pont; t depends on the poston and angular velocty of the wheel. As a consequence, the use of the RD contact frame for the defnton of the creep forces can lead to problems n the tracton and brakng scenaros when the magntude of the angular velocty becomes small or n the case of velocty dscontnuty due to mpact between the vehcle components (Goldsmth, 1960). The swtch to the drecton of the relatve velocty, when the moton becomes predomnantly sldng, can also lead to numercal problems and drecton dscontnutes. A second choce for the contact frame s to use the normal and the longtudnal tangent (LT) whch s determned by dfferentatng the ral surface equaton wth respect to the longtudnal surface parameter. Ths choce does not depend on wheel moton varables, and depends only on the ral geometry at the contact pont. For ths reason, the LT drecton s well defned regardless of the moton of the wheel wth respect to the ral. Usng the cross product of the normal to the surface and the LT vector; the lateral tangent, whch represents the drecton of the lateral creep force, and consequently, the LT contact frame can be determned n a straght forward manner. A thrd choce of the contact frame s to use the surface normal wth one of the axes of the contact ellpse (CE) used n the Hertz contact theory. The axs closer to the rollng drecton

21 6 can be selected. In Hertz theory (Hertz, 1882; Johnson, 1985), the drectons of the axes of the contact ellpse are determned usng the prncpal drectons assocated wth the prncpal curvatures. As shown n the lterature (Johnson, 1985; Shabana et al., 2008), the drectons of the axes of the contact ellpse are functons of the wheel and ral prncpal drectons that depend, respectvely, on the geometry of the wheel and ral surfaces. The coeffcents of the frst and second fundamental forms of surfaces are used to determne the prncpal curvature and prncpal drectons (Kreyszg, 1991). Therefore, the defnton of the CE frame does not also depend on wheel moton varables such as the angular velocty vector. The RD, LT, and CE contact frames can be dfferent, partcularly, n the case of sgnfcant lateral devatons or n the case of rals wth varable profles. The RD contact frame, as prevously mentoned, s the only frame of the three contact frames dscussed n ths secton that depends on moton varables. Whle ths s the frame whch s employed n Kalker s wheel/ral creep theory, Kalker and others assume that the rollng drecton s the same as one of the axes of the contact ellpse when Hertz contact theory s used. There are moton scenaros n whch the rollng drecton can be sgnfcantly dfferent from the drecton of the axes of the contact ellpse as demonstrated n ths nvestgaton. Nonetheless, the dependence of the rollng drecton on wheel moton varables such as the angular velocty vector represents a serous lmtaton n scenaros such as tracton and brakng. Furthermore, n the case of mpacts between the vehcle components the angular veloctes of the bodes can be dscontnuous, whle geometrc poston varables reman contnuous. For ths reason, t s mportant to nvestgate the accuracy of usng other alternatves of contact frames n the formulaton of the wheel/ral tangental creep force equatons.

22 Wheel/ral Contact Formulatons and Algorthms The wheel/ral contact analyss has been studed by many researchers n the dynamc modelng of ralroad vehcle systems (Kalker, 1990; Shabana et al., 2008). Most contact approaches developed n the prevous studes such as the constrant and elastc contact formulatons allow for only one or two contact ponts (Shabana, 2005). Moreover, there have been several nvestgatons on multpont wheel/ral contact n the lterature. Pascal and Sauvage (1991) proposed a new method to reduce mult-contact wheel/ral problem to one equvalent contact patch when the contact ponts are close to each other. Smlar studes were carred out by Potrowsk and Chollet (2005) as well as Ayasse et al. (2000). A nodal search approach was developed by Shabana et al. (2005) but that algorthm was lmted to two contact patches. Also, because of geometry dscretzaton adopted n ths approach, dscontnutes could be encountered n the results. Sugyama et al. (2009) nvestgated the wheel/ral contact n the smulaton scenaros that nclude ralroad swtches usng an onlne contact search algorthm. A lnear nterpolaton method was used n ths study to model ral profle varatons, whch may not be accurate n the case of sgnfcant profle varatons. Pombo and Ambroso (2008) developed a method that can be appled for the two contact pont scenaros. However, ths method can be used n three-dmensonal wheel/ral contact analyss Ar Brake Modelng One of the most mportant longtudnal tran forces that has a sgnfcant effect on the wheel/ral contact forces s the brakng force. For ths reason, the ralroad vehcle brakes have been consdered n many nvestgatons (Obara et al., 1995; Balon and Aznbud, 1989; Gulloux, 1984; Sakamoto et al., 2009). Many studes have focused on the pneumatc or ar brake that s

23 8 commonly used n the trans n North Amerca (Abdol-Hamd, 1986; Banster, 1979; Bharath et al., 1990; Funk and Robe, 1970; Ho, 1981; Lmbert, 1991; We and Ln, 2009). Shute et al. (1979) nvestgated the effect of leakage on brake ppe gradents and flow rates. In all of these studes, dfferent aspects of the tran ar brake were nvestgated usng one-dmensonal flow assumpton. Gauther (1977) and Wrght (1979) studed the pneumatc control valve systems usng a smlar assumpton. On the other hand, there have been studes that nvestgated the effect of the ar brake on the tran longtudnal dynamcs. Nasr and Mohammad (2010) studed the effect of brake delay tme on the tran longtudnal forces. In ths research, one-dmensonal moton assumpton was employed for the tran dynamcs and a smple model was used for the ar brake. Sanborn et al. (2005) used a smple brake model n whch a constant propagaton speed s assumed for the brake sgnal. Such an assumpton of constant ar propagaton speed cannot be justfed n many applcatons and does not allow for accurately predctng car coupler forces n severe brakng scenaros Scope, Objectves and Organzaton of the Thess The scope, objectves, and organzaton of ths thess are summarzed as follows: In Chapter 2, one of the objectves of ths nvestgaton, whch s to dscuss the fundamental dfferences between and compare the smulaton results obtaned by usng the RD, LT and CE contact frames s presented. Because of the mportance of the geometry n the analyss dscussed n Chapter 2 of ths thess, a bref revew of the geometry s frst presented. The CE frame s functon of the prncpal drectons assocated wth the prncpal curvatures of the wheel and ral. For ths reason, t s mportant to understand the relatonshp between the prncpal

24 9 curvatures of the ral surface and the curvatures of the profle and the ral space curves. As shown n Chapter 2, the curvatures of the ral profle and the ral space curve cannot be used n general as the prncpal curvatures of the ral at the contact pont. Ths s partcularly true n the case of the spral regon of the ral or when the ral profle changes as a functon of the arc length of the ral space curve. Ths fundamental geometry problem s dscussed n more detal when the CE contact frame s defned n Chapter 2. In order to show the fundamental dfferences between the RD, LT and CE frames; the equatons that defne these frames are presented and compared. Numercal examples are presented n order to compare the results obtaned usng the three dfferent frames. These results are obtaned usng two dfferent track models; one s a track wth sgnfcant lateral devatons, whle the other s a track whch has rals that have varable profles. In Chapter 3, a new multpont contact search algorthm that can be appled to general three-dmensonal wheel/ral contact problems s developed. The algorthm s capable of fndng any number of wheel/ral contact ponts. Furthermore, the algorthm can be used wth both constrant and elastc contact approaches. These contact formulatons and other assumptons used n the development of the search algorthm are revewed n Chapter 3. The results obtaned usng the constrant and elastc contact approaches are compared n the case of multpont wheel/ral contact. The man steps used n the proposed search algorthm to fnd all possble contact ponts are summarzed.

25 10 Chapters 4 and 5 develop a new ar brake model and explan how ths model can be ntegrated t wth effcent tran longtudnal force algorthms that are based on the trajectory coordnate formulatons. The ar brake model developed n Chapter 4 conssts of the locomotve automatc brake valve, ar brake ppe, and car control unt (CCU). The proposed ar brake force model accounts for the effect of the ar flow n long tran ppes as well as the effect of leakage and branch ppe flows. For ths reason, ths model can be used to study the dynamc behavor of the ar flow n the tran ppe and ts effect on the longtudnal tran forces durng brake applcaton and release. The governng equatons of the ar pressure flow are developed usng the general flud contnuty and momentum equatons, smplfed usng the assumptons of one dmensonal sothermal flow. Usng these assumptons, two coupled ar velocty/pressure partal dfferental equatons that depend on tme and the longtudnal coordnate of the brake ppes are obtaned. The partal dfferental equatons, whch are converted to a set of frst order ordnary dfferental equatons, are solved smultaneously wth the tran second order nonlnear dynamc dfferental equatons of moton. A new method s presented n ths nvestgaton to formulate the fnte element model for the brake ppe ar, whch leads to a constant matrx of coeffcents. A detaled model of the locomotve automatc brake valve s presented n Chapter 4 and used to defne the nputs to the ar brake ppe durng the smulaton. A smplfed model of ths valve s also proposed n order to reduce the computatonal tme of the smulaton.

26 11 The detaled CCU formulaton s presented n Chapter 5. Chapter 5 also focuses on the ntegraton of a tran ar brake model wth a nonlnear tran dynamc model that employs the trajectory coordnate formulatons. The tran ar brake forces obtaned n Chapter 5 depend on dfferent ar brake system components, ncludng the locomotve automatc brake valve, the brake ppe, and the car control unt (CCU). The car brake forces, whch depend on the locomotve automatc brake valve handle poston and are appled to the wheels usng the CCU located along the brake ppe, enter nto the formulaton of the nonlnear tran dynamc equatons n addton to other external forces. In order to develop an effcent computatonal procedure, smplfed valve models, wth more straghtforward operaton modes, are consdered here n order to reduce the computatonal overhead. The CCU model used n Chapter 5 has a control valve connected to three man pneumatc components; the auxlary reservor, the emergency reservor, and the brake cylnder. It s also assumed that the CCU modeled n ths study has the emergency component that enables applyng emergency brake, ncludng the effect of the CCU emergency vent valve. The relatonshp between the man components of the ar brake system and the tran dynamcs s dscussed n Chapter 5, and the fnal set of dfferental equatons that ncludes the two models are presented. Furthermore, three dfferent computer smulaton scenaros are consdered n order to nvestgate the effect of the ar brake forces on the tran longtudnal dynamcs n the case of dfferent brakng modes. The numercal results obtaned n Chapter 5 are compared wth expermental results publshed n the lterature.

27 12 The thess s concluded wth Chapter 6 that presents the summary and conclusons drawn from ths nvestgaton.

28 2. TANGENTIAL CONTACT FORCE The wheel/ral contact forces have a sgnfcant effect on the nonlnear dynamcs, and stablty of ralroad vehcle systems. In general, the wheel/ral contact force can be represented by a normal component and two components that le on the plane tangent to the wheel and ral at the contact pont (Kalker, 1990, 1967; Pascal and Zaazaa, 2007; Shabana et al., 2008). If the wheel/ral relatve moton s domnated by rollng, the creep effect must be taken nto consderaton. The wheel/ral creep forces must be correctly evaluated and used n the nonlnear dynamc equatons of ralroad vehcle systems for an accurate analyss of the vehcle dynamcs. In some of the wheel/ral creep theores used n ralroad vehcle smulatons, the drecton of the tangental creep forces s assumed to be the wheel rollng drecton (RD). When Hertz theory s used, an assumpton s made that the rollng drecton s the drecton of one of the axes of the contact ellpse. In prncple, the rollng drecton depends on the wheel moton, whle the drecton of the axes of the contact ellpse (CE) s determned usng the prncpal drectons whch depend only on the geometry of the wheel and ral surfaces and do not depend on the moton of the wheel. The RD and CE drectons can also be dfferent from the drecton of the ral longtudnal tangent (LT) at the contact pont. In ths nvestgaton, the dfferences between the contact frames that are based on the RD, LT, and CE drectons, that enter nto the calculaton of the wheel/ral creep forces and moments, are dscussed. The choce of the frame, n whch the contact forces are defned, can be determned usng one longtudnal vector and the normal to the ral at the contact pont. Whle the normal vector s unquely defned, dfferent choces can be made for the longtudnal vector ncludng the RD, LT, and CE drectons (See Fg. 1). In the case of pure rollng or when the slppng s small, the RD drecton can be defned usng the cross 13

29 14 product of the angular velocty vector and the vector that defnes the locaton of the contact pont. Therefore, ths drecton does not depend explctly on the geometry of the wheel and ral surfaces at the contact pont. The LT drecton s defned as the drecton of the longtudnal tangent obtaned by dfferentaton of the ral surface equaton wth respect to the ral longtudnal parameter (arc length). Such a tangent does not depend explctly on the drecton of the wheel angular velocty nor does t depend on the wheel geometry. The CE drecton s defned usng the drecton of the axes of the contact ellpse used n Hertz theory. In the Hertzan contact theory, the contact ellpse axes are determned usng the prncpal drectons assocated wth the prncpal curvatures. Therefore, the CE drecton dffers from the RD and LT drectons n the sense that t s functon of the geometry of the wheel and ral surfaces. Fgure 1. The RD, CE, and LT contact frames The objectve of ths chapter s to develop mathematcal defntons for the RD, LT and CE contact frames, dscuss the fundamental dfferences between them, and compare the smulaton results obtaned by usng these contact frames. Because of the mportance of the geometry n the analyss presented n ths study, a bref revew of the geometry s frst presented.

30 15 The CE frame s functon of the prncpal drectons assocated wth the prncpal curvatures of the wheel and ral. For ths reason, t s mportant to understand the relatonshp between the prncpal curvatures of the ral surface and the curvatures of the profle and the ral space curves. As shown n ths chapter, the curvatures of the ral profle and the ral space curve cannot be used n general as the prncpal curvatures of the ral at the contact pont. Ths s partcularly true n the case of the spral regon of the ral or when the ral profle changes as a functon of the arc length of the ral space curve. Ths fundamental geometry problem s dscussed n more detal when the CE contact frame s defned n ths chapter. In order to show the fundamental dfferences between the RD, LT and CE frames; the equatons that defne these frames are presented and compared. Numercal examples are presented n order to compare the results obtaned usng the three dfferent frames. These results are obtaned usng two dfferent track models; one s a track wth sgnfcant lateral devatons, whle the other s a track whch has rals that have varable profles Wheel and Ral Geometry In order to accurately model the wheel/ral nteracton usng nonlnear computatonal multbody system algorthms, t s necessary to express the wheel and ral surfaces n a parametrc form. Before dscussng the wheel and ral surface geometry, some basc dfferental geometry defntons and denttes that wll be used n ths study are revewed (Kreyszg, 1991) Surface Geometry The geometry of a surface can be completely defned usng two ndependent surface parameters s 1 and s 2 as shown n Fg. 2. The poston vector x of an arbtrary pont on the surface can be wrtten n terms of these parameters as

31 16 x s, s ) x ( s, s ) x ( s, s ) x ( s, s ) ( 2.1) ( where x, 1,2,3, s component of the vector x. The tangent and the unt normal vectors of the surface are obtaned from the followng equaton (Kreyszg, 1991): T x x x x, x, n x,1,2,1,2 s1 s2 x,1 x,2 ( 2.2) Fgure 2. Surface Geometry The unt normal vector n enters nto the defnton of all contact frames (RD, LT, and CE) consdered n ths Chapter. The unt tangent t 1 x,1 x,1 and the normal n are used to defne the LT contact frame, where the thrd axs s defned usng the cross product. The frst fundamental form of the surface s defned as

32 17 I T ) 2Fds1ds2 G( 2 ) dx dx E( ds ds ( 2.3) where the coeffcents n ths equaton can be wrtten n terms of the tangents of Eq. 2.2 as T, 1,1 T,1,2 T,2 E x x, F x x, G x x ( 2.4),2 The second fundamental form of the surface can be defned as II d 2 T 1 2 x n L( ds ) 2Mds ds N( ds ) ( 2.5) where the coeffcents of the second fundamental form are T T T L x, 11n, M x,12n, N x, 22n ( 2.6) The coeffcents of the frst and second fundamental forms are used to determne the prncpal curvatures and prncpal drectons by solvng the followng homogeneous equaton: L k E M k F M k F d N kg d , 2 ( 2.7) k, are the prncpal curvatures (egenvalues), and d d where, 1, T d s the egen vector assocated wth the egenvalue k (Kreyszg, 1991). The prncpal drectons can be wrtten n terms of the egenvectors as e x x ( 2.8) d 1,1d2,2, 1,2 The prncpal drectons are requred when Hertz contact theory s used. These drectons defne the drectons of the axes of the contact ellpse. One of these axes s used wth the unt normal n of Eq. 2.2 to defne the CE contact frame axs, t ce, whch can be used n the formulaton of the tangental creep forces. Let ψ be the angle between the prncpal axes of the ral and the prncpal axes of the wheel. The angle α between prncpal axes of the ral and the

33 18 contact ellpse axs, s obtaned usng the formula tan 2 sn 2 (cos 2 B ) where k B k ( k r x k r y ) ( k w x k w y ) and l k j (l=w,r and j=x,y) s the prncpal curvature along the ndcated drecton obtaned usng Eq The defnton of the contact frames that defne the drectons of the tangental creep forces wll be dscussed n the followng sectons of ths chapter Ral Geometry The coordnate systems used for the ral n the multbody system algorthm employed n ths r study are shown n Fg. 3. The ral surface s descrbed usng the two parameters s 1 r (longtudnal) and s 2 (profle). The frame XYZ s the global coordnate system, rr rr rr X Y Z s the rght ral coordnate system, and rl rl rl X Y Z s the left ral coordnate system. The formulaton used n ths study allows the rght and left rals to be treated as two separate bodes, each of whch can have ndependent dsplacements; or they can be treated as one body that represents the track. Let rp rp rp X Y Z be a profle frame, where rp X s n the drecton of the longtudnal tangent to the ral space curve. Wthout any loss of generalty, one can then wrte the locaton of a pont on the profle n the profle coordnate system as u rp 0 s r 2 f( s r 1, s r 2) T ( 2.9) In ths equaton, ( r r f s1, s 2) s a functon that defnes the rp Z coordnate of the pont n terms of the ral surface parameters as shown n Fg. 4. The precedng equaton represents a general profle that can change along the track. If the profle shape does not change as functon r of the longtudnal surface parameter s 1, the precedng equaton can be wrtten as

34 19 u rp 0 s r 2 f( s r 2) T ( 2.10) Fgure 3. The coordnate systems used to defne track geometry Fgure 4. The surface parameters at the contact pont of the rght ral

35 20 Fgure 5. Space curve and the horzontal plane The locaton of the contact pont wth respect to the track or the ral body coordnate system can be wrtten as (Berzer et al, 2000) u r r rp rp R A u ( 2.11) where r R and rp A are, respectvely, the locaton of the orgn and the transformaton matrx that defnes the orentaton of the profle frame rp rp rp X Y Z wth respect to the track or ral body coordnate system. The transformaton matrx rp A can be expressed n terms of Euler angles as (Shabana et al., 2008) cos cos sn cos cos sn sn sn sn cos sn cos rp A sn cos cos cos sn sn sn cos sn sn sn cos ( 2.12) sn cos sn cos cos where the three Euler angles,, and represent three successve rotatons about Z, -Y, and X, respectvely. In the multbody system computatonal algorthm, the locaton and orentaton of the profle frame rp rp rp X Y Z at ponts along the track space curve can be defned usng three

36 21 nputs (Berzer et al, 2000; Dukkpat and Amyot, 1988). These nputs are the horzontal curvature defned by the projecton of the curve on the horzontal plane (see Fg. 5), the grade defned by the angle, and the superelevaton that depends on the bank angle. Usng ths nput, the relatonshp between the angle and the horzontal curvature H C H can be wrtten as ds d CH ds ( 2.13) R where R H and S are the radus of curvature and the projected arc length, respectvely. Usng Fg. 5, the relatonshp between the actual arc length s and the projected arc length S can be wrtten as ds ds (2.14) cos In the computer algorthm used n ths chapter for the smulaton of multbody ralroad vehcle systems, a preprocessor computer code s used to generate track geometry data based on the nputs prevously descrbed. The preprocessor output fle ncludes data that are used as nput to the man processor computer code SAMS/Ral that s used to solve the nonlnear dynamc equatons of the multbody vehcle system (Shabana et al, 2008) Wheel Geometry Snce the wheel surface s consdered as a surface of revoluton, the poston vector of an arbtrary pont on the wheel profle wth respect to the wheel body coordnate system can be defned as u w x0 g( s w 1 )sn( s w 2 ) y w 0 s1 z0 g( s w 1 )cos( s w 2 ) T ( 2.15)

37 22 w where s 1 s the lateral surface parameter that defnes the geometry of the wheel profle as gven by the functon w 1 w g s, s 2 s the angular surface parameter as shown n Fg. 6, and x 0, y 0, and z 0 are the coordnates of the poston vector of the orgn of the profle frame wth respect to the wheel or wheelset body coordnate system. As prevously mentoned, the prncpal drectons assocated wth the prncpal curvatures are used to defne the drectons of the axes of the contact ellpse. These axes can be used wth the normal to defne the CE contact frame whch can be used n the formulaton of the creep forces. Fgure 6. The wheel surface parameters 2.2. Rollng Drecton (RD) Frame Whle the wheel/ral normal contact force s always defned along the normal to the contact surfaces, the drecton of the creep forces s defned by the theory used n the formulaton of these forces. Dfferent sets of two perpendcular axes can be defned n the tangent plane. In Kalker s nonlnear theory (Kalker, 1990), the RD contact frame n whch the creep forces are defned s determned usng the unt normal and a unt vector along the rollng drecton; wth the

38 23 thrd axs defned by the cross product of these two vectors. Kalker and several other authors assume that the rollng drecton s the drecton of one of the axes of the contact ellpse when Hertz theory s used. In general, the rollng drecton, whch depends on the moton of the wheel wth respect to the ral, can be sgnfcantly dfferent from the drecton of the axes of the contact ellpse whch are determned usng the prncpal drectons of the wheel and ral surfaces. The prncpal drectons as dscussed n the precedng secton are not functon of any wheel moton varables. In ths secton, the rollng drecton s defned n a way that s ndependent of Hertz theory. The rollng drecton used n ths secton s assumed to be the drecton of the velocty component due to the rotaton of the wheel whch s predomnantly due to the ptch rotaton about the wheelset axs. The rollng drecton enters nto the defntons of the creepages and creep contact forces. Let w v c and r v c be the absolute veloctes of the wheel and the ral at the contact pont, and let w ω and r ω be the absolute angular veloctes of the two bodes. If the vector u w c defnes the locaton of the contact pont on the wheel, the rollng drecton (RD) s defned by the unt vector t u ω ω w w r c rd w w r uc ω ω ( 2.16) Note that the defnton of ths unt vector s ndependent of the normal vector n, and therefore, there s no guarantee that ths vector les n the tangent plane. Furthermore, ths defnton s not an explct functon of the wheel and ral geometry, and can lead to an llcondtoned defnton f the wheel angular velocty becomes small as n the case of tracton and brakng scenaros. The defnton of Eq can also lead to numercal problems n the case of mpact between the vehcle components. Impacts lead to jump dscontnutes n the veloctes,

39 24 and as a consequence, the use of the precedng equaton n the formulaton of the creepages and creep forces can be a source of problems n some ralroad vehcle smulaton scenaros. Usng Eq. 2.16, the orentaton of the RD frame can be defned usng the followng transformaton matrx: A rd t n rd t n rd ( 2.17) Note that, n general, there s no guarantee that the vector t rd les n the tangent plane. For ths reason, ths vector can be replaced by another vector obtaned by the cross product trd ntrd n ntrd n. Usng the RD frame defnton, the vector rd lr n t t defnes the drecton of the lateral creepage and lateral creep force. In ths case, the creepages can be wrtten as w r T w r T w r T ( vc vc) trd ( vc vc) tlr ( ω ω ) n x, y, V V V ( 2.18) where x and y are the longtudnal and lateral creepages, respectvely; s the spn creepage; and V s the forward velocty of the wheel. The creepage expressons of Eq can be used wth the normal force, the wheel and ral materal propertes, and the contact ellpse sem-axes to evaluate the tangental creep forces and the spn moment whch are defned along the axes of the RD frame Contact Ellpse (CE) Frame The CE frame s formed usng the normal vector and one of the axes of the contact ellpse used n Hertz contact theory. The axs closer to the drecton of the forward moton of the wheel can be selected. Let a unt vector along the axs of the contact ellpse be denoted as t ce whch was

40 25 prevously defned. Usng ths unt vector and the unt normal n, the thrd axs can be defned usng the unt vector t n t ( 2.19) lc ce Ths vector defnes the drecton of the lateral creepage and the lateral creep force. Therefore, the transformaton matrx that defnes the orentaton of the CE frame can be wrtten as A ce t n ce t n ce ( 2.20) The creepage expressons when the CE frame s used can be wrtten as w r T w r T w r T ( vc vc) tce ( vc vc) tlc ( ω ω ) n x, y, V V V (2.21) In order to accurately defne the axes of the contact ellpse used to defne the CE frame ntroduced n ths chapter t s necessary to determne onlne the prncpal drectons assocated wth the prncpal curvatures of the wheel and ral at the contact pont. The ral, n partcular, can have complex geometry n regons such as the sprals and also due to the varaton of the ral profle n the longtudnal drecton due to wear or the exstence of swtches and turnouts. The prncpal curvatures of the ral, as shown n the remander of ths secton, cannot be assumed n general to be equal to the curvature of the ral profle and the curvature of the ral longtudnal curve at the contact pont. Therefore, t s necessary to determne the prncpal curvatures by solvng, at every tme step, the egenvalue problem defned n the prevous secton of ths thess. Ths egenvalue problem s formulated usng the coeffcents of the frst and second fundamental forms of the ral surface, as dscussed n the precedng secton. The condtons under whch the

41 26 ral prncpal curvatures reduce to the curvature of the profle and the curvature of the ral longtudnal curve at the contact pont are also presented n ths secton. The two tangents to the ral surface can be defned usng Eq as follows: u R A u A u, u A u (2.22) r r rp rp rp rp r rp rp,1,1,1,1,2,2 In general, these two tangents are not orthogonal. Ther dot product s gven by u u R A u u A A u u u rt r rt rp rp rpt rpt rp rp rpt rp,1,2,1,2,1,2,1,2 ( 2.23) The frst term n the rght hand sde of the precedng equaton vanshes snce the vectors r R,1 and A rp rp u,2 are always perpendcular, whle the second term can be smplfed usng the propertes of the transformatons matrx, leadng to u rp T A rp T,1 A rp u rp,2 rp T ~ u σ u (2.24) rp,2 where ~ σ A rp T A rp,1 s a skew symmetrc matrx. The vector σ assocated wth σ ~ s defned n the ral body coordnate system. Usng the precedng two equatons, one has u u u σu u u u ( σ u ) u u r T r rpt rp rpt rp rp rp rp rp,1,2,2,1,2,2,1,2 x s f ( s, s ) f( s, s ) f ( s, s ) f ( s, s ) r r r r r r r r r 2, ,1 1 2,2 1 2 ( 2.25) The terms that appear n the rght hand sde of ths equaton are not n general equal to zero. In the specal case n whch the profle does not change ts shape along the track, that s,,1( r r f s1, s2 ) 0, the precedng equaton reduces to rt r,1,2 r 2,2 ( r 1, r 2 ) ( r 1, r x s f s s f s s2 ) u u, whch shows that the two tangents are perpendcular n ths specal case f and only f 0. Ths condton s satsfed f x x, 1,1 sn 0 (2.26)

42 27 Usng ths condton, one can evaluate the coeffcents of the frst and second fundamental forms and determne the prncpal curvatures and prncpal drectons. One can r show that, n ths specal case, the prncpal drectons are the same as the ral unt tangents t 1 r and t 2. The prncpal curvatures and the assocated prncpal drectons n ths specal case are L r r k1, e1 t1 E N r r k2, e 2 t2 G ( 2.27) Therefore, f the profle does not change ts shape, one has the followng results for dfferent types of the track segments (tangent, curve or spral): r r 1. In the case of a tangent track,,1 0 and the two unt tangents t 1 and t 2 are perpendcular. The prncpal curvatures of a tangent track segment reduce to the curvatures of the ral space curve and the profle curve. The results are k 0, k r r f ( s ) t r,22 2 r r 2, e t r 2 (1 [ f,2( s2)] ) e ( 2.28) 2. In the case of a curve segment, one has, 1 0 whle,1 s constant. Agan, for such a r r segment, the two unt tangents t 1 and t 2 are perpendcular and the second prncpal curvature k 2 reduces to the curvature of the ral profle curve. The results n ths specal case, assumng R s the radus of curvature of the ral space curve segment, are

43 28 r f,2( s2) r r k1, e 2 1 t1 r r ( Rs2 ) 1 [ f,2( s2)] r f,22( s2) r r k2, e r t 2 (1 [ f,2( s2)] ) ( 2.29) 3. In the case of a spral segment, the condtons used for the tangent and curve segments cannot be appled because of the more complex spral geometry. In fact, the soluton for a spral s a clothod. Wthn a spral, the Euler angles change wth respect to the arc length such that n r r general the two unt tangents t 1 and t 2 do not reman perpendcular. In ths case, the prncpal r r drectons are a lnear combnaton of the two surface tangents t 1 and t 2. One can also develop smlar condtons for the wheel. The wheel tangents are defned as follows: w w w T w w w,1 g,1 s2 g,1 s2,2 g s2 g s2 u sn( ) 1 cos( ), u cos( ) 0 sn( ) ( 2.30) Usng these defntons, one can show that the dot product of the two unt tangents w w w w w w t1 u,1 u,1 and 2,2,2 t u u s equal to zero regardless of the wheel profle gs w 1. It can be T shown that for the wheel surface, the coeffcent M of the second fundamental form s also equal w to zero. Therefore, the prncpal drectons are the same as the drectons the unt tangents t 1 and w t 2, and one has the followng results (Shabana et al., 2001): w g,11( s1 ) w w k1, e t1 w (1 [ g,1( s1 )] ) 1 w w k2, e w w 2 2 t 2 gs ( 1 ) 1 [ g,1( s1 )] ( 2.31) These results apply only to unworn wheels.

44 Longtudnal Tangent (LT) Frame r Another choce for the contact frame s to use the unt normal n and the longtudnal tangent t 1 as defned n the precedng secton to form the LT contact frame. In ths case, the orentaton of the contact frame s defned by the transformaton matrx A lt t n 1 t n 1 ( 2.32) The vector nt1 t l1 defnes the lateral tangent along whch the lateral creepage and lateral creep force are defned. The creepage expressons when the LT frame s used can be wrtten as w r T w r T w r T ( vc vc) t1 ( vc vc) tl1 ( ω ω ) n x, y, V V V ( 2.33) The defnton of the LT frame does not nvolve the angular velocty vector, and f Hertz contact theory s not used, the evaluaton of the prncpal curvatures and prncpal drectons s not requred. Furthermore, the LT contact frame dffers from the other two frames prevously dscussed n the sense that t does not depend on the wheel moton or geometry. It depends only on the ral geometry at the contact pont. r r As prevously ponted out, the two tangents t 1 and t 2 are not n general perpendcular. Therefore, one can defne another contact frame usng the unt normal vector n and the lateral r tangent t 2. The longtudnal axs that completes the frame can be determned usng the cross product. Numercal expermentaton conducted n ths study have shown that the results obtaned r usng the contact frame formed by n and t 2 are n a good agreement wth the results obtaned r usng the LT contact frame. Therefore, the contact frame based on the n - t 2 combnaton wll not be dscussed further n ths chapter.

45 Multbody Contact Formulatons In order to study the wheel/ral dynamc nteracton, the normal force at the contact pont must be calculated and used wth other materal and geometrc parameters to determne the tangental creep forces and spn moment. There are two general approaches that can be used to study the wheel/ral nteracton; these are the constrant and elastc approaches (Shabana et al, 2008). In the constrant approach, t s assumed that there s no penetraton or separaton between the wheel and the ral at the contact pont. Snce four geometrc parameters are ntroduced to descrbe the wheel and ral surface geometry, the followng fve constrant equatons can be mposed: r w r t1.( rp r ) P r w r t2.( rp rp) r w r n.( rp rp) 0 w r t1. n w r 2. t n ( 2.34) where l r P s the absolute poston vector that defnes the locaton of the contact pont P on the body l (l=w, r). When the constrant formulaton s used, the normal force at the contact pont s consdered as a reacton force whch can be determned usng the technque of Lagrange multplers. In the case of the constrant contact approach, the constrants defned n the precedng equaton must be satsfed at the poston, velocty, and acceleraton levels. On the other hand, n the elastc approach that wll be used n ths study, the wheel has sx degrees of freedom wth respect to the ral, and therefore, wheel/ral penetraton and separaton are allowed. In ths approach, the thrd constrant n the precedng equaton, whch specfes the relatve moton of the two surfaces along ther common normal, s not mposed whle the other

46 31 four nonlnear algebrac equatons are used to solve for the four geometrc surface parameters. In the elastc approach, the normal force s defned as a functon of the penetraton usng the Hertzan contact formulaton. The penetraton can be calculated as T w r r wr r ( r r ) n r n ( 2.35) P P P Usng ths defnton for the penetraton, the normal contact force can be determned. A dampng coeffcent can also be ntroduced to the force-penetraton relatonshp to take nto account the effect of the energy dsspaton. Therefore, the normal force can be wrtten as follows (Shabana et al 2005): F n F F K 2 C h d h 3 ( 2.36) where K h s the Hertzan constant (Johnson, 1985), C s the dampng constant, s the ndentaton velocty obtaned from the dot product of the wheel/ral relatve velocty at the contact pont and the normal vector to the surface at ths pont. The expressons of the wheel/ral normal and tangental creep forces can be ntroduced to the nonlnear dynamc equatons of moton of the ralroad vehcle system as generalzed forces. These equatons can be solved for the system acceleratons and jont forces. The acceleratons can be ntegrated forward n tme n order to determne the coordnates and veloctes as descrbed n detal n the lterature (Shabana et al., 2008) Numercal Results In ths secton, smulaton results of suspended wheelset examples usng the RD, LT, and CE contact frames are presented and compared. These smulatons nclude rals wth constant profle shape as well as profles that change ther shape along the ral arc length. Whle n both cases a

47 32 tangent track s used; n the frst case, a track wth sgnfcant lateral devaton s used n order to compare the results obtaned usng dfferent contact frames. Ultmately, n order to show the dfference between the three frames n extreme cases, the track devaton n the frst smulaton, s ncreased to a large devaton and the results are also presented n ths secton Track wth Lateral Devaton In ths example, a wheelset attached to a frame that s constraned to move along a tangent track s consdered. It s assumed that there s a lateral devaton startng at dstance m (50 ft) wth ampltude of m (0.6 n) and wdth of m (5 ft). Ths devaton causes the prncpal drectons of the ral to change. Snce the wheelset moton s affected by the devaton, the angle between the prncpal drectons of the wheel and the ral n ths porton of the track s expected to be much hgher as compared to the remanng sectons of the track. The wheelset s assumed to have a constant forward velocty of 10 m/s (22.4 mph) along the track. The results are obtaned for three models that correspond to the three dfferent contact frames dscussed n ths study. Fgures 7 and 8 show, respectvely, the lateral dsplacement and the angle of attack (rght wheel) of the wheelset when the three frames are used.

48 33 Fgure 7. Lateral dsplacement of the wheelset (CE frame:, LT frame:, RD frame: ) Fgure 8. Angle of attack at the rght wheel (CE frame:, LT frame:, RD frame: )

49 34 Fgures 9-11 show the normal and tangental creep forces. Fgure 12 shows the angle between the rollng drecton of the RD frame and the prncpal drecton used to defne one of the axes of the contact ellpse, whle Fg. 13 shows the angle between the prncpal drecton and the longtudnal tangent at the contact pont. Fgure 9. Normal force at the rght wheel (CE frame:, LT frame:, RD frame: )

50 35 Fgure 10. Longtudnal force at the rght wheel (CE frame:, LT frame:, RD frame: ) Fgure 11. Lateral force at the rght wheel (CE frame:, LT frame:, RD frame: )

51 36 Fgure 12. The angle between the RD and the CE frames Fgure 13. The angle between the LT and the CE frames Fgures show the global components of the contact forces. Fgures 17 and 18 show the longtudnal and lateral creepages defned n the contact frame.

52 37 Fgure 14. X component of the rght contact force n the global coordnate system (CE frame:, LT frame:, RD frame: ) Fgure 15. Y component of the rght contact force n the global coordnate system (CE frame:, LT frame:, RD frame: )

53 38 Fgure 16. Z component of the rght contact force n the global coordnate system (CE frame:, LT frame:, RD frame: ) Fgure 17. Longtudnal creepage at the rght wheel (CE frame:, LT frame:, RD frame: )

54 39 Fgure 18. Lateral creepage at the rght wheel (CE frame:, LT frame:, RD frame: ) The percentage of the average devaton of the RD and LT models from the CE model, when the wheelset passes over the devaton, s shown n Table I.

55 40 Table I. The average devaton of the RD and LT models from the CE model Parameter LT model RD model Rght longtudnal creepage 1.5% 5% Lateral axs of the left contact ellpse 0.002% 0.003% Rght longtudnal force 1.5% 5% Rght lateral force 0.7% 2.2% Left spn creepage 0.02% 0.05% Rght spn moment 0.6% 2% Rght contact area 0.005% 0.005% Rght lateral creepage 3% 11% X component of the rght creepage n the global coordnate system 3% 5% Y component of the rght creepage n the global coordnate system 1% 1% Left normal force 0.007% 0.008% X component of the rght contact force n the global coordnate system 0.3% 1.3% Y component of the rght contact force n the global coordnate system 0.4% 0.7% Z component of the rght contact force n the global coordnate system 0.007% 0.008% The results obtaned show that the contact force results of dfferent models are n a good agreement, whle dscrepances n the creepage results can be observed. Smulaton results have shown that the dscrepancy n the lateral creepage does not have a sgnfcant effect on the creep force due to the domnant effect of the spn creepage n the lateral force calculaton Track wth Varable Profle In ths example, the suspended wheelset s constraned to move along the tangent track wth a forward velocty of 10 m/s (22.4 mph). The track profle s assumed to vary along the track. The profles used to defne the rght ral are shown n Fg. 19. The order of these profles are as

56 41 follows: the cross-sectons of the rght ral at the dstances 0, 20 m (65.6 ft), 120 m (393.7 ft) and 220 m (721.8 ft) are assumed to have the shapes descrbed by the profles wth numbers 1, 1, 2, and 3, respectvely. Fgure 19. Rght ral profles (Profle #1:, Profle #2:, Profle #3: ) These profles are also mrrored and used at the same locatons for the left ral. Lnear nterpolaton s used to defne the profle shape at a dstance between the locatons of the profles shown n Fg. 19. Fgure 20 shows the lateral tangental creep forces as defned n the contact frames of the three dfferent models. Numercal results obtaned n ths chapter showed that the normal force and longtudnal tangental creep force obtaned usng the models based on the three frames are n a good agreement. Furthermore, the solutons for the wheel knematcs, such as the wheel dsplacements and yaw angles, as well the dmensons of the contact ellpse are n a good agreement. Fgures 21 and 22 show the angles between the axes of the contact frames. The results obtaned for ths track model show that the profle varaton does not have a sgnfcant

57 42 effect on the creepages and contact forces of the RD, CE, and LT models. The devatons of the RD and LT model results from the CE model results s less than 0.1% for the track model used n ths study. Fgure 20. Lateral force at the rght wheel (CE frame:, LT frame:, RD frame: )

58 43 Fgure 21. The angle between the RD and the CE frames Fgure 22. The angle between the LT and the CE frames Extreme Case The smulatons used to obtan the results prevously dscussed n ths secton, represent realstc scenaros. As observed, n such smulaton scenaros, the results obtaned usng all contact

59 44 frames are n a good agreement. However, n order to show the dfferences n the results obtaned usng the three frames n extreme cases, an uncommon track model, s used. In the new track model, the wdth and ampltude of the devaton are changed to m (2 ft) and m (1 n), respectvely. Table II. The average devaton of the RD and LT models from the CE model Parameter LT model RD model Rght longtudnal creepage 7% 21% Rght lateral force 10% 22% Left spn moment 9% 18% Rght longtudnal force 7% 21% Rght spn creepage 0.5% 2% Rght contact area 1.4% 3.3% Rght lateral creepage 9% 23% The results obtaned usng ths model show that maxmum angle between the RD and the CE frames s as large as o whle ths maxmum angle between the LT and the CE frames s 4.26 o. Table II shows the average devaton of the RD and LT models from the CE model for dfferent contact parameters. It s observed that the results obtaned usng the LT frame are n better agreement wth those of the CE frame as compared to the ones obtaned usng the RD frame. Ths s manly because both the CE and the LT frames, accordng to ther defnton, drectly take nto account the effect of the track devaton whle ral geometry s not used n the defnton of the RD frame Concludng Remarks In wheel/ral creep force formulatons, the rollng drecton s used wth the normal to the contact surfaces to construct the contact frame n whch the tangental creep forces are defned. When

60 45 Hertz theory s used, an assumpton s made that the rollng drecton s the same as the drecton of one of the axes of the contact ellpse. The rollng drecton, however, depends on the moton of the wheel wth respect to the ral; whle the drectons of the axes of the contact ellpse depend only on the geometry of the wheel and ral surfaces and they are determned usng the prncpal drectons. Therefore, the rollng drecton can be sgnfcantly dfferent from the drecton of the axes of the contact ellpse n some smulaton scenaros. Ths chapter amed at studyng ths fundamental problem and comparng the results obtaned usng three dfferent contact frames. These frames are the rollng drecton (RD) frame, the contact ellpse (CE) frame and the longtudnal tangent (LT) frame. The RD frame s defned by the normal and the axs along the rollng drecton, whch s defned n ths study usng the cross product of the angular velocty vector and the poston vector of the contact pont wth respect to the wheel or wheelset body coordnate system. In defnng the rollng drecton, t s assumed that the rotaton of the wheelset or wheel s predomnantly ptch rotaton. The RD frame s rarely used n the ralroad vehcle smulatons and can lead to problems n tracton and brakng scenaros and also n the case of velocty dscontnutes due to mpact between the vehcle components. The CE frame s defned by the normal vector and one of the axes of the contact ellpse. Whle none of the axes of the contact ellpse defne n general the rollng drecton, the CE frame s the one used n many nvestgatons on ralroad vehcle systems. Some authors have also employed the LT frame whch s ndependent of Hertz theory and does not nvolve any moton varables such as the angular veloctes. The results obtaned usng the three contact frames are compared and t was shown that the results obtaned usng the CE and LT contact frames are n a very good agreement. However, there are smulaton scenaros n whch the rollng drecton as defned by

61 46 the angular velocty vector can sgnfcantly dffer from the axes of the contact ellpse. The numercal results obtaned n ths study show that whle there can be dfferences n the creepage results of dfferent contact frame models, the contact forces obtaned usng all these models, n the case of realstc scenaros, are n a good agreement despte the sgnfcant orentaton dfference between dfferent frames. Such a good agreement for the force results can be attrbuted to the domnant effect of the spn creepage n the lateral force calculatons. However, n some extreme cases, there can be dfferences between the results obtaned usng dfferent contact frames, n partcular the results of the RD model can sgnfcantly dffer from the results of the CE and LT models. Ths s can be attrbuted to the fact that the RD model s dfferent from the two other models n the sense that t does not use ral geometry to defne ts contact frame.

62 3. MULTIPOINT CONTACT SEARCH ALGORITHM In ralroad vehcle wheel/ral contact analyss, t s crucal to determne the contact forces accurately n order to be able to correctly predct the dynamc response of the vehcles. A frst step to acheve ths goal s to detect all possble wheel/ral contact ponts. Usng the contact formulatons, the contact forces can then be evaluated based on the poston of the contact ponts, the creepages, the wheel and ral geometres and ther materal propertes (Shabana et al., 2008). Most ralroad smulaton formulatons allow for only one or two wheel/ral contact ponts. As dscussed n Chapter 2, methods such as the constrant contact method or the elastc contact method can be used to fnd the frst pont of contact. There are, however, scenaros n whch there are mult ponts of contact between the wheel and the ral. A general contact algorthm capable of fndng all possble wheel/ral contact ponts needs to be developed. It s, therefore, the objectve of ths chapter to develop a multpont contact search algorthm that can be appled to general wheel/ral contact problems wthout any restrcton on the number of contact ponts. Ths algorthm can be used wth ether the constrant or elastc contact methods whch can be used to determne the frst pont of contact. For the other contact ponts, an elastc approach wll be used. Ths s due to the fact that the constrant contact method mposes knematc constrants that can lead to ndetermnate system f such an approach s appled at multple contact ponts Contact Formulatons The contact formulatons used n ths nvestgaton are general formulatons that can be appled to multpont contact problems. The man assumptons used here are as follows: All the contacts are non-conformal. 47

63 48 The frst contact pont can be determned ether by the constrant contact method or the elastc contact method. However, the other ponts are only determned by the elastc method. The locaton and surface parameters assocated wth the frst pont of contact are determned before searchng for the other contact ponts. The contact areas are ellptcal. There s no overlap of contact ellpses assocated wth dfferent contact ponts. There s no lmtaton on the number of contact ponts that can be determned. Snce n the formulatons used n ths nvestgaton, the wheel and ral geometres are descrbed usng surface parameters, the goal of the multpont contact search algorthm s to fnd the surface parameters assocated wth these ponts. The surface parameters of contact pont k can be wrtten as elements of the vector k s as follows: k wk wk rk rk T s s s s s ] k 2,.., m ( 3.1) [ where m s total number of contact ponts, and s wk 1 and s wk 2 are the wheel surface parameters whle rk s 1 and s rk 2 are the ral surface parameters as descrbed n Chapter 2. Note that n the developed algorthm t s assumed that the surface parameters of the frst contact pont ( k 1) are known. Usng the assumptons of non-conformal contact, fve nonlnear algebrac equatons must be satsfed at any of the contact ponts when the constrant contact formulaton s used. These equatons result from the facts that the global postons of the contact pont on the wheel and the ral are the same and the wheel and ral have a common normal at the contact pont. When the elastc method s used for the multpont contact search algorthm, only four nonlnear algebrac

64 49 equatons need to be used as dscussed n Chapter 2. These four algebrac equatons are as follows:,.., m k rk wk rk wk rk P wk P rk rk P wk P rk 2.. ).( ).( n t n t r r t r r t ( 3.2) The four surface parameters at the contact ponts can be determned by solvng the nonlnear algebrac equatons of Eq. 3.2 usng a Newton-Raphson scheme. To ths end, the followng system of algebrac equatons s teratvely solved: r w r w wr r wr r r r w w r s w r s w r w s r w s r s w r s w r w s r w s r r wr r s r r wr r s w r w r r r wr r s r r wr r s w r w r s s s s r r w w r r w w r r r r n t n t r t r t n t n t n t n t n t n t n t n t t t r t t t r t t t t t t t r t t t r t t t t t , 2, 2 2, 2,, 1, 1 1, 1, 2 2 2, 1 2 2, , 1 1 1, ( 3.3) The wheel/ral penetraton at the contact pont k s,.., m k rk P wk P rk k 2 ).( r r n ( 3.4) and the assocated complant normal force at ths contact pont can be wrtten as follows (Shabana et al 2005):,.., m k C K F F F k k k k k h k d k h k n 2 ) ( 2 3 ( 3.5) where k h K s the Hertzan constant, k C s the dampng coeffcent, and k s the tme rate of the penetraton. It should be noted agan that the algorthm s developed such that the frst contact pont can be determned ether by the elastc method or the constrant method. However, one of the conclusons made n ths chapter s that the constrant contact approach can be used only for the

65 50 frst contact pont. The man dfferences between the constrant and elastc methods when multpont contact pont scenaros are consdered are as follows: In the constrant method, the frst contact pont s determned usng a formulaton dfferent from the one used for other contact ponts whch are determned usng the elastc method as prevously mentoned. The constrant method, unlke the elastc method, allows no penetraton or separaton at the frst contact pont. Due to ths fact, f the contact constrants are mposed at one locaton, unrealstc results can be obtaned f the algorthm does not allow for swtchng the constrant contact locaton n the case of multpont contact search. In multpont contact problems, n general, the penetraton at one pont can sgnfcantly affect the knematcs and forces at other contact ponts. In the constrant method, the zero penetraton at the frst contact pont can nfluence the penetraton at other contact ponts as wll be demonstrated n the second example of ths chapter Search Algorthm The multpont contact search algorthm developed n ths chapter employs the followng steps: 1. At the begnnng of the smulaton, the wheel and ral profles are dscretzed to a selected number of segments; the profle surface parameters ( s w 1 and s r 2 ) of at the nodes that defne these segments are stored. These nodal ponts are used to determne potental ponts of contact durng the smulaton. It s assumed that before the search

66 51 starts at any nstant of tme durng the smulaton, the frst contact pont s already determned for that specfc tme. 2. Durng the smulaton and after the locaton of the frst contact pont s determned at a gven tme pont, the search algorthm uses the saved profle surface parameters (Step 1) to calculate the postons of the nodal ponts. 3. The poston of these ponts are calculated usng the saved profle surface parameters and the other two surface parameters w s 2 and r s 1. These two latter parameters change durng the smulaton whle the profle surface parameters at the nodal ponts do not change. Snce the surface parameters of the frst contact pont s known, ts angular surface parameter w1 2 s can be used for other potental contact ponts. Smlarly, the ral longtudnal surface parameter can be calculated usng that of the frst contact pont r r r1 s by makng the adjustment of s ( s s ) tan, where s the yaw r rotaton of the wheel wth respect to the track frame. Knowng the four surface parameters, the global poston vector of the nodal ponts can be calculated. The ponts close to the frst contact pont (based on a gven tolerance) are elmnated n order to mprove the computatonal effcency. 4. The dstances between the ponts of the wheel and those of the ral are calculated and the par of ponts that are close to wthn a specfc tolerance are selected as potental ponts of contact between the wheel and the ral. 5. Once all the potental ponts of contact are determned at gven tme pont, the surface parameters assocated wth each pont are used as the ntal guess for an teratve Newton-Raphson method that employs Eq The converged Newton-Raphson

67 52 solutons are checked usng the penetraton condton n order to determne the pars of contact ponts. Converged solutons that do not satsfy the penetraton condton are not consdered. 6. The dstances between all the contact ponts, ncludng the frst contact pont, are calculated. If the dstance between two contact ponts s larger than a specfed tolerance, the two contact ponts are consdered as new contact ponts. Otherwse, the contact pont wth the larger penetraton s only consdered, and the second pont s gnored. 7. If the algorthm does not fnd a second pont of contact, another search begns to fnd possble lead and lag contacts, whch can be encountered n the case of large yaw rotatons. To ths end, one can use r w s 1 R and s w arctan( tan ) where 2 w R s wheel radus and s the contact angle for the flange contact (Escalona et al., 2003), to determne a pont on the flange. If there s a wheel/ral penetraton at ths pont, a search s performed n the neghborhood of ths pont by usng postve and negatve small ncrements based the above mentoned equatons to fnd a pont that leads to penetraton. The surface parameters of ths potental contact pont are used n the Newton-Raphson teratve algorthm as an ntal guess. The resultng contact pont defned by the converged soluton s checked to make sure that t s not n the vcnty of one of the prevously determned contact ponts. Once all the contact ponts are dentfed and the assocated penetratons are calculated usng Eq. 3.4, the normal contact force can be determned usng Eq The creepages at each of the contact ponts are also calculated. The creepages, the normal contact forces, the wheel and

68 53 ral materal propertes, and contact ellpse dmensons are used to calculate the tangental creep forces and spn moment. The normal and creep contact forces are used to determne generalzed contact forces assocated wth the generalzed coordnates of the wheel and ral. Because of the generalty of ths search algorthm, there s no lmtaton on the number of contact ponts that can be found Numercal Results In ths secton, the numercal results obtaned usng the proposed algorthm are presented for two smulaton scenaros. The smulaton scenaros are desgned to have multple wheel/ral contacts. A comparson between the results obtaned usng the elastc and constrant contact methods n the case of mult contact pont scenaros s also presented n ths secton Sngle Truck Model Subjected to Lateral Forces In ths example, a sngle truck model negotatng a tangent track s consdered. The model, as shown n Fg. 23, conssts of a frame, two wheelsets, two equalzers, and the rals. The bodes are connected by bushngs and bearngs. For ths smulaton, the elastc method s used for the frst pont of contact.

69 54 Fgure 23. Sngle truck model subjected to two lateral forces As shown n Fg. 23, two lateral forces F 1 and F 2 are, respectvely, appled on the front and rear wheelsets. The magntudes of the forces are the same and they are assumed to be functons of tme accordng to the followng equaton: 0 t 8 or t 11 F 1 F2 ( 3.6) 6000( t 8) 8 t 11 The applcaton of these forces causes the truck to move to the left, whch ultmately, results n multple contact ponts for the left wheels of the truck and the left ral due to the shape of ther profles that are shown n Fgs. 24 and 25.

70 55 Fgure 24. The wheel profle Fgure 25. The ral profle

71 56 The shapes of the profles allow for three wheel/ral contact ponts. The algorthm should be able to capture the one, two, and three contact pont scenaros. The approxmate locatons of the contact ponts are depcted n Fg. 26. Fgure 26. The approxmate locatons of contact ponts The results of ths smulaton are shown n Fgs Fgures 27 and 28 show the Y- coordnate of the contact ponts for the front and rear contacts, respectvely. Also, n Fgs. 29 and 30, the ral profle surface parameters of these contact ponts are shown. By examnng the wheel and ral profle shapes shown n Fgs. 24 and 25, t s clear that the proposed algorthm successfully found the ponts at the expected locatons. The correspondng normal forces for these contact ponts are shown n Fgs. 31 and 32.

72 57 Fgure 27. Y-coordnate of the left front contact ( Frst contact pont, Second contact pont) Fgure 28. Y-coordnate of the left rear contact ( Frst contact pont, Second contact pont)

73 58 Fgure 29. Ral profle surface parameters of the left front contact ( Frst contact pont, Second contact pont) Fgure 30. Ral profle surface parameters of the left rear contact ( Frst contact pont, Second contact pont)

74 59 Fgure 31. Normal force of the left front contact ( Frst contact pont, Second contact pont) Fgure 32. Normal force of the left rear contact ( Frst contact pont, Second contact pont)

75 Suspended Wheelset Model on a Curve Track In ths example, the multpont contact search algorthm s used n the smulaton of a suspended wheelset on a curved track. The model, as shown n Fg. 33, conssts of three bodes ncludng a frame, a wheelset, and rals. The frame and wheelset are connected by lnear sprngs and dampers whle the frame s constraned to move along the track wth a forward velocty of m/s (40 mph). Fgure 33 Suspended wheelset model In the prevous example, the external lateral forces caused wheel/ral contact at multple ponts. In ths example, however, t s the track geometry, whch leads to three contact ponts. The curved track used for ths smulaton s shown n Fg. 34. The rght wheel and rght ral, whch are, respectvely, shown n Fgs. 35 and 36, have specfc shapes that can result n three contact ponts when the vehcle moves on the track.

76 61 Fgure 34. The top vew of the track used for the suspended wheelset model Fgure 35. The wheel profle

77 62 Fgure 36. The ral profle Smlar to the prevous example, the elastc method s used for the frst pont of contact. However, after presentng the results of ths smulaton, the smulaton wll be carred out agan usng the constrant method n order to compare the results obtaned usng these two methods. The results show that the search algorthm successfully fnds all the expected contact ponts. Fgure 37 shows the profle surface parameters of the rght ral whle the normal contact forces assocated wth these ponts are shown n Fg. 38.

78 63 Fgure 37. Ral profle surface parameters of the rght contact ( Frst contact pont, Second contact pont, Thrd contact pont) Fgure 38. Normal force of the rght contact ( Frst contact pont, Second contact pont, Thrd contact pont)

79 64 If the constrant contact method s used for ths smulaton, the search algorthm agan determnes the same three contact ponts. The normal contact forces, however, are much dfferent from those of the elastc contact method as shown n Fg. 39. The results demonstrate that the change of the contact formulatons used for the frst contact pont can consderably affect the force dstrbuton among the other contact ponts. Fgure 39. Normal force of the rght contact (constrant contact method) ( Frst contact pont, Second contact pont, Thrd contact pont) 3.4. Concludng Remarks A multpont contact search algorthm was developed n ths chapter. The developed algorthm mposes no lmtaton on the method used to fnd the frst pont of contact or on the number of contact ponts. The algorthm s capable of fndng the contact ponts wth a good precson because of the use of an teratve Newton-Raphson algorthm. The developed algorthm successfully fnds multple wheel/ral contact ponts. Ths was demonstrated by two examples n

80 65 ths chapter. A comparson between the results obtaned usng the constrant and elastc contact methods n the case of multpont contact scenaro was made and the dfferences between these methods n predctng the contact force dstrbutons were dscussed. It was demonstrated by an example that the change of the contact method used for predctng the frst contact pont n the case of multpont contact scenaros can consderably affect the normal force dstrbuton among the contact ponts.

81 4. AIR BRAKE FORCE MODEL Most trans n North Amerca use pneumatc brakes or ar brakes. For the most part, the automatc ar brake system of a freght tran conssts of a locomotve control unt, a car control unt located n each car, and a ppe connectng all these elements as shown n Fg. 40. Fgure 40. Man ar brake components Ths ppe that both transfers arflow and brake sgnals wll be referred to as brake ppe, whle the locomotve automatc brake valve wll be called automatc brake valve n ths chapter. The functon of the locomotve automatc brake valve s to control the ar pressure n the brake ppe for on car compressed ar storage as well as brake applcaton and release of all cars. Such a control acton provdes the pressure control sgnal that propagates along the brake ppe serally reachng one car after another. In addton to the brake ppe pressure, the controllng locomotve equalzng and emergency reservor pressures are also controlled. By varyng the poston of the automatc brake valve handle, dfferent scenaros and states of the brake system can be acheved. These states nclude the brake release mode n whch the pressure n the brake ppe s ncreased n order to release the brakes and recharge the CCU compressed ar storage reservors, the servce mode n whch the pressure must be reduced n order to apply the brakes wth a pressure reducton (at a servce rate) that s determned by the handle poston wth respect to the full 66

82 67 servce poston, and the emergency mode that allows the brake ppe pressure to be quckly reduced by ventng the brake ppe ar to the atmosphere (The latter two, servce and emergency modes, are referred to as the brake applcaton mode). Therefore, an accurate ar brake model must be able to predct the response of the ar flow n the brake ppe to the changes made n the locomotve automatc brake valve handle poston. The 26C valve also has the ndependent brake valve functon that s not modeled n ths study. The objectve of ths chapter s to ntegrate a dynamc ar brake model wth effcent nonlnear tran longtudnal force algorthms based on trajectory coordnate formulatons. The proposed ar brake force model used n ths chapter employs the contnuty and momentum equatons and accounts for the effect of the ar flow n long tran ppes as well as the effect of leakage and branch ppe flows. Ths model can be effectvely used to study the dynamc behavor of the ar flow n the tran ppe and ts effect on the longtudnal tran forces durng brake applcaton and release. The contnuty and momentum equatons are smplfed by usng the assumptons of one dmensonal sothermal flow, leadng to two coupled ar velocty/pressure partal dfferental equatons that depend on tme and the longtudnal coordnate of the brake ppe. The resultng partal dfferental equatons are converted to a set of ar brake frst order ordnary dfferental equatons usng the fnte element dscretzaton. These frst order ordnary dfferental equatons are solved smultaneously wth the tran second order nonlnear dynamc dfferental equatons of moton that are based on the trajectory coordnates. In ths chapter, the tran car dynamcs s defned usng a body track coordnate system that follows the car moton. The translaton and orentaton of ths coordnate system are defned n terms of one geometrc trajectory parameter that descrbes the dstance travelled by the car. The confguraton of the car

83 68 wth respect to ts track coordnate system s descrbed usng two translaton coordnates and three Euler angles (Shabana et al., 2008). The nonlnear trajectory coordnate formulaton used n ths study allows for the use of arbtrary track geometry. The operaton of the brake system, ncludng brake applcaton and release, s controlled by the locomotve automatc brake valve that defnes the nput to the ar brake system durng the dynamc smulaton. A smplfed valve model s also proposed n order to reduce the smulaton computatonal tme. The procedure for couplng the brake ppe ar flow, locomotve automatc brake valve, CCU, and tran equatons s establshed and used n the smulaton of the nonlnear dynamcs of long trans Ar Flow Equatons In ths secton, the basc contnuum mechancs equatons used n ths chapter to study the ar flow dynamcs n tran brake ppes are presented. These equatons nclude the momentum, contnuty, and consttutve equatons. In the followng secton, the general three-dmensonal equatons presented n ths secton are smplfed to the case of one dmensonal sothermal flow. It should be noted that f the process takes place rapdly, the adabatc process assumpton s more realstc than the sothermal process. Nonetheless, the actual process s nether of these processes. In fact, the actual process s polytropc. However, n ths nvestgaton, smlar to some prevous studes, the sothermal flow assumpton s used (Abdol-Hamd, 1986) Contnuty Equaton The general contnuty equaton for a flud can be wrtten as (John and Keth, 2006; Shabana, 2008; Whte, 2008) dv. ds 0 t vn ( 4.1) V S

84 69 In ths equaton, V s the volume whch s assumed to reman constant for the brake ppe, and therefore, no dstncton s made between the volumes n the reference and current confguratons; S s the surface area, s the mass densty, v s the velocty vector, and n s the normal to the surface. Usng the dvergence theorem, the contnuty equaton can be wrtten as v 0 ( 4.2) t In ths equaton, s the dvergence operator. Snce ar flow s consdered n ths study, the densty cannot be treated as a constant, and therefore, the assumpton of ncompressblty s not used n ths study Momentum Equaton The momentum equaton or the partal dfferental equaton of equlbrum can be wrtten n the followng form (Shabana, 2008): adv σnds f dv ( 4.3) v s v b In ths equaton, a s the acceleraton vector, σ s the symmetrc Cauchy stress tensor, s and v are, respectvely, the area and volume n the current confguraton, and f b s the vector of body forces per unt volume. Applyng the dvergence theorem, and assumng that v brake ppe, the precedng equaton leads to T b V V V V for the adv σ dv f dv ( 4.4) Ths equaton can also be wrtten n the followng dfferental form:

85 70 a σ T f b ( 4.5) The frst term of ths equaton, whch represents the nerta force, can be rewrtten n a dfferent form, whch s more convenent to use when dealng wth compressble fluds. Multplyng the contnuty equaton of Eq. 4.2 by the velocty vector v, one obtans t v v v 0 ( 4.6) Usng the expresson for the total dervatve of the velocty vector v, the nerta term a can be wrtten as dv dt v a= t v v ( 4.7) Substtutng Eq. 4.7 nto Eq. 4.5 and addng Eq. 4.6, one obtans Ths equaton can be wrtten as v vv vvv σ T f b ( 4.8) t t v vvv σ T f b ( 4.9) t Ths equaton s an alternate form of the momentum equaton of Eq Flud Consttutve Equaton In the case of sotropc fluds, the consttutve equatons that dfferentate one flud from another and defne the flud characterstcs can be wrtten as

86 71 σ p(, T) (, T)tr D I2 (, T) D ( 4.10) where p s the hydrostatc pressure, s the mass densty, T s the temperature, Ds the rate of deformaton tensor, and and are vscosty coeffcents that depend on the flud densty and temperature Naver-Stokes Equatons In order to obtan the flud equatons of moton, the consttutve equatons of Eq are substtuted nto the partal dfferental equatons of equlbrum of Eq Ths leads to v T vvv pitr DI2D f b ( 4.11) t Ths equaton represents the general three-dmensonal partal dfferental equatons of moton for sotropc fluds. If the flud s assumed to be Newtonan, that s the shear stress s proportonal to the rate of the shear stran, the precedng equaton reduces to t v p tr 2 v v v I D I D f ( 4.12) Equaton 4.11 governs the moton of general sotropc flud, whle Eq governs the moton n the specal case of Newtonan flud. T b 4.2. One-dmensonal Model The assumpton of one-dmensonal ar flow used n ths study mples that the flow, at any cross secton, has only one drecton along the longtudnal axs of the ppe, that s, the velocty components n the other drectons are not consdered. Furthermore, the magntude of the flow velocty s assumed to be unform at any cross secton. Consequently, shear stresses are

87 72 neglected, and as a result, the off-dagonal elements of the Cauchy stress tensor are assumed to be zero. In the case of one dmensonal ar flow, the contnuty equaton of Eq. 4.2 reduces to u L 0 t x ( 4.13) In ths equaton, x s the longtudnal spatal ppe coordnate, L s the ar leakage, and u s the velocty component along the longtudnal x coordnate of the brake ppe. Note that the precedng equaton s a partal dfferental equaton that depends on both tme t and the spatal coordnate x. The effect of ar flow through the ppe branches can also be ntroduced systematcally to the contnuty equaton n order to account for the mass flow rate. In the case of multple branches connected to the man ar ppe, a term can be added to the contnuty equaton as dscussed n the prevous chapter where the car control unt model s developed. In the case of one dmensonal flow, the Naver-Stokes equaton of Eq becomes u f 2 b u p u t x x x ( 4.14) If the flud s assumed to be nvscd, that s, the effect of shear s neglected, the second term on the rght hand sde of the precedng equaton can be neglected, leadng to 2 u u p t x x f b ( 4.15) Usng the assumpton of sothermal flow, one has the followng relatonshp (Whte, 2008):

88 73 p Rg ( 4.16) In ths equaton, R g s the gas constant whch has unts J/(Kg K), and s the local temperature ( K). The relatonshp of Eq can be used to elmnate the ar densty from the contnuty and momentum equatons leadng to the followng system of pressure/velocty coupled equatons: p pu tl 0 t x 2 pu pu p t t fb t x x ( 4.17) In ths equaton, t R g. Gven the boundary and ntal condtons, the precedng system of coupled partal dfferental equatons can be solved for the pressure and velocty dstrbutons usng numercal methods as dscussed n the followng secton Fnte Element Formulaton In ths study, a fnte element procedure s used to transform the partal dfferental equatons of Eq to a set of coupled frst order ordnary dfferental equatons. These ordnary dfferental equatons can be solved usng the method of numercal ntegraton to determne the pressure and velocty for dfferent brakng scenaros. Let q Eq can be rewrtten as pu be a new varable. Usng ths defnton, p q tl t x q qu p t t f b t x x ( 4.18)

89 74 In the fnte element analyss, the brake ppe s assumed to consst of m fnte elements. e e e The doman of the element s defned by the spatal coordnate x x,0 x l, where l e s the length of the fnte element. Over the doman of the fnte element, the varables p and q are nterpolated usng the followng feld: e e e e e e p xt, Sp, q xt, Sq, e1,2,, m ( 4.19) p q where e S p and S e q are approprate shape functons, and p e and q e are the vectors of nodal e coordnates. Multplyng the frst equaton n Eq by the vrtual change p and the second equaton by the vrtual change e q, ntegratng over the volume, usng the relatonshp dv A dx, where e e e e A s the cross secton area; and usng Eq. 4.19; one obtans the followng system of frst order ordnary dfferental equatons for the fnte element e : e e e Me Q, e1,2,, m ( 4.20) In ths equaton, e e e e e p Q e p M e pp M pq e,, e Q e M e e q Qq Mqp Mqq ( 4.21) where e e l l e e et e e e e e et e Mpp A Sp Spdx, Mpq Mqp 0, Mqq A Sq Sqdx, 0 0 e e e l l S e e et q e e et e e p A p dx A p dx t L Q, S q + x S 0 0 e e e l l e S e e et q e e e et e u e Qq A q u dx A q q dx S q x S S q x 0 0 S e e e l l e et p e e et t A Sq dx p + A q dx t f b x S 0 0 ( 4.22)

90 75 The fnte element equatons of Eq can be assembled usng a standard fnte element assembly procedure. Ths leads to the frst order ordnary dfferental equatons of the brake ppe system whch can be wrtten n the followng matrx form: Me Q ( 4.23) In ths equaton, e s the vector of nodal coordnates, M s the brake ppe global coeffcent matrx that results from assemblng the e M element matrces, and Q s the rght hand sde vector that results from the assembly of the e Q element vectors. Usng the poston of the locomotve automatc brake valve handle, the ntal condtons and nputs for Eq can be defned and used wth numercal ntegraton methods to solve for the pressure and the velocty dstrbuton. Usng the approach descrbed n ths secton n whch the new varable q ntroduced, one obtans constant symmetrc pu s e M and M matrces. Therefore, one needs to defne the LU factors of M only once at the start of the smulaton. The effect of the ar leakage n the fnte element formulaton presented n ths secton can be consdered by ntroducng ths effect at the nodal ponts usng the sotropc approach or the average densty approach (Abdol-Hamd, 1986). The latter s the approach adopted n ths nvestgaton Alternate Formulaton Another alternate approach s to use the same assumptons and combne the contnuty equaton of Eq wth the one dmensonal form of the momentum equaton of Eq. 4.5 to obtan the followng coupled system of frst order partal dfferental equatons: u L 0 t x u u p u f b t x x ( 4.24)

91 76 Usng Eq ( p ), one obtans t p pu tl 0 t x u u p p pu t t f b t x x ( 4.25) As an alternatve to ntroducng the varable q pu, one can use p and u nstead of usng p and q. One can, however, show that the use of Eq nstead of Eq wll not lead to a constant coeffcent matrx. In general, a constant coeffcent matrx can always be obtaned f the number of varables s not reduced usng the gas law of Eq Nonetheless, the advantage of the form of Eq s that the pressure and veloctes can be nterpolated ndependently Wall Frcton Forces A smple expresson for the ppe wall frcton force F S s used n ths study. In the case of duct flow, one may assume that the force F S s only related to the wall shear stress as (Abdol-Hamd, 1986) FS d ( 4.26) where d s the hydraulc dameter, and s the one dmensonal flow shear stress. They are defned as (Abdol-Hamd, 1986) xx u u d Aavg Adx, fw x 8 u ( 4.27) x where A avg s the average area, and the wall frcton factor f w s a functon of the local Reynolds number Re. The followng relaton between f w and Re s often used:

92 77 fw b a Re ( 4.28) where a and b are selected to gve a good ft to data for dfferent flow regmes; lamnar, transton and turbulent. The expresson for the frcton force F S can be ntroduced to the momentum equaton. If the area of the ppe s constant, then Aavg A constant Tran Nonlnear Dynamc Equatons The mathematcal models of the ar flow n the brake ppe and the commands of the locomotve automatc brake valve can be used to defne the brakng scenaros that affect the tran longtudnal dynamcs. In ths secton, the nonlnear dynamc equatons of the tran cars are developed usng the trajectory coordnates. It s assumed that ral vehcle dynamcs has no effect on the ar flow n the brake ppe, whle the brakng forces can have a sgnfcant effect on the tran longtudnal forces Poston, Velocty, and Acceleraton In order to develop the nonlnear dynamc equatons of moton of the tran, the global poston vector of an arbtrary pont on a car body s frst defned. The poston vector r of an arbtrary pont on body wth respect to the global coordnate system can be defned as shown n Fg. 41 as (Shabana, 2010) r R Au ( 4.29) where R s the global poston vector of the orgn of the body coordnate system, u s the poston vector of the arbtrary pont on the body wth respect to the local coordnate system, and A s the rotaton matrx that defnes the orentaton of the local coordnate system wth respect to the global system. In rgd body dynamcs, u s constant and does not depend on tme.

93 78 Fgure 41. Coordnate systems Dfferentatng Eq wth respect to tme, one obtans the absolute velocty vector defned as r R u ( 4.30) where s the absolute angular velocty vector defned n the global coordnate system, and u = Au. The absolute acceleraton vector s obtaned by dfferentatng the precedng equaton wth respect to tme, leadng to r R u u ( 4.31) where s the angular acceleraton vector of body. The precedng equaton can also be wrtten n the followng alternate form: r R A u u ( 4.32)

94 79 where =A and ω =Aω. The angular velocty vectors defned, respectvely, n the global and body coordnate systems can be wrtten n terms of the tme dervatves of the orentaton coordnates as follows: G, G ( 4.33) where G and G can be expressed n terms of the orentaton parameters (Shabana et al., 2008) Trajectory Coordnates The trajectory coordnate formulaton s suted for the study of the tran longtudnal force dynamcs snce the car body degrees of freedom can be systematcally reduced to a set that can be related to the track geometry. A centrodal body coordnate system s ntroduced for each of the ralroad vehcle components. In addton to the centrodal body coordnate system, a body/track coordnate system that follows the moton of the body s ntroduced. The locaton of the orgn and the orentaton of the body/track coordnate system are defned usng one geometrc parameter s that defnes the dstance travelled by the body along the track. The body coordnate system s selected such that t has no dsplacement n the longtudnal drecton of moton wth respect to the body/track coordnate system. Two translatonal coordnates, r y and r r r z ; and three angles,,, and r, are used to defne the poston and orentaton of the body coordnate system wth respect to the body/track coordnate system t t t X Y Z, as shown n Fg. 42.

95 80 Fgure 42. Trajectory coordnates Therefore, for each body n the system, the followng sx trajectory coordnates can be used: r r r r r T p [ s y z ] ( 4.34) In terms of these coordnates, the global poston vector of the center of mass of body can be wrtten as t t r R R A u ( 4.35) where system, r u s the poston vector of the center of mass wth respect to the body/track coordnate t R s the global poston vector of the orgn of the trajectory coordnate system, and s the matrx that defnes the orentaton of the body/track coordnate system and s a functon of t t t three predefned Euler angles,, and whch are used to defne the track geometry. The vector t R and the matrx t A are functons of only one tme dependent arc length parameter t A s. For a gven s, one can also determne the three Euler angles

96 81 s s s s T t t t t that enter nto the formulaton of the rotaton matrx t (Shabana et al., 2008). The vector The matrx r u can be wrtten as r r r T u [0 y z ] ( 4.36) r A that defnes the orentaton of the body coordnate system wth respect to the body/track coordnate system can be expressed n terms of the three tme dependent Euler r r r r T angles [ ] prevously defned. A Equatons of Moton A velocty transformaton matrx that relates the absolute Cartesan acceleratons to the trajectory coordnate acceleratons can be systematcally developed. Usng the velocty transformaton and the Newton-Euler equatons that govern the spatal moton of the rgd bodes, the equatons of moton of the car bodes expressed n terms of the trajectory coordnates can be developed. The followng form of the Newton-Euler equatons s used n ths chapter (Shabana, 2010): m I 0 0 I R α M e ω F e ( I ω ) ( 4.37) where m s the mass of the rgd body; I s a 3 3 dentty matrx; I s the nertal tensor defned wth respect to the body coordnate system; F e s the resultant of the external forces appled on the body defned n the global coordnate system; and M e s the resultant of the external moments actng on the body defned n the body coordnate system. The forces and moments actng on the body nclude the effect of the gravty, brakng forces, coupler forces, and tractve effort and moton resstng forces. If a s the vector of absolute Cartesan acceleratons

97 82 of the body, one can use the knematc descrpton gven n ths secton to wrte the Cartesan acceleratons n terms of the trajectory acceleratons as a Bp γ ( 4.38) In ths equaton, T T T, a R α B s a velocty transformaton matrx, and γ s a quadratc velocty vector (Shabana et al., 2008). Substtutng Eq nto Eq and premultplyng by the transpose of the velocty transformaton matrx B, one obtans the dynamc equatons expressed n terms of the trajectory coordnates, as descrbed n detal n (Shabana et al., 2008) Locomotve Valve Model The ar pressure n the brake ppe system s controlled by the brake locomotve automatc brake valve. The mathematcal model of the 26C locomotve valve developed by Abdol-Hamd (1986) s used n ths chapter. The valve and ts man components are shown n Fg. 43.

98 83 Fgure C valve scheme The prmary functon of the automatc brake valve s to control ar pressure n the brake ppe allowng for the applcaton or release of the tran and locomotve brakes. By changng the