Use of Non-inertial Coordinates and Implicit Integration for Efficient Solution of Multibody Systems AHMED K. ABOUBAKR

Transcription

1 Use of Non-nertal Coordnates and Implct Integraton for Effcent Soluton of Multbody Systems BY AHMED K. ABOUBAKR B.Sc., Caro Unversty, 2004 M.Sc., Caro Unversty, 2009 THESIS Submtted as partal fulfllment of the requrements for the degree of Doctor of Phlosophy n Mechancal Engneerng n the Graduate College of the Unversty of Illnos at Chcago, 2014 Chcago, Illnos Defense Commttee: Professor Ahmed A. Shabana, Char and Advsor, Mechancal and Industral Engneerng Behrooz Fallah, Mechancal Engneerng, Northern Illnos Unversty Crag D. Foster, Cvl and Materals Engneerng Thomas J. Royston, Boengneerng Antono M. Recuero, Mechancal and Industral Engneern

2 Ths thess s dedcated to my famly.

3 ACKNOWLEDGEMENTS I would lke to express my grattude to my advsor, Dr. Ahmed A. Shabana, for hs support, patence, and encouragement throughout my graduate studes at the Unversty of Illnos at Chcago. Hs nvaluable advce and gudance was essental to the completon of ths work and has taught me nnumerable lessons and nsghts on the workngs of academc research n general. I would also lke to thank the members of my thess commttee: Dr. Behrooz Fallah, Dr. Crag D. Foster, Dr. Antono M. Recuero, and Dr. Thomas J. Royston for ther nvaluable advce. I would also lke to thank my frends and colleagues from the Dynamc Smulaton Laboratory wth whom I have worked wth over the years. Each of them has helped me n ther own way whether t be advce drectly pertanng to my work or one of the many dscussons we have had that have expanded my academc knowledge. Fnally, I would lke to thank my famly for ther contnuous support and encouragement. For my parents, who rased me wth a love of scence and supported me n all my pursuts. I am partcularly grateful to my lovng, supportve, encouragng, and patent wfe Maram and to my babes Rawd and Saja wthout your support, sound, and smle I could never have accomplshed ths work.

4 TABLE OF CONTENTS 1. INTRODUCTION Motvaton and Objectves Inertal and Non-nertal Coordnates Motvaton for the Use of Non-nertal systems Important Applcaton MBS Dynamcs and Longtudnal Tran Dynamcs Implct Integraton for Stff Dfferental /Algebrac Equatons Scope and Organzaton of the Thess NON-INERTIAL COORDINATES Inertal and Non-nertal Coordnates Dfferental and Algebrac Equatons Algebrac Force Equatons Coupler Knematcs Knematcs of Fxed Ponts Knematcs of Sldng Blocks Translaton and Rotaton Coupler Generalzed Forces Draft Gear/EOC Forces Shank Forces Shank and Draft Gear/EOC Attachment Inertal and Non-Inertal Coordnate Generalzed Forces Soluton Algorthm Numercal Examples Tangent Track Model Curved Track Model Concludng Remarks VERIFICATION OF THE NON-INERTIAL SYSTEMS Scope and Objectve Inertal systems v

5 TABLE OF CONTENTS (contnued) 3.3 Coupler Knematcs Usng Redundant Coordnates Coupler Force Equatons and Algorthm Draft Gear/EOC Connecton Knuckle Force Model Two-Car Model Smulaton Results Egen Value and Frequency Doman Analyss Ten-Car Model Curved Track Smulaton Concludng Remarks IMPLEMENTATION OF THE TLISMNI METHOD MBS Dfferental/Algebrac Equatons (DAEs) MBS Equatons of Moton Generalzed Coordnate Parttonng Algorthm and Sparse Matrx Implementaton TLISMNI Algorthm Krylov Subspace and Inexact Newton Method Krylov Subspace Projecton Method Jacoban-Free Newton-Krylov Methods TLISMNI Newton-Krylov Algorthm Implementaton Convergence Crtera Low Order Integraton Formulas Park Method BDF Method Trapezodal Method HHT/Newmark Method Error Control and Tme Step Selecton Explct Adams Method Numercal Examples v

6 TABLE OF CONTENTS (contnued) Pendulum Example Rgd Tracked Vehcle Model Flexble Tracked Vehcle Model Pantograph/ Catenary Ralroad Vehcle Example Concludng Remarks CONCLUSIONS AND FUTURE WORK Concluson Future Work APPENDIX A APPENDIX B CITED LITERATURE VITA v

7 LIST OF TABLES Table 1. Car parameters Table 2. Coupler parameters Table 3: Model parameters Table 4: Car parameters Table 5: Contact parameters v

8 LIST OF FIGURES Fgure 1: Knuckle coupler... 5 Fgure 2: Conventonal auto-coupler package... 5 Fgure 3: Concept of the non-nertal coordnates Fgure 4: Two-car system Fgure 5: Forward poston Fgure 6: EOC and shank knuckle sprng deformatons Fgure 7: EOC and shank knuckle sprng deformaton rates Fgure 8: Draft gear and shank knuckle sprng deformatons Fgure 9: Draft gear and shank knuckle sprng deformaton rates Fgure 10: Curved track geometry Fgure 11: EOC sprng deformaton Fgure 12: EOC sprng deformaton rates Fgure 13: Torsonal sprng deformaton of car Fgure 14: Torsonal sprng deformaton rate of car Fgure 15: Torsonal sprng deformaton of car j Fgure 16: Torsonal sprng deformaton rate of car j Fgure 17: Forward poston Fgure 18: EOC and shank knuckle relatve dsplacements (loaded car scenaro) Fgure 19: EOC and shank knuckle relatve veloctes (loaded car scenaro) Fgure 20: EOC and shank knuckle relatve dsplacements (empty car scenaro) Fgure 21: EOC and shank knuckle relatve veloctes (empty car scenaro) v

9 LIST OF FIGURES (contnued) Fgure 22: Mode shapes Fgure 23: Frequency doman analyss of the acceleraton Fgure 24: Lagrange multpler frequences Fgure 25: Ten-car model (ATTIF nterface) Fgure 26: EOC and shank knuckle relatve dsplacements (5 th coupler) Fgure 27: EOC and shank knuckle relatve veloctes (5 th coupler) Fgure 28: Curved track Fgure 29: Forward poston Fgure 30: Torsonal sprng deformaton Fgure 31. Flexble pendulum ntal confguraton Fgure 32: Nodal deformaton for md-node Fgure 33: M113 tracked vehcle model Fgure 34: Suspenson system layout of M113 tracked vehcle Fgure 35: Chasss forward poston Fgure 36: Chasss forward velocty Fgure 37: Road-wheel Euler parameter Fgure 38: Road-wheel vertcal acceleraton Fgure 39: Track lnk vertcal poston Fgure 40: Track lnk vertcal velocty Fgure 41: Chasss forward poston Fgure 42: Chasss forward velocty Fgure 43: Road-wheel angular velocty x

10 LIST OF FIGURES (contnued) Fgure 44: Chasss forward poston Fgure 45: Chasss forward velocty Fgure 46: Chasss vertcal acceleraton Fgure 47: Road-wheel angular velocty Fgure 48: Road-wheel angular acceleraton Fgure 49: Vertcal global nodal poston Fgure 50: Longtudnal global nodal poston Fgure 51: Vertcal global nodal velocty Fgure 52: Longtudnal global nodal velocty Fgure 53: Pantograph/catenary ralroad vehcle model Fgure 54: Artculated pantograph System Fgure 55: Catenary model Fgure 56: Wheelset angular velocty Fgure 57: Longtudnal global nodal poston Fgure 58: Longtudnal global nodal velocty x

11 SUMMARY Development of computatonal methods, formulatons, and algorthms to study nterconnected bodes that undergo large deformaton, translatonal, and rotatonal dsplacements has been the focus of prevous multbody system (MBS) nvestgatons. MBS dynamcs s used n the study and dynamc modelng of varous systems such as structural, mechancal, and bologcal systems, as a result, t has a wde range of applcatons n ndustry. For ths reason, the development of effcent and accurate computatonal methods, formulatons, and algorthms to study MBS dynamcs, s the man focus of ths thess. The thess dscusses the concept of non-nertal coordnates usng ralroad examples. Furthermore the thess presents effcent mplementaton of mplct ntegraton methods for the soluton of stff MBS dfferental/algebrac equatons, and demonstrates ths mplementatons usng complex examples that nclude rgd and flexble bodes such as tracked vehcles. The use of the non-nertal coordnates can be advantages n many applcatons, as wll be ntroduced n ths thess. Inertal coordnates have generalzed nerta forces assocated wth them, whle the non-nertal coordnates have no generalzed nerta forces. In order to avod havng a sngular nerta matrx and/or hgh frequency oscllatons, the second dervatves of the non-nertal coordnates are not used when formulatng the system equatons of moton n ths study. In ths case, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates, leadng to a formulaton smlar to the one used n the case of non-generalzed coordnates. The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons are used to determne the nertal acceleratons whch can be ntegrated to determne the nertal coordnates and veloctes. The non-nertal coordnates are determned by usng an x

12 SUMMARY(contnued) teratve algorthm to solve a set of nonlnear algebrac force equatons obtaned usng quas-statc equlbrum condtons. The non-nertal veloctes are determned by solvng these algebrac force equatons at the velocty level. The non-nertal coordnates and veloctes enter nto the formulaton of the generalzed forces assocated wth the nertal coordnates. Usng the concept of non-nertal coordnates and the resultng dfferental/algebrac equatons obtaned n ths thess leads to sgnfcant reducton n the numbers of state equatons, system nertal coordnates, and constrant equatons; and allows avodng a system of stff dfferental equatons that can arse because of the relatvely small mass. One of the mportant areas n whch the concept of the non-nertal coordnates can be effcently appled s the longtudnal tran dynamcs. The development of accurate nonlnear longtudnal tran force models s necessary n order to better understand ralroad vehcle dynamc scenaros that nclude brakng, tracton, deralments and other types of accdents. Car coupler forces have sgnfcant effects on the longtudnal tran forces and tran stablty. Usng the concept of non-nertal coordnates enables the development of a more detaled coupler model that captures the coupler knematcs wthout sgnfcantly ncreasng the number of state equatons and the dmenson of the problem. The proposed coupler model n ths thess, allows for the car bodes to have arbtrary dsplacements, also avods havng a stff system of dfferental equatons that can result from the use of relatvely small masses. By assumng the coupler nerta neglgble compared to the car body nerta, one can dentfy two dstnct sets of coordnates; nertal and non-nertal coordnates. The nertal coordnates descrbe the moton of the car bodes and have nertal forces x

13 SUMMARY(contnued) assocated wth them, whle the non-nertal coordnates have no nerta forces assocated wth them. Gven the nertal coordnates and veloctes, the nonlnear coupler force equatons are solved teratvely for the coupler non-nertal coordnates. The obtaned non-nertal coordnates are used n the formulaton of the generalzed forces actng on the car bodes. Numercal examples are presented n order to demonstrate the use of the concept of non-nertal coordnates for effcent modelng of the three-dmensonal coupler n the analyss of tran longtudnal moton. In order to demonstrate the use of the concept of non-nertal coordnates, the effect of neglectng the coupler nerta on the accuracy of the soluton and the computatonal effcency s examned, therefore an nertal coupler model s developed usng the general multbody system (MBS) algorthms. The results obtaned usng the nertal and non-nertal coupler models are compared. The numercal results obtaned n ths study showed that the neglect of the coupler nerta does not have a sgnfcant effect on the accuracy of the soluton. On the other hand, the neglect of ths nerta leads to sgnfcant mprovement n the computatonal effcency. The results obtaned showed that the longtudnal tran dynamcs (LTD) mplementaton that neglects the effect of the coupler nerta becomes more effcent as the number of cars ncreases. Where the relatvely small nerta of the coupler can lead to hgh frequency oscllatons n the soluton; requrng the use of smaller ntegraton tme steps and sgnfcantly ncreasng the CPU tme. An egenvalue analyss and FFT are used to dentfy the frequences assocated wth the coupler nerta. As dscussed n ths thess, these hgh frequences do not appear when the non-nertal coupler model s used. Large and complex multbody systems that nclude flexble bodes, and contact/mpact pars are governed by stff equatons. The use of explct ntegraton for solvng the stff equatons x

14 SUMMARY(contnued) can be very neffcent. Therefore the use of mplct numercal ntegraton n the case of stff MBS equatons has been recommended. Therefore, t s one of the man objectves of ths work s to develop a new and effcent mplementaton of the two-loop mplct sparse matrx numercal ntegraton (TLISMNI) method proposed for the soluton of constraned rgd and flexble multbody system (MBS) dfferental and algebrac equatons. The TLISMNI method has desrable features that nclude avodng numercal dfferentaton of the forces, allowng for an effcent sparse matrx mplementaton, and ensurng that the knematc constrant equatons are satsfed at the poston, velocty and acceleraton levels. In ths method, a sparse Lagrangan augmented form of the equatons of moton that ensures that the constrants are satsfed at the acceleraton level s frst used to solve for all the acceleratons and Lagrange multplers. The generalzed coordnate parttonng or recursve methods can be used to satsfy the constrant equatons at the poston and velocty levels. In order to mprove the effcency and robustness of the TLISMNI method, the smple teraton and the Jacoban-Free Newton-Krylov approaches are used n ths nvestgaton. The new mplementaton s tested usng several low order formulas that nclude Hlber Hughes Taylor (HHT) method, whch ncludes numercal dampng, L- stable Park, A-stable Trapezodal, and A-stable BDF methods. Dscusson on whch method s more approprate to use for a certan applcaton s provded. The thess also dscusses TLISMNI mplementaton ssues ncludng the step sze selecton, the convergence crtera, the error control, and the effect of the numercal dampng. The use of the computer algorthm descrbed n ths thess s demonstrated by solvng complex rgd and flexble tracked vehcle models, ralroad vehcle models, and very stff structure problems. The results, obtaned usng these low order formulas, are compared wth the results xv

15 SUMMARY(contnued) obtaned usng the explct Adams-predctor-corrector method. Usng the TLISMNI method, whch does not requre numercal dfferentaton of the forces and allows for an effcent sparse matrx mplementaton, for solvng complex and stff structure problems leads to sgnfcant computatonal cost savng. xv

16 CHAPTER 1 INTRODUCTION 1.1 Motvaton and Objectves Development of computatonal methods, formulatons, and algorthms to study nterconnected bodes that undergo large deformaton, translatonal, and rotatonal dsplacements has been the subject of prevous multbody system (MBS) nvestgatons. MBS dynamcs s used n the study and vrtual prototypng of varous systems such as structural, mechancal, and bologcal systems; as a result, t has a wde range of applcatons n ndustry. An mportant example of such applcatons s ralroad vehcles where the vehcles can be treated as multbody systems. Regardng the crucal role of the ralroad ndustry n passenger and freght transportaton n many countres, t s mportant to effectvely make use of MBS dynamcs n order to mprove safety and rde comfort, and reduce the mantenance cost of the ral systems. Another example of these MBS applcatons, n whch accurate modelng of the system s necessary, s tracked vehcles. Hgh moblty tracked vehcle such as mltary tanks and armored personal carrers are desgned for operaton over rough and off-road terrans. Investgatons on the dynamc analyss of such tracked vehcles have been lmted because of the complexty of the forces resultng from the nteracton between the vehcle components. These forces are mpulsve n nature, and are the source of hgh frequences. Capturng the effect of these forces accurately requred the use of sophstcated numercal algorthms. To ths end, the development of effcent and accurate computatonal methods, formulatons, and algorthms to study MBS dynamcs, s the man objectve of ths thess. Ths s accomplshed n ths thess by employng the concept of non-nertal 1

17 coordnates and an effcent mplementaton of mplct ntegraton methods for the soluton of the stff MBS dfferental/algebrac equatons. 1.2 Inertal and Non-nertal Coordnates The concept of the nertal and non-nertal coordnates s developed n ths thess. Inertal coordnates have generalzed nerta forces assocated wth them, whle the non-nertal coordnates have no generalzed nerta forces. In order to avod havng a sngular nerta matrx and/or hgh frequency oscllatons, the second dervatves of the non-nertal coordnates are not used when formulatng the system equatons of moton n ths study. In ths case, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates, leadng to a formulaton smlar to the one used n the case of non-generalzed coordnates (Shabana and Sany, 2001). The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons are used to determne the nertal acceleratons whch can be ntegrated to determne the nertal coordnates and veloctes. The non-nertal coordnates are determned by usng an teratve algorthm to solve a set of nonlnear algebrac force equatons obtaned usng quas-statc equlbrum condtons. The non-nertal veloctes are determned by solvng these algebrac force equatons at the velocty level. The non-nertal coordnates and veloctes enter nto the formulaton of the generalzed forces assocated wth the nertal coordnates. 2

18 1.3 Motvaton for the Use of Non-nertal systems One of the applcatons that the concept of non-nertal coordnates can be used for s the analyss of longtudnal tran dynamcs. The study and understandng of longtudnal tran dynamcs was probably frst motvated by the desre to reduce longtudnal oscllatons n passenger trans and n so dong mprove the general comfort of passengers. The study of longtudnal tran dynamcs has been addressed n several papers (Duncan and Webb, 1989 ; Jolly and Ssmey, 1989; Van Der Meulen, 1989; P. Belfort et al., 2008). From these studes an understandng of the force magntudes and an awareness of the need to lmt these forces wth approprate drvng strateges was developed. Impact n-tran forces are assocated wth run-n and run-out occurrences due to changes n locomotve power and brakng settngs, changes n grade and undulatons (Scown et al., 2000). The need to control, and where possble reduce, n-tran forces resulted n the development of longtudnal tran smulators for both engneerng analyss and drver tranng. More recent research nto longtudnal tran dynamcs was started n the early 1990s, motvated not ths tme by equpment falures and fatgue damage, but deralments. It stands to reason that as trans get longer and heaver, n-tran forces get larger. Wth larger n-tran forces, lateral and vertcal components of these forces resultng from coupler angles on horzontal and vertcal curves are also larger; at some pont these components wll adversely affect wagon stablty. The frst known work publshed addressng ths ssue was that of (El-Sabe, 1993) whch looked at the relatonshp between lateral coupler force components and wheel unloadng. Further modes of nteracton were reported and smulated (McClanachan et al., 1999). Coupler elements that connect tran cars have a sgnfcant effect on longtudnal tran forces and can sgnfcantly nfluence the stablty, deralments, and accdents of ralroad vehcle systems. It s known that longtudnal tran forces durng brakng, tracton, and curve negotatons heavly depend on the 3

19 desgn of these couplers, ther degrees of freedom, and ther ablty to absorb mpact forces and dsspate energy. Most nvestgatons that are concerned wth the coupler forces are focused on the force/dsplacement nonlneartes wth less attenton gven to the geometrc nonlneartes due the relatve knematc moton between the coupler components (Cole and Sun, 2006; Duncan and Webb, 1989; El-Sbae, 1993; Bentley, 1993). Capturng these coupler geometrc nonlneartes usng dynamc smulatons wll lead to better understandng of the causes of accdents and deralments. More detals about the automatc tran couplers used n North Amerca wll be dscussed n the followng subsecton, as a demonstratng applcaton for the non-nertal system Important Applcaton The automatc coupler s one of the mportant applcatons to employ the use of the concept of non-nertal coordnates. The use of the automatc coupler was made compulsory n North Amerca n 1893 and t became standard n 1917 wth the ntroducton of the knuckle coupler, also called buckeye coupler or Janney coupler (Janney, 1873), that replaced the Mller Hook, whch was never wdely used n freght vehcles to replace the lnk and pn couplng system (Dawson, 1997). The knuckle coupler, shown schematcally n Fg.1, conssts of a knuckle attached to the end of a shank fastened to a housng mechansm on the car. The assembly s desgned to allow for some lateral play n order to avod deralments durng curve negotatons. Cars are automatcally coupled by engagng the open knuckle on one car to a closed knuckle on the other car. A conventonal autocoupler package s shown n Fg. 2 (Cole and Sun, 2006). The energy-absorbng unt of the knuckle coupler s the draft gear/eoc whch provdes stffness, and hydraulc or frcton dampng. Frcton s more common n North Amerca and Australa, partcularly for freght trans. The draft gear employs dry frcton usng frcton wedges, whle the EOC devce ncludes ol-based dashpots 4

20 (dampers) that produce dampng force that depends on the moton of the devce (Cole and Sun, 2006; Dural and Shadmehr, 2003). Fgure 1: Knuckle coupler Passenger cars often use draft gears desgned specfcally for passenger servce; some of the passenger cars use twn cushon draft gears. In addton to the draft gears behnd the couplers, many models use frcton buffers n order to keep the couplers stretched and to allow for free slack (Dawson, 1997). As prevously mentoned, n order to be able to develop accurate longtudnal tran force models, t s mportant to take nto account the relatve moton between the coupler components ncludng the relatve rotatons, the slack acton sldng, and the moton due to the deformaton of the coupler complant components. Fgure 2: Conventonal auto-coupler package 5

21 Usng the concept of non-nertal coordnates the thess focuses on developng a formulaton that captures the geometrc nonlneartes due to the three-dmensonal knematc moton of the car bodes as well as the moton of the coupler components. In order to develop an effcent formulaton and avod ncreasng the number of coordnates and dfferental equatons of moton, the effect of the nerta of the coupler s neglected. The use of ths assumpton s also necessary n order to avod havng a stff system of dfferental equatons due to the relatvely small coupler mass. Usng ths assumpton, one can dentfy two dstnct sets of coordnates; nertal and non-nertal coordnates. Inertal coordnates, whch descrbe three-dmensonal arbtrary dsplacements of the car bodes, has nerta forces and coeffcents assocated wth them. On the other hand, no nerta forces are assocated wth the non-nertal coordnates, whch are used to descrbe the coupler knematcs. Usng the prncple of vrtual work, one obtans a coupled system of dfferental/algebrac equatons that must be solved smultaneously for the nertal and nonnertal coordnates. The dfferental equatons govern the moton of the car bodes, whle the resultng algebrac force equatons are the result of the quas-statc equlbrum of the coupler. The car body dfferental equatons, whch are expressed n terms of the nertal and non-nertal coordnates, can be ntegrated forward n tme to determne the nertal coordnates and veloctes. Knowng the nertal coordnates and veloctes, the algebrac coupler force equatons can be solved for the non-nertal coordnates usng an teratve Newton-Raphson algorthm. It s also mportant to pont out that the parttonng of the coordnates as nertal and non-nertal leads to a formulaton smlar to the one used n the case of non-generalzed coordnates (Shabana and Sany, 2001), these non-generalzed coordnates can nclude surface parameters used n the nonconformal contact constrants between rgd bodes. These surface parameters, however, are not the same as the non-nertal coordnates used n ths thess. Surface parameters are treated as non- 6

22 generalzed coordnates because they do not have any types of forces assocated wth them, and they do not lead to an ncrease n the number of degrees of freedom; ths s despte the fact that an analytcal approach smlar to the one adopted n ths s used to obtan the equatons of moton n the case of non-generalzed coordnates. The non-nerta coordnates have forces assocated wth them and are determned by the quas-statc force analyss whch s not employed n the case of non-conformal contact formulaton. 1.4 MBS Dynamcs and Longtudnal Tran Dynamcs As prevously mentoned, accurate predcton of coupler forces s necessary n the analyss of tran longtudnal dynamcs and stablty. In order to acheve hgher degree of computatonal effcency, longtudnal tran dynamcs (LTD) algorthms tend to be smpler as compared to the more general MBSalgorthms whch requre the soluton of a system of dfferental/algebrac equatons (DAE s). The algebrac equatons that relate redundant coordnates n MBS algorthms must be satsfed at the poston, velocty, and acceleraton levels. The use of redundant coordnates n MBS algorthms leads to a sparse matrx structure that can be exploted n order to effcently solve the dynamc equatons of moton. Most LTD algorthms, on the other hand, do not employ a DAE s solver, and they use ndependent coordnates nstead of redundant coordnates. The equatons of moton are expressed n terms of the system degrees of freedom by systematcally elmnatng dependent varables. Nonetheless, the mplementaton of effcent force elements may necesstate ntroducng algebrac equatons n LTD algorthms as wll be dscussed n ths thess. Whle n many nvestgatons on tran longtudnal dynamcs, the couplers are represented by massless dscrete sprng-damper elements, MBS algorthms allow for the development of detaled models that capture the coupler geometrc nonlneartes resultng from the relatve rotatons of the 7

23 coupler components. In order to develop such detaled MBS coupler and tran models, most MBS algorthms take nto account the effect of the coupler nerta. However, because couplers have hgh stffness and small nerta n comparson wth the tran car nerta, ncludng the effect of the coupler nerta n the dynamc model can lead to hgh frequency oscllatons that can adversely affect the computatonal effcency. One example of a coupler that has such a hgh stffness couplng element s the knuckle-type automatc coupler, whch connects frmly when one car s bumped aganst another. As mentoned before, n most computer formulatons, the coupler s modeled as a sprngdamper element wth no knematc degrees of freedom (Sanborn et al., 2007). The force n ths sprng-damper element s functon of the relatve dsplacements between the two cars connected by ths coupler. Ths smplfed approach does not take nto account the effect of the coupler degrees of freedom and fals to capture the coupler geometrc nonlneartes that can nfluence the car moton. Developng a more accurate coupler model requres ncludng coupler knematc degrees of freedom that account for the geometrc nonlneartes resultng from the moton of the coupler components. For such a detaled coupler model, two fundamentally dfferent approaches that requre the use of two dfferent soluton procedures can be used. In the frst approach, the nerta of the coupler s taken nto account. In such a model, called nertal coupler model, the algebrac equatons, f they are present n the model, descrbe connectvty constrant equatons whch must be satsfed at the poston, velocty, and acceleraton levels snce the coupler acceleratons appear n the equatons of moton. The coupler degrees of freedom have generalzed nerta forces assocated wth them; and therefore, ther second dervatves appear n the system equatons of moton. Ths leads to an ncrease n the number of the system state equatons that must be ntegrated numercally. 8

24 The second approach, on the other hand, the nerta of the coupler, assumed small compared to the car body nerta, s neglected. In ths model, called the non-nertal coupler model, no generalzed nerta forces are assocated wth the coupler degrees of freedom, and as a consequence, the second dervatves of these coordnates do not appear n the fnal form of the equatons of moton. The use of such non-nertal coupler model can lead to sgnfcant reducton n the number of state equatons and can also contrbute to elmnatng the hgh frequency oscllatons that mght result from the hgh coupler stffness and ts relatvely small nerta. A set of quas-statc coupler equlbrum condtons can be developed and used to defne a set of nonlnear algebrac equatons that can be solved teratvely for the coupler non-nertal coordnates. These coupler non-nertal coordnates enter nto the formulaton of the equatons of moton of the tran cars (Shabana et al., 2010), as wll be dscussed n the thess. In order to examne the effect of neglectng the coupler nerta forces, a comparatve study between the two approaches wll be presented n the thess. 1.5 Implct Integraton for Stff Dfferental /Algebrac Equatons Another approach that can be used to mprove computatonal effcency s the mplct ntegraton method. The numercal soluton of constraned MBS problems requres the numercal ntegraton of dfferental algebrac equatons (DAE). Potra (1994) and Negrut (1998) dscussed many technques for the soluton of DAE of MBS dynamcs. Large and complex multbody systems that nclude flexble bodes and contact/mpact lead to stff problems. Because explct ntegraton methods can be very neffcent and often fal n the case of stff problems, the use of mplct numercal ntegraton methods s recommended. A good understandng of the process of convertng the DAE system to a second order ordnary dfferental equatons (ODE), along wth 9

25 the ablty to effcently generate the needed dervatve nformaton for mplct ntegraton, allowed developng robust mplct numercal methods (Negrut et al., 2007; Pogorelov, 1998). Nonetheless, exstng mplct numercal ntegraton methods used for the soluton of MBS applcatons have several drawbacks. Frst, most of these methods do not ensure that the nonlnear algebrac constrant equatons are satsfed at all levels (poston, velocty, and acceleraton). Second, exstng mplct methods requre the use of numercal dfferentaton of the force vectors n order to solve a nonlnear system of algebrac equatons usng a Newton Raphson algorthm; analytcal dfferentaton cannot n general be used wth general MBS algorthms that allow for the use of tabulated and Splne data to defne the forces and constrants. Thrd, many of the exstng mplct MBS ntegraton methods are not suted for an effcent sparse matrx mplementaton because they are based on recursve formulatons that lead to dense matrces. In order to address these lmtatons, the two-loop mplct sparse matrx numercal ntegraton (TLISMNI) method was proposed (Shabana and Hussen, 2009). The TLISMNI method ensures that the constrant equatons are satsfed at the poston, velocty, and acceleraton levels, does not requre the use of numercal or analytcal dfferentaton of the forces, and allows for effcent sparse matrx mplementaton. In the case of constraned multbody systems, the algebrac knematc constrant equaton, whch descrbe mechancal jonts and specfed moton trajectores that restrct relatve and absolute moton of MBS components, must be consdered. Dfferent technques have been proposed n the lterature for the numercal ntegraton of DAE systems. These technques nclude the drect ntegraton method, the generalzed coordnates parttonng method, and the constrant stablzaton method (Haug, 1989; Negrut, 1998). The drect ntegraton method employs a numercal ntegraton method of ordnary dfferental equatons to ntegrate the dfferental 10

26 algebrac equatons wthout any modfcaton of the ntegraton algorthm or dynamc equaton (Negrut, 1998). Ths ntegraton method, whch ntegrates generalzed coordnates wthout consderng ther dependency, s smple, easy to mplement, and computatonally fast, but t suffers from a lack of error control on the constrants and may lead to erroneous results (Shabana, 2010). The generalzed coordnates parttonng method was proposed by Wehage and Haug (1982). In ths approach, generalzed coordnates are parttoned nto dependent and ndependent coordnates, and only the ndependent acceleratons are ntegrated to defne the state of the system at the next tme step. Dependent coordnates are obtaned by solvng the nonlnear algebrac constrant equatons usng a Newton-Raphson algorthm. The advantage of ths method s that t ensures that the constrant equatons are satsfed to wthn a user specfed tolerance. For a good predcton of the dependent generalzed coordnates can mprove the effcency of the method snce such a predcton can satsfy the constrants wthout the Newton-Raphson teratons. Numercal expermentaton has shown that, wth a good predcton of the dependent varables, Newton- Raphson teratons are needed only for few tme steps durng the dynamc smulaton. Baumgarte (1972) proposed a dfferent approach based on a constrant stablzaton method. In ths approach, the constrant equatons and ther dervatves were added to the second dervatve of the constrant equatons. Wth ths modfcaton, the constrant stablzaton ntegraton algorthm requres drect ntegraton of all the acceleratons. Ths approach can be more stable, effcent and accurate as compared to the elementary drect ntegraton algorthm. The dsadvantage of ths method, however, s that there s no general and unformly accepted method for selectng the coeffcents of the constrant equatons and ther dervatves that are added the second dervatves of the constrant equatons. Improper selecton of these coeffcents can lead to wrong solutons and make the method less robust. 11

27 As prevously mentoned, the mplct ntegraton methods can be more effcent than explct ntegraton methods n solvng stff DAE systems (Petzold, 1982). Implct methods requre the soluton of a system of nonlnear algebrac equatons at each tme step. Newton s method or modfed Newton s method appears to be the best general approach to be used wth mplct methods (Negrut, 1998). For large problems, most of the calculatons requred for the mplct ntegraton are assocated wth the computaton of the Jacoban and the lnear algebra operatons. Ths s n addton to the need of a relatvely large core memory for the storage of the coeffcent matrx and ts decomposton. Gear and Saad (1983) proposed the use of Krylovsubspace projecton method known as the ncomplete orthogonalzaton method (IOM) and Arnold s algorthm whch are teratve methods for the soluton of lnear systems (Saad, 2003). Arnlod s algorthm and IOM do not requred the soluton of the coeffcent matrx n any form and hence requre less storage than the drect methods. Brown and Hndmarsh (1986) referred to the combned stff ODE method and Krylov method as a matrx-free method and dscussed the theoretcal and computatonal aspects of the combned algorthm. Brown and Hndmarsh (1986) vewed the combned Newton-IOM and Newton-Arnold s method as an nexact newton method, whch belongs to a class of methods n whch the lnear system of Newton teraton s solved approxmately. The mplementatons of the Arnold s algorthm and IOM for solvng lnear system of equatons are dscussed by Saad (2003). 1.6 Scope and Organzaton of the Thess Ths thess s organzed n fve chapters ncludng ths ntroductory chapter. In ths secton, the organzaton and scope of the thess as well as a summary of the contents and contrbutons of ts chapters are presented. 12

28 Chapter 2 In ths chapter, the concept of the non-nertal coordnates s ntroduced and used to develop a new three-dmensonal non-lnear tran car coupler model that takes nto account the geometrc nonlneartes due to the coupler and car body dsplacements. These geometrc nonlneartes that cannot be captured usng the exstng smpler models (Shabana et al., 2011). The use of ths concept leads to sgnfcant reducton n the numbers of state equatons, system nertal coordnates, and constrant equatons; and allows avodng a system of stff dfferental equatons that can arse because of the relatvely small coupler mass. The use of the concept of the non-nertal coordnates and the resultng dfferental/algebrac equatons obtaned n ths study s demonstrated usng the knuckle coupler wdely used n North Amerca. Chapter 3 The objectve of ths chapter s to verfy the non-nertal coordnate models and prove ther effcency. Ths s accomplshed by examnng the effect of the coupler nerta on the dynamcs and computatonal effcency of tran models (Massa et al., 2011). To ths end, the results obtaned usng the two models, nertal and non-nertal coupler models, are compared n order to examne the assumpton of neglectng the coupler nerta. In the case of the non-nertal coupler model, the quas-statc coupler condton used n ths thess do not requre havng a velocty dependent terms as compared to methods prevously publshed n the lterature (Arnold et al., 2010). Only algebrac equatons are requred n order to be able to use the procedure employed n ths thess. Nonetheless, velocty dependent forces can stll be ncluded n the non-nertal coupler formulaton used n ths thess. A frequency doman analyss s also performed n order to dentfy the frequences n the soluton assocated wth the coupler nerta. Chapter 4 Ths chapter dscusses TLISMNI mplementaton ssues ncludng the step sze selecton, the error control, and the effect of the numercal dampng and proposes a new effcent and robust TLISMNI-based soluton procedure for constraned MBS equatons. Furthermore, the 13

29 chapter examnes the use of the generalzed coordnate parttonng and the recursve methods n the TLISMNI framework soluton, n order to satsfy the constrant equatons at the poston and velocty levels. In both cases, the constrants are also satsfed at the acceleraton level. The use of the computer mplementaton descrbed n ths Chapter s demonstrated by solvng complex rgd and flexble body tracked vehcle models, ral road models, and very stff structure problems. Several second order formulas such as Hlber Hughes Taylor (HHT) method, whch ncludes numercal dampng,l- stable Park formula, A-stable Trapezodal formula, and A-stable BDF formula are used n ths nvestgaton as the ntegraton formulas, and recommendatons are made wth regard to the approprateness of each of these ntegraton formula for partcular MBS applcaton. The effcent ntegraton of Krylov subspace projecton method wth these ntegraton formulas wthn the TLISMNI framework soluton procedure s one of the man contrbutons of ths chapter. The results, obtaned usng these ntegraton methods, are compared wth the results obtaned usng the explct Adams predctor-corrector method. Usng the TLISMNI method, whch does not requre numercal dfferentaton of the forces, allows for an effcent sparse matrx mplementaton for solvng complex and very stff structure problems, and ensures that the constrant at the poston, velocty, and acceleraton levels, sgnfcantly mproves the computatonal effcency as demonstrated n ths Chapter usng dfferent examples. The thess s ended by a summary and concluson presented n Chapter 5. Some more detals are gven n the thess appendces. 14

30 CHAPTER 2 NON-INERTIAL COORDINATES In ths chapter, the concept of the non-nertal coordnates s ntroduced and used to develop a new three-dmensonal non-lnear tran car coupler model that takes nto account the geometrc nonlnearty due to the coupler and car body dsplacements. By assumng the nerta of the coupler components neglgble compared to the nerta of the car body, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates. The nertal coordnates that descrbe the car moton have nerta forces assocated wth them. The non-nertal coupler coordnates, on the other hand, descrbe the coupler knematcs and have no nerta forces assocated wth them. Vrtual work s used to obtan expressons for the generalzed forces assocated wth the car body and coupler coordnates. The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons, whch depend on the coupler non-nertal coordnates, govern the moton of the tran cars; whle the algebrac force equatons are the result of the quas-statc equlbrum condtons of the massless coupler components. Ths approach leads to sgnfcant reducton n the numbers of state equatons, system nertal coordnates, and constrant equatons; and allows avodng a system of stff dfferental equatons that can arse because of the relatvely small coupler mass. The proposed non-lnear coupler model allows for arbtrary three-dmensonal moton of the car bodes and captures knematc degrees of freedom that are not captured usng exstng smpler models. The coupler knematc equatons are expressed n terms of the car body coordnates as well as the relatve coordnates of the coupler wth respect to the car body. Gven the nertal coordnates and veloctes, the quas-statc coupler algebrac 15

31 force equatons are solved teratvely for the non-nertal coordnates usng a Newton-Raphson algorthm. The use of the concept of the non-nertal coordnates and the resultng dfferental/algebrac equatons obtaned n ths study s demonstrated usng the knuckle coupler wdely used n North Amerca. Numercal results of smple tran models are presented n order to demonstrate the use of the formulaton developed n ths chapter. 2.1 Inertal and Non-nertal Coordnates The concept of the nertal and non-nertal coordnates s ntroduced n ths secton. Inertal coordnates have generalzed nerta forces assocated wth them, whle the non-nertal coordnates have no generalzed nerta forces. In order to avod havng a sngular nerta matrx and/or hgh frequency oscllatons, the second dervatves of the non-nertal coordnates are not used when formulatng the system equatons of moton n ths study. In ths case, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates, leadng to a formulaton smlar to the one used n the case of non-generalzed coordnates (Shabana and Sany, 2001). The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons are used to determne the nertal acceleratons whch can be ntegrated to determne the nertal coordnates and veloctes. The non-nertal coordnates are determned by usng an teratve Newton-Raphson algorthm to solve a set of nonlnear algebrac force equatons obtaned usng quas-statc equlbrum condtons. The non-nertal veloctes are determned by solvng these algebrac force equatons at the velocty level. The non-nertal coordnates and veloctes enter nto the formulaton of the generalzed forces assocated wth the nertal coordnates. 16

32 2.2 Dfferental and Algebrac Equatons The concept of non-nertal coordnates can be ntroduced usng the smple two-mass system shown n Fg. 3. The two masses m and j m are connected by a sprng that has stffness coeffcent k. Ths smple system, n whch the effect of frcton s neglected, has two degrees of freedom x and x j. If the two masses are subjected to external forces F F x, x j, t and j j j F F x, x, t whch can be lnear or nonlnear, the two equatons of moton that govern the moton of ths system are mx F k x j x, and mx j j F j k x j x. If the mass j m s assumed to be neglgble, one can set the nerta force n the second equaton equal to zero leadng j j to the followng algebrac equaton: F k x x 0 j. Gven the external force F, the j j deformaton of the sprng can be determned as x x F k ths deformaton can be substtuted nto the equaton of the frst mass n order to determne determne the coordnate and velocty of the mass m. x whch can be ntegrated to Fgure 3: Concept of the non-nertal coordnates For the smple example shown n Fg. 3, one can wrte the coordnate of the mass j m n terms of the coordnate of the mass m as j j x x d, where j d defnes the locaton of mass j m wth 17

33 j respect to mass m. Usng the defntons of d, the two equatons of moton of the system can be wrtten as mx F kd j, m j x d j F j kd j (2.1) One can also use the vrtual work prncple whch states that the vrtual work of the system nerta forces W s equal to the vrtual work of the system appled forces We, that s, W We. Usng ths prncple, one obtans m x x m j x d j x d j F kd j x F j kd j x d j (2.2) Groupng the coeffcents of x and d j, and assumng that the coordnates x and j d are ndependent, one obtans the followng alternate form of the equatons of moton:, mx m j x d j F F j m j x d j F j kd j (2.3) Note the frst equaton n Eq. 2.3 does not explctly nclude the sprng force. Nonetheless, the two systems of Eqs. 2.1 and 2.3 are equvalent snce the frst equaton n Eq. 2.3 can be smply obtaned by addng the two equatons of moton of Eq If the nerta of the mass j m s neglected, the second equaton of Eq. 2.3 leads to F j kd j. Assumng that the coordnate x s known, the equaton F j j j kd represents a lnear or nonlnear algebrac equaton that can be solved for d. Substtutng ths result nto the frst equaton of Eq. 2.3 wth the assumpton of neglgble mass m, one obtans the equaton of the frst mass as mx F kd j from the vrtual work of Eq. 2.2 by neglectng the nerta of the mass. Ths result can also be obtaned j m, leadng to mx x F kd j x F j kd j x j (2.4) j Ths vrtual work equaton leads to the same dfferental equaton j mx F kd and to the same algebrac equaton F j j j kd prevously obtaned by neglectng the nerta of m. 18

34 In summary, the quas statc equlbrum of for the non-nertal relatve coordnate known. Once the non-nertal coordnate j m leads to an algebrac equaton that can be solved j d wth the assumpton that the nertal coordnate x s j d s determned, t can be substtuted nto the equaton of moton of m. Ths equaton of moton can be ntegrated to determne x and x. Ths procedure wll be generalzed n ths chapter to study the three-dmensonal moton of coupled tran cars. 2.3 Algebrac Force Equatons The dfferental and algebrac equatons of multbody systems, ncludng ralroad vehcle systems, can be obtaned usng the prncple of vrtual work whch states that the vrtual work of the system nerta forces equal to the vrtual work of the system appled forces. That s, W We, where W s the vrtual work of the system nerta forces, and We s the vrtual work of the appled forces. As prevously mentoned, ncludng the relatvely small nerta of the coupler leads to a stff system of dfferental equatons and can sgnfcantly ncrease the problem dmenson by ncreasng the number of coordnates and the number of dfferental equatons. For ths reason, the nerta of the coupler components s neglected. If the nerta of the components of the couplers s neglected, the vrtual work of the nerta and appled forces can be wrtten, respectvely, as W Q T q, W Q T qq T q (2.5) e e n n In ths equaton, q s the vector of the system nertal coordnates (coordnates whch have nerta assocated wth them), q n s the vector of the system non-nertal coordnates (coordnates whch have zero nerta assocated wth them), Q s the vector of generalzed nerta forces, vector of generalzed appled forces assocated wth the nertal coordnates, and Q e s the Q n s the vector 19

35 of generalzed appled forces assocated wth the non-nertal coordnates. Usng the prncple of vrtual work, W We and Eq. 2.5, one has T T Q q Q q 0 (2.6) e n n In ths equatonq e Q Q e. Assume that the system s subjected to knematc constrants that descrbe trajectory and jont constrants defned by the equaton Cqq,, t 0 (2.7) n A vrtual change n the coordnates leads to, C q C q 0 (2.8) q qn n Multplyng ths equaton by Lagrange multplers λ and addng the results to Eq. 2.6, one obtans T T T T e q n q n 0 Q C λ q Q C λ q (2.9) n Snce Lagrange multplers are used to account for the constrant forces, all the coordnates can be treated as ndependent. Usng the generalzed coordnate parttonng as descrbed n the lterature (Shabana, 2010), the precedng equaton leads to T T Q C λ 0, Q C λ 0 (2.10) e q n qn LetQ Mq, where M the system mass matrx s; ths s wth the understandng that the gyroscopc forces are ncluded nq. Dfferentatng Eq. 2.7 twce wth respect to tme, one obtans the constrant equatons at the acceleraton level, e Cq C q Q (2.11) q qn n d 20

36 where Q d s the quadratc velocty vector that arses from the dfferentaton of the knematc constrant equatons twce wth respect to tme. By combnng Eqs and 2.11, one has M 0 C q Q (2.12) T q e T 0 0 Cq n qn Qn Cq Cq 0 d n λ Q In the specal case n whch the non-nertal coordnates are not subjected to constrants, the precedng equaton leads to T M C q q Qe, Qn q, qn, q, q n, t 0 (2.13) Cq 0 λ Qd Furthermore, f an embeddng technque s used to elmnate the dependent nertal coordnates such that all the components of the vector q are ndependent, the precedng equatons reduce to Mq Qe, Qn qq, n, qq, n, t 0 (2.14) In ths chapter, the system of Eq s consdered for three-dmensonal moton of coupled tran cars. The numercal soluton procedure used to solve ths system to determne the nertal and nonertal coordnates s dscussed n a followng secton. 2.4 Coupler Knematcs As shown n Fg.4, the coupler model developed n ths nvestgaton s assumed to consst of several components that can experence relatve dsplacements wth respect to each others. Therefore, t s mportant to develop the knematc equatons of ponts that are fxed on a rgd 21

37 body as well as ponts that can move wth respect to a rgd body. These knematc equatons wll be used to capture the dsplacements of the coupler EOC/draft gear and shank. Fgure 4: Two-car system Knematcs of Fxed Ponts The global poston vector of an arbtrary pont P fxed on the car body can be wrtten as r R A u (2.15) P Po where R s the global poston vector of the orgn of the car body coordnate system, orthogonal transformaton matrx that defnes the orentaton of the car body, and 22 A s the u Po s the local poston vector of the arbtrary pont wth respect to the orgn of the car body coordnate system (Shabana, 2010). The transformaton matrx parametersθ. The vrtual change n the poston vector of Eq s A can be expressed n terms of a set of orentaton

38 r R A u Gθ L q (2.16) P Po P Where u Po s the skew symmetrc matrx assocated wth the vector u Po, G s the matrx that relates the angular velocty vector ω defned n the car body coordnate system to the tme dervatves of the orentaton parameters generalzed coordnates of the car body, and θ, that s, ω G θ, T T T q R θ s the vector of L P I A u PoG (2.17) The absolute velocty vector of the fxed pont P can be wrtten as r L q (2.18) P P The knematcs of a fxed pont on the car body wll be used to descrbe the knematcs of the pont on the car body that s connected to the draft gear or the EOC devce, as shown n Fg Knematcs of Sldng Blocks In the case of a block B that sldes wth respect to the car body n a certan drecton, as shown n Fg. 4, the global poston vector of the center of the block s defned as In ths equaton, T ˆ ˆ r R A u u R R d d (2.19) B Po d o B B u Po s the local poston vector of pont P, u ˆ d lo d B along whch the block s sldng, l o s the ntal dstance between pont block B, dˆ A d ˆ, B B d, d ˆ B s a unt vector P and the center of the R o s the global poston of the orgn of the car body coordnate system before dsplacement, and d s the dstance that measures the locaton of the block along d ˆ B. Because T d n Eq s defned as ˆ d d d R R o d B, where d d s the relatve 23

39 dsplacement of the sldng block wth respect to the car body, the last term n Eq s ntroduced to elmnate the effect of the translaton of the car body n the drecton of d ˆ B on the dsplacement of the slder block. Note that f famlar equaton l d d s used nstead of d, Eq reduces to the d ˆ r R A u d. The motvaton for usng the form of Eq B Po o d B s to allow for the calculaton of the relatve poston of the slder wth respect to the car body. Snce no dynamc equatons of moton are assocated wth the non-nertal coordnates, the value of d from the prevous tme step wll be stored and used to determne the current value of the sprng deformaton d. That s, at step k 1, the deformaton of the sprng that enters nto the sprng d force formulaton s determned usng the equatond d T d k1 k o B R R d ˆ. Usng ths equaton to defne the draft gear/eoc force n the teratve procedure ntroduced n ths Chapter allows for determnng the current value of k 1 d d that can be used n the calculaton of the forces T assocated wth the nertal coordnates. The relatonshp ˆ dfferentated n order to determne d d R R d can also be d o B d d that enters nto the formulaton of the dampng force. The use of the nformaton from the prevous tme step s necessary snce non-nertal coordnates are treated as ndependent varables; and the non-nertal acceleratons are not consdered n the coupler quas-statc formulaton developed n ths chapter, and therefore, no non-nertal acceleratons are ntegrated to determne the ndependent non-nertal coordnates and veloctes. The last term of Eq can also be wrtten usng the outer product notatons as ˆ ˆ B B o d d R R. It follows that the vrtual change n ths term can be wrtten as 2 ˆ ˆ T ˆ ˆ d d R R R d d. Ths leads to B B o B B 24

40 T ˆ 2 dˆ dˆ R R dˆ dˆ R R R dˆ Ad Gθ B B o B B o B B (2.20) The vrtual dsplacement of the poston vector of Eq can then be wrtten as r R A u u Gθ Ad ˆ d L q B Po d B Rd L L q L d B Rd Bd (2.21) where, ˆ LB, I A u Po u d G LBd A db T ˆ ˆ ˆ 2 ˆ (2.22) LRd db db R Ro db A dbg The absolute velocty vector of the sldng block can be wrtten as d B Rd Bd r L L q L (2.23) The knematc equatons of the slder block wll be used to defne the draft gear and EOC knematcs Translaton and Rotaton The coupler shank, shown n Fg. 4, experences, wth respect to the car body, base moton defned by the sldng d d as well as rotaton defned by the angle. The locaton of the end pont Q of the rod wth respect to the orgn of the car body coordnate system s defned by the vector u u u u, where as shown n Fg. 4, Q Po d r u r s a vector defned along the rotatng arm whch s assumed to have length l r. It follows that cos sn 0l r cos u r sn cos 00 lr sn (2.24)

41 In ths equaton, the transformaton matrx, denoted as A r, that depends on the angle defnes the orentaton of the rod wth respect to the car body coordnate system. One can then wrte the global poston vector of pont Q shown n Fg. 4 as T ˆ ˆ r R Au R R d d (2.25) Q Q o B B The vrtual change n ths poston vector can be wrtten as where In ths equaton, l sn cos 0 be wrtten as u r r r L L q L d L (2.26) Q Q Rd Bd r L I Au G L A u (2.27), Q Q r r Q Q Rd Bd r T. The absolute velocty vector of pont Q can r L L q L d L (2.28) The knematc equaton of the rod developed n ths secton wll be used to descrbe the knematcs of the coupler shank. 2.5 Coupler Generalzed Forces In ths secton, the vrtual work s used to determne the generalzed forces assocated wth the car body nertal and the coupler non-nertal coordnates. These generalzed force expressons wll be used to defne a set of nonlnear algebrac equatons that can be solved for the coupler nonnertal coordnates usng a Newton-Raphson algorthm. The coupler slack acton can always be accounted for by usng approprate defntons of the stffness and dampng coeffcents that appear n the force expressons developed n ths secton. 26

42 2.5.1 Draft Gear/EOC Forces The force of the sprng between the car body and ts coupler slder component that represent the EOC or the draft gear s denoted as as j f B. The forces f B and j f B can be wrtten as f B. Smlarly the force assocated wth car body j s denoted kd cd ˆ, kd cd j j j j j ˆ j f d f d (2.29) B B d B d B B B d B d B In ths equaton, k B and j k B, and c B and j c B are the stffness and dampng coeffcents assocated wth the draft gear or EOC of car bodes and j, respectvely. Note that the sprng deformatons d d and j d d can be expressed n terms of d and j d as dscussed n Secton 4. Slack actons or free travel can be accounted for by usng zero a stffness coeffcent n the precedng equaton. The vrtual work of the forces gven n the precedng equaton can be wrtten as f r f r f r f r (2.30) j T T jt j jt j W B B P B B B P B B Usng the expressons of the vrtual dsplacements, the precedng equaton leads to j T j T j j T j B B B B n n W Q q Q q Q q (2.31) In ths equaton, j q n s the vector of the coupler non-nertal coordnates defned as T j j j q n dd dd, (2.32) and 27

43 T T 0 0 T j j j j j QB LP LB LRd fb, QB LP LB LRd fb, j T j T j Q B n LBd fb LBd fb (2.33) Note that from the defntons of L Bd and j Bd j j j j j B kd B d cd B d kd B d cd B d j L, the force vector B Q reduces to Q 0 0 T n. Draft gear frcton force can be ntroduced to the model presented n ths secton usng the normal force actng on the draft gear and the wedge angle. Furthermore, nonlnear stffness and dampng coeffcents obtaned usng measurements can also be ncluded n the draft gear/eoc force model usng a Splne functon representaton (Sanborn et al., 2007). n Shank Forces The vrtual work of the force of the sprng that connects the two coupler s denoted as force vector can be wrtten as f j f rr. Ths k l l c l d ˆ (2.34) j j j j j j j rr rr r ro rr r rr In ths equaton, k and c j are the sprng stffness and dampng coeffcents, l j s the current j rr rr r sprng length, l j ro s the un-deformed length of the sprng, and d ˆ j s a unt vector along the lne connectng ponts can be wrtten as Q and j Q, that s, ˆ j j j rr Q Q Q Q rr d r r r r. The vrtual work of the force f j rr f r f r (2.35) j jt jt j W rr rr Q rr Q Usng the knematc equatons developed n Secton 4, the precedng equaton can be wrtten as 28

44 j T j T j j T j rr rr rr rr n n W Q q Q q Q q (2.36) In ths equaton, T T j j j j j Qrr LQ LRd frr, Qrr LQ LRd frr, j T j T j j T j j T j T Qrr n LBd frr Lr frr LBd frr Lr frr (2.37) As n the case of the draft gear/eoc forces, nonlnear stffness and dampng coeffcents obtaned usng measurements can be used wth the shank force model developed n ths secton usng a Splne functon representaton (Sanborn et al., 2007) Shank and Draft Gear/EOC Attachment In order to account for the stffness at the jont between the coupler shank and the draft gear/eoc of each car, a torsonal sprng stffness k r and dampng c can be ntroduced at ths jont. Let d ˆ and d ˆ j r be a unt vector along the shank and draft gear/eoc jont axs of car bodes and j, respectvely. It follows that the generalzed forces assocated wth the non-nertal coordnates j and can be wrtten, respectvely, as, j j j j j j r r o r r r o r r Q k c Q k c (2.38) r In ths equaton, and o j o defne the ntal angles. The precedng equaton leads to the followng generalzed force vector assocated wth the non-nertal coupler coordnates: 0 Q 0 j r n r r T Q Q (2.39) 29

45 Splne functon representaton can also be used to defne nonlnear torque/angle relatonshp, as prevously mentoned Inertal and Non-Inertal Coordnate Generalzed Forces Usng the expressons of the forces developed n ths secton, the generalzed coupler forces assocated wth the nertal coordnates of the car bodes and j can be wrtten as (see Eqs. 2.33, 2.37, and 2.39) T T T j Qc QB Qrr LP LB LRd fb LQ LRd frr, j j j j j j j j j T j Q c QB Qrr LP LB LRd fb LQ LRd frr (2.40) The forces assocated wth the non-nertal coupler coordnates are T j LBd fb f rr T j Lr frr Qr j T j j LBd fb frr jt j j Lr frr Qr Qc QB Qrr Qr n n n n (2.41) The generalzed forces assocated wth the car body nertal coordnates of Eq can be ntroduced to the dynamc equatons of moton, whle the four-dmensonal force vector assocated wth the coupler non-nertal coordnates can be used to solve for the non-nertal coordnates d,, d, and j d d j usng an teratve procedure as wll be descrbed n the followng secton. Note also that n Eq. 2.40, the matrx LP L B wll contrbute only moment actng on the car body. A smlar comment apples to the car body j. 30

46 2.6 Soluton Algorthm In ths secton, the algorthm used to solve the system equatons of Eq s dscussed. Recall that the assumpton of quas-statc non-nertal system s consdered n ths study n order to avod havng a stff system of dfferental equatons of moton and also to avod a sgnfcant ncrease n the problem dmensonalty. Usng the quas-statc equlbrum assumpton the terms that depend on the nertal and non-nertal veloctes n the second equaton of Eq are neglected. Therefore, knowng the nertal coordnates; the second equaton of Eq can be consdered as a system of nonlnear algebrac equatons n the non-nertal coordnates. The force vector Q e, on the other hand, can be functon of the nertal and non-nertal coordnates and veloctes. By assumng that the nertal veloctes are know from the numercal ntegraton of the equatons of moton, one can obtan the non-nertal veloctes by consderng the second equaton of Eq at the velocty level. The steps of the numercal algorthm used n ths chapter to solve Eq can be summarzed as follows: 1. For a gven set of ntal condtons q 0 and q 0 assocated wth the nertal coordnates, the second equaton of Eq. 2.14,,,,, Qn q qn q qn t 0, can be consdered as a nonlnear system of algebrac equatons n the non-nertal coordnates q n. If ths system depends on the veloctes, that s,,,, Qn q qn q qn t 0, the effect of the nertal and non-nertal veloctes s neglected, thereby defnng the system,, Qn q qn t Gven the nertal coordnates q and the nertal veloctes q, the system,, Qn q qn t 0 can be solved for q n usng a Newton-Raphson algorthm that teratvely solves the followng system of equatons: 31

47 Q n q n Q n q, q n, t q n (2.42) In ths equaton, norm of 3. If the vector e q n s the vector of Newton dfferences. Convergence s acheved f the q n or the norm of Q n s smaller than a specfed tolerance. Q s functon of the non-nertal veloctes, that s, Q Q qq,, qq,,. e e n n t An approxmate soluton for the non-nertal veloctes can be obtaned by solvng the second equaton of Eq at the velocty level. In ths case, the followng smplfed system,, Qn q qn t 0 s dfferentated, defnng the followng lnear system n the nonnertal veloctes: Q q Q q Q (2.43) q n n n n n q t The soluton of ths system can be used to determne the non-nertal veloctes q n. In obtanng Eq. 2.43, an assumpton s made that the nertal coordnates and veloctes are assumed to be known. The dervatves requred for the evaluaton of the rght hand sde of Eq are presented n Appendx A of ths thess. Note also that f the vector an explct functon of tme, then Qn t 0. Q n s not 4. Knowng the nertal coordnates and veloctes and the non-nertal coordnates and veloctes from the prevous two steps, the frst equaton of Eq. 2.14, Mq Q q, q, q, q,, can be constructed and solved for the nertal acceleratons q. e n n t 5. The nertal acceleratons q can be ntegrated forward n tme usng drect numercal ntegraton method n order to determne the nertal coordnates q and the nertal veloctes q. 32

48 6. The smulaton stops f the end tme s reached; otherwse the procedure s repeated startng wth Step 2. For the presented three-dmensonal coupler model, n order to solve for the non-nertal coordnates, the coeffcent matrx n Eq must be evaluated. The evaluaton of ths matrx requres the followng dervatves: j Lr Lr j,, j 0 ur 0 0 j u r qn q n j j j drr ˆ ˆ j j lr 1 j T drr B r B r, rr, j d L d L d j j j qn qn lr q n j j j j j j ˆ j l drr r rr n rr r n fb, ˆ k B B, j j d q q d q d l q j l 2 r n n j j j f ˆ j B j ˆ j f rr j, ˆ j lr j j d rr k B B rr rr r ro, j j j j 0 0 d 0 k d l l qn qn qn qn j Qr Q r j 0 kr 0 0, k j j r q n qn (2.44) k k k In ths equaton, ur A u r, k, j. Usng the defntons of Eq. 2.44, one can show that the 4 4 matrx assocated wth the coupler j that can be used to construct the coeffcent matrx n Eq s defned as L j T B rr Bd j j qn qn j T rr jt r r Lr f j j rr j j q n n qn qn j j j n j T B rr LBd j j qn qn jt r f f f L Q Q q f f L f L Q q q q j j j rr jt r r f j rr j j n n n (2.45) 33

49 j Ths matrx s non-sngular n the case of non-zero stffness coeffcents k, k, and mportant to pont out, as prevously mentoned n Secton 4, that snce B B d and j k rr. It s also j d are consdered as ndependent non-nertal coordnates, these coordnates are predcted when evaluatng the rght hand sde of Eq and the force vector assocated wth the nertal coordnates usng the equaton k k k T ˆ k ˆ k kˆ k ˆ k u R R d d l d d, k, j. d o B B o B B T As prevously mentoned, the method used n ths chapter allows for arbtrary dsplacements for the car bodes; that s, the car bodes can have up to sx degrees of freedom. The moton of the car bodes can be descrbed usng the trajectory coordnate system (Shabana et al., 2008). The method also allows for developng smpler tran models by systematcally elmnatng car degrees of freedom usng the embeddng technque. Therefore, the formulaton presented n ths chapter dffers from some exstng tran longtudnal dynamcs formulatons that allow only for one longtudnal degree of freedom for each car body. 2.7 Numercal Examples In ths secton, a smple two-car vehcle model, as the one shown n Fg. 4, s used to demonstrate the use of the nonlnear coupler formulaton presented n ths nvestgaton. The trajectory coordnates are used to formulate the dynamc equatons of moton of the vehcle (Sanborn et al, 2007; Shabana et al., 2008). Each car s assumed to have one longtudnal degree of freedom wth the nerta propertes and ntal coordnates presented n Table 1. Two smulaton scenaros, one wth tangent track whle the other wth curved track, are consdered n ths secton. For the frst scenaros, two dfferent couplers, one wth an EOC devce and the other wth a draft 34

50 gear, are consdered. Draft gears employ dry frcton elements whch are desgned to have strong hysteress characterstcs, whle EOC devces employ ol-based dashpots (dampers). Table 1. Car parameters Parameter Car Car j m kg I kg.m 2 xx I kg.m 2 yy I zz kg.m 2 I xy, Ixz, I yz 0 kg.m 2 Intal coordnates X, Y, Z (0, 0, 1.85) m (13.976, 0, 1.85) m Intal angles (roll, yaw, ptch) (0, 0, 0) m Intal veloctes (roll, yaw, ptch) (0, 0, 0) m/s The same symbols are used to defne the coeffcents of the two coupler types. The coupler parameters are shown n Table 2, where k, c, and are, respectvely, the sprng, dampng, and B B frcton coupler coeffcents assocated wth car ; k r and c r are, respectvely, the torsonal stffness and dampng coeffcents of the coupler shank of car ; l 0 s the ntal length of the sprng k B ; j k rr and j c rr are, respectvely, the sprng and dampng coeffcents that descrbe the complant connecton at the coupler heads and the knuckle; and l r s the length of the coupler shank (Dawson, 1997). The data are selected for the Janney coupler and for the SL-76 draft gear. The use of the curved track n the second scenaro allows examnng the coupler geometrc nonlneartes due to the coupler non-nertal knematc degrees of freedom. 35

51 Table 2. Coupler parameters Coupler parameter value Coupler parameter l m j l o k B c B N/m N. s /m r j l o j k rr j 0.25 c rr l m r k r l m c j o j k B j c B j N/m N. s /m r j k r j c r value m m N/m N. s /m Nm/rad N. s m/rad Nm/rad N. s m/rad Tangent Track Model In ths scenaro, car j s consdered as the locomotve that pulls car whch represents a freght car. The ntal forward velocty for each car s assumed to be 0 m/s. Two coupler models are consdered n ths scenaro, one wth an EOC and the second wth draft gear. The smulaton results for ths model are presented n Fgs Fgure 5 shows the forward poston of car, slder blocks and j that represent EOC devces, and car j as the result of the applcaton of constant locomotve force of N. It s clear from the results presented n ths fgure that car follows the moton of the locomotve. In ths case of constant tracton force, the veloctes of the two cars ncrease lnearly. Fgures 6 and 7 show the EOC and shank knukcle sprng deformatons and deformaton rates. When the two cars have the same forward velocty, the sprng deformaton becomes constant because of the constant tractve force and constant acceleraton, whle the relatve velocty becomes zero. For the moton scenaro consdered, the non-nertal j coordnates and reman constant. In the case of usng a draft gear nstead of EOC devces, the overall moton of the two cars are approxmately the same as prevously reported. The 36

52 deformaton of the draft gear sprng, however, s dfferent from the deformaton of the EOC sprng. Draft gears employ Coulomb frcton for energy dsspaton, whle EOC devces use vscous dampng. Fgures 8 and 9 show the deformatons and deformaton rates of the draft gears and shank knukcle Curved Track Model In the example of the tangent track dscussed n ths secton, and j reman constant. In order to examne the geometrcal nonlneartes of the coupler resultng from ts knematc degrees of freedom, d, d, and d j d j ; the vehcle model consdered n ths secton s assumed to negotate the curved track shown n Fg. 10. Ths track s assumed to consst of m tangent segment, m spral segment, 3-degree constant curve of length m, spral segment of length m, tangent segment of length m, spral segment of length 91.44, -3-degree curve of length of 91.44, spral segment of m, and tangent segment of length A 76.2 mm super-elevaton s assumed for the frst two spral segments, and mm for the last two spral segments. In ths example, car j s consdered as the locomotve that pulls car whch represents a freght car. The ntal forward velocty for each car s assumed to be 20 m/s, also the smulaton tme s ncreased to be 40 s n order to capture the nonlnear behavor of the coupler on dfferent track sectons. Fgures 11 and 12 show the EOC sprng deformatons and deformaton rates. Fgures 13 and 14 show the torsonal sprng deformaton and torsonal sprng deformaton rates of the torsonal sprng of car, smlarly Fgs. 15 and 16 show the torsonal sprng deformatons and deformaton rates of the torsonal sprng of car j. 37

53 Forward poston (m) Fgure 5: Forward poston ( car, slder block, slder block j, car j ) Tme (s) Sprng deformaton (m) Tme (s) Fgure 6: EOC and shank knuckle sprng deformatons ( d, d, d ) d j d j d 38

54 Relatve velocty (m/s) Tme (s) Fgure 7: EOC and shank knuckle sprng deformaton rates ( d, d, d ) d j d j d Sprng deformaton (m) Tme (s) Fgure 8: Draft gear and shank knuckle sprng deformatons j j ( d d, d d, d d ) 39

55 Relatve velocty (m/s) Tme (s) Fgure 9: Draft gear and shank knuckle sprng deformaton rates ( d j d, d j d, d d ) Curved regon Global Y coordnate (m) Spral regon Tangent regon Spral regon Tangent regon Curved regon Spral regon Global X coordnate (m) Fgure 10: Curved track geometry 40

56 Sprng deformaton (m) Tme (s) Fgure 11: EOC sprng deformaton j ( d d, d d ) Relatve velocty (m/s) Tme (s) Fgure 12: EOC sprng deformaton rates ( d j d, d d ) 41

57 (rad) Tme (s) Fgure 13: Torsonal sprng deformaton of car a (rad/s) Tme (s) Fgure 14: Torsonal sprng deformaton rate of car 42

58 j (rad) Tme (s) Fgure 15: Torsonal sprng deformaton of car j a j (rad/s) Tme (s) Fgure 16: Torsonal sprng deformaton rate of car j 43

59 2.8 Concludng Remarks The concept of the nertal and non-nertal coordnates s developed n ths chapter. Inertal coordnates have generalzed nerta forces assocated wth them, whle the non-nertal coordnates have no generalzed nerta forces. In order to avod havng a sngular nerta matrx and/or hgh frequency oscllatons, the second dervatves of the non-nertal coordnates are not used when formulatng the system equatons of moton n ths study. In ths case, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates. The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons are used to determne the nertal acceleratons whch can be ntegrated to determne the nertal coordnates and veloctes. The non-nertal coordnates are determned by usng an teratve Newton-Raphson algorthm to solve a set of nonlnear algebrac force equatons obtaned usng quas-statc equlbrum condtons. The non-nertal veloctes are determned by solvng these algebrac force equatons at the velocty level. The non-nertal coordnates and veloctes enter nto the formulaton of the generalzed forces assocated wth the nertal coordnates. Usng the concept of non-nertal coordnates and the resultng dfferental/algebrac equatons obtaned n ths chapter leads to sgnfcant reducton n the numbers of state equatons, system nertal coordnates, and constrant equatons; and allows avodng a system of stff dfferental equatons that can arse because of the relatvely small mass. The development of accurate nonlnear longtudnal tran force models s necessary n order to better understand ralroad vehcle dynamc scenaros that nclude brakng, tracton, deralments and other types of accdents. Car coupler forces have sgnfcant effects on the longtudnal tran forces and tran stablty. Usng the concept of nonnertal coordnates enables the development of a more detaled coupler model that captures the 44

60 coupler knematcs wthout sgnfcantly ncreasng the number of state equatons and the dmenson of the problem. The proposed coupler model n ths thess allows for the car bodes to have arbtrary dsplacements, also avods havng a stff system of dfferental equatons that can result from the use of relatvely small masses. By assumng the coupler nerta neglgble compared to the car body nerta, one can dentfy two dstnct sets of coordnates; nertal and non-nertal coordnates. The nertal coordnates descrbe the moton of the car bodes and have nertal forces assocated wth them, whle the non-nertal coordnates have no nerta forces assocated wth them. Gven the nertal coordnates and veloctes, the nonlnear coupler force equatons are solved teratvely for the coupler non-nertal coordnates. The obtaned non-nertal coordnates are used n the formulaton of the generalzed forces actng on the car bodes. Numercal examples are presented n order to demonstrate the use of the concept of non-nertal coordnates for effcent modelng of the three-dmensonal coupler n the analyss of tran longtudnal moton. In order to demonstrate the effcency of non-nertal coordnate models, a verfcaton of the non-nertal coordnates wll be presented n Chapter 3. 45

61 CHAPTER 3 VERIFICATION OF THE NON-INERTIAL SYSTEMS As prevously mentoned, as the coupler nerta s relatvely small n comparson wth the car nerta; the hgh stffness assocated wth the coupler components can lead to hgh frequences that adversely mpact the computatonal effcency of tran models. Portons of ths chapter were publshed before by Massa et al. ( 2012) and used n ths chapter wth permsson from Journal of Nonlnear Dynamcs,whch s provded n Appendx B. The objectve of ths chapter s to demonstrate the effcency of the non-nertal coordnate models, by studyng the effect of the ther nertas on the dynamcs and on the computatonal effcency as measured by the smulaton tme. To ths end, two dfferent models are developed for the car couplers; one model, called the nertal coupler model, ncludes the effect of the coupler nerta, whle n the other model, called the nonnertal model, the effect of the coupler nerta s neglected as descrbed n chapter 2. Both nertal and non-nertal coupler models used n ths chapter are assumed to have the same coupler knematc degrees of freedom that capture geometrc nonlneartes and allow for the relatve translaton of the draft gears and end of car cushonng (EOC) devces as well as the relatve rotaton of the coupler shank. In both models, the coupler knematc equatons are expressed n terms of the car body and coupler coordnates. Both the nertal and non-nertal models used n ths study lead to a system of dfferental and algebrac equatons that are solved smultaneously n order to determne the coordnates of the cars and couplers. In the case of the nertal model, the coupler knematcs s descrbed usng the absolute Cartesan coordnates, and the algebrac equatons descrbe the knematc constrants mposed on the moton of the system. It s mportant to menton that, n case of the nertal model, the constrant equatons are satsfed at the poston, velocty, and acceleraton levels. In the case of the non-nertal model, the equatons of moton are 46

62 developed usng the relatve jont coordnates, thereby elmnatng systematcally the algebrac equatons that represent the knematc constrants. As Shown n Chapter 1, a quas-statc force analyss s used to determne a set of coupler nonlnear force algebrac equatons for a gven car confguraton. These nonlnear force algebrac equatons are solved teratvely to determne the coupler non-nertal coordnates whch enter nto the formulaton of the equatons of moton of the tran cars. The results obtaned n ths study showed that the neglect of the coupler nerta elmnates hgh frequency oscllatons that can negatvely mpact the computatonal effcency. The effect of these hgh frequences that are attrbuted to the coupler nerta on the smulaton tme s examned usng frequency and egenvalue analyses. Whle the neglect of the coupler nerta leads, as demonstrated n ths chapter, to a much more effcent model, the results obtaned usng the nertal and non-nertal coupler models show good agreement, demonstratng that the coupler nerta can be neglected wthout havng an adverse effect on the accuracy of the soluton. 3.1 Scope and Objectve In most ralroad vehcle computer formulatons, the coupler s modeled as a sprng-damper element wth no knematc degrees of freedom (Sanborn et al., 2007). The force n ths sprngdamper element s functon of the relatve dsplacements between the two cars connected by ths coupler. Ths smplfed approach does not take nto account the effect of the coupler degrees of freedom and fals to capture the coupler geometrc nonlneartes that can nfluence the car moton. Developng a more accurate coupler model requres ncludng coupler knematc degrees of freedom that account for the geometrc nonlneartes resultng from the moton of the coupler components. For such a detaled coupler model, two fundamentally dfferent approaches that requre the use of two dfferent soluton procedures can be used. In the frst approach, the nerta 47

63 of the coupler s taken nto account. In such a model, called nertal coupler model, the algebrac equatons, f they are present n the model, descrbe connectvty constrant equatons whch must be satsfed at the poston, velocty, and acceleraton levels snce the coupler acceleratons appear n the equatons of moton. The coupler degrees of freedom have generalzed nerta forces assocated wth them; and therefore, ther second dervatves appear n the system equatons of moton. Ths leads to an ncrease n the number of the system state equatons that must be ntegrated numercally. In the second approach, on the other hand, the nerta of the coupler, assumed small compared to the car body nerta, s neglected. In ths model, called the non-nertal coupler model, no generalzed nerta forces are assocated wth the coupler degrees of freedom, and as a consequence, the second dervatves of these coordnates do not appear n the fnal form of the equatons of moton. The use of such non-nertal coupler model can lead to sgnfcant reducton n the number of state equatons and can also contrbute to elmnatng the hgh frequency oscllatons that mght result from the hgh coupler stffness and ts relatvely small nerta. A set of quas-statc coupler equlbrum condtons can be developed and used to defne a set of nonlnear algebrac equatons that can be solved teratvely for the coupler non-nertal coordnates. These coupler non-nertal coordnates enter nto the formulaton of the equatons of moton of the tran cars (Stronat, 2010; Shabana et al., 2010). It s, therefore, the objectve of ths chapter to examne the effect of the coupler nerta on the dynamcs and computatonal effcency of tran models. The results obtaned usng the two models, nertal and non-nertal coupler models, are compared n order to examne the assumpton of neglectng the coupler nerta. In the case of the non-nertal coupler model, the quas-statc coupler condton used n ths nvestgaton do not requre havng a velocty dependent terms as compared 48

64 to methods prevously publshed n the lterature (Arnold et al., 2010). Only algebrac equatons are requred n order to be able to use the procedure employed n ths study. Nonetheless, velocty dependent forces can stll be ncluded n the non-nertal coupler formulaton used n ths Chapter. A frequency doman analyss s also performed n order to dentfy the frequences n the soluton assocated wth the coupler nerta. 3.2 Inertal systems If all the coordnates of the bodes n a MBS are treated as nertal coordnates, the equatons of moton of a body can be wrtten as (Roberson and Schertassek, 1988; Shabana et al., 2008) Mq Q Q Q (3.1) e c where M s the mass matrx of the body, q R θ T T T s the vector of the acceleratons of the body wth vector of external forces, R defnng the body translaton and θ defnng the body orentaton, Q e s the Q c s the vector of the constrant forces whch can be wrtten n terms of Lagrange multplers λ as T Q C λ, q c q C s the constrant Jacoban matrx assocated wth the coordnates of body, and Q s the vector of the nerta forces that absorb terms that are quadratc n the veloctes (Shabana, 2010). The nonlnear algebrac knematc constrant equatons can be wrtten n the vector form Cq,t 0, where q s the vector of the system generalzed coordnates, and t s tme. The constrant equatons at the acceleraton level can be wrtten as Cq Q, where Qd s a vector that absorbs frst dervatves of the coordnates. Usng Eq. 2.1 q d wth the constrant equatons at the acceleraton level, one obtans 49

65 T M C q q Qe Q Cq 0 λ Q d (3.2) Ths matrx equaton, whch ensures that the constrant equatons are satsfed at the acceleraton level, can be solved for the acceleratons and Lagrange multplers. In order to ensure that the algebrac knematc constrant equatons are satsfed at the poston and velocty levels, the ndependent nertal acceleratons q are dentfed and ntegrated forward n tme n order to determne the ndependent veloctes q and ndependent coordnates q. Knowng the ndependent coordnates from the numercal ntegraton, the dependent coordnates q d can be determned from the nonlnear constrant equatons usng an teratve Newton-Raphson algorthm that requres the soluton of the system Cq q d d C, where q d s the vector of Newton dfferences, and C q s the constrant Jacoban matrx assocated wth the dependent coordnates. d Knowng the system coordnates and the ndependent veloctes, the dependent veloctes q d can be determned by solvng a lnear system of algebrac equatons that represents the constrant equatons at the velocty level. Ths lnear system of equatons n the veloctes can be wrtten as q d d q t; where q C q C q C C s the constrant Jacoban matrx assocated wth the ndependent coordnates, and t t C C s the partal dervatve of the constrant functons wth respect to tme. Lagrange multplers, on the other hand, can be used to determne the constrant forces. For a gven jont k, the generalzed constrant forces actng on body, connected by ths jont, can be obtaned from the equaton T T T T c k k k k Q C q λ F T (3.3) k 50

66 where F k and T k are the generalzed jont forces assocated, respectvely, wth the translaton and orentaton coordnates of body. Usng the results of Eq. 3.3, the reacton forces at the jont defnton pont can be determned usng the concept of the equpollent system of forces. 3.3 Coupler Knematcs Usng Redundant Coordnates As prevously mentoned, n most coupler models reported n the lterature, the coupler s represented usng a dscrete massless sprng-damper element whch has no knematc degrees of freedom. Ths smple model does not capture geometrc nonlneartes resultng from the relatve moton between the coupler components as well as the relatve moton of the coupler wth respect to the car body. In ths chapter, the coupler knematc degrees of freedom are consdered. There are two methods for the descrpton of the coupler knematcs. In the frst method, the coupler nonlnear knematc equatons are formulated n terms of redundant coordnates that are related by algebrac constrant equatons. Ths approach s used n most general purpose MBS computer algorthms and t leads to a sparse matrx structure of the equatons of moton. Nonetheless, the redundant coordnate approach requres the soluton of a system of dfferental and algebrac equatons and the use of the technque of Lagrange multplers that can be used to determne the generalzed constrant forces. In the second approach, the knematc equatons of the coupler are expressed n terms of the coupler degrees of freedom as prevously descrbed n Chapter 2. In ths secton, the knematcs usng the redundant coordnate approach s dscussed. As prevously shown n Fg. 4, the coupler conssts of a shank and a draft gear or an End-of-car cushonng (EOC) devce unt. The draft gear or EOC devce s connected to the car body usng a prsmatc jont, whle the shank s connected to the draft gear or EOC devce usng a pn jont that allows for the relatve 51

67 rotaton of the shank wth respect to the car body. The draft gear produces dry frcton force, whle the EOC devce produces vscous dampng force; both oppose the sldng moton of the coupler wth respect to the car body. As prevously mentoned, n the coupler model consdered n ths study, the draft gear or end of car cushonng are assumed to be connected to the car body usng a prsmatc (translatonal) jont that allows for relatve translaton along the jont axs. In general MBS algorthms, the nonlnear algebrac equatons that defne the prsmatc jont are expressed n terms of the absolute coordnates of the two bodes and j connected by the jont. The fve algebrac constrant equatons that elmnate fve degrees of freedom can be wrtten n terms of the absolute Cartesan coordnates of the two bodes as (Shabana, 2010), P 2 C q q j v T v j v T v j v T r j v T r j h T h j T P 0 (3.4) where h and h j are two orthogonal vectors drawn perpendcular to the jont axs, defned respectvely on bodes and j ; v and v j are two vectors defned along the jont axs on bodes T T and j, respectvely; v, v1, v 2 form an orthogonal trad defned on body ; and r r r R Au R Au (3.5) j j j j j P P P P P In ths equaton, A and and j, respectvely; and j A are the transformaton matrces that defne the orentaton of bodes u P and to bodes and j, respectvely. Ponts j u P are the local poston vectors of ponts P and P and j P wth respect j P are defned on the axs of the prsmatc jont on bodes and j, respectvely. One can show that the Jacoban matrx of the prsmatc jont constrants s defned as 52

68 jt T j v H1 v 1H jt T j v H2 v 2H jt T T j C q C C j rp H1v 1HP v 1H q q P jt T T j rp H2 v 2HP v 2HP jt T j h Hh h Hh (3.6) where C and C j are the constrant Jacoban matrces assocated wth the coordnates of bodes q q and j, respectvely; and other vectors and matrces that appear n the precedng equaton are T j j jt j v1 v2 HP I Au P G, HP I A u P G, H1 A v1, H2 A v2, q q q q j j v j j T H j j A v, H h 0 Ah G, and q q j j jt j H h 0 A h G s the skew symmetrc matrx assocated wth the vector h.. In these equatons, h The shank of the coupler s assumed to be connected to the draft gear or EOC devce usng a pn jont. In ths model, only one relatve rotaton degree of freedom about the Z axs s consdered (Dawson, 1997). In the case of the pn jont, the algebrac constrant equatons n terms of the absolute Cartesan coordnates are the same as the equatons of the prsmatc jont gven by Eq. jt j 3.4 except of replacng the last equaton by the constrant equaton r r 0, where k s a P P k constant (Shabana, 2010), and r j r r j. Usng the algebrac constrant equatons of the pn P P P jont, one can show that the Jacoban matrx of the revolute jont constrants can be wrtten as jt T j v H1 v 1H jt T j v H2 v 2H jt T T j C q C C j rp H1v 1HP v 1H q q P jt T T j rp H2 v 2HP v 2HP jt jt j 2 P P 2 r H rp HP (3.7) 53

69 where the vectors and matrces that appear n ths equaton are the same as prevously defned n ths secton. 3.4 Coupler Force Equatons and Algorthm In ths secton, the formulatons of the forces resultng from the connecton between the car body and shank wth the draft gear or EOC devce; and the forces resultng from the coupler knuckle connecton are presented. The generalzed forces assocated wth the absolute Cartesan coordnates that are used n general MBS algorthms are obtaned. The computatonal algorthm used wth the redundant coordnate formulaton s also summarzed before concludng ths secton n order to shed lght on the fundamental dfferences between general MBS algorthms used n ths chapter for developng the nertal coupler model and the smpler LTD algorthms used for developng the non-nertal coupler model Draft Gear/EOC Connecton The car body s assumed to be connected to the draft gear or EOC devce usng a complant force element that has stffness k B and dampng coeffcent c B. The force due to ths complant force element can be wrtten as B B 0 B F k ll c l (3.8) where l s the current sprng length, l0 s the ntal sprng length, and l s the tme rate of deformaton of the sprng. Usng the vrtual work, W Fl B, and the defnton of l as l r r, where and j refer to the two components connected by the sprng, one can wrte jt B j B 54

70 the vrtual change n the sprng length n terms of the vrtual changes n the absolute Cartesan coordnates as 1/2 1 jt j jt j jt j l rb rb rb rb rb rb l jt rb j j j j B B l R u G θ R u G θ (3.9) where r r r defnes the poston vector of the sprng attachment pont j j B B B B on the draft gear or EOC devce wth respect to the sprng attachment pont j B on the car body; G and G j are the matrces that relate the angular velocty vector to the dervatves of the orentaton parameters; u B and j u B are the skew symmetrc matrces assocated wth the vectors Au B and j j Au B that defne the locatons of the sprng attachment ponts wth respect to the orgn of the coordnate systems of bodes and j, respectvely; and A and A j are the transformaton matrces that defne the orentaton of the two bodes. Defnng the drecton of the force F B by the unt vector rˆj r, the vrtual work can be wrtten as j l B B W F ˆjT j j j j Br B R ubgθ R ubg θ Q R Q θ Q R Q θ T T jt j jt j R R (3.10) whch can be rewrtten n a compact form as W Q q Q q, where generalzed coordnates of bodes and j, and T jt j B B q and j q are the Q B j j j Q ˆ ˆ R FBr B j Q R FBr B, T T j QB j jt j T j Q F ˆ ˆ BG u B rb Q FBG u B rb (3.11) 55

71 Ths equaton defnes the generalzed forces assocated wth the absolute Cartesan coordnates due to the connecton of the draft gear or EOC devce wth the car body Shank Connecton In ths chapter, the shank s assumed to be connected to the draft gear or EOC devce by a pn jont whch allows yaw rotaton. The restorng torque resultng from the relatve shank rotaton s ntroduced usng a rotatonal sprng-damper element. Ths torque s assumed to take the followng form: T k c (3.12) j j j r r j where represents the relatve rotaton of the shank. The vrtual change n ths rotaton can be defned as j h jt G θ G j θ j (3.13) where G and vrtual work, one has j G are as defned before, and h j s a unt vector along the jont axs. Usng the WT Q q Q q (3.14) j j T jt j r r T where Q T j T j r 0 T G h and j j jt j r T nonlnear functons of the relatve shank rotaton. T Q T 0 G h. These generalzed forces are Knuckle Force Model The connecton between the couplers of two cars s represented by a complant element that defne the knuckle force model. Followng a procedure smlar to the one used prevously n ths secton, 56

72 one can defne the followng vectors of generalzed forces assocated wth the absolute Cartesan coordnates of the shanks of the two couplers that connect two cars: Q rr j j j Q F ˆ ˆ Qr R Q Q F R Qr Q, T T j Qrr j jt j T j F ˆ Q ˆ Q Q Q F G u r Q QG u Q rq (3.15) where the force s defned as FQ krrlrr crrl rr (3.16) and k rr and c rr are, respectvely, the stffness and dampng coeffcents of the knuckle; l rr s the relatve dsplacement between the two shanks; and l rr s the rate of the relatve dsplacement Soluton Algorthm The nertal coupler model s developed usng a general MBS algorthm based on the redundant coordnates. The dynamc equatons of moton of the tran system are augmented wth the nonlnear algebrac equatons that descrbe mechancal jonts and specfed moton trajectores. These algebrac equatons are satsfed at the poston, velocty, and acceleraton levels. As prevously mentoned, LTD algorthms tend to be much smpler than general MBS algorthms that requre the use of a procedure for the soluton of dfferental and algebrac equatons. The smpler LTD algorthms s prevously descrbed n Chapter 2. General MBS algorthms employ the technque of Lagrange multplers and they are, for the most part, based on Eq These algorthms also explot sparse matrx technques n order to obtan effcent soluton for the nonlnear dynamc equatons of moton. Usng the developments presented n Secton 2, the followng algorthm s used for developng the nertal coupler model: 57

73 1. An estmate of the ntal condtons that defne the ntal confguraton of the MBS s made. The ntal condtons that represent the ntal coordnates and veloctes must be a good approxmaton of the exact ntal confguraton. 2. Usng the ntal coordnates, the constrant Jacoban matrx C q can be constructed, and based on the numercal structure of ths matrx an LU factorzaton algorthm can be used to dentfy a set of ndependent coordnates. 3. Usng the values of the ndependent coordnates, the constrant equatons Cq,t 0 can be consdered as a nonlnear system of algebrac equatons n the dependent coordnates. Ths system can be solved teratvely usng a Newton-Raphson algorthm and sparse matrx technques. 4. Assumng that the ndependent coordnates and veloctes are known from the numercal ntegraton and the dependent coordnates are determned from the prevous step, one can construct the constrant equatons at the velocty level, q C t, where t C q C s the partal dervatve of the constrant functons wth respect to tme. Ths lnear system of algebrac equatons has a number of scalar equatons equal to the number of dependent veloctes. 5. Havng determned the coordnates and veloctes, Eq. 2.2 can be constructed and solved for the acceleratons and Lagrange multplers. The vector of Lagrange multplers can be used to determne the generalzed reacton forces. 6. The ndependent acceleratons can be dentfed and used to defne the state space equatons whch can be ntegrated forward n tme usng a drect numercal ntegraton method. The numercal soluton of the state equatons defnes the ndependent coordnates and veloctes, whch can be used to determne the dependent coordnates and veloctes as dscussed n steps 3 and 4. 58

74 7. Ths process contnues untl the desred end of the smulaton tme s reached. The soluton algorthm used for developng the non-nertal coupler model n Chapter 2 dffers sgnfcantly from the soluton algorthm dscussed n ths secton. Nonetheless, both algorthms requre the soluton of dfferental and algebrac equatons. The algebrac equatons n the case of the non-nertal coupler model, however, are obtaned from force equatons nstead of knematc constrant equatons. It can be shown that, the specalzed LTD algorthms tend to be smpler as compared to general MBS algorthms. LTD algorthms do not n general requre the soluton of DAE s system and are based on formulatons that employ ndependent coordnates. They are desgned to solve effcently long tran conssts for the purpose of analyzng the longtudnal forces actng on the consst cars. These longtudnal forces can nclude tractve, brakng, resstance, and coupler forces. Effcent mplementaton of some of these force elements such as couplers may requre modfyng LTD algorthms to allow for solvng algebrac equatons smultaneously wth the system equatons of moton. 3.5 Two-Car Model The man goal of ths study s to study the use of non-nertal coordnates by examne the effect of the coupler nerta on the soluton accuracy and computatonal effcency usng smple tran models. To ths end, two dfferent models are developed. The frst model s developed usng a general MBS algorthm that employs redundant coordnates and allows for the soluton of DAE systems. Ths model, whch accounts for the coupler nerta, s called the nertal coupler model. The second model s developed usng the LTD algorthms mplemented n ATTIF code and descrbed n Chapter 2. Ths algorthm s modfed to allow for the soluton of the quas-statc coupler algebrac equatons wth the dfferental equatons of moton. In ths second model, whch 59

75 s referred to as the non-nertal coupler model, the effect of the coupler nerta s neglected. The type of coupler consdered n ths numercal chapter s called long shank E-type, whch has propertes reported n the lterature (Dawson, 1997). The smple two-car tran model, shown n Fg. 4, s used to obtan the numercal results reported n ths numercal nvestgaton. The coupler stffness and dampng coeffcents are assumed as k k k, j B B B c c c, j B B B k k k, j r r r c c c, j r r r k rr k and j rr c rr c, wth the values shown n Table 3. j rr Table 3: Model parameters Coupler Coupler Value parameter parameter l (m) c, c (N.m.s) 0 k B, c B, k r, j k B (N/m) j c B (N.s/m) j k r (N.m) r r0 j r j k rr (N/m) Value j c rr (N.s/m) l (m) The masses and mass moments of nerta of the coupler components are assumed as mshank 140 kg, 2 Izz, shank 8 kg m, and EOC 50 m kg. In order to examne the effect of the coupler nerta, two dfferent smulaton scenaros are consdered n ths nvestgaton; the frst s wth empty cars, whle the second s wth full loaded cars. In both scenaros the tracton force s appled to the rear 5 car wth a magntude of 3 10 N. For both scenaros, the ntal veloctes and angles are assumed to be zero. The ntal coordnates of the center of mass of the two cars along the longtudnal global X axs are assumed to be 0 m for the rear car and m for the leadng car; whle the vertcal Z coordnate of the center of mass s assumed to be 1.85 m for the full loadng condton and m for the empty condton. The nertal propertes of the cars n the empty and full cases 60

76 are reported n Table 4 (Shabana et al., 2010). In case of full loaded car, the rato of the mass of the coupler to the mass of the car s ; whle n the case of empty car, ths rato s Note that ths rato s less than 1%, even n the case of the empty car. The coupler model consdered n ths nvestgaton s assumed to have EOC unt. Table 4: Car parameters Parameter Full loadng Empty m(kg) I xx (kg.m 2 ) I (kg.m 2 ) yy I zz (kg.m 2 ) Ixy, Ixz, I yz (kg.m 2 ) Smulaton Results Fgure 17 shows the forward poston of the two cars, for the two scenaros of empty and loaded cars. Ther veloctes ncrease lnearly due to the applcaton of the constant force. Fgure 18 shows the EOC and shank knuckle relatve dsplacements d d and j d rr respectvely for the full loaded car scenaro usng both the MBS and LTD algorthms, whle Fg.19 shows the tme dervatves of these relatve dsplacements. Fgure 20 shows the EOC and shank knuckle relatve dsplacements d d and j d rr respectvely for the empty car scenaro usng both the MBS and LTD algorthms, whle Fg. 21 shows the tme dervatves of these relatve dsplacements. Each of these fgures shows the effect of neglectng the coupler nerta. 61

77 Forward poston (m) Tme (s) Fgure 17: Forward poston ( Rear car loaded, Leadng car loaded, Rear car empty, Leadng car empty ) Sprng deformaton (m) Fgure 18: EOC and shank knuckle relatve dsplacements (loaded car scenaro) ( d nertal, d non-nertal, d nertal, d non-nertal) d d Tme (s) j rr j rr 62

78 Relatve velocty (m/s) Tme (s) ( Fgure 19: EOC and shank knuckle relatve veloctes (loaded car scenaro) d nertal, d non-nertal, d nertal, d non-nertal) d d j rr j rr Sprng deformaton (m) Fgure 20: EOC and shank knuckle relatve dsplacements (empty car scenaro) ( d nertal, d non-nertal, d nertal, d non-nertal) d d Tme (s) j rr j rr 63

79 Relatve velocty (m/s) Fgure 21: EOC and shank knuckle relatve veloctes (empty car scenaro) ( d nertal, d non-nertal, d nertal, d non-nertal) d d Tme (s) j rr j rr The results presented n ths secton demonstrate that neglectng the coupler nerta does not have a sgnfcant effect on the accuracy of the soluton. Nonetheless, the neglect of the coupler nerta leads to sgnfcant mprovement n the computatonal effcency. The LTD model that does not take nto account the effect of the coupler nerta was found to be more than 85 tmes faster than the MBS model that takes nto account the effect of the coupler nerta. The same explct numercal ntegraton method s used to obtan the results of the nertal and non-nertal coupler model. The effect of changng the stffness coeffcent k B on the dsplacement between the two shanks of the two cars was also examned. It was found that ncreasng ths stffness coeffcent leads to a decrease n the phase shft, but can lead to an ncrease n the ampltude of the relatve oscllatons between the shanks due to the ncrease n the force transmtted. 64

80 3.5.2 Egen Value and Frequency Doman Analyss Fgure 22 shows the mode shapes and frequences obtaned by solvng the egenvalue problem of the nertal coupler model (Genta, 2009; Shabana, 1997), for the full loaded car model. Usng the results presented n ths fgure, t s possble to recognze some frequences that are assocated wth the small nerta of the coupler components, ncludng the modes that descrbe the rotaton of the shanks or relatve dsplacement between the slder and the car. The frst mode wth a low frequency of 2.8 Hz s the only mode that s present n both the nertal and non-nertal coupler models. Other modes presented n Fg. 22 are manly due to the coupler nerta. It s clear that the second mode, for example, whch has frequency of approxmately Hz corresponds to the longtudnal moton of the coupler wth respect to the car body. Ths frequency can be easly calculated by usng the nerta of the coupler and the stffness coeffcentk B. It s also mportant to pont out that the hghest frequency s assocated wth the rotatonal moton of the shanks. The mode assocated wth ths frequency depends on the coupler nerta, and therefore, ths mode does not appear n the non-nertal system. In order to have a better understandng of the effect of these frequences, a Fast Fourer Transform (FFT) of the acceleraton s performed (Brgham, 1974). The acceleratons are drectly related to the coupler forces and can be used as good FFT measure. Fgure 23 shows the frequency doman analyss of the acceleraton of the rear coupler EOC unt whch has the same frequency content as the shank acceleraton. The results presented n ths fgure are obtaned for dfferent values of the dampng coeffcents c B of the rear coupler. Fgure 24 shows the frequency doman results of Lagrange multpler assocated wth longtudnal generalzed reacton of the revolute jont constrants of the nertal coupler. The results presented n ths fgure show clearly the effect of the frst mode (2.8 Hz) and there s nearly no effect from the other hgher modes. 65

81 Fgure 22: Mode shapes Fgure 23: Frequency doman analyss of the acceleraton 66

82 Ampltude (N) Zoomed Ampltude (N) Hz Frequency (Hz) Frequency (Hz) Fgure 24: Lagrange multpler frequences 3.6 Ten-Car Model In order to check the robustness of usng the non-nertal coordnate system mplemented n ATTIF (Analyss of Tran/Track Interacton Forces), a 10-car model s developed. Ths tran model, shown n Fg. 25, conssts of ten fully loaded cars; the cars and EOC devces have the same specfcatons and propertes as the fully loaded cars and EOC devces used n the 2-car model dscussed n the precedng secton, except for the EOC dampng coeffcent whch s assumed to be c c c j 5 B B B 510 N s rad. The value of B c has been chosen n order to reach steady state 6 n a reasonable tme. The tractve force s assumed to be 3 10 N, and the smulaton s performed for 20 seconds. Fgures 26 and 27 show the relatve dsplacements and veloctes of the 5 th coupler usng MBS and LTD algorthm. The results presented n these fgures show that the LTD and MBS solutons are n a good agreement. Nonetheless, the LTD usng model s more than 300 tmes faster 67

83 than the MBS model, ndcatng that as the number of cars ncreases, the LTD model wll become more effcent as compared to the MBD model Fgure 25: Ten-car model (ATTIF nterface) Sprng deformaton (m) Tme (s) ( Fgure 26: EOC and shank knuckle relatve dsplacements (5 th coupler) d nertal, d non-nertal, d nertal, d non-nertal) d d j rr j rr 68

84 Relatve velocty (m/s) Tme (s) ( Fgure 27: EOC and shank knuckle relatve veloctes (5 th coupler) d nertal, d non-nertal, d nertal, d non-nertal) d d j rr j rr 3.7 Curved Track Smulaton In the examples consdered n the precedng two sectons, a tangent track was used, and therefore, and j reman nearly constant. In order to examne the effect of the geometrc nonlneartes resultng from the coupler knematc degrees of freedom j and ; the full loaded two-car model s consdered and t s assumed to negotate the curved track shown n Fg. 28. Ths track s assumed to consst of m tangent segment, m spral segment, 3-degree constant curve of length m, spral segment of length m, tangent segment of length m, spral segment of length 91.44, -3-degree curve of length of 91.44, spral segment of length m, and tangent segment of length A 76.2 mm super-elevaton s assumed for the frst two spral segments, and mm for the last two spral segments. In ths example, car j s consdered as the 69

85 locomotve that pulls car whch represents a freght car. The length of the shank knuckle lr0 s assumed to be 0 m n the smulaton scenaro consdered n ths secton. The ntal forward velocty of each car s assumed to be 20 m/s, and the smulaton tme s ncreased to 40 s n order to capture the nonlnear behavor of the coupler on dfferent track sectons. Fgure 29 shows the forward poston durng the frst 5 seconds of car, slder blocks and j that represent EOC devces, and car j as the result of the applcaton of constant forward velocty 20 m/s usng both the MBS and LTD algorthms. Fgure 30 shows the rotatonal deformaton of the torsonal sprng of car for both MBS and LTD models. The results presented n these fgures show that the LTD and MBS solutons are n a good agreement. Nonetheless, the LTD model s more than 200 tmes faster than the MBS model, ndcatng that the LTD model s accurate and more effcent as compared to the MBD model n the case of curved track smulatons Curved regon Global Y coordnate (m) Spral regon Tangent regon Spral regon Tangent regon Curved regon Spral regon Global X coordnate (m) Fgure 28: Curved track 70

86 Forward poston (m) Tme (s) Fgure 29: Forward poston ( Rear car nertal, Slder block nertal, Slder block j nertal, Leadng car nertal, Rear car non-nertal, Slder block non-nertal, Slder block j non-nertal, Leadng car non-nertal ) (rad) Fgure 30: Torsonal sprng deformaton ( Inertal, Non-nertal) 71 Tme (s)

87 3.8 Concludng Remarks The objectve of ths chapter s to demonstrate the use of the non-nertal coordnate concept and prove ts effcency n the analyss of longtudnal tran dynamcs. Ths s accomplshed by studyng the effect of neglectng the coupler nerta whch can lead to sgnfcant deteroraton of the computatonal effcency. The relatvely small nerta of the coupler can lead to hgh frequency oscllatons n the soluton; requrng the use of smaller ntegraton tme steps and sgnfcantly ncreasng the CPU tme. In order to address ths problem, a non-nertal coupler model s developed by replacng the coupler dfferental equatons wth quas-statc nonlnear algebrac force equatons. These algebrac equatons are teratvely solved usng a Newton- Raphson method n order to determne the non-nertal coordnates. The non-nertal veloctes are determned by solvng these quas-statc equatons at the velocty level. The non-nertal coordnates and veloctes are then used n formulaton of the generalzed coupler forces assocated wth the coordnates of the car bodes. In order to examne the effect of neglectng the coupler nerta on the accuracy of the soluton and the computatonal effcency, an nertal coupler model s developed usng MBS algorthms. The results obtaned usng the nertal and non-nertal coupler models are compared. The numercal results obtaned n ths study showed that the neglect of the coupler nerta does not have a sgnfcant effect on the accuracy of the soluton. On the other hand, the neglect of ths nerta leads to sgnfcant mprovement n the computatonal effcency. The results obtaned showed that the LTD mplementaton that neglects the effect of the coupler nerta becomes more effcent as the number of cars ncreases. An egenvalue analyss and FFT are used to dentfy the frequences assocated wth the coupler nerta. As dscussed n ths study, these hgh frequences do not appear when the non-nertal coupler model s used. LTD algorthms tend to be smpler as compared to MBS algorthms that requre the use of DAE solver. LTD are 72

88 desgned for the effcent soluton and analyss of longtudnal tran forces of long conssts. Car coupler forces have a sgnfcant effect on the behavor of trans n response to brakng, tractve, and resstance forces. For ths reason, t s mportant to develop effcent coupler models that can be used n LTD algorthms. The development of such an effcent coupler model requres modfyng exstng LTD algorthms to nclude a procedure for solvng dfferental and algebrac equatons smultaneously as demonstrated n ths chapter. Another approach that can be used for the effcent soluton of MBS dynamcs s to use the mplct numercal ntegraton methods. To ths end, Chapter 4 presents a new mplementaton for the two-loop mplct sparse matrx numercal ntegraton (TLISMNI) method proposed for the soluton of constraned MBS dfferental and algebrac equatons. 73

89 CHAPTER 4 IMPLEMENTATION OF THE TLISMNI METHOD The dynamcs of large and complex multbody systems that nclude flexble bodes and contact/mpact pars s governed by stff equatons. Explct ntegraton methods can be very neffcent and often fal n the case of stff problems. The use of mplct numercal ntegraton methods s recommended n ths case. Ths chapter presents a new and effcent mplementaton of the two-loop mplct sparse matrx numercal ntegraton (TLISMNI) method proposed for the soluton of constraned rgd and flexble multbody system (MBS) dfferental and algebrac equatons(aboubakr and Shabana, 2014). The TLISMNI method has desrable features that nclude avodng numercal dfferentaton of the forces, allowng for an effcent sparse matrx mplementaton, and ensurng that the knematc constrant equatons are satsfed at the poston, velocty and acceleraton levels. In ths method, a sparse Lagrangan augmented form of the equatons of moton that ensures that the constrants are satsfed at the acceleraton level s frst used to solve for all the acceleratons and Lagrange multplers. The generalzed coordnate parttonng or recursve methods can be used to satsfy the constrant equatons at the poston and velocty levels. In order to mprove the effcency and robustness of the TLISMNI method, the smple teraton and the Jacoban-Free Newton-Krylov approaches are used n ths nvestgaton. The new mplementaton s tested usng several low order formulas that nclude Hlber Hughes Taylor (HHT) method, whch ncludes numercal dampng, L- stable Park, A-stable Trapezodal, and A-stable BDF methods. Dscusson on whch method s more approprate to use for a certan applcaton s provded. The chapter also dscusses TLISMNI mplementaton ssues ncludng the step sze selecton, the convergence crtera, the error control, and the effect of the numercal 74

90 dampng. The use of the computer algorthm descrbed n ths chapter s demonstrated by solvng complex rgd and flexble tracked vehcle models, ralroad vehcle models, and very stff structure problems. The results, obtaned usng these low order formulas, are compared wth the results obtaned usng the explct Adams-predctor-corrector method. Usng the TLISMNI method, whch does not requre numercal dfferentaton of the forces and allows for an effcent sparse matrx mplementaton, for solvng complex and stff structure problems leads to sgnfcant computatonal cost savng. 4.1 MBS Dfferental/Algebrac Equatons (DAEs) In ths secton the constraned MBS equatons of moton and the soluton algorthm used to solve these equatons are brefly dscussed. The defntons provded n ths secton wll be used repeatedly n ths chapter MBS Equatons of Moton As prevously mentoned, n MBS dynamcs, the constrant relatonshps are used wth the dfferental equatons of moton to solve for the unknown acceleratons and constrant forces. Whle ths approach leads to a sparse matrx structure, t has the drawback of ncreasng the problem dmensonalty and t requres more sophstcated numercal algorthms to solve the resultng DAE system. Usng the generalzed coordnates, the equatons of moton of a body can be wrtten as (Roberson and Schwertassek, 1988; Shabana, 2010) Mq Q Q Q (4.1) e c 75

91 where M s the mass matrx of the body, q R θ T T T s the vector of the acceleratons of the body, R defnes the body translaton and θ s a set of parameters that defne the body orentaton, Q e s the vector of external forces, Q c s the vector of the constrant forces whch can T be wrtten n terms of Lagrange multplers λ as Q c C λ, C q q s the constrant Jacoban matrx assocated wth the coordnates of body, and Q s the vector of the nerta forces that absorb terms that are quadratc n the veloctes. The constrant equatons at the acceleraton level can be wrtten asc q q Q d, where Q d s a vector that absorbs frst dervatves of the coordnates. Usng Eq. 4.1 wth the constrant equatons at the acceleraton level, one obtans the system equatons of moton wrtten n the segmented form as T M C q q Qe Q Cq 0 λ Qd (4.2) The symbols that appear n ths equaton (wthout the superscrpt ) refer to system vectors and matrces obtaned by standard MBS assembly of body vector and matrces. The precedng matrx equaton, whch ensures that the constrant equatons are satsfed at the acceleraton level, can be solved for the acceleratons and Lagrange multplers. Lagrange multplers, on the other hand, can be used to determne the constrant forces. For a gven jontk, the generalzed constrant forces actng on body, connected by ths jont, can be obtaned from the equaton (Shabana, 2010) T T T T c k k k k Q C q λ F T (4.3) k 76

92 As prevously mentoned, F k and T k are the generalzed jont forces assocated, respectvely, wth the translaton and orentaton coordnates of body. Usng the results of Eq. 4.3, the reacton forces at the jont defnton pont can be determned usng the concept of the equpollent systems of forces Generalzed Coordnate Parttonng In order to ensure that the algebrac knematc constrant equatons are satsfed at the poston and velocty levels, the ndependent acceleratons q are dentfed and ntegrated forward n tme n order to determne the ndependent veloctes q and ndependent coordnatesq ( Shabana, 2010). Knowng the ndependent coordnates from the numercal ntegraton, the dependent coordnates q d can be determned from the nonlnear constrant equatons usng an teratve Newton-Raphson algorthm that requres the soluton of the system Cq q d d C, where q d s the vector of Newton dfferences, and C q s the constrant Jacoban matrx assocated wth the dependent coordnates. d Knowng the system coordnates and the ndependent veloctes, the dependent veloctes q d can be determned by solvng a lnear system of algebrac equatons that represents the constrant equatons at the velocty level. Ths lnear system of equatons n the veloctes can be wrtten as q d d q t; where q C q C q C C the constrant Jacoban matrx s assocated wth the ndependent coordnates, and t t C C s the partal dervatve of the constrant functons wth respect to tme. The selecton of the set of ndependent coordnates s an mportant step n numercal soluton usng the generalzed coordnate parttonng. Ths selecton can have a sgnfcant effect on the stablty of the soluton and also on reducng the accumulaton of the numercal errors when the 77

93 algebrac constrant equatons are solved for the dependent varables. Also numercal problems can be encountered when usng mplct ntegratons wth the generalzed coordnate parttonng, partcularly when a large tme step s selected by the ntegrator. In ths case, the teratve Newton- Raphson algorthm may fal to converge. On the other hand, the generalzed coordnate parttonng technque s suted for the sparse matrx mplementaton and can be used for MBS applcatons that nclude rgd and flexble bodes and systems that suffer from sngularty problems such as closed chans Algorthm and Sparse Matrx Implementaton As prevously mentoned, n order to ensure that the constrant equatons are satsfed at the poston level, the dependent coordnates are determned by solvng the nonlnear algebrac constrant equatons usng the teratve Newton-Raphson procedure. In the sparse matrx mplementaton one can use the followng system of algebrac equatons n Newton teratons (Wehage, 1980, Shabana, 2010): Cq -C Δq= (4.4) I d 0 In ths equaton C q s the constrant Jacoban matrx, q s the vector of Newton dfferences, and I d s a Boolean matrx that has zeroes and ones only; wth the ones n the locaton that correspondng to the ndependent coordnates n order to ensure that Δq 0. The square coeffcent matrx n Eq. 4.4 s sparse, and therefore, sparse matrx technques can be used to effcently solve the precedng system of equatons for the dependent coordnates. Once the 78

94 dependent coordnates are determned, the dependent veloctes can be obtaned usng the followng lnear sparse system that defnes the constrant equatons at the velocty level: Cq -Ct q= I d q (4.5) The rght hand sde of Eq. 4.5 s assumed to be known snce ntegraton, and q s determned from the numercal Ct depends on tme and the coordnates that are assumed to be known from the poston analyss. One of the advantages of ths algorthm that f the set of ndependent coordnates change durng the smulaton tme one has only to change the locatons of the nonzero entres of the matrx I d, whle the structure of the Jacoban matrx C q remans the same. Once the generalzed coordnates and veloctes are determned, the augmented form of Eq. 4.2 can be constructed and solved for the acceleratonq, and Lagrange multplers λ as prevously mentoned. The man steps for a numercal algorthm usng the generalzed coordnate parttonng can then be summarzed as follows: 1. An estmate of the ntal condtons that defne the ntal confguraton of the multbody system s made. The ntal condtons that represent the ntal coordnate and veloctes must be a good approxmaton of the system ntal confguraton. 2. Usng the coordnates, the constrant Jacoban matrx C q can be constructed, an LU factorzaton algorthm can be used to dentfy a set of ndependent coordnates. 3. Usng the values of the ndependent coordnates, the constrant equatons Cq,t 0 can be consdered as a nonlnear system of algebrac equatons n the dependent coordnates 79

95 (Shabana, 2010). Ths system can be solved teratvely usng Eq. 4.4 and a Newton- Raphson algorthm and sparse matrx technques. 4. Assumng that the ndependent coordnates and veloctes are known from the numercal ntegraton and the dependent coordnates are determned from the prevous step, one can construct the constrant equatons at the velocty level (Eq. 4.5), ths lnear system of algebrac equatons can be solved for the dependent veloctes. 5. Havng determned the generalzed coordnates and veloctes, Eq. 4.2 can be constructed and solved for the acceleratons and Lagrange multplers. The vector of Lagrange multplers can be used to determne the generalzed reacton forces. 6. The ndependent acceleratons can be dentfed and used to defne the state space equatons whch can be ntegrated forward. The numercal soluton of the state equatons defnes the ndependent coordnates and veloctes. 7. If the end of the smulaton s not reached, the algorthm returns to Step 2. In the next secton, the TLISMNI mplementaton s dscussed. 4.2 TLISMNI Algorthm The TLISMNI method was recently proposed for the soluton of the MBS dfferental/algebrac equatons (Shabana and Hussen, 2010). As mentoned before, ths method ensures that the algebrac constrant equatons are satsfed at the poston, velocty, and acceleraton levels; does not requre numercal or analytcal dfferentaton of the forces, and allows for effcent sparse matrx mplementaton. In ths secton, the soluton steps for a modfed TLISMNI algorthm that allows for the mplementaton of dfferent low order ntegraton formulas s presented. The performance of ths algorthm wll be tested n a later secton of the chapter usng complex flexble 80

96 and rgd body tracked vehcle models. The two-loop mplct procedure proposed n ths study has an teratve outer loop that nvolves equatons whch are lnear n acceleratons, and Lagrange multplers. The use of the generalzed coordnate parttonng leads to another teratve nner Newton Raphson loop that nvolves the knematc constrant equatons whch are nonlnear functons n the dependent coordnates, n addton to a system of lnear equaton that can be solved for the dependent velocty. In case of usng the recursve approach nstead of the generalzed coordnates parttonng the nner Newton Raphson loop s avoded and the procedure has only the teratve outer loop that nvolves equatons whch are lnear n acceleratons and Lagrange multplers, despte the fact that these equatons are nonlnear n the ndependent coordnates and veloctes. The TLISMNI computatonal algorthm that employs the generalzed coordnate parttonng can be summarzed as follows: 1. Assumng that the state of the system s known at tme t n, the constrant Jacoban matrx C q can be constructed, and used wth Gaussan elmnaton procedure to determne a set of system ndependent coordnatesq. Therefore, the vector of system generalzed coordnates q can be parttoned to dependent and ndependent coordnates as T T T q q q d, where q d s the set of system dependent coordnates. ( k ) 2. A predcton of the ndependent coordnates q at tme tn 1, usng an explct ntegraton n, 1 formula can be made and used to predct the ndependent velocty q usng an mplct ( k ) n, 1 ntegraton formula such as HHT, BDF, Park, or Trapezodal method. k n, 1 3. Knowng the ndependent coordnates ( ) q, the constrant equatons Cq 81,t 0 can be consdered as a nonlnear system of algebrac equatons n the dependent coordnates ( k q ) dn, 1

97 . Ths system can be solved teratvely, usng Newton-Raphson algorthm as n Eq. 4.4 employng a sparse matrx technques, for the dependent coordnates ( k q ). dn, 1 4. Usng the predcted ndependent coordnates q and veloctes ( k ) n, 1 q from Step 2 and ( k ) n, 1 the dependent coordnates q determned from the prevous step; one can construct the ( k) dn, 1 constrant equatons at the velocty level (Eq. 4.5). The soluton of ths system of lnear equatons defnes the dependent veloctes q. ( k) dn, 1 5. Havng estmate all the coordnates and veloctes at tme tn 1, Eq. 4.2 can be constructed and solved usng sparse matrx technques for the acceleratons ( k q ) n1 and Lagrange multplers ( k λ ) n1 (Shabana, 2010). 6. The ndependent acceleratons can be dentfed and used to defne the state space equatons whch can be ntegrated forward n tme usng lower order formulas such as HHT, BDF, Park, or Trapezodal method to obtan q ( k1) n, 1, and q. ( k1) n, 1 7. Usng the convergence and error crtera ntroduced n a later secton of ths chapter, one can judge whether or not the ntegraton s successful. If the soluton satsfes the convergence and error crtera, the coordnates, veloctes, acceleratons, and Lagrange multplers are accepted. In ths case, one needs to calculate the new step sze n order to advance the ntegraton. If, on the other hand, convergence s not acheved or the error exceeds the specfed error tolerance, then more teratons are requred usng a tme step sze that acheves convergence. In ths case, the algorthm returns to Step 2 and the process contnues untl the convergence and error crtera s satsfed. 8. Ths process contnues untl the desred end of the smulaton tme s reached. 82

98 It s clear from the two computatonal algorthms presented n ths chapter that TLISMNI method has sparse coeffcent matrces that requre mnmum storage. Furthermore, no numercal dfferentaton of the external or nerta forces wth respect to the coordnates and veloctes s requred. The algorthm outlned above s equvalent to performng fxed pont teraton for the soluton of nonlnear system of equatons for q n1, and q n1 usng an mplct ntegraton formula (Graca et al., 1994; Shampne, 1981). The TLISMNI algorthm employs the mplct ntegraton formulas whch are more effcent than the explct formulas partcularly n the case of stff problems. 4.3 Krylov Subspace and Inexact Newton Method One of the man contrbutons of ths chapter s to ntegrate Krylov subspace projecton method n the TLISMNI algorthm. In the TLISMNI algorthm descrbed n ths chapter, one can consder the outer loop as a set of ODE ntal value problem that can be wrtten as y f(, t y) where y( t ) 0 0 y T T T, and y q q. Assumng these ODE s are stff and there s a need to use an mplct ntegraton formula to obtan a soluton, one can wrte the general formula for mplct ntegraton methods as y ψhf( t, y ), h0 (4.6) n1 n1 n1 Where ψ s a vector that has varables prevously computed, h s the tme step sze, s a constant that depends on the mplct formula used, and n 1 n 1 f(, y ) s the rght hand sde of the system of dfferental equatons. In ths chapter, an equvalent form of Eq. 4.6 wll be used. Ths form can be wrtten n terms of t 83

99 (4.7) x hy( t, y ) ( y ψ)/ n1 n1 n1 n1 Typcally, Newton and modfed Newton algorthms are employed to solve Eq. 4.7 for x n1, and ths generates a set of lnear systems to be solved at each tme step, where Eq. 4.7 can be wrtten Fx ( ) as n1 0. There are standard methods to solve Eq. 4.7, one method s to approxmate the k k Jacoban matrx JF / y 1 at tme tn 1 and then solve the followng teratve equaton: n ( Ih J) x F(, y ) (4.8) k n1 tn 1 n1 If we consder J 0, the teratve procedure wll be smple and n ths case s called smple teratons, and t wll be equvalent to performng fxed pont teraton for the soluton of the nonlnear equatons for y n1 as follows: y ψh f( t, y ) (4.9) k1 k n1 n1 n1 Equaton 4.9 represents the outer loop for the algorthm descrbed n Secton 2. In case J 0 expensve computatons are requred for the numercal calculatons or approxmatons for J, matrx decomposton, and large storage space wll be requred, especally when solvng large scale problems. In addton, the numercal dfferentaton requred for a general MBS mplementaton can be a source of numercal errors that may lead to non-accurate solutons as the problem dmensonalty ncreases. Therefore, for such complex and large scale problems, the use of a method that can approxmately solve Eq. 4.8 at reasonable cost and also reduces the core memory requred s desrable. Incomplete orthogonalzaton method (IOM) and Arnold s algorthm are examples of such methods that wll be dscussed n a followng subsecton. IOM and Arnold s algorthm are methods for the approxmate soluton of a lnear system Ax = b n Fx ( ) The use of such methods n the soluton of the nonlnear system n1 N R (Saad, 2003). 0 by Newton s method 84

100 gves rse to what s called nexact Newton method. An nexact Newton method has the followng form (Brown and Hndmarsh,1986): Set x (0) as an ntal guess For m 0,1, 2... untl convergence Fnd n some unspecfc manner a vector s( m) satsfyng F( x( m)) s ( m) = F( x( m)) r ( m) Set x( m 1) x( m) s ( m) where the resdual r ( m ) represents the amount by whch the vector s( m) fals to satsfy the Newton equaton (Eq. 4.8). The theoretcal foundaton and convergence of the nexact Newton method s dscussed by Brown and Hndmarsh (1986) Krylov Subspace Projecton Method In ths subsecton the teratve Arnold s algorthm for the soluton of lnear system Ax = b s dscussed (Saad, 2003). For the nonlnear problem gven n Eq. 4.7 and can be wrtten as Fx ( ) 0, the vector b represents F( x m), the matrx A represents ( m) sm xm x m.gven the ntal value 0 ncrement 1 1 an orthogonal projecton method whch takes 0 where 0 0 Fx and the vector x represents the x to the orgnal lnear system, one consders κκ (, ) m Ar wth κ ( Ar, ) span r, Ar, Ar,..., A r (4.10) m 2 m r bax. The objectve of ths method s to obtan an approxmate soluton m affne subspace 0 m x from the x κ m of dmenson m by mposng the Galerkn condton that m b Ax s orthogonal to κ m. More detals about the method can be found n (Saad, 2003; Brown and Hndmarsh, 1986). The Arnold s algorthm can be descrbed as follows (Saad, 2003): 85

101 1. Computer 0 bax 0, : r 0, and v 2 1: r 0 2. Defne H m hj, set H 0, j 1,..., m m 3. for j 1,..., m, Do 4. Compute wj: Av j 5. for 1,..., j, Do 6. hj ( wj, v ) 7. wj wjhjv 8. End Do 9. Compute h j 1, j w, f h j j 1, j 0 2 set m : j and go to Compute vj 1 w j hj 1, j 11. End Do Compute ym Hm( e 1), and set xm x0 Vmy m Here ms the dmenson of the Krylov subspace, (.,.) s the Eucldean nner product,. s the 2 Eucldean norm, and 1 [1,0,0,...,0] T R m e. v,..., 1 v m s an orthonormal bass for κmand the matrx VAV s the upper Hessenberg matrx T m m Hm whose nonzero elements are the hj defned n the above algorthm. In the prevous algorthm as m gets large, a consderable amount of the work nvolved s n makng the vector v m1 orthogonal to all the prevous vector, v 1... v m. Saad (1983, 2003) proposed a modfcaton of Arnold s algorthm n whch the vector v m1 s only requred to be orthogonal to the p vectors v... 1 v. Ths leads to an algorthm called mp ncomplete orthogonalzaton method (IOM). IOM dffers from Arnold s algorthm n the modfed Gram-Schmdt orthogonalzaton process n Step 5; nstead of startng wth 1, one starts wth 0 where 0 max(1, mp 1). All the features and propertes of Alrnold s algorthm stll hold for IOM (Saad, 2003). m 86

102 The Hessenberg matrx obtaned n the above algorthm has a band structure wth a bandwdth m 1. Due to the structure of the upper Hessenberg matrx H, there s a convenent way to obtan m the LU factorzaton of H m by usng the LU factors of m1 H ( m 1). Therefor the approxmate soluton can be gven as x x V U L ( e ) (4.11) 1 1 m 0 m m m 1 where Hm LmU m, and L m s a unt lower bdagonal matrx, and U m s an upper trangular matrx. Defnng P V U, and 1 m m m z L ( e ), Eq can be wrtten as m 1 m 1 x x Pz (4.12) m 0 m m Due to the structure of U m, the vector p m can be calculated usng the prevous p s and m v as follows: m1 1 m m um u mm 1 p v p (4.13) Smlarly, because of the structure ofl m, the vector z m can be updated usng z m1 as z m1 z m (4.14) m where m lm, mm 1. The approxmate soluton can be updated at each teraton as x x p (4.15) m m1 m m One of the mportant practcal consderatons of the prevous algorthm s the choce of m, whch amounts to a stoppng crteron. An nterestng feature of the algorthm s that one does not have to obtan x m n order to compute baxm and t s easy to show that (Saad, 2003) 2 87

103 m bax m h (4.16) 2 m1, m umm The Arnold algorthm wth stoppng crtera can be effcently mplemented as follows: 1. Choose x 0, and computer 0 bax 0, : r 0, and v 2 1: r 0 2. For m 1,..., untl convergence Compute hm, 1... m, and v m1 as prevously descrbed. for m 1 ) otherwse m lm, mm 1 3. Update LU factorzaton of H m, and compute m( m 4. Update pm usng Eq. 4.13, and x musng Eq Compute m m bax m h 2 m1, m umm, f m go to 6.Otherwse go to 2 6. End Do Jacoban-Free Newton-Krylov Methods An extensve lterature about the Jacoban-free Newton-Krylov method can be found n (Knoll and Kenyes, 2004). It can be shown for the algorthms descrbed n the prevous subsecton that the matrx A s not needed explctly. One needs to compute the matrx-vector product Av only. Snce for the stff ODE, t s assumed that A F( x) I h J, where x s an approxmaton to the root of Fx ( ) by dfferent quotents of the form 0.Then the matrx-vector product Av n the above algorthm can be replaced 88 F( x) v F( xv) F( x ) (4.17) where s a scalar, the resultng algorthm can be referred as a fnte- dfference projecton method or Jacoban-free Newton Krylov Method. The choce of s mportant, f s too large the dervatves s poorly approxmated, and f t s too small the results of the fnte dfference s contamnated by floatng pont round off error. Approaches for choosng are dscussed by Knoll, and Keyes (2004). Brown presented the convergence theory for the combned nexact-

104 Newton/fnte-dfference projecton methods. The Arnold s or IOM algorthm usng the fnte dfference method (Jacoban-free) s presented as follows: 1. Choose x 0, and computeq F( x x ) F( x ) Setr 0 bq 0, : r 0, and v 2 1 : r 0, and q1: v 1 3. For m 1,..., untl convergence m m m m m Compute hm, 1... m, and m1 4. Update LU factorzaton of H m, and compute m( m 5. Update pm usng Eq. 4.13, and x musng Eq Compute Av q F( x v ) F( x ) v as prevously descrbed. for m 1 ) otherwse m lm, mm 1 6. Compute m m bax m h 2 m1, m umm, f m go to 6.Otherwse go to 3 7. End Do TLISMNI Newton-Krylov Algorthm Implementaton In ths secton, the use of Jacoban-free Newton-Krylov approach for solvng stff DAE s presented. The proposed algorthm wll employ the scaled Arnold s or IOM method (Brown and Hndmarsh, 1986), where the weght assocated wth component y of y durng the teraton s w RTOL y ATOL (4.18) n1 where RTOL, and ATOL are the relatve and absolute error tolerances, respectvely. In order to avod a bas when dealng wth smlar ODE systems of dfferent szes, we use the root mean square D dag(d,...,d ) (RMS) norm nstead of the Eucldean norm. Gven a dagonal matrx 1 N, where d w N, and N s the length of a vector y, the weghted RMS of a vector yapproxmatng y n 1 can be wrtten as y D y. Based on the scalng n the Arnold s algorthm or IOM RMS 2 algorthm, the TLISMNI-Newton-Krylov algorthm can be descrbed as follows: 89

105 1. Assumng that the state of the system s known at tme t n, the constrant Jacoban matrx C q can be constructed, and used wth Gaussan elmnaton procedure to determne set of system ndependent coordnatesq. Therefore, the vector of system generalzed coordnates q can be parttoned to dependent and ndependent coordnates as T T T q q q d, where q d s the set of dependent coordnates. 2. A predcton of the ndependent coordnates ( k q ) at tme tn 1, usng an explct ntegraton formula can be obtaned. Usng an mplct ntegraton formula such as HHT, n, 1 BDF, Park, or Trapezodal formula, one can predct the ndependent velocty q. ( k ) n, 1 3. Usng the values of the ndependent coordnates ( k q ) n, 1, the constrant equatons Cq,t 0 can be consdered as a nonlnear system of algebrac equatons n the dependent coordnates q ( k) dn, 1. Ths system can be solved teratvely usng sparse matrx technques and Newton-Raphson algorthm as descrbed by Eq. 4.4 for the dependent coordnates q. ( k) dn, 1 4. Usng the predcted ndependent coordnates q and veloctes ( k ) n, 1 q from step 2 and ( k ) n, 1 the dependent coordnates q determned from the prevous step; one can construct the ( k) dn, 1 constrant equatons at the velocty level (Eq. 4.5). The soluton of ths system of lnear ( k) equatons defnes the dependent veloctes q. dn, 1 90

106 5. Knowng all the coordnates and veloctes at tme tn 1, Eq. 4.2 can be constructed and solved usng sparse matrx technques for the acceleratons q and Lagrange multplers ( k) n1 λ. ( k) n1 6. The ndependent acceleratons can be dentfed and used to defne the state space equatons whch can be ntegrated forward usng the Jacoban-free Newton-Krylov algorthm descrbed before. If convergence s acheved go to Step 7 otherwse update to obtan q ( k1) n, 1, and q and go to Step 3. In case the convergence s not acheved wthn a specfc ( k1) n, 1 number of teratons, the tme step s reduced and the algorthm s restarted. 7. If the convergence crtera proposed n the followng secton s satsfed and the error s less than the user specfed tolerance, the coordnates, veloctes, and the soluton for the acceleraton and Lagrange multplers are accepted. In ths case, one needs to update hstory, and calculate the new step sze n order to advance the ntegraton. If convergence s not acheved or the error exceeds the specfed error tolerance, then the tme step should be reduced and the algorthm s restarted. In ths case, the algorthm goes to step 2 agan untl the convergence and error crtera are satsfed. 8. Ths process contnues untl the desred end of the smulaton tme s reached. The proposed algorthm takes advantages of the Jacoban-Free Newton-Krylov algorthm, and stll explots the sparse matrx structure to acheve mnmum storage. Furthermore, the constrants are satsfed at the poston, velocty, and acceleraton levels. Furthermore, no numercal dfferentaton of the external or nerta forces wth respect to the coordnates and veloctes s requred to obtan the Jacoban matrx. 91

107 4.4 Convergence Crtera The TLISMNI method has an teratve outer loop to solve for the coordnates, veloctes, acceleratons, and Lagrange multplers. The steps of the TLISMNI outer loop can be summarzed as follows: 1- Defne the system coordnates q 0 n1, and veloctes 0 q n1 2- Knowng acceleratons q q 0 n1 0 n1, and 0 q n1, one can construct the augmented form n Eq. 4.2 and solve for, and Lagrange multplers 0 λ n1 3- Usng smple teraton algorthm(tlsmni), or the Jacoban-Free Newton-Krylov an mplct ntegraton formula can be used to obtan q 1 n1, and 1 q n1 4- Check f the soluton converges, for example, one can check f specfed tolerance. q q k1 k n1 n1 k 1 q n1 less than the 5- If the convergence s acheved,then go to the next step otherwse go to step 3 for more teratons k 1, 2,... n The algorthm outlned above s equvalent to performng fxed pont teraton for the soluton of nonlnear equatons for q n1, and q n1 usng an mplct ntegraton formula (Graca et al., 1994; Shampne, 1981). It follows that the condton h J 1,where J s the maxmum norm of the Jacoban must hold for successve convergence. Fulfllng ths condton guarantees the convergence of the soluton. In other words, the TLISMNI s outer loop has a lnear convergence and to guarantee the convergence of teraton x m to obtan the exact soluton * x of the equatons Fx ( ) m 0, one has a lnear convergence as x x x x, where 0C 1, and. * * m 1 m C 92

108 denotes an N R norm. As * x s unknown, t s approprate to assume a lnear convergence wth a Cm m m m m convergence rate estmated as x 1x x x 1. Therefore, one can assume convergence s acheved for x m satsfyng * xm x, whch s approxmately satsfed f C x x, where CC/(1 C). More detals on the lnear convergence analyss can be m m1 found n (Brown and Hndmarsh, 1986). The condton used n ths secton mples that the teraton converges n a suffcent small ball around the root. It can be shown that TLISMNI s teraton s much less expensve than the Newton teratons and allows for much more rapd varaton of the step sze n order to acheve convergence. In order to ensure rapd convergence n a practcal mplementaton, the convergence rate C s selected to be much smaller than 1, for example 0.3. The goal s to have an algorthm that takes less than four teratons to converge, partcularly n the case of large scale problems. On the other hand, ths restrcton can slow TLISMNI smple teraton method and the mplct formula can lose ts propertes such as numercal dampng n case of HHT (Hussen and Shabana, 2010). In such a case, t s recommended to use the Jacoban-Free Newton- Krylov method (Knoll and Keyes, 2003) nstead of the TLISMNI smple teraton method. The convergence procedure for both the TLISMNI smple teraton method or the TLISMNI Newton- Krylov method can be descrbed as follows (Brown and Hndmarsh, 1986): 1. At the begnng of the teratons (tme step t n ) set C After number of teratons m 1, compute C x 1x x x 1 m m m m m 3. Update C max(0.2 CC, m ) 4. Check f x 1 x mn(1,1.5c) constant, then convergence acheved. Otherwse reduce the m m tme step and restart the teratons. The constant n Step 4 depends only on the order of the mplct ntegraton formula used. 93

109 4.5 Low Order Integraton Formulas As prevously mentoned, mplct ntegraton methods transform the MBS dfferental equatons of moton to a set of nonlnear algebrac equatons. These nonlnear algebrac equatons can be solved teratvely for the requred soluton. Explct methods can be very neffcent or fal to solve stff problems whch are characterzed by wdely separated egenvalues because of hgh frequency contents. Hgher order ntegraton formulas cannot be used effectvely for the soluton of such stff problems. For these problems, low order A-stable ntegraton formulas can be more effcent. Usng A-stable low order ntegraton formula, the restrcton on the sze of the tme step to mantan absolute stablty s no longer requred. In ths secton, dfferent low order ntegraton formulas wll be dscussed and recommendatons are made on the approprateness of each method for a partcular problem Park Method The frst ntegraton formula consdered n ths secton s Park method whch has order 2 and was proposed by Park (1975) as an mproved stffly stable method for drect ntegraton of nonlnear structural dynamc equatons. By combnng the Gear s two-step and three-step method, a superor stffly stable method was developed (Park, 1975). Park method can be appled to both stff and non-stff problems. The results ndcate that the Park method s second best after the trapezodal rule for non-stff problems and t appears to be stable for stff problems that nclude frctonal contact/mpact phenomena, as wll be demonstrated n the numercal results secton. Pogorelov, 1998 proposed the use of Park method for the soluton of stff constraned MBS applcatons. Park method does not requre any hstory dervatve nformaton, whch can cause numercal nstablty n nonlnear dynamcs problems even though the methods are uncondtonally stable. The 94

110 equatons that defne the generalzed coordnates and veloctes at tme tn 1 n Park method are, respectvely, gven by qn1 qn qn 1 qn2 hq (4.19) n and q n1 q n q n1 q n2 hq (4.20) n where h s the tme step, and n refers to vectors at tme t n. The local truncaton error of Park method n terms of the acceleratons s 1 2 δn 1 h q n1 q n (4.21) BDF Method The second ntegraton method consdered n ths nvestgaton s the second order backward dfferentaton (BDF2) method. Ths method was proposed by Gear (1971). The BDF2 method s a stffly A-stable method that has been wdely used for the soluton of stff problems due to ther good stablty propertes for such problems. The BDF2 equatons for the generalzed coordnates and veloctes are qn1 qn qn 1 hq (4.22) n and q n1 q n q n1 hq (4.23) n As t s clear from these equatons, the BDF2 method does not requre any hstory dervatve nformaton; therefore, t wll be stable wth stff problems such as t s the case wth Park method (Park, 1975). The local truncaton error for the BDF2 method s 95

111 2 2 δ n1 h q n1 q n (4.24) 9 Comparng the BDF2 method s truncaton error and the park method s truncaton error, one can see that park method can acheve the same accuracy as the BDF2 method for a larger tme step sze by a factor of approxmately Trapezodal Method The thrd ntegraton method that wll be consdered n ths nvestgaton s the trapezodal method whch s consdered the most accurate second order A-stable methods. The method does not damp out any frequency content from the system and t s uncondtonally stable for lnear problems, and condtonally stable for nonlnear systems (Cardona et al., 1989; Gourley, 1970). More detals about the stablty and the accuracy of the trapezodal method can be found n (Hughes, 1987). The trapezodal method equatons do requre hstory dervatve nformaton; therefore the method suffers from stablty problems wth stff problems. The trapezodal algebrac equatons that represent the generalzed coordnates and veloctes are gven, respectvely, as 1 q h n1 qn ( q n q n1) (4.25) 2 2 and 1 q h n1 q n ( qn q n1) (26) 2 2 The truncaton error usng the trapezodal method can be estmated as 1 2 δn 1 h q n1 q n (4.27) HHT/Newmark Method Solvng numercally stff ODE obtaned usng fnte element dscretzaton, requres the use of numercal methods wth good stablty propertes and controlled numercal dsspaton such as 96

112 Hlber Hughes Taylor (HHT) method. The foundaton of the HHT method s the Newmark method whch was proposed by Newmark (1959). The Newmark equatons can be used to defne, respectvely, the generalzed coordnates q n1 and the generalzed veloctes q n1 as follows: and 2 h qn 1qnhqn 12qn2q n1 (4.28) 2 n1 n n n1 q q h 1 q q (4.29) where and are constants. The dfferental equatons of moton and the constrant equatons can be wrtten n the form T t,, Mqq Cλ Q q q (4.30) q The Nemark method s frst order method that produces a second order accurate method when 12, and 14only, ths choce leads to the trapezodal method (Negrut; 2002). The Nemark method s uncondtonally stable when 0.5 2, where the hgh frequency dsspaton s obtaned when (Newmark; 1959). The HHT method ntroduces a numercal dampng parameter to the equatons of moton n order to allow for energy dsspaton and retan the order and stablty condton of the method. The modfed equatons of moton can be wrtten as 1 Mq C λ Q C λ Q 0 (4.31) T T q q n1 1 n1 1 n 97

113 In order to obtan a stable soluton usng the mplct HHT method, the followng relatons should be satsfed 0.3 0, 0.5 and The local truncaton error usng HHT method s 1 2 δn 1 h q n1q n (4.32) 6(1 ) 4.6 Error Control and Tme Step Selecton The error crtera and the tme step-sze selecton for the proposed TLISMNI algorthm are dscussed n ths secton. Usng the estmate of the truncaton errors δ n1 gven n the precedng secton for each ntegraton formula, the followng maxmum norm for the vector of truncaton error can be used as an estmate of the error at the current tme step tn 1 ; e δ Y n1 (28) where Y s a weghted vector such that Y max(1, q ). In order to accept the soluton at the tme step tn 1 the error must be less than the user specfed tolerance, such that e. The tme step-sze selecton s an mportant ssue n the mplementaton of effcent numercal ntegraton algorthms. A very small tme step-sze leads to unnecessary calculatons that may exceed the user requred accuracy and at the same tme negatvely mpact the computatonal effcency. On the other hand, a large tme step leads to large number of teratons for the teratve method to acheve convergence and at the same tme can have a negatve mpact on the accuracy. The new tme step sze s selected such that the truncaton error assocated wth each ntegraton method s wthn the user specfed tolerance. Ths selecton s made accordng to 98

114 h new e sh 0.5 e sh e e (4.29) where s s a safety factor, h s the current tme step sze. In case the tme step sze does not satsfy the user specfed error e, then the tme step sze s reduced and the teraton step s restarted. 4.7 Explct Adams Method Adams method s an explct predctor-corrector method for the numercal soluton of frst order ordnary dfferental equatons. Ths method cannot be used to drectly solve a system of dfferental and algebrac equatons. The algebrac equatons must be frst elmnated usng the generalzed coordnate parttonng technque, whch s n prncple equvalent to the embeddng technque (Roberson and Schwertasek, 1988; Shabana, 2010). The embeddng technque, that elmnates the reacton forces and the dependent coordnates, leads to a mnmum set of dfferental equatons expressed n terms of the ndependent coordnates (degrees of freedom) only. Usng the generalzed coordnate parttonng, one can wrte the system coordnates as T T T q q q where d q s the vector of ndependent coordnates or degrees of freedom, and q d s the vector of dependent coordnates. One can rewrte the equatons of motons n terms of the ndependent coordnates as follows (Roberson and schwertassek, 1988; Shabana, 2010): BMBq BMQ BQ (4.30) T T T d d d d d 99

115 T T -1 where Bd I Cq C -1 q s the velocty transformaton matrx, and d q d d Note that d tt 2 t q q q T Q 0 C Q d. Q C C C q q. Note that the augmented formulaton prevously dscussed n ths Chapter can be used to determne all the acceleraton. The ndependent acceleratons can then be dentfed and ntegrated forward n tme. Therefore, the explct Adams method can be used wth both the augmented formulaton and the embeddng technque. In order to use the explct Adams method, one has to transform the second order ordnary dfferental equatons to a system of frst order ordnary dfferental equatons (ODE). Ths s T accomplshed by ntroducng a new vector T T T y q q.takng the tme dervatve of ths vector and substtutng the value of q from the soluton of Eq. 30 yelds q y 1 T (4.31) B T T dmbd BdQ BdMQd Note that the rght hand sde of ths equaton s a functon of problem s reformulated as y fy,t q, q andt. Therefore, the orgnal, whch s the standard form of a frst order ODE that can be solved by Adams method. The explct Adams method used n ths study s the Adams predctorcorrector method documented n the book by Shampne and Gordon (1975). In ths method, a p predcted value y n1 at the tme step tn 1 based on the soluton at tme step t n and several prevous values of f s obtaned by usng the Adams-Bashforth formula: tn1 p n1 n m, n tn t y y P dt (4.32) where mn, t P s a Lagrange nterpolatng polynomal that nterpolates prevous m values of fy,t, that s, 100

116 wth n1 m m ttn 1 k Pmn, t f n 1 j j1 k1 tn 1 j t (4.33) n 1 k k j p p p y avalable, one can then fnd the value of 1 1, t n n n1 f y. The corrector Adams-Moulton p formula s then used to fnd a corrected value of y n1, whch s denoted by y n1 : tn1 * n1 n m, n tn y y P t dt. (4.34) where * mn, t P s an nterpolatng polynomal that nterpolates the prevous m ponts n addton p to the predcted value f n1, that s m m m P * p tt n 1 k ttn 1 k mn, t f n1 n1j k1 tn 1t f n1k j1 k0 tn 1j t n1k (4.35) k j One can then use the corrected value y n1 to evaluate the functon fn 1whch s used n the next tme step. A full documentaton of the procedures used n ths explct method for error control and selecton of the order and tme step can be found n the lterature (Shampne, and Gordon, 1975). 4.8 Numercal Examples In ths secton, numercal results, obtaned usng a smple pendulum, M113 tracked vehcle model, and ralroad vehcle models, are used to demonstrate the use of the TLISMNI algorthm proposed n ths nvestgaton for solvng large and complex stff systems that ncludes flexble bodes and contact/mpact forces. The results obtaned usng the TLISMNI algorthm and the explct Adams predctor-corrector method, are compared n terms of effcency and accuracy of the results. In addton, recommendatons are made on the approprateness of each ntegraton formula for a partcular problem. 101

117 4.8.1 Pendulum Example The pendulum used n ths example s assumed to be ntally horzontal, and fall under the effect of gravty forces. The pendulum model s developed usng the fnte element absolute nodal coordnate formulaton (ANCF). The beam n ths pendulum s dvded nto two-dmensonal fnte beam element along ts length as shown n Fg. 31 (Hussen et al., 2008). The pendulum s assumed to have undeformed length 0.4 m, cross sectonal area m the mass densty s assumed to be Kg m, the modulus of elastcty s assumed to be Nm, and the Poson s rato s assumed to be 0.3. In order to compare the performance of the TLISMNI algorthm and the explct Adams method, a dynamc smulaton of the stff flexble pendulum s performed by usng the two ntegraton methods. Fgure 32 shows that the mplct TLISMNI method damped out some hgh frequency oscllatons and ths ntegraton method produces a smoother soluton than the one obtaned usng the explct Adams method. Wth regard to the effcency, the TLISMNI method s 35 tmes faster than the Adams method for ths example. Also t s mportant to menton that there s no sgnfcant dfference between the two reference moton solutons, and therefore, the numercal dampng of the TLISMNI method does not have a sgnfcant effect on the rgd body moton of the beam, as wll be demonstrated usng more complex examples. Y Gravty force Cross secton X Fgure 31. Flexble pendulum ntal confguraton 102

118 4.0x10-7 Y-nodal deformaton (m) 2.0x x x Tme (s) Fgure 32: Nodal deformaton for md-node ( Adams, HHT) Rgd Tracked Vehcle Model The tracked vehcle model shown n Fg. 33 s for a challengng model that represents the M113 armored personnel carrer whch conssts of a chasss, 2 dlers, 2 sprockets, 10 road-wheels, and 128 total track lnks (64 for each track). Fgure 5 shows the engagement of the track lnks wth some of the vehcle components. The vehcle has a suspenson system that conssts of road arms placed between the road wheels and the chasss as well as shock absorbers connected to each road arm as shown n Fg

119 Fgure 33: M113 tracked vehcle model Fgure 34: Suspenson system layout of M113 tracked vehcle Table 5 shows the stffness and the dampng coeffcents of the contact models and the suspenson system used for ths model. The road arms and the sprockets are connected to the chasss by revolute jonts, and the road arms are connected to the road-wheels by revolute jonts. Table 5: Contact parameters Parameters Sprocket-Track Contact Roller-Track Contact Ground-Track Contact k N/m N/m N/m c N s/m N s/m N s/m μ

120 The track lnks are connected to each other usng revolute jonts. Tensoners are added to the system, each dler s connected to a tensoner wth a revolute jont and the tensoner s connected to the chasss wth a prsmatc jont to ensure only relatve translaton. The model n ths example s subjected to prescrbed sprocket angular velocty that ncreases lnearly untl t reaches a constant value after 8 seconds. Both generalzed coordnate parttonng approach and the recursve approach are used for the soluton of ths tracked vehcle model. The number of equatons used n the two formulatons s 2184 and 648, respectvely. Park ntegraton method was found to be more effcent for ths example compared to BDF2, HHT, and Trapezodal ntegraton methods. The 6 smulaton s carred out for 10 seconds wth error tolerance 110 for the explct Adams method 7 and 1 10 for Park Integraton method. Fgures show the results of the model usng the recursve approach wth the explct Adams method, and TLISMNI Park method. The results show very good agreement wth maxmum error dfference less than 2% n the acceleraton/force results and less than 1% n both the poston and velocty results. For ths example, the total smulaton usng the TLISMNI algorthm wth Park method was at least fve tmes faster than Adams method. All the smulatons were performed on Wndows 7, 3.40 GHZ CPU computer. It s mportant to menton that as the smulaton tme ncreases as the sprocket angular velocty ncreases, the TLISMNI algorthm wth Park ntegraton method becomes more effcent compared to the Adams explct method. 105

121 12 Forward poston (m) Tme (s) Fgure 35: Chasss forward poston ( Adams, Park) Forward velocty (m/s) Tme (s) Fgure 36: Chasss forward velocty ( Adams, Park) 106

122 Euler Parameter Vertcal acceleraton (m/s) Tme (s) Fgure 37: Road-wheel Euler parameter ( Adams, Park) Tme (s) Fgure 38: Road-wheel vertcal acceleraton ( Adams, Park) 107

123 1.0 Vertcal poston (m) Tme (s) Fgure 39: Track lnk vertcal poston ( Adams, Park) Vertcal velocty (m/s) Tme (s) Fgure 40: Track lnk vertcal velocty ( Adams, Park) 108

124 35 30 Forward poston (m/s) Tme (s) Fgure 41: Chasss forward poston ( Adams, Park, BDF2) Forward velocty (m/s) Tme (s) Fgure 42: Chasss forward velocty ( Adams, Park, BDF2) 109

125 5 Angular velocty (rad/s) Tme (s) Fgure 43: Road-wheel angular velocty ( Adams, Park, BDF2) In wth the case of 10 rad/sec sprocket angular velocty, TLISMNI algorthm wth Park was found to be more than 10 tme faster than Adams explct method. The chasss forward poston and velocty are shown n Fgs. 35 and 36, respectvely. The results presented n these fgures show a very good agreement between TLISMNI Park ntegraton method and Adams explct method. Euler parameter and vertcal acceleraton of one of the road-wheels are shown n Fgs. 37 and 38, respectvely, whle the vertcal poston and velocty for one of the track lnks are shown n Fgs. 39 and 40, respectvely. Fgures 41 and 42 show the forward poston and velocty of the M113 tracked vehcle model wth 25 rad/s sprocket angular velocty usng the generalzed coordnate 5 parttonng approach wth error tolerance 110 for explct Adams, Park, and BDF2 Integraton methods, whle Fg. 43 shows the angular velocty of one of the road-wheels. The results show very good agreement for the poston, velocty, and acceleraton wth maxmum dfference less than 3% n the acceleraton results. The results obtaned also show that the TLISMNIN BDF2 110

126 results have a better agreement than the TLISMNIN Park results wth Adams results. The CPU tme usng TLISMNI Park and BDF2 methods was found to be at least sx tmes faster than the explct Adams method on the same machne. It s also mportant to menton that the recursve approach was found to be more effcent than the augmented formulaton for such a complex chan problem Flexble Tracked Vehcle Model In ths model, The M113 vehcle model shown n Fg. 33 s used, where a new complant contnuum-based jont formulaton s used for the jont formulaton between the track lnks. In the numercal nvestgaton presented n ths Chapter, a three-dmensonal cable element s used to model the flexblty of the chan lnks (Walln et al, 2013; Hamed et al, 2014; Gerstmayr and Shabana, 2006). As reported by Hamed et al. (2014), the use of the new ANCF fnte element mesh makes the CPU tmes of the augmented formulaton and the recursve approach approxmately the same because the chan revolute jont constrants are elmnated at a preprocessng stage. The most effcent ntegraton method to be used wth ths model was found to be the HHT ntegraton method that allows flterng out hgh frequences. The number of equatons for that model s 2192, and the length of the smulaton tme s 10 sec wth error tolerance wth Adams method and 110 wth TLISMNI HHT method. It was observed that n order to capture correctly the rotatonal coordnates usng the TLISMNI HHT ntegraton method, the HHT error tolerance should be three orders of magntude tghter than the Adams error tolerance wth. The results show very good agreement wth maxmum dfference less than 0.1% n the acceleraton results, whle the smulaton tme sgnfcantly reduced usng the TLISMNI HHT method. The smulaton tme usng TLISMNI HHT method was at least 8 tmes faster than that of the explct Adams. It s mportant to menton that as the smulaton tme and the angular velocty 111

127 of the sprocket ncreases, the TLISMNI HHT method becomes more effcent. Fgures 44 and 45 show the chasss forward poston and velocty, whle Fg. 46 shows the chasss vertcal acceleraton. The angular velocty and acceleraton of one of the road-wheels are shown n Fgs. 47 and 48, respectvely, whle Fgures show the global nodal poston and velocty for a certan node n the ANCF fnte element mesh. It can be shown from these results that although the TLISMNI HHT algorthm employs numercal dampng to flter out the hgh frequences, such a method does not damp out any of the frequences assocated wth the rgd body modes or any mportant deformaton modes n ths partcular example. Numercal expermentatons show that decreasng the numercal dsspaton parameter wll ncrease the numercal dsspaton, whle at the same tmes reduces the tme step as can be demonstrated from Eq Therefore, t s mportant to select the proper value for the numercal dsspaton n order to ncrease the tme step Forward poston (m) Tme (s) Fgure 44: Chasss forward poston ( Adams, HHT) 112

128 7 6 Forward velocty (m/s) Tme (s) Fgure 45: Chasss forward velocty ( Adams, HHT) 26 Vertcal acceleraton (m/s 2 ) Tme (s) Fgure 46: Chasss vertcal acceleraton ( Adams, HHT) 113

129 5 Forward velocty (m/s) Tme (s) Fgure 47: Road-wheel angular velocty ( Adams, HHT) Angular acceleraton (rad/s 2 ) Tme (s) Fgure 48: Road-wheel angular acceleraton ( Adams, HHT) 114

130 Vertcal poston (m) Tme (s) Fgure 49: Vertcal global nodal poston ( Adams, HHT) Longtudnal poston(m) Tme (s) Fgure 50: Longtudnal global nodal poston ( Adams, HHT) 115

131 10 Vertcal velocty (m/s) Longtudnal velocty (m/s) Tme (s) Fgure 51: Vertcal global nodal velocty ( Adams, HHT) Tme (s) Fgure 52: Longtudnal global nodal velocty ( Adams, HHT) 116

132 In the flexble tracked vehcle example dscussed n ths secton, the ANCF three-dmensonal cable element was used to model the flexblty of the chan lnks. Ths element, however, does not capture the cross secton and sheer deformatons. Several smulatons have were carred out usng the fully parameterzed three-dmensonal beam element to model the flexblty of the chan lnks. The general contnuum mechancs approach and the elastc lne approach are used to formulate the structural element elastc forces. The use of the general contnuum mechancs approach leads to the ANCF coupled deformaton modes ncludng the Posson modes. These modes couple the cross secton deformaton, bendng, and extenson of the structural elements. On the other hand, n the elastc lne approach, all the deformaton modes are defned along the beam centerlne, and the curvature expresson s used to defne the bendng strans. Usng the elastc lne approach to formulate the elastc forces for the three-dmensonal fully parameterzed beam element used n modelng the chan of the tracked vehcle example, the HHT TLISMNI method was found to be at least 18 tmes faster than the explct Adams method wth very good agreement n the results. Usng the contnuum mechancs approach t was dffcult to obtan the soluton usng the explct Adams method due to the stffness of the equatons, whle usng the HHT TLISMNI methods the results are obtaned effcently and accurately compared to the elastc lne approach Pantograph/ Catenary Ralroad Vehcle Example The pantograph/catenary model shown n Fg. 53, s ntegrated wth a ralroad vehcle model that conssts of 14 rgd bodes ncludng the track, four wheelsets, two frames, four equalzers, two bolsters, and the car body. The model has two revolute jonts connectng the frames and bolsters, 48 bushng elements connectng the bodes, and eght bearng elements between the wheelsets and the equalzers. The pantograph s modeled after the CX pantograph and s composed of sx rgd bodes: a lower arm, upper arm, lower lnk, upper lnk, plunger, and a panhead as shown n Fg. 117

133 54. The pantograph system has three revolute jonts and four sphercal jonts connectng ts bodes to each other and to the ralroad vehcle. More detals on the model ncludng the nerta propertes as well as the ntal global postons and orentatons of the bodes for the pantograph model are reported by Patel et al. (2014). The catenary contact wre s supported by dropper (sprng/damper) elements from a fully constraned messenger wre as shown n Fg. 55. Fgure 53: Pantograph/catenary ralroad vehcle model Fgure 54: Artculated pantograph System 118

134 Fgure 55: Catenary model The contact wre s modeled usng 16 fully parameterzed three-dmensonal ANCF beam elements wth the followng propertes; densty modulus of rgdty N/m 3 11 Kg/ m, modulus of elastcty N/m, and.the number of equatons of that system s 744, whle the length of the smulaton tme s 3.87 seconds. The car body s constraned to move wth 20 m/s 5 over a tangent track, the error tolerance used wth Adams method s 1 10 and wth TLISMNI- 7 HHT method s1 10. The results obtaned usng both Adams and TLISMNI-HHT method show very good agreement wth maxmum dfference 1% as shown n Fgs The angular velocty of one of the wheelsets s shown n Fg. 56, whle Fgs. 57 and 58 show the forward global poston and velocty of one of the nodes on the catenary, respectvely. The tme requred for TLISMNI HHT smulaton was found to be about 8 tmes faster than the tme requred by Adams method. 119

135 Angular velocty (rad/s) Tme (s) Fgure 56: Wheelset angular velocty ( Adams, HHT) Poston (m) Tme (s) Fgure 57: Longtudnal global nodal poston ( Adams, HHT) 120

136 2 1 Velocty (m/s) Tme (s) Fgure 58: Longtudnal global nodal velocty ( Adams, HHT) 4.9 Concludng Remarks The objectve of ths chapter s to ntegrate the Newton-Krylov projecton method n a MBS algorthm based on two-loop mplct sparse matrx numercal ntegraton (TLISMNI) procedure wth the goal to mprove the effcency and robustness of the TLISMNI method when used for the numercal soluton of constraned complex rgd and flexble MBS dfferental and algebrac equatons. The smple teratons and Jacoban-Free Newton-Krylov approaches are used n the TLISMNI mplementaton. The TLISMNI method does not requre numercal dfferentaton of the forces, allows for an effcent sparse matrx mplementaton, and ensures that the algebrac constrant equatons are satsfed at the poston, velocty, and acceleraton levels. In the augmented formulaton and recursve method used n ths nvestgaton, the constrant equatons are satsfed at all levels. Dfferent low order ntegraton formulas such as HHT, whch ncludes numercal dampng, Park, Trapezodal, and BDF2 methods were used and recommendatons on the 121

137 approprateness of each method for a partcular problem are made. TLISMNI mplementaton ssues ncludng step sze selecton, convergence crtera, error control, and the effect of the numercal dampng were dscussed. Smple pendulum, complex rgd and flexble tracked vehcle, and ralroad vehcle models were used to demonstrate the use of the proposed TLISMNI method. A comparson between the results obtaned usng the TLISMNI algorthm and the explct Adams predctor-corrector method s presented and show good agreement. On the other hand, usng TLISMNI method whch does not requre numercal dfferentaton of the forces and allows for an effcent sparse matrx mplementaton for solvng complex and very stff structure problems sgnfcantly mproves the smulaton tme. For the rgd body model consdered n ths nvestgaton, the TLISMNI s at least fve tmes faster than the explct Adams method. Usng the TLISMNI algorthm wth ntegraton formulas that employ numercal dampng such as HHT n the smulaton of flexble body models consdered n ths study can acheve up to thrty fve tmes faster smulaton compared to Adams method. Nonetheless, t s mportant to menton that there are cases of non-stff problems n whch the use of explct Adams method can be more effcent than the TLISMNI methods. The use of the Jacoban-Free Newton-Krylov approach nstead of the smple teraton approach mproves the convergence and accuracy of the TLSMNI method. 122

138 CHAPTER 5 CONCLUSIONS AND FUTURE WORK 5.1 Concluson Development of computatonal methods, formulatons, and algorthms to study nterconnected bodes that undergo large deformaton, translatonal, and rotatonal dsplacements s the man focus for ths thess. MBS dynamcs s used n the analyss and vrtual prototypng of varous systems such as structural, mechancal, and bologcal systems, consequently, MBS algorthms has a wde range of applcatons n ndustry. Therefore, the development of effcent and accurate computatonal methods, formulatons, and algorthms to study MBS dynamcs s necessary. Ths thess dscusses the use of the concept of non-nertal coordnates and mplct numercal ntegratons methods to solve stff MBS dfferental/algebrac equatons. Complex MBS examples that consst of rgd and flexble bodes are used as examples n order to demonstrate the use of these developed algorthms. One of the man contrbutons of ths thess s to ntroduce the concept of the nertal and non-nertal coordnates to obtan an effcent soluton for practcal MBS applcatons. Inertal coordnates have generalzed nerta forces assocated wth them, whle the non-nertal coordnates have no generalzed nerta forces. In order to avod havng a sngular nerta matrx and/or hgh frequency oscllatons, the second dervatves of the non-nertal coordnates are not used when formulatng the system equatons of moton n ths study. In ths case, the system coordnates are parttoned nto two dstnct sets; nertal and non-nertal coordnates. The use of the prncple of vrtual work leads to a coupled system of dfferental and algebrac equatons expressed n terms of the nertal and non-nertal coordnates. The dfferental equatons are used 123

139 to determne the nertal acceleratons whch can be ntegrated to determne the nertal coordnates and veloctes. The non-nertal coordnates are determned by usng an teratve algorthm to solve a set of nonlnear algebrac force equatons obtaned usng quas-statc equlbrum condtons. The non-nertal veloctes are determned by solvng these algebrac force equatons at the velocty level. The non-nertal coordnates and veloctes enter nto the formulaton of the generalzed forces assocated wth the nertal coordnates. Usng the concept of non-nertal coordnates and the resultng dfferental/algebrac equatons obtaned n ths thess leads to sgnfcant reducton n the numbers of state equatons, system nertal coordnates, and constrant equatons; and allows avodng a system of stff dfferental equatons that can arse because of the relatvely small mass. The development of accurate nonlnear longtudnal tran force models s necessary n order to better understand ralroad vehcle dynamc scenaros that nclude brakng, tracton, and deralments. Car coupler forces have sgnfcant effects on the longtudnal tran dynamcs and stablty. Usng the concept of non-nertal coordnate developed n ths thess allows developng of a more detaled coupler model that captures the coupler knematcs wthout sgnfcantly ncreasng the number of state equatons and the dmenson of the problem. The coupler model proposed n ths thess allows for the car bodes to have arbtrary dsplacements, also avods havng a stff system of dfferental equatons that can result from the use of relatvely small masses. By assumng the coupler nerta neglgble compared to the car body nerta, one can dentfy two dstnct sets of coordnates; nertal and non-nertal coordnates. The nertal coordnates descrbe the moton of the car bodes and have nertal forces assocated wth them, whle the non-nertal coordnates have no nerta forces assocated wth them. Gven the nertal coordnates and veloctes, the nonlnear coupler force equatons are solved teratvely for the coupler non-nertal 124

140 coordnates. The obtaned non-nertal coordnates are used n the formulaton of the generalzed forces actng on the car bodes. In order to n order to examne the effcency of usng the concept of non-nertal coordnates, a comparatve study of the nertal and non-nertal coordnate coupler models s conducted. In ths comparatve study the effect of neglectng the coupler nerta s examned. The relatvely small nerta of the coupler can lead to hgh frequency oscllatons n the soluton; requrng the use of smaller ntegraton tme steps and sgnfcantly ncreasng the CPU tme. In order to address ths problem, a non-nertal coupler model s developed by replacng the coupler dfferental equatons wth quas-statc nonlnear algebrac force equatons. These algebrac equatons are teratvely solved usng a Newton-Raphson method n order to determne the nonnertal coordnates. The non-nertal veloctes are determned by solvng these quas-statc equatons at the velocty level. The non-nertal coordnates and veloctes are then used n formulaton of the generalzed coupler forces assocated wth the coordnates of the car bodes. In order to examne the effect of neglectng the coupler nerta on the accuracy of the soluton and the computatonal effcency, an nertal coupler model s developed usng MBS algorthms. The results obtaned usng the nertal and non-nertal coupler models are compared. The numercal results obtaned n ths study showed that the neglect of the coupler nerta does not have a sgnfcant effect on the accuracy of the soluton. On the other hand, the neglect of ths nerta leads to sgnfcant mprovement n the computatonal effcency. The results obtaned showed that the LTD mplementaton that neglects the effect of the coupler nerta becomes more effcent as the number of cars ncreases. An egenvalue analyss and FFT are used to dentfy the frequences assocated wth the coupler nerta. As dscussed n ths study, these hgh frequences do not appear when the non-nertal coupler model s used. LTD algorthms tend to be smpler as compared to 125

141 MBS algorthms that requre the use of DAE solver. LTD are desgned for the effcent soluton and analyss of longtudnal tran forces of long conssts. Car coupler forces have a sgnfcant effect on the behavor of trans n response to brakng, tractve, and resstance forces. For ths reason, t s mportant to develop effcent coupler models that can be n LTD algorthms. The development of such an effcent coupler model requres modfyng exstng LTD algorthms to nclude a procedure for solvng dfferental and algebrac equatons smultaneously as demonstrated n ths thess. The dynamcs of large and complex multbody systems that nclude flexble bodes and contact/mpact pars s governed by stff equatons. Explct ntegraton methods can be very neffcent and often fal n the case of stff problems. The use of mplct numercal ntegraton methods s recommended n ths case. To ths end, the thess presents a new and effcent mplementaton of the two-loop mplct sparse matrx numercal ntegraton (TLISMNI) method proposed for the soluton of constraned rgd and flexble multbody system (MBS) dfferental and algebrac equatons. Another contrbuton of ths thess s to ntegrate the Newton-Krylov projecton method n a MBS algorthm based on two-loop mplct sparse matrx numercal ntegraton (TLISMNI) procedure wth the goal to mprove the effcency and robustness of the TLISMNI method when used for the numercal soluton of constraned complex rgd and flexble MBS dfferental and algebrac equatons. The smple teratons and Jacoban-Free Newton-Krylov approaches are used n the TLISMNI mplementaton. The TLISMNI method does not requre numercal dfferentaton of the forces, allows for an effcent sparse matrx mplementaton, and ensures that the algebrac constrant equatons are satsfed at the poston, velocty, and acceleraton levels. In the augmented formulaton and recursve method used n ths nvestgaton, the constrant equatons are satsfed at all levels. Dfferent low order ntegraton formulas such as 126

142 HHT, whch ncludes numercal dampng, Park, Trapezodal, and BDF2 methods were used and recommendatons on the approprateness of each method for a partcular problem are made. TLISMNI mplementaton ssues ncludng step sze selecton, convergence crtera, the error control, and effect of the numercal dampng were dscussed. Smple pendulum, complex rgd and flexble tracked vehcle, and ralroad vehcle models were used to demonstrate the use of the proposed TLISMNI method. A comparson between the results obtaned usng the TLISMNI algorthm and the explct Adams predctor-corrector method s presented and show good agreement. On the other hand, usng TLISMNI method whch does not requre numercal dfferentaton of the forces and allows for an effcent sparse matrx mplementaton for solvng complex and very stff structure problems sgnfcantly mproves the smulaton tme. For the rgd body model consdered n ths nvestgaton, the TLISMNI s at least fve tmes faster than the explct Adams method. Usng the TLISMNI algorthm wth ntegraton formulas that employ numercal dampng such as HHT n the smulaton of- the flexble body models consdered n ths study can acheve up to thrty fve tmes faster smulaton compared to Adams method. Nonetheless, t s mportant to menton that there are cases of non-stff problems n whch the use of explct Adams method can be more effcent than the TLISMNI methods. The use of the Jacoban-Free Newton-Krylov approach nstead of the smple teraton approach mproves the convergence and accuracy of the TLSMNI method. 5.2 Future Work Based on the work presented n ths thess, the followng ssues need to be addressed n future nvestgatons: 1. Recently, the concept of nertal and non-nertal coordnates s mplemented wth a recursve formulaton. The concept can be expended to be used n general MBS 127

143 formulaton n order to avod a system of stff dfferental/algebrac equatons that can arse because of the relatvely small nerta coeffcent n some MBS applcatons such as vehcle systems. 2. Precondtoned and parallelzaton technques need to be explored n a computatonal framework based on the TLISMNI Newton-Krylov approach, whch shows a good performance n mprovng the convergence for the stff equatons. 3. The use of the generalzed coordnate parttonng proved to be effcent n the case of dfferental/algebrac equaton problems. More nvestgatons are requred n order to develop better crtera for the best selecton of the ndependent coordnates and the tme to change these coordnates. Ths s necessary n order to avod sngularty problems arsng n applcatons that nclude closed chans, and mprove the performance of the TLISMNI method. 128

144 APPENDIX A In order to determne the non-nertal veloctes, one needs to determne the dervatves of the force algebrac equatons used n the poston analyss wth respect to tme. The dfferentaton of the forces wth respect to tme wll lead to terms that depend on the non-nertal veloctes and terms a a q q that depend on the nertal veloctes. For a gven vector a, we wll use the notaton to denote the part of the tme dervatve of a wth respect to the nertal coordnate of the car body. It follows that L Bd LBd q ω L Bd ω d B, q Lr L r q ω L r q (A.1) In ths equaton, for car body j. One also has ω s the angular velocty of the car body. Smlar equatons can be developed d R ω u R ω u (A.2) j j j j rr Q Q Usng Eq. A.2, one can wrte l 1, ˆ l l d d d d d j l j j j j j j T j j r rr r rr r rr rr rr r j l 2 r (A.3) The part of the dervatves of the forces assocated wth the nertal coordnates can then be wrtten as 129

145 APPENDIX A (contnued) ˆ fb kd B d ω d B (A.4) j j j ˆj j j j ˆj frr krrlr drr krr lr l ro drr 130

146 APPENDIX B 131