Dynamical Analysis of Vehicle Systems: Theoretical Foundations and Advanced Applications (CISM International Centre for Mechanical Sciences)

CISM COURSES AND LECTURES Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider...

Author: W. Schiehlen | W. Schiehlen

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CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schrefler - Padua

Executive Editor Paolo Serafini - Udine

The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 497

DYNAMICAL ANALYSIS OF VEHICLE SYSTEMS THEORETICAL FOUNDATIONS AND ADVANCED APPLICATIONS EDITED BY WERNER SCHIEHLEN UNIVERSITY OF STUTTGART, GERMANY

This volume contains 226 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2007 by CISM, Udine Printed in Italy SPIN 12068067

All contributions have been typeset by the authors.

ISBN 978-3-211-76665-1 SpringerWienNewYork

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W. Schiehlen

43

O>KR @HH= KH:= ?:BK KH:= ;:= KH:=

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44


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W. Schiehlen

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46

Vehicle and Guideway Modelling: Suspension Systems BGINM

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48


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W. Schiehlen

49

!HK A:KFHGB< >Q :?HK>F>GMBHG>= LM:G=:K=L @BO> : IK> K>E:MBHG MA:M LNBML P>EE ?HK O>AB<E> =RG:FB A:KFHGB< :K:MBHG / B LBG B

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50


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W. Schiehlen

51

>QIHLNK> MBF>

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/

+ *! * * ) =

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52


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6 D B / B 6 D9 B D B 5 / B

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W. Schiehlen

53

/A> LA:I> EM>K H? L>K A A:L MA> ?HEEHPBG@ ?HKF :KL / A / A A ;

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54

Vehicle and Guideway Modelling: Suspension Systems *%& "'&38

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6HE FUNDAMENTAL GUIDANCE SYSTEM OF RAILWAYS CONSISTS OF A ANGED WHEEL SET ROLLING ON A TRACK WITH TWO RAILS # CLEARANCE BETWEEN THE OUTER EDGE OF THE ANGE OF THE WHEEL AND THE INNER FACE OF THE RAIL IS PROVIDED TO PREVENT SQUEEZING OF THE WHEELS BETWEEN THE RAILS # PLANE THROUGH THE WHEELSET CONTAINING ITS CENTRE LINE WILL INTERSECT THE WHEEL TREAD IN A CURVE CALLED 6*' 9*''. 241.' 6HE ANGLE BETWEEN THE TANGENT TO THE WHEEL PROLE AND THE CENTRE LINE OF THE WHEEL SET MEASURED IN RADIANS IS CALLED 6*' %106#%6 #0).' +T IS MADE POSITIVE IN MOST POINTS ON THE WHEEL TREAD 6HE POSITIVE CONTACT ANGLE MAKES THE WHEEL SET KINEMATICALLY SELFCENTERING 0EGATIVE CONTACT ANGLES THAT MAY DEVELOP AS A RESULT OF WEAR MAKE THE MOTION OF THE WHEEL SET UNSTABLE AND SHOULD BE AVOIDED #S LONG AS THE WHEEL SET ROLLS ALONG A STRAIGHT TRACK THE ANGES OUGHT NOT CONTACT THE RAILS &ISTURBANCES IN THE TRACK GEOMETRY WILL PUSH THE WHEEL SET AWAY FROM ITS CENTRED POSITION BUT OWING TO THE STABILIZING EECT OF THE WHEEL PROLE THE WHEEL SET WILL THEN OSCILLATE HORIZONTALLY AROUND THE CENTRE LINE OF THE TRACK (LANGE CONTACT WILL ONLY OCCUR

* %" !#%* %$(" % %,("* -% &( * !( %%!# )+''&(* & * '('(*!&% & * ) #*+( %&*)

76

Dynamics of Railway Vehicles and Rail/Wheel Contact

IF THE DISTURBANCES ARE SUCIENTLY LARGE 9HEN IT OCCURS THE RESTORING FORCE SUDDENLY GROWS VERY FAST 2ROBLEMS WILL OBVIOUSLY ARISE WHEN A VEHICLE WITH A XED WHEEL BASE MUST NEGOTIATE A CURVE 6HE LONGER THE WHEEL BASE THE SMALLER THE CURVATURE MUST BE IN ORDER TO PREVENT ANGE CONTACT AND SQUEEZING 1N THE OTHER HAND IT IS ALSO OBVIOUS THAT THE LONGER THE XED WHEEL BASE IS THE SMALLER WILL THE YAW MOTION BE OF A VEHICLE RUNNING ON A STRAIGHT BUT IMPERFECT TRACK 6HE LONGER WHEEL BASE THEREFORE LEADS TO AN IMPROVEMENT IN RIDING QUALITY ON A STRAIGHT TRACK +N AN EORT TO SOLVE THE CONICT BETWEEN GOOD CURVING BEHAVIOUR AND RIDING QUALITY THE RAILWAY COMPANIES INTRODUCED RAIL VEHICLES WITH TWO OR MORE WHEEL SETS IN UNDERCARRIAGES ONE UNDERCARRIAGE UNDER EACH END OF THE LONG VEHICLE INSTEAD OF THE SINGLE WHEEL SETS SEE (IGURE +N 0ORTH #MERICA WHERE THEY WERE RST INTRODUCED IN THE MIDDLE OF THE NINETEENTH CENTURY THE UNDERCARRIAGES ARE CALLED 647%-5 +N 'UROPE THEY ARE CALLED $1)+'5 BO GIZ 6HE CAR BODY OF BOGIE VEHICLES IS SUPPORTED BY THE FRAME OF THE BOGIES AND THE BOGIE FRAME IS SUPPORTED BY THE WHEEL SETS 5YSTEMS OF SPRINGS AND DAMPERS ARE BUILT INTO THE SUPPORTS TO IMPROVE THE RIDING QUALITY IN THE CAR BODY 6HE SUPPORT SYSTEM BETWEEN THE WHEEL SETS AND THE BOGIE FRAME IS CALLED 6*' 24+/#4; 5752'05+10 AND THE SUPPORT SYSTEM BETWEEN THE FRAME AND THE CAR BODY IS CALLED 6*' 5'%10 5752'05+10 (REIGHT WAGONS OFTEN HAVE ONLY ONE SUPPORT SYSTEM 6HE #MERICAN THREEPIECEFREIGHT TRUCK HAS ONLY A SECONDARY SUSPENSION 6HE 'UROPEAN STANDARD TWOAXLE FREIGHT WAGON HAS NO BOGIES AND THEREFORE ONLY ONE PRIMARY SUSPENSION SYSTEM

(&2/$ # FOURAXLE BOGIE PASSENGER CAR 6HE RADIUS OF THE WHEELS DECREASE FROM THE ANGE TO THE ELD SIDE AT LEAST WHEN THEY ARE NEW 6HE STABILIZING EECT OF THE WHEEL PROLE IS DUE TO THE DIERENCE IN ROLLING RADII WHEN THE WHEEL SET RUNS O CENTRE 9HEN ONE OF THE WHEELS OF A WHEEL SET RUNS ON A LARGER RADIUS THAN THE OTHER WHEEL THEN THE WHEEL SET WILL TURN BACK TOWARDS THE EQUILIBRIUM POSITION WHERE THE ROLLING RADII ARE EQUAL 9ITHOUT DAMPING IE IN THE PURELY KINEMATIC SITUATION THE WHEEL SET WILL OVERSHOOT THE EQUILIBRIUM POSITION AND THE WHEEL SET WILL START YAW OSCILLATIONS 6HE IMPORTANT INUENCE OF THIS SOCALLED EECTIVE CONICITY HAS BEEN RECOGNIZED FOR SOME TIME 0OTICE THAT THE EECTIVE CONICITY DEPENDS ON THE LATERAL AND YAW POSITIONS OF THE WHEEL SET AND IT VARIES WITH THE DISPLACEMENTS 6HE STABILIZING ACTION OF THE WHEEL PROLE IS THE CAUSE OF A KINEMATIC INSTABILITY WHICH WAS TREATED BY -LINGEL IN A CELEBRATED PAPER 6HE OSCILLATION THAT RESULTS FROM

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THIS KINEMATIC INSTABILITY IS 0'764#..; 56#$.' 14 ;#27018 56#$.' 6HESE CONCEPTS WILL BE DENED LATER 6HE -LINGEL INSTABILITY IS A KINEMATIC PRODUCT AND IT MUST NOT BE CONFUSED WITH THE SOCALLED *706+0) WHICH IS AN ASYMPTOTICALLY STABLE OSCILLATING MOTION THAT IS A PRODUCT OF THE DYNAMICS +N GENERAL HOWEVER THE EECTIVE CONICITY HAS A DESTABILIZING EECT MEANING THAT 6*' %4+6+%#. 52''& IT IS THE LOWEST SPEED OF THE VEHICLE WHERE HUNTING IS POSSIBLE WILL DECREASE WITH INCREASING VALUES OF THE EECTIVE CONICITY FOR XED VALUES OF ALL OTHER PARAMETERS 6HE WHEEL PROLE THEREFORE HAS A STRONG INUENCE ON THE DYNAMICS OF THE VEHICLE 9ITH WHEEL WEAR THE EECTIVE CONICITY INCREASES #TTEMPTS HAVE BEEN MADE TO DESIGN WHEEL PROLES WITH AN EECTIVE CONICITY THAT IS INDEPENDENT OF OR VERY LITTLE DEPENDENT ON THE WEAR 6HESE WHEEL PROLES MUST HAVE A PROLE THAT RESEMBLES THE RAIL PROLE IN ORDER TO DISTRIBUTE THE WEAR MORE EVENLY +T MEANS THAT THE NEW PROLE IS MADE TO LOOK LIKE A WORN ONE OR IN OTHER WORDS IN ORDER TO PREVENT A DESTABILIZING EECT OF THE WEAR THE STABILITY PROPERTIES ARE AGGRAVATED TO BEGIN WITH

+N THE IDEAL CASE OF RIGID WHEELS AND RAILS A WHEEL AND A RAIL WILL HAVE CONTACT IN ONE OR MORE POINTS 9HEEL LIFT O OCCASIONALLY HAPPENS IN REALITY BUT IT SHOULD BE AVOIDED SINCE IT CAN LEAD TO A DERAILMENT 9HEEL LIFT O IS AN EXCEPTION THAT IS NOT CONSIDERED IN THESE NOTES 5INCE THE EECTIVE CONICITY HAS A BIG INUENCE ON THE DYNAMICS OF THE VEHICLE ITS INSTANTANEOUS VALUE MUST BE KNOWN 6HE VALUES CHANGE WITH THE LATERAL AND YAW POSITION OF THE WHEEL RELATIVE TO THE RAIL 9HEN THE RAIL AND WHEEL PROLES ARE KNOWN THE POSITION OF A CONTACT POINT CAN BE CALCULATED AS A FUNCTION OF THE LATERAL POSITION AND THE YAW ANGLE OF THE WHEEL SET 6HE GEOMETRIC VARIABLES THAT DEPEND ON THE LATERAL POSITION ! OF THE WHEEL RELATIVE TO THE RAIL ARE

R INSTANTANEOUS ROLLING RADIUS OF THE LEFT WHEEL R INSTANTANEOUS ROLLING RADIUS OF THE RIGHT WHEEL Z INSTANTANEOUS HEIGHT OF THE CONTACT POINT ON THE LEFT RAIL Z INSTANTANEOUS HEIGHT OF THE CONTACT POINT ON THE RIGHT RAIL CONTACT ANGLE OF THE LEFT WHEEL CONTACT ANGLE OF THE RIGHT WHEEL ROLL ANGLE OF THE WHEEL SET WITH RESPECT TO THE PLANE OF THE RAILS

(&2/$ 6HE WHEELRAIL PARAMETERS REAR VIEW

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6HESE CONSTRAINED VARIABLES ARE ILLUSTRATED ON (IGURE 6HE VARIATIONS OF THE POSITION OF THE RAILS WILL OF COURSE INUENCE THE DYNAMICS OF THE VEHICLE 5INCE THE VARIATIONS USUALLY ARE SMALL AND THE FEEDBACK NEGLIGIBLE THE VARIATIONS WILL ENTER THE MODEL EQUATIONS AS AN EXTERNAL FORCING

6+ *("0

+N ORDER TO MODEL THE DYNAMICS WE SHALL ADOPT TWO DIERENT POINTS OF VIEW 9HEN WE DERIVE THE RELATIONS BETWEEN THE CONTACT FORCES AND THE CONTACT GEOMETRY IT IS NECESSARY TO TAKE THE FOLLOWING ASPECTS INTO ACCOUNT 6HE CONTACT BETWEEN THE WHEEL AND THE RAIL OCCURS OVER A NITE SURFACE THAT MAY NOT BE PLANE 6HIS IS DUE TO THE COMPLIANCE OF THE BODIES IN CONTACT 6HE RELATIVE MOTION OF DIERENT POINTS IN THIS SURFACE MAY DIER 6HE STRESS DISTRIBUTIONS ACROSS THE SURFACE ARE NOT UNIFORM 1N THE OTHER HAND WE WANT TO AVOID TO TREAT THE TWO BODIES AS CONTINUA +T IS THEREFORE DESIRABLE TO CONSIDER THE CONTACT AS IF THE CONTACT OCCURS AT A SINGLE POINT THE WHEEL AND THE RAIL HAVE A RELATIVE VELOCITY AT THIS POINT THIS RELATIVE VELOCITY LIES IN A PLANE 6*' %106#%6 2.#0' THAT IS TANGENT TO THE BODIES AT THE CONTACT POINT AS LONG AS THE TWO BODIES REMAIN IN CONTACT +N GENERAL THE CONTACT POINT LIES IN A VERTICAL PLANE THAT CONTAINS THE CENTRE LINE OF THE WHEEL SET AND THE CONTACT PLANE IS PERPENDICULAR TO THAT PLANE 9HEN THE WHEEL SET YAWS THE CONTACT POINT MOVES AWAY FROM THE VERTICAL PLANE 6HE DISPLACEMENT IS THE SOCALLED 8ORVERLAGERUNG FROM )ERMAN THAT CAN BE CALCULATED BY *EUMANN S METHOD *EUMANN 6HE DYNAMIC EECT OF THE 8ORVERLAGERUNG IS IMPORTANT IN CURVES WHEN THE WHEEL SET YAWS 'XPERIENCE HAS SHOWN HOWEVER THAT THE 8ORVERLAGERUNG MAY BE NEGLECTED WHEN THE RADIUS OF THE CURVE IS LARGER THAN AROUND M WITH THE WHEEL AND RAIL PROLES THAT ARE USED TODAY 9HEN THE VEHICLE MOVES THERE IS ROLLING CONTACT IN THE WHEELRAIL CONTACT SURFACE 7NDER THE LOAD BOTH THE WHEEL AND THE RAIL WILL DEFORM *ERTZ S THEORY OF DEFORMATION HAS YIELDED SATISFACTORY RESULTS FOR THE CALCULATION OF THE CONTACT SURFACE THE NORMAL FORCE AND THE MUTUAL PENETRATION OF THE WHEEL AND THE RAIL #5 .10) #5 6*' %748#674' 1( 6*' %106#%6 574(#%' +5 0').+)+$.' ,/* ) ,/"$0 *ERTZ CONSIDERED TWO CONVEX ELASTIC BODIES THAT ARE PRESSED TOGETHER BY A FORCE *E ASSUMED THAT THE DIMENSIONS OF THE CONTACT SURFACE ARE SMALL COMPARED WITH THE RADII OF CURVATURE OF THE TWO BODIES IN A NEIGHBOURHOOD OF THE CONTACT POINT *ERTZ THEN SHOWED THAT THE CONTACT SURFACE WOULD BE ELLIPTIC WITH THE MAJOR SEMIAXIS AND THE MINOR SEMIAXIS 6HE ESSENCE OF *ERTZ S RESULTS IS THAT THE RATIO ONLY DEPENDS ON THE CURVATURES OF THE TWO BODIES INDEPENDENT OF THE TANGENTIAL STRESS DISTRIBUTION THE MUTUAL PENETRATION

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6HE CONTACT ANGLES OF THE RIGHT AND LEFT WHEEL RESPECTIVELY CAN ONLY ASSUMED TO BE EQUAL WHEN THE WHEEL SET IS CENTRED IN A STRAIGHT TRACK WITH ZERO YAW +N GENERAL THEY WILL BE DIERENT AND THE SUM OF THE NORMAL CONTACT FORCES WILL THEN HAVE A LATERAL RESULTANT 6HIS SUM OF THE LATERAL COMPONENTS OF THE NORMAL FORCES IS OFTEN REFERRED TO AS 6*' )4#8+6#6+10#. 56+0'55 +T IS APPROXIMATELY EQUAL TO THE AXLE LOAD MULTIPLIED BY HALF THE DIERENCE OF THE CONTACT ANGLES PLUS THE WHEEL SET ROLL ANGLE FOR SMALL CONTACT ANGLES #T LARGER CONTACT ANGLES TRIGONOMETRIC FUNCTIONS OF THE ANGLES ENTER THE EXPRESSIONS FOR THESE TERMS THEREBY MAKING THE LATERAL RESULTANT FORCE A NONLINEAR FUNCTION OF THE CONTACT ANGLES +N MOST CASES THE GRAVITATIONAL STINESS FORCE ACTS IN A DIRECTION TO ACCELERATE THE WHEEL SET TOWARDS ITS CENTRED POSITION +F THE GRAVITATIONAL STINESS IS LARGE OWING TO A LARGE CHANGE OF THE CONTACT ANGLE DIERENCE WITH LATERAL DISPLACEMENT SIGNICANT RESTORING FORCES WILL RESULT 6HUS THE GRAVITATIONAL STINESS IN GENERAL HAS A STABILIZING INUENCE 9HEN ONE OF THE BODIES IS CONCAVE IN A NEIGHBOURHOOD OF THE CONTACT POINT THEN THE CONTACT SURFACE IS NOT A PLANE SURFACE AND *ERTZ S THEORY MUST BE USED WITH CAUTION +T MAY BE USED AS A RST APPROXIMATION TO DETERMINE QUALITATIVE FEATURES OF THE MOTION OF THE VEHICLE AND ITS COMPONENTS 6HE CALCULATED STRESSES IN THE CONTACT SURFACE WILL HOWEVER MOST LIKELY BE ERRONEOUS 4ECENT WORK BY 2IOTROWSKI AND -IK TAKES THE CURVATURE OF THE CONTACT SURFACE INTO ACCOUNT IN THE STRESS CALCULATIONS *OMANN MADE IN HIS THESIS AN INTERESTING COMPARISON BETWEEN THE CALCULATIONS OF THE DYNAMICS OF A TWOAXLE FREIGHT WAGON WITH APPLICATION OF RST *ERTZ S THEORY AND SECOND -IK S 45)'1 ROUTINE THAT TAKES THE NONELLIPTICITY OF THE CONTACT SURFACE INTO ACCOUNT *E FOUND THAT THE RESULTS WERE QUALITATIVELY VERY SIMILAR BUT THE DISPLACEMENTS USING -IK S ROUTINE 45)'1 WERE UP TO THREE TIMES LARGER THAN THE DISPLACEMENTS FOUND USING *ERTZ S THEORY 6HE -IK RESULTS ARE ACCORDING TO EARLIER EXPERIENCE VERY CLOSE TO THE REAL VALUES +F TWO CONTACT POINTS EXIST SIMULTANEOUSLY IN A WHEELRAIL COMBINATION THE CALCULATION OF THE CONTACT STRESSES BECOMES MORE DICULT 5AUVAGE AND 2ASCAL AND 2ASCAL AND 5AUVAGE HAVE FORMULATED APPROXIMATIVE METHODS TO HANDLE THE PROBLEM WHEREBY THE PROBLEM IS REDUCED TO THE DETERMINATION OF THE STRESSES IN ONE CTIVE CONTACT POINT 6HE CONTACT POINT MAY ALSO JUMP FROM ONE POSITION TO ANOTHER UNDER THE INUENCE OF A LATERAL DISPLACEMENT OF THE WHEEL SET 6HE JUMP INTRODUCES A DISCONTINUITY IN THE DYNAMIC SYSTEM AND WE SHALL TREAT THE NUMERICAL HANDLING OF SUCH NONSMOOTH PROBLEMS IN SECTION +F HOWEVER THE COMPLIANCE OF THE WHEEL AND THE RAIL IS TAKEN INTO ACCOUNT THEN THE GROWTH OF THE SECOND CONTACT SURFACE AND THE DECAY OF THE RST ARE CONTINUOUS FUNCTIONS OF THE DISPLACEMENT 6HE DISCONTINUOUS RIGID BODY PROBLEM IS THEN REDUCED TO THE AFOREMENTIONED TWOPOINT PROBLEM 5EVERAL NUMERICAL ROUTINES HAVE BEEN MADE THAT TAKE THE NONELLIPTICITY AND TWO POINT CONTACT INTO ACCOUNT -ALKER S PROGRAM %106#%6 HAS WON INTERNATIONAL APPROVAL AS THE MOST ACCURATE ROUTINE FOR THE DETERMINATION OF THE CONTACT SURFACES AND CALCULATION OF THE CONTACT FORCES +TS DISADVANTAGE IS THE LONG %27TIME NEEDED TO RUN THE PROGRAM 1THER PROGRAMS THAT MAY BE LESS ACCURATE BUT NEED LESS %27TIME ARE COMMERCIALLY AVAILABLE +T IS HIGHLY RECOMMENDED TO USE THESE ROUTINES OR %106#%6 FOR VEHICLE DYNAMIC INVESTIGATIONS IN FUTURE + SHALL NOT GIVE ANY SPECIC RECOMMENDATIONS BUT ONLY REFER TO THE EXISTING LITERATURE AND THE SOFTWARE PROVIDERS

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+N A SURVEY LECTURE 2IOTROWSKI AND %HOLLET DESCRIBED THE APPROXIMATIVE SOLUTION METHODS AS WELL AS LATER WORK IN THE AREA 6HE MOST RECENT CONTRIBUTION TO THE HANDLING OF THE CONTACT PROBLEM IS BY 3UOST ET AL +&$+1( ) ,/"$0 +N ROLLING NOT SLIDING CONTACT THE CONTACT SURFACE IS DIVIDED INTO TWO PARTS THAT DIER IN THE KIND OF CONTACT 5EE (IGURE +N THE RST PART THE TWO BODIES STICK TO EACH OTHER AND THE TANGENTIAL DEFORMATIONS ARE PURELY ELASTIC +N THE SECOND PART THE TWO BODIES SLIDE AGAINST EACH OTHER AND %OULOMB S FRICTION LAW APPLIED IN THIS PART YIELDS SATISFACTORY RESULTS FOR THE TANGENTIAL STRESSES IN THAT PART OF THE CONTACT SURFACE 6HE ELASTIC DEFORMATION AND THE SLIDING TOGETHER RESULT IN A NITE SLIDING VELOCITY IN THE CONTACT SURFACE +T IS DENOTED 6*' %4''2 6HE CREEP NORMALIZED WITH THE SPEED OF THE VEHICLE IS DENOTED 6*' %4''2#)' 6HE EECT OF THE CREEP IS THAT THE RESULTING RELATIVE SPEED OF THE WHEEL IN THE CONTACT SURFACE ALWAYS IS DIERENT FROM ZERO WHEN THE VEHICLE MOVES 6HAT OF COURSE IS THE SOURCE OF THE WEAR ON BOTH THE RAIL AND THE WHEEL TREADS AND ANGES

(&2/$ 6HE WHEELRAIL CONTACT SURFACE 9HILE THE NORMAL LOAD IS VERY IMPORTANT FOR THE LONGTERM DYNAMIC DETERIORATION OF THE TRACK IT IS THE TANGENTIAL FORCES THAT ARE MOST IMPORTANT FOR THE VEHICLE DYNAMICS 6HE CREEP CREATES TANGENTIAL FORCES THAT ARE DENOTED 6*' %4''2 (14%'5 6HE CREEP FORCES HAVE TWO COMPONENTS IN THE CONTACT SURFACE 1NE IS LONGITUDINAL IN THE DIRECTION OF THE MOTION OF THE WHEEL SET AND THE OTHER IS LATERAL +T IS NOT NECESSARILY HORIZONTAL DUE TO THE CONTACT ANGLE 2ERPENDICULAR TO THE CONTACT SURFACE IS A THIRD COMPONENT 6HE SPIN CREEP WHICH EXERTS A TORQUE / SEE GURE 6HE TORQUE IS CREATED BY THE RESULTANT OF THE CREEP FORCES WHICH HAS A POINT OF ATTACK THAT IN GENERAL DIERS FROM THE CENTRE OF THE CONTACT ELLIPSE AND DEPENDS ON THE CONTACT ANGLE 9HEN THE CONTACT ANGLE IS ZERO THEN THE SPIN CREEP IS ZERO 9HEN THE CONTACT ANGLES OF THE LEFT AND RIGHT WHEEL OF THE WHEEL

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SET ARE EQUAL THEN THE SPIN CREEPS OF THE WHEELS ARE EQUAL AND OPPOSITE 6HE RESULTING SPIN CREEP OF THE WHEEL SET IS THEN ZERO

(&2/$ 6HE WHEELRAIL CREEPAGES AND CREEP FORCES 6HE PROBLEM OF CALCULATING THE CREEPAGE AND THE CREEP FORCES IS HIGHLY NONLINEAR AND IT HAS BEEN TREATED BY SEVERAL SCIENTISTS 6HE MAIN REFERENCE TO THIS PROBLEM IS -ALKER +N A SERIES OF PUBLICATIONS STARTING WITH HIS 2H& THESIS -ALKER HAS FORMULATED AND SOLVED THE ROLLING CONTACT PROBLEM AND HE HAS MADE COMPUTER PROGRAMS OF INCREASING ACCURACY AND %27 TIME FOR THE NUMERICAL COMPUTATION OF THE STRESSES THE RESULTING CONTACT FORCES AND THE SPIN TORQUE -ALKER S MOST ACCURATE PROGRAM IS THE EARLIER MENTIONED PROGRAM %106#%6 +T IS GENERALLY ACCEPTED AS THE MOST ACCURATE PROGRAM FOR THE CALCULATION OF THE WHEELRAIL CONTACT FORCES AND TORQUE # FORMULATION OF THE CLASSICAL DYNAMIC CONTACT PROBLEM HAS BEEN MADE BY %HUDZIKIEWICZ # FORMULATION OF AN EXTENSION OF THE CLASSICAL THEORY TO 0ON5TEADY5TATE AND 0ON*ERTZIAN PROBLEMS IS GIVEN BY -NOTHE ET AL 1THER MODELS OF THE CONTACT FORCES VERSUS THE CREEPAGE HAVE BEEN FORMULATED 6HE MOST WIDE SPREAD ONE IS THE METHOD BY 5HEN *EDRICK AND 'LKINS SINCE IT IS EASY TO IMPLEMENT +T YIELDS GOOD RESULTS AS LONG AS THE SPIN CREEP IS SMALL 5HEN ET AL PRESENT A SURVEY OF THESE MODELS # MORE RECENT SURVEY OF THE MODELS HAS BEEN WRITTEN BY -ALKER # RECENT CONTRIBUTION BY 2OLACH THAT IS NOT INCLUDED IN -ALKER DESCRIBES A FAST AND FOR MOST APPLICATIONS SUCIENTLY ACCURATE COMPUTER PROGRAM FOR THE CALCULATION OF THE

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(&2/$ 6HE GENERAL FORM OF THE DEPENDENCE OF THE NORMALIZED CREEP FORCE ( ON THE CREEPAGE C

WHEELRAIL CONTACT FORCES 6HE GENERAL SHAPE OF THE CREEP FORCE NORMALIZED BY THE %OULOMB FRICTION FORCE ( VERSUS THE CREEPAGE C IS SHOWN ON GURE +T IS CLEARLY NONLINEAR AND HAS THE FORM OF A CHARACTERISTIC OF A WEAKENING DAMPER 6HE CURVE APPROACHES THE %OULOMB FRICTION LIMIT FOR LARGE CREEPAGE WHICH CORRESPONDS TO PURE SLIDING OR WHEEL SLIP (OR STILL LARGER CREEPAGES THE CURVE DROPS O AGAIN 6HIS IS THE CAUSE OF A HYSTERESIS EECT THAT IS WELL KNOWN FROM WHEEL SETS UNDER HIGH TRACTIVE TORQUES 7NDER TRACTION SOCALLED MACROSLIP OSCILLATIONS DEVELOP AND UNDER BRAKING THE VEHICLE BRAKING FORCE DECREASES WHEN THE WHEELS ARE BLOCKED 6HE LATERAL ACCELERATIONS OF THE VEHICLES ARE THE MOST ANNOYING DISTURBANCES +N ORDER TO CALCULATE THE LATERAL DISPLACEMENTS AND ACCELERATIONS WE MUST OF COURSE KNOW THE LATERAL CREEP FORCES +T MUST BE REMEMBERED HOWEVER THAT THE LONGITUDINAL CREEPAGE HAS A RECIPROCAL EECT ON THE LATERAL CREEP FORCES (URTHERMORE THE SPIN CREEPAGE PRODUCES A LATERAL FORCE 6HIS CONTRIBUTION TO THE TOTAL LATERAL FORCE REDUCES THE LONGITUDINAL PART OF THE TOTAL CREEP FORCE THAT IS NEEDED FOR THE TRACTION SOMETIMES QUITE SUBSTANTIALLY 6HIS IS THE CAUSE OF THE WHEEL SLIP OF DRIVEN AXLES AT HIGH SPEED THAT HAS BEEN OBSERVED IN REAL LIFE 6HE AXLE TORQUE AT HIGH SPEED OF THE VEHICLE IS SMALLER THAN UNDER THE INITIAL ACCELERATION WHERE HIGH LONGITUDINAL CREEP FORCES ARE PRODUCED SO THE HIGH SPEED WHEEL SLIP HAS PUZZLED THE ENGINEERS FOR DECADES #LTOGETHER THERE EXISTS A RATHER COMPLICATED NONLINEAR RELATIONSHIP BETWEEN ALL THE CREEPAGES AND THE COMPONENTS OF THE CREEP FORCES #LTHOUGH -ALKER S THEORY IS CALLED EXACT IT IS IMPORTANT TO BE AWARE OF -ALKER S ASSUMPTIONS $IRKEDAL 0IELSEN DISCUSSES -ALKER S ASSUMPTIONS AND EXTENDS THE THEORY BUT $IRKEDAL S OWN RESULTS ARE ONLY TWODIMENSIONAL $IRKEDAL INVESTIGATES THE EECT OF A SPEED DEPENDENT ADHESION COECIENT *IS POLYNOMIAL MODEL OF THE CONTACT SURFACES ALLOWS VERY SMALL CHARACTERISTIC WAVELENGTHS OF THE TRACK IRREGULARITIES SUCH AS THE DREADED SMALL WAVELENGTH CORRUGATIONS *IS MODEL ALSO CAN BE USED TO INVESTIGATE 010

H. True

83

56'#&; TWODIMENSIONAL ROLLING CONTACT PROBLEMS 6HE EXTENSION TO THREEDIMENSIONAL CONTACT PROBLEMS 9+6*176 52+0 %4''2 IS CONCEPTIONALLY SIMPLE +T IS DONE BY EXTENDING THE SURFACE INTEGRAL IN $IRKEDAL S THEORY OVER THE ENTIRE CONTACT ELLIPSE 6HE APPLICATION OF THE THEORY TO THE GENERAL THREEDIMENSIONAL CONTACT PROBLEMS IS HOWEVER STILL AN OPEN PROBLEM 4ECENT CONTRIBUTIONS TREAT THE WHEEL AND THE RAIL AS ELASTIC CONTINUA AND SOLVE THE THEORETICAL CONTACT PROBLEM NUMERICALLY USING NITE ELEMENTS 6HEREBY AN IMPROVED CORRESPONDANCE BETWEEN THE THEORETICAL AND MEASURED RESULTS AS WELL AS A DEEPER UNDER STANDING OF THE PROCESS HAVE BEEN ACHIEVED 6HE COSTS MEASURED IN %27TIME HOWEVER ARE SO HIGH THAT THE PROGRAMS ARE LESS USEFUL FOR MOST VEHICLE DYNAMICS INVESTIGATIONS 6HE INTERESTED READER IS REFERRED TO &AMME ET AL AND THE REFERENCES THEREIN 6HE WHEELRAIL CONTACT GEOMETRY AND THE WHEELRAIL CONTACT FORCES CONSTITUTE THE BASIS FOR ALL RAILWAY VEHICLE DYNAMIC INVESTIGATIONS INCLUDING WEAR AND FATIGUE 6HE PROBLEMS TO BE SOLVED ARE SO COMPLICATED THAT A NUMERICAL ANALYSIS IS MANDATORY FOR EVEN THE APPROXIMATE SOLUTIONS 6HE METHODS THAT USE NITE ELEMENT METHODS AND -ALKER S PRO GRAM %106#%6 CONSUME SO MUCH %27 TIME THAT THEY HARDLY CAN BE USED IMPLICITLY AS SUBROUTINES IN ANY RAILWAY DYNAMIC PROGRAMS 5EVERAL APPLICATIONS OF %106#%6 EXIST HOWEVER IN WHICH THE POSSIBLE RAILWHEEL CONTACT CONGURATIONS AND FORCES VERSUS THE INDEPENDENT VARIABLES ARE CALCULATED AND TABULATED A PRIORI 6HE POSITIONS AND THE FORCES THAT ARE NEEDED IN THE DYNAMIC SYSTEM FOR THE CALCULATION OF THE NEXT TIME STEP ARE THEN FOUND BY INTERPOLATIONS OF THE TABULATED VALUES AND USED AS EXTERNAL INPUTS IN THE DYNAMIC SYSTEM +T IS RECOMMENDED TO USE EITHER SPLINES OR SIMILAR POLYNOMIAL FORMS FOR THE INTERPOLATION SINCE THE NONSMOOTH LINEAR INTERPOLATION MAY CREATE NUMERICAL PROBLEMS # HIGHER ACCURACY CAN BE ACHIEVED THROUGH THE USE OF IMPLICIT INTERPOLATION SCHEMES 6HE APPROXIMATIONS BY 5HEN ET AL AND 2OLACH ARE EASY TO IMPLEMENT AND WORK SO FAST THAT THEY EASILY MAY BE IMPLEMENTED IMPLICITLY IN A NUMERICAL PROGRAM FOR THE DETERMINATION OF THE DYNAMICS OF A RAILWAY VEHICLE 6HE INTERESTED READER MAY ND THE EXPRESSIONS TO BE PROGAMMED OR EVEN THE ROUTINE ITSELF IN THE REFERENCES 5HEN ET AL AND 2OLACH

%$"!$' !"+, !" ,$#!(

+1/,#2"1(,+

6HE VEHICLE DYNAMIC PROBLEMS ARE PARAMETER DEPENDENT 'XAMPLES OF PARAMETERS ARE THE ADHESION COECIENT IN THE RAILWHEEL CONTACT SURFACE SPRING STINESSES IN THE SUSPENSIONS AND THE SPEED OF THE VEHICLE 6HE DYNAMICS OF PARAMETER DEPENDENT SYSTEMS DEPEND ON THE VALUES OF THE PARAMETERS +F THE PARAMETERS VARY DURING OPERATION THEN THE DYNAMICS CAN CHANGE QUALITATIVELY AT CERTAIN %4+6+%#. PARAMETER VALUES +N RAILWAY VEHICLE DYNAMICS THE SPEED 8 OF THE VEHICLE IS AN EXTERNALLY CONTROLLED PARAMETER THAT VARY DURING THE OPERATION 6HE INVESTIGATION OF THE DYNAMICS OF A VEHICLE THEREFORE MUST START WITH A SEARCH FOR THE POSSIBLE STEADY STATES OF THE DYNAMIC SYSTEM AND THE CRITICAL SPEEDS WHERE THE DYNAMICS MAY CHANGE QUALITATIVELY #LL RAILWAY VEHICLES KNOWN TO THE AUTHOR HAVE A %4+6+%#. 52''& 6HEY MAY HAVE MORE

84


BUT IT IS ONLY THE LOWEST WHICH IS CALLED %4+6+%#. #T SPEEDS BELOW THE CRITICAL SPEED THE IRREGULAR MOTIONS OF THE VEHICLE WILL EXCLUSIVELY BE RESPONSES TO THE TRACK GEOMETRY AND ITS IRREGULARITIES 9E NEGLECT AERODYNAMIC FORCES IN THESE NOTES #BOVE THE CRITICAL SPEED THE VEHICLE OSCILLATES IN THE PLANE OF THE TRACK 6HROUGH COUPLING IN THE SUSPENSIONS THE OSCILLATIONS MAY SPREAD TO ROLL AND PITCH MOTIONS 6HIS IS THE WELLKNOWN *706+0) 6HE CRITICAL SPEED DEPENDS ON MANY PARAMETERS AND ITS DETERMINATION IS NOT TRIVIAL 6HE RAILWAY INDUSTRY TRIES TO DESIGN THE VEHICLES IN SUCH A WAY THAT THE CRITICAL SPEED LIES WELL ABOVE THE MAXIMUM OPERATIONAL SPEED OF THE VEHICLE 6HE QUESTIONS OF INITIAL COSTS MAINTENANCE COSTS AND ROBUSTNESS DO HOWEVER PLAY AN IMPORTANT ROLE 6HE SUSPENSIONS THAT GUARANTEE A HIGH CRITICAL SPEED ARE VERY SOPHISTICATED NOWADAYS ACTIVE SYSTEMS THAT DO NOT SATISFY THE COST CRITERIA FOR EVERY TYPE OF VEHICLE 6HEREFORE THE CHEAP DESIGNS FOR FREIGHT WAGONS MOST OFTEN HAVE AN OPERATIONAL SPEED RANGE THAT INCLUDES THE CRITICAL SPEED +T HAS BEEN ASSUMED FOR MORE THAN A CENTURY THAT THE DETERMINATION OF A CRITICAL PARAMETER VALUE IN DYNAMIC PROBLEMS CAN BE FORMULATED MATHEMATICALLY AS A STABILITY PROBLEM 6HIS IS NOT TRUE IN GENERAL AND FOR RAILWAY VEHICLES AS A GROUP OF DYNAMIC SYSTEMS IN PARTICULAR 6HAT IS CONNECTED WITH THE NONLINEAR NATURE OF VEHICLE DYNAMICS AND WE MUST THEREFORE TAKE A LOOK AT THE BASICS OF NONLINEAR DYNAMICS IN THE FOLLOWING CHAPTERS

+ +,+)(+$ / #6+ *(" -/,!)$*0 (+ &$+$/ )

+N GENERAL ALL KINEMATICAL AND CONSTITUTIVE RELATIONS IN VEHICLE SYSTEM DYNAMICS ARE NONLINEAR 6HE MODELLING OF SUCH SYSTEMS THEREFORE LEADS TO NONLINEAR DYNAMIC SYSTEMS WHICH MUST BE INVESTIGATED WITH RESPECT TO THEIR PROPERTIES AND EVOLUTION +T IS CHARACTERISTIC FOR VEHICLE DYNAMIC SYSTEMS THAT THEY CONTAIN ELEMENTS WHICH DISSIPATE ENERGY 6HEREFORE WE CONSIDER 10.; &+55+2#6+8' #0& 010%105'48#6+8' 5;56'/5 IN THESE NOTES 6HE TRADITIONAL WAY OF SOLVING A NONLINEAR DISSIPATIVE DYNAMIC PROBLEM IS TO SOLVE

THE CORRESPONDING LINEAR DYNAMIC PROBLEM RST AND THEN APPLY PERTURBATION THEORY BY MAKING AN EXPANSION OF THE SOLUTION OF THE FULLY NONLINEAR PROBLEM AROUND THE SOLUTION TO THE LINEAR PROBLEM 6HIS METHOD HAS OFTEN LED TO SATISFACTORY RESULTS ALTHOUGH THE SOLUTION IS APPROXIMATE AND THE ERROR WILL GROW WITH THE EXPANSION PARAMETER OR COOR DINATE +T IS TACITLY ASSUMED THAT THE SOLUTION CAN BE APPROXIMATED TO THE RST ORDER BY THE UNIQUE SOLUTION OF A LINEAR PROBLEM AND THE NONLINEAR INUENCE CAN BE FOUND BY ADDITION OF A CORRECTION THE TERMS IN THE PERTURBATION EXPANSION TO THE SOLUTION OF THE LINEAR PROBLEM 6HE RST ASSUMPTION IS PRINCIPALLY WRONG # NONLINEAR DYNAMIC PROBLEM MAY HAVE SEVERAL DIERENT STEADY STATE SOLUTIONS EVEN THOUGH IT IS DISSIPATIVE 5EE AN EXAMPLE OF MULTIPLE STEADY STATE SOLUTIONS ON (IGURE +N CONTRAST THE SOLUTION OF A LINEAR DISSIPATIVE SYSTEM WITHOUT FORCING WILL ALWAYS TEND TO ZERO 9ITH FORCING IT TENDS TO THE PARTICULAR INTEGRAL DENED BY THE FORCING INDEPENDENTLY OF THE INITIAL CONDITIONS 6HEREFORE THE SOLUTION TO THE LINEAR PROBLEM MAY BE WRONG AS A RST APPROXIMATION TO THE NONLINEAR PROBLEM 6HE OTHER ASSUMPTION IS THAT THE SOLUTION TO THE NONLINEAR PROB LEM CAN BE APPROXIMATED BY A SUM OF TWO PARTS 6HE SOLUTION TO THE LINEAR PROBLEM AND A SERIES EXPANSION 6HAT METHOD WILL LEAD TO A GOOD APPROXIMATION OF A SOLUTION TO

H. True

85

THE NONLINEAR PROBLEM IN A CERTAIN PARAMETER INTERVAL PROVIDED THE SOLUTION OF THE LINEAR PROBLEM IS A GOOD RST APPROXIMATION IE THE RST ASSUMPTION HOLDS AND THE CORRECT PERTURBATION METHOD IS USED +N VEHICLE SYSTEM DYNAMICS THE DYNAMIC PROBLEM IS MOST OFTEN RST TO ND THE STEADY STATES NEXT TO DETERMINE THEIR STABILITY AND THIRD TO DETERMINE THE BEHAVIOUR OF THE VEHICLE WHEN CHANGES BETWEEN THE STEADY STATES EG BY BRAKING A VEHICLE OCCUR 6HERE FORE THE FUNDAMENTAL PROBLEM IS RST TO ND THE STATIONARY AND OTHER STEADY STATES AND THEIR STABILITY *' #0#.;5+5 1( # 010.+0'#4 &;0#/+% 241$.'/ /756 6*'4'(14' #.9#;5 $')+0 9+6* # 5'#4%* (14 #.. 6*' 56'#&; 56#6'5 &'2'0&+0) 10 6*' 2#4#/'6'45 +0 6*' 241$.'/ $; 51.76+10 1( 6*' (7.. 010.+0'#4 241$.'/ +N AN APPROPRIATE FRAME AT LEAST ONE OF THE STEADY STATES CAN OFTEN BE FOUND EASILY BUT THAT IS ONLY THE BEGINNING +N THE SIMPLEST NONLINEAR PROBLEMS SOME STEADY STATE SOLUTIONS CAN BE FOUND ANALYTI CALLY BUT MOST OFTEN NUMERICAL METHODS MUST BE USED 6HE USE OF NUMERICAL METHODS WILL BE DISCUSSED IN SECTION +F MULTIPLE STEADY STATES EXIST THEN THEY FORTUNATELY VERY OFTEN DEVELOP THROUGH SO CALLED $+(74%#6+105 OR BRANCHING O FROM OTHER STEADY STATES WHEN A CONTROL PARAMETER IN THE PROBLEM IS VARIED 6HERE ARE SEVERAL TYPES OF BIFURCATIONS AND THE ONES MOST RELEVANT FOR VEHICLE DYNAMIC SYSTEMS ARE DESCRIBED IN CHAPTER 6HIS FACT OERS US A POSSIBILITY TO ND MULTIPLE STEADY STATES BY TRACING ALREADY FOUND STEADY STATE SOLUTIONS BY VARYING THE CONTROL PARAMETER THROUGH THE VALUES WHERE BIFURCATIONS TAKE PLACE AND THEN CHANGE TO FOLLOW THE NEW SOLUTION FROM THE BIFURCATION IN DEPENDENCE ON THE CONTROL PARAMETER 6HE FUNDAMENTAL STEPS IN THE SOLUTION OF A NONLINEAR DYNAMIC PROBLEM THEREFORE CONSIST OF NDING AT LEAST ONE STEADY STATE SOLUTION IN A SUCIENTLY LARGE PARAMETER INTERVAL AND DETERMINE ITS STABILITY IN DEPENDENCE ON THE PARAMETER 6HEN ND BIFURCATION POINTS WHERE DIERENT STEADY SOLUTIONS WILL BE FOUND 6HE NEW SOLUTIONS AND THEIR STABILITY MUST THEN BE DETERMINED IN DEPENDENCE ON THE CONTROL PARAMETER 6HE NEW SOLUTIONS MAY CREATE NEW BIFURCATION POINTS WHICH MUST BE FOUND AND THE GAME STARTS ALL OVER IN THESE NEW BIFURCATION POINTS 7NFORTUNATELY THERE IS NO GUARANTEE THAT #.. THE STEADY STATE SOLUTIONS WILL BE DETERMINED IN THIS WAY BUT EXPERIENCE SHOWS THAT THE METHOD IS VERY ECIENT 6HE METHOD IS DESCRIBED IN DETAIL IN SECTION )OOD ENGINEERING KNOWHOW IS STILL NEEDED TO JUDGE WHETHER ALL POSSIBLE STATES HAVE BEEN FOUND OR IT IS NECESSARY TO USE OTHER STRATEGIES TO ND POSSIBLE MISSING STATES 0ONLINEAR MECHANICS DEALS WITH THE FORMULATION OR MODELLING OF PROCESSES WHILE NON LINEAR DYNAMICS IS A MATHEMATICAL DISCIPLINE 6HE TOPICS THEREFORE ARE DIERENT 9E SHALL DEAL WITH 010.+0'#4 #0& &'6'4/+0+56+% &;0#/+%5 IN THESE NOTES 5TOCHASTIC VARIABLES ARE EXCLUDED +N NONLINEAR PROBLEMS 6*' ':+56'0%' OF SOLUTIONS IS THE PRIMARY OBJECT OF ANALYSIS IN CONTRAST TO 56#$+.+6; WHICH COMES SECOND 9HEN SEVERAL STABLE STEADY STATE SOLUTIONS OF A DISSIPATIVE PROBLEM COEXIST EACH AND EVERYONE WILL HAVE A SET OF ITS OWN INITIAL CONDITIONS WITH THE PROPERTY THAT ALL SOLUTIONS DETERMINED BY INITIAL CONDITIONS FROM THAT SET WILL TEND TO THAT PARTICULAR STABLE SOLUTION 5UCH A SET OF INITIAL CONDITIONS IS CALLED A $#5+0 1( #664#%6+10 6HIS FACT HAS AN IMPORTANT IMPLICATION +F THE DYNAMIC PROBLEM HAS MULTIPLE STEADY STATES AND THE CORRESPONDING PHYSICAL VEHICLE IN THE REAL WORLD IS SUBJECT TO UNCONTROLLED INITIAL CONDITIONS THEN NITE PROBABILITIES EXIST FOR THE VEHICLE TO TEND TO ANY ONE OF THE STABLE STEADY STATES +N OTHER WORDS 9HEN SEVERAL STABLE STEADY STATE SOLUTIONS OF A DISSIPATIVE DYNAMIC PROBLEM COEXIST THEN THE VEHICLE IS

86


7056#$.' 61 0+6' &+5674$#0%'5 6HE MATHEMATICAL PROPERTIES OF NONLINEAR SYSTEMS ARE FUNDAMENTALLY DIERENT FROM THE MATHEMATICAL PROPERTIES OF LINEAR SYSTEMS AND THE NONLINEAR SYSTEMS MUST BE TREATED ACCORDINGLY #N IMPORTANT DIERENCE BETWEEN THE PROPERTIES OF LINEAR AND NONLINEAR SYS TEMS IS THAT THE PRINCIPLE OF SUPERPOSITION DOES NOT APPLY TO NONLINEAR SYSTEMS 6HE PRINCIPLE OF SUPERPOSITION IS THE BASIS OF INTEGRAL TRANSFORM SOLUTION METHODS #NOTHER DIERENCE IS THE EECT OF THE INITIAL CONDITIONS ON THE STEADY STATE SOLUTION OF AN INI TIAL VALUE PROBLEM # LINEAR DISSIPATIVE PROBLEM HAS ONLY ONE STEADY STATE SOLUTION INDEPENDENT OF THE INITIAL CONDITIONS BUT A NONLINEAR PROBLEM MAY HAVE MORE STEADY STATE SOLUTIONS WHERE EACH ONE DEPENDS UNIQUELY ON THE APPLIED INITIAL CONDITIONS 6HE EXISTENCE OF MORE STEADY STATE SOLUTIONS OF A NONLINEAR PROBLEM IS DEMONSTRATED IN THE FOL LOWING EASY EXAMPLE OF A NONLINEAR INITIAL VALUE PROBLEM (ROM 6HOMPSON AND 5TEWART P %ONSIDER THE NONLINEAR DIERENTIAL EQUATION

!

WITH THE INITIAL CONDITIONS ! !

! !

AND

6HE TWO DIERENT INITIAL VALUE PROBLEMS AND LEAD TO TWO DIF FERENT STEADY SOLUTIONS CALLED #664#%6145 6HEY ARE FOUND NUMERICALLY USING THE 4UNGE -UTTA SOLVER RK FROM /#2.' 6HE ATTRACTORS ARE DEPICTED ON GURE

(&2/$ 6HE TWO SOLUTIONS OF AND +T IS EASILY SEEN THAT ALTHOUGH THE EXCITATION IS A PERIODIC FUNCTION WITH THE AMPLITUDE AND THE SYSTEM IS DAMPED THEN THERE EXISTS A STEADY STATE LARGE AMPLITUDE SOLUTION TO THE DIERENTIAL EQUATION IN ADDITION TO THE EXPECTED STEADY STATE SOLUTION WITH AM PLITUDE LESS THAN TWO 6HIS IS AN EXAMPLE OF %1':+56+0) 56'#&; 56#6' 51.76+105 9HEN THE

H. True

87

CORRESPONDING LINEAR PROBLEM IS SOLVED WITHOUT THE TERM ONLY A LOW AMPLITUDE STEADY STATE PERIODIC SOLUTION EXISTS FOR BOTH SETS OF INITIAL CONDITIONS (URTHERMORE OSCILLATION FREQUENCIES ARE AMPLITUDE DEPENDENT IN NONLINEAR DYNAMICS #S A CONSEQUENCE HEREOF 6*' %10%'26 1( 4'510#0%' 57%* #5 +6 +5 75'& +0 .+0'#4 6*'14; /756 $' 4'%105+&'4'& #0& 4'8+5'& +N ORDER TO SIMPLIFY THE NONLINEAR DYNAMIC PROBLEM THE SYSTEM MAY BE IDEALIZED WITHIN CERTAIN LIMITS 6HE SIMPLEST ASSUMPTION IS THAT ALL THE PARAMETERS ARE INDEPENDENT OF THE STATE OF THE SYSTEM BUT IT IS WRONG FOR ALL REAL MECHANICAL SYSTEMS +N MANY CASES HOWEVER IT IS POSSIBLE TO ND INTERVALS OF THE ACTION IN WHICH CERTAIN PARAMETERS IN PRACTICE IE WITHIN A GIVEN SMALL TOLERANCE REMAIN CONSTANT +T MAY FOR EXAMPLE HAPPEN THAT THE ACTION OF A DAMPER IN THE PROBLEM OF INTEREST IS LIMITED TO SUCH A SMALL VELOCITY INTERVAL THAT ITS NONLINEAR CHARACTER MAY BE NEGLECTED 5UCH SIMPLICATIONS IN THE THEORETICAL MODEL MUST HOWEVER BE MADE WITH GREAT CAUTION BECAUSE THEY MAY CHANGE THE SOLUTIONS OF THE PROBLEM 37#.+6#6+8'.; #0& 14 37#06+6#6+8'.; AND IMPORTANT PROPERTIES OF THE SOLUTION MAY BE LOST +N THE FOLLOWING WE SHALL INTRODUCE THE BASIC NOTIONS OF A 56#6' 52#%' AND 64#,'%614+'5 BECAUSE THEY LEAD TO A VISUALIZATION OF THE SOLUTIONS OF DYNAMIC PROBLEMS THEIR UNIQUENESS OR LACK OF SAME AND STABILITY 9E SHALL ALSO PRESENT THE CONDITIONS UNDER WHICH A LINEARIZATION OF A NONLINEAR OPERATOR PROBLEM IS PERMITTED AND INDICATE HOW THE SOLUTION OF THE LINEARIZED PROBLEM CAN GIVE VALUABLE INFORMATION ABOUT THE LOCAL PROPERTIES OF THE NONLINEAR PROBLEM 9E SHALL DENE AND EXAMINE STATIONARY AND OTHER STEADY STATE SOLUTIONS THEIR STABILITY AND BIFURCATIONS (INALLY PROPERTIES OF SOLUTIONS THAT CANNOT EXIST IN LINEAR SYSTEMS SUCH AS CHAOS WILL BE DESCRIBED AND THEIR EECTS BRIEY DISCUSSED

'$ ,/*2) 1(,+ ,%

()4 6 $'(")$ 6+ *(" /,!)$*

6HE PROBLEMS IN THEORETICAL VEHICLE DYNAMICS ARE FORMULATED AS NONLINEAR DYNAMIC MULTIBODY PROBLEMS 6HE ROLLING WHEELRAIL CONTACT AS WELL AS THE MORE RARELY APPLIED SLIDING CONTACT ARE THE MOST IMPORTANT FEATURES OF A VEHICLE DYNAMIC SYSTEM AND THE CON STITUTIVE LAWS FOR THESE CONTACTS ARE NONLINEAR +N A MULTIBODY SYSTEM THE SINGLE BODIES ARE INTERCONNECTED AND SEVERAL OF THE CONNECTIONS EG SPRINGS AND DAMPERS OBEY NONLINEAR CONSTITUTIVE LAWS )UIDES INTRODUCE CONSTRAINTS WHICH OFTEN ARE FORMULATED BY ALGEBRAIC OR TRANSCENDENTAL EQUATIONS SO THE PROBLEMS IN VEHICLE DYNAMICS MUST OFTEN BE FORMULATED AS A SYSTEM OF DIERENTIALALGEBRAIC EQUATIONS &#' $UMPER STOPS AND DRY FRICTION CONTACT ARE IMPORTANT NONLINEARITIES THAT INTRODUCE DISCONTINUITIES 6HE DIS CONTINUITIES VIOLATE THE ASSUMPTIONS OF THE MATHEMATICAL THEOREMS WHICH FORM THE BASIS OF MANY METHODS THAT ARE APPLIED FOR THE SOLUTION OF SYSTEMS OF DIERENTIALALGEBRAIC EQUATIONS +N VEHICLE DYNAMICS THE ANALYSIS OF THE PROBLEM IS MOST OFTEN FORMULATED AS AN INITIAL VALUE PROBLEM FOR A SYSTEM OF ORDINARY DIERENTIAL EQUATIONS WITH THE TIME AS THE SINGLE INDEPENDENT VARIABLE 6HE SYSTEM OF DIERENTIAL EQUATIONS MAY BE COMBINED WITH ALGEBRAIC OR TRANSCENDENTAL EQUATIONS THAT EXPRESS CERTAIN KINEMATICAL CONSTRAINTS EG THE GEOMETRIC CONTACT BETWEEN A WHEEL SET AND THE RAILS 6HE ALGEBRAIC EQUATIONS ARE IN GENERAL NONLINEAR AND THEY MUST BE SOLVED TOGETHER WITH THE SYSTEM OF DIERENTIAL EQUATIONS AS ONE SYSTEM OF DIERENTIALALGEBRAIC EQUATIONS $OUNDARY VALUE PROBLEMS

88


FOR PARTIAL DIERENTIAL EQUATIONS OR EQUATIONS WITH TIME LAG MAY HOWEVER ALSO ENTER THE DYNAMIC SYSTEM BUT THEY WILL NOT BE CONSIDERED IN THESE NOTES

$ ) %#)', % %$"!$' *)%$%#%*( ,$#! ,()#(

,*$ $+(1(,+0

6HE GEOMETRIC THEORY OF NONLINEAR DYNAMIC SYSTEMS PROVIDES THE BEST ACCESS TO AN UNDERSTANDING OF THE BEHAVIOUR OF SUCH SYSTEMS # DYNAMIC SYSTEM IS A SYSTEM OF RST ORDER ORDINARY DIERENTIAL EQUATIONS THAT OFTEN IS CONNECTED WITH A SYSTEM OF ALGEBRAIC OR TRANSCENDENTAL EQUATIONS WHICH DENE CONSTRAINTS IN THE SYSTEM *' 56#6' 1( 6*' 5;56'/ IS DENED AS THE SET OF POSITIONS AND VELOCITIES OF EACH AND EVERY ELEMENT OF THE VEHICLE IN A SUITABLE COORDINATE SYSTEM $OTH THE POSITIONS AND THE VELOCITIES ARE TREATED AS DEPENDENT VARIABLES IN THE DYNAMIC SYSTEM WITH TIME AS THE INDEPENDENT VARIABLE 6HE DIMENSION 0 OF THE STATE SPACE EQUALS THE NUMBER OF DEPENDENT VARIABLES OF THE DYNAMIC SYSTEM 6HE MATHEMATICAL PROBLEM CONSISTS OF NDING THE STATE OF THE SYSTEM UNDER EXTERNAL ACTION AS A FUNCTION OF THE TIME AND IN DEPENDENCE ON THE PARAMETERS +N THE FOLLOWING SECTION WE SHALL ASSUME THAT THE PARAMETERS ARE ALL CONSTANT 6HE DEPENDENT VARIABLES ! ARE CALLED 56#6' 8#4+#$.'5 AND THEY ARE ALL REAL 5 IS THE 0DIMENSIONAL VECTOR ! 6HE TIME IS THE INDEPENDENT VARIABLE +F THE TIME DOES NOT ENTER THE DYNAMIC SYSTEM EXPLICITLY THE SYSTEM IS CALLED #76101/175 +F THE TIME ENTERS THE DYNAMIC SYSTEM EXPLICITLY OR THE PARAMETERS DEPEND ON TIME THEN THE DYNAMIC SYSTEM WILL BE 010#76101/175 #N 0DIMENSIONAL NONAUTONOMOUS DYNAMIC SYSTEM CAN BE CONVERTED INTO AN 0DIMENSIONAL AUTONOMOUS SYSTEM BY THE INTRODUCTION OF THE TIME AS AN ADDITIONAL DEPENDENT VARIABLE BY THE PHASE EQUATION ! AND CONSIDER THE NEW INDEPENDENT TIME VARIABLE 9E SHALL THEREFORE ONLY CONSIDER AUTONOMOUS SYSTEMS IN THE FOLLOWING +F THE PARAMETERS DEPEND ON THE STATE OF THE DYNAMIC SYSTEM THEN THE PARAMETERS WILL GIVE RISE TO ADDITIONAL NONLINEARITIES 9E DENOTE BY ? AND CORRESPONDINGLY BY 5 ! J!/ DENOTES A SET OF / REAL VALUED PARAMETERS # 2#4#/'6'4 &'2'0&'06 010.+0'#4 &;0#/+% 5;56'/ CAN THEN BE WRITTEN IN THE FORM 5 ! 5

IS AN 0DIMENSIONAL REAL VALUED @SUCIENTLY SMOOTH NONLINEAR VECTOR FUNCTION OF THE 0 DEPENDENT VARIABLES $Y (70%6+10 WE ALWAYS MEAN AN EXPRESSION FROM WHICH THE VALUE IS DETERMINED 70+37'.; BY THE ARGUMENTS +F UNIQUENESS IS NOT GUARANTEED WE SPEAK OF A 4'.#6+10 #N EXAMPLE OF A RELATION IS THE EXPRESSION ! SINCE THERE EXIST TWO REAL VALUES OF FOR EVERY POSITIVE VALUE OF 9HEN WE DEAL WITH FUNCTIONS WE CALL THE SET OF ALL PERMISSIBLE VALUES OF THE ARGUMENTS 6*' &1/#+0 1( 6*' (70%6+10 6HE SET OF VALUES THAT THE FUNCTION CAN ASSUME FOR ALL PERMISSIBLE VARIATIONS OF THE ARGUMENTS IS CALLED 6*' 4#0)' 1( 6*' (70%6+10 # MECHANICAL SYSTEM WITH - DEGREES OF FREEDOM - &1( WILL NORMALLY BE MODELLED MATHEMATICALLY BY A SYSTEM OF - ORDINARY DIERENTIAL EQUATIONS OF SECOND ORDER IN THE DEPENDENT VARIABLES 6HE TRIVIAL SUBSTITUTION ! AND ? ! WILL HOWEVER

H. True

89

TRANSFORM THE SYSTEM OF - SECOND ORDER DIERENTIAL EQUATIONS INTO A SYSTEM OF - RST ORDER DIERENTIAL EQUATIONS AND WE THEN SET 0 EQUAL TO - 9E NOW INTRODUCE A CARTESIAN COORDINATE SYSTEM WITH THE STATE VARIABLES AS THE COORDINATES +T WILL BE 0DIMENSIONAL AND IT IS DENOTED 6*' 56#6' 52#%' OR 2*#5' 52#%' 9E CAN PLOT THE VALUES THAT SATISFY EQN FOR XED PARAMETER VALUES IN THE PHASE SPACE 9E THEN OBTAIN A SET OF CURVES DEPENDING ON THE INITIAL CONDITIONS WHICH ARE CALLED 64#,'%614+'5 OR +06')4#. %748'5 CORRESPONDING TO EQN IS A CURVE PARAMETER AND ASSOCIATED WITH EACH TRAJECTORY IS A DIRECTION SHOWING HOW THE STATE OF THE SYSTEM CHANGES AS TIME INCREASES 6HE COMPLETE GURE IS CALLED 6*' 2*#5' &+#)4#/ OR 2*#5' 21464#+6 FOR THE SYSTEM # POINT IN THE PHASE SPACE IS CALLED AN +/#)' 21+06 AND THE TRAJECTORY THROUGH AN IMAGE POINT IS OFTEN TERMED A 2*#5' 64#,'%614; IN ORDER TO DISTINGUISH IT FROM THE TRAJECTORY OF THE MOTIONS IN THE PHYSICAL WORLD 6HE SPEED OF AN IMAGE POINT IS CALLED 6*' 2*#5' 52''& NOT TO BE CONFUSED WITH THE PHYSICAL SPEED OF THE ELEMENTS IN THE SYSTEM +N AUTONOMOUS SYSTEMS 64#,'%614+'5 %#0 016 %4155 '#%* 16*'4 MEANING THAT THE MOTION OF THE SYSTEM IS DETERMINED 70+37'.; BY ANY POINT ON A TRAJECTORY #N '37+.+$4+7/ 21+06 IS AN IMAGE POINT THAT DOES NOT CHANGE WITH TIME SO THE PHASE TRAJECTORY IS JUST A POINT +T CORRESPONDS TO A 56'#&; 56#6' OF THE DYNAMIC SYSTEM AND IT IS A 56#6+10#4; 51.76+10 # POINT FOR WHICH THE SLOPE OF THE TRAJECTORY IS NOT DENED IS CALLED A 5+0)7.#4 21+06 ALL OTHER POINTS ARE CALLED 4')7.#4 21+065 #N EQUILIBRIUM POINT IS A SINGULAR POINT 5INCE A PERIODIC SOLUTION WILL ALWAYS RETURN TO ANY GIVEN STATE WITH ITS PERIOD 6 6*' 64#,'%614; 1( # 2'4+1&+% 51.76+10 /756 $' # %.15'& %748' &ISSIPATIVE AUTONOMOUS SYSTEMS ARE %1064#%6+0) +0 6*' 2*#5' 52#%' IN THE SENSE THAT A SMALL VOLUME BOUNDED BY A BUNDLE OF TRAJECTORIES AND LYING BETWEEN AND WILL

(&2/$ 'XAMPLES OF TRAJECTORIES OF A DISSIPATIVE DYNAMIC SYSTEM IN STATE SPACE

90


DECREASE WHEN IT MOVES ALONG ITS BUNDLE OF TRAJECTORIES WITH GROWING SEE GURE FOR TWO EXAMPLES

1 !()(16 ,+"$-10

9E NOW INTRODUCE THE FOUR MOST IMPORTANT CONCEPTS OF STABILITY (IRST WE DENE THE STABILITY OF EQUILIBRIUM POINTS OR STATIONARY SOLUTIONS )IVEN 5 ! 5

WITH THE EQUILIBRIUM POINT +T MEANS THAT !

# SIMPLE COORDINATE TRANSFORMATION WILL TAKE ANY ONE POINT IN THE STATE SPACE TO (OR EQUILIBRIUM POINTS WE RST DENE 1%#. 6#$+.+6; OR ;#27018 6#$+.+6; $+(1(,+ ( 6*'4' (14 #.. ':+56 61 '8'4; 6*'4' %144'5210&5 #0 57%* 6*#6 5 (14 #.. 6 9*'4' &'016'5 #0 +0+6+#. %10&+6+10 #0& &'016'5 # 57+6#$.' 014/ 14 /'#574' 6*'0 6*' 21+06 +5 LOCALLY STABLE OR .YAPUNOV STABLE 6HE DENITION STATES THAT A TRAJECTORY WHICH STARTS INSIDE A BALL WITH RADIUS AND CENTRE IN THE DOMAIN NEEDS NOT BE SPHERICAL NEVER WILL LEAVE THE BALL WITH RADIUS AND CENTRE IN THE DOMAIN IS NOT NECESSARILY SPHERICAL SEE (IGURE $+(1(,+ ( +5 .1%#..; 56#$.' (14 #.. 64#,'%614+'5 6*#6 56#46 +05+&' # &1/#+0 9*+%* %106#+05 #0& 6*' (1..19+0) 4'.#6+10 +0 #&&+6+10 *1.&5 5 FOR T 6*'0 +5 ASYMPTOTICALLY STABLE

(&2/$ 6RAJECTORIES NEAR THE STABLE POINT .EFT .OCALLY STABLE AND RIGHT #SYMP TOTICALLY STABLE

H. True

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+T IS THE SECOND AND MOST USED CONCEPT OF STABILITY FOR DISSIPATIVE SYSTEMS 6HE DE NITION STATES THAT IS ASYMPTOTICALLY STABLE IF ALL TRAJECTORIES THAT START IN OR ENTER ULTIMATELY WILL COME ARBITRARILY CLOSE TO 0EXT WE DENE 4$+6#. 6#$+.+6; 9E NOW CONSIDER AN ORBIT OR CLOSED TRAJECTORY AND A TUBE OF RADIUS AROUND 2LEASE REMEMBER THAT A CLOSED TRAJECTORY IN PHASE SPACE REPRESENTS A PERIODIC SOLUTION IN THE PHYSICAL SPACE $+(1(,+ ( #.. 2#6*5 56#46+0) (41/ +0+6+#. 21+065 9+6*+0 # &+56#0%' (41/ #0; 21+06 10 4'/#+0 2'4/#0'06.; 9+6*+0 # 67$' #4170& 1( 4#&+75 (14 6 6*'0 6*' 14$+6 +5 ORBITALLY STABLE +F ALL THE PATHS COME ARBITRARILY CLOSE TO FOR T THEN THE ORBIT IS #5;/2616+%#..; 14$+6#..; 56#$.' SEE (IGURE 6HIS DENITION CAN BE EXTENDED TO OTHER STEADY STATE SOLUTIONS OF THE DYNAMIC SYSTEM

(&2/$ #N ASYMPTOTICALLY ORBITALLY STABLE TRAJECTORY 2OINTS OR PATHS THAT ARE NOT STABLE LOCALLY OR ORBITALLY ARE 7056#$.' 6HEY ARE UNSTABLE UNDER +00+6'5+/#..; 5/#.. &+5674$#0%'5 #LL THE STABILITY CONCEPTS MENTIONED ABOVE REFER TO STABILITY WITH RESPECT TO DISTUR BANCES OF +0+6+#. %10&+6+105 6HE LAST CONCEPT OF STABILITY WE SHALL USE REFER TO DISTUR BANCES OF THE VECTOR FUNCTION 5 9E THEREFORE NALLY DENE 5647%674#. 56#$+.+6; $+(1(,+ &;0#/+% 5;56'/ +5 STRUCTURALLY STABLE +( 5/#.. &+5674$#0%'5 1( 6*' 8'% 614 (70%6+10 5 .'#& 61 '37+8#.'06 2*#5' &+#)4#/5 #N OBSERVABLE PHENOMENON SHOULD ALWAYS BE MODELLED BY A STRUCTURALLY STABLE SYS TEM BECAUSE SMALL ERRORS IN THE MODELLING AND UCTUATIONS IN THE PHYSICAL SYSTEM ARE UNAVOIDABLE # CONSERVATIVE DYNAMIC SYSTEM IS STRUCTURALLY UNSTABLE ACCORDING TO THIS DENITION SINCE THE ADDITION OF AN EVER SO SMALL DAMPING WILL CHANGE THE PHASE DIAGRAM QUALITATIVELY

92


(OR THE INVESTIGATIONS OF ASYMPTOTIC STABILITY OF EQUILIBRIUM POINTS ;#270185 456 /'6*1& MAY BE APPLIED 9ITH NO LOSS OF GENERALITY WE SHALL AGAIN ASSUME THAT THE EQUILIBRIUM POINT IS )IVEN THE AUTONOMOUS SYSTEM 5 ! 5

WITH ! 9E ASSUME THAT THE ,ACOBIAN

! =

>

! 0

EXISTS AND PROVIDES THE MATRIX IN 5 ! 9E THEN WRITE EQN IN THE FORM 5 ! 5 ! 5 5

!

6HE EQUATION 5 ! 5

IS THE EQN LINEARIZED AROUND AND 5 IS A REMAINDER WHICH CONTAINS ALL THE NONLINEAR CONTRIBUTIONS '$,/$* .ET BE AN EQUILIBRIUM POINT OF 5 ! 5 ( 6*' #%1$+#0 *#5 0'+6*'4 6RUE AND 6RZEPACZ INVESTIGATED THE SAME WHEELSET BUT THEY INTRODUCED A REALISTIC WHEELRAIL GEOMETRY AND DRY FRICTION YAW DAMPERS IN #SMUND S MODEL 6HE LATERAL COMPONENT OF THE NORMAL FORCES IN THE WHEELRAIL CONTACT SURFACE WAS IGNORED (OR A LARGE RANGE OF SPEEDS THE WHEELSET OSCILLATED SO MUCH THAT THE AXLE BOXES HIT THEIR GUIDANCES 6HE OSCILLATIONS WERE CHAOTIC 6HE IMPACTS WERE MODELLED AS IDEALLY ELASTIC IMPACTS BUT IN ONE SERIES OF SIMULATIONS THE IMPACTS WERE MODELLED BY LINEARLY ELASTIC DYNAMIC SYSTEMS WHERE THE CONTACT FORCES WERE CALCULATED 6HE DYNAMICS WAS HARDLY AECTED BUT THE COMPUTATION TIME EXPLODED *OMANN NDS IN HIS WORK ON THE DYNAMICS OF TWOAXLE FREIGHT WAGONS WITH THE STANDARD 7+% SUSPENSION THAT THE *OPF BIFURCATION FROM THE STABLE SET OF STATIONARY SOLUTIONS IS A CLASSICAL ONE 6HIS IS IN SPITE OF THE FACT THAT THE PROBLEM CONTAINS MUCH NONSMOOTHNESS *OMANN GIVES THE FOLLOWING REASON 6HE MOTION FOR SMALL LATERAL AND YAW DISPLACEMENTS AROUND EACH OF THE STATIONARY STATES IS SMOOTH (OR SUCH SMALL MOTIONS THE LINKS IN THE SUSPENSION ROLL ON EACH OTHER AND THE SUSPENSION ACTS AS A LINEAR SPRING WITHOUT STICK AND DISSIPATION 6HESE EXAMPLES EMPHASIZE THAT THE DYNAMICS OF NONSMOOTH SYSTEMS MUST BE INVES TIGATED WITH GREAT CARE

H. True

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$ ) *#'!" $",(!( % !" ,$#! ,()#(

'$ 2*$/(" ) ,)21(,+ ,% *,,1' /,!)$*0

+N THIS SECTION WE SHALL BRIEY SURVEY THE NUMERICAL SOLUTION OF VEHICLE DYNAMIC PROB LEMS 5PECIAL ATTENTION WILL BE PAID TO PROBLEMS WITH COEXISTING ATTRACTORS AND TO NON SMOOTH PROBLEMS 9E SHALL DEMONSTRATE HOW SUCH PROBLEMS MUST BE TREATED IN ORDER TO OBTAIN RELIABLE DYNAMIC RESULTS 9HEN COEXISTING ATTRACTORS EXIST IN A NONLINEAR DYNAMIC PROBLEM THE STEADY STATE SOLUTIONS OF A SPECIC INITIAL VALUE PROBLEM DEPEND ON THE INITIAL VALUES 6HIS IS IN CONTRAST TO LINEAR DYNAMIC PROBLEMS WHERE COEXISTING ATTRACTORS CANNOT EXIST 1N THE OTHER HAND THE SOLUTION OF A GIVEN NONLINEAR INITIAL VALUE PROBLEM IS UNIQUE PROVIDED CERTAIN CONTINUITY CONDITIONS ARE SATISED +N OTHER WORDS 6HE TRAJECTORY IN THE STATE SPACE IS UNIQUELY DENED +F HOWEVER TWO TRAJECTORIES THAT CONVERGE TOWARDS TWO DIERENT ATTRACTORS ARE SUCIENTLY CLOSE IN SOME PART OF THE STATE SPACE THEN NUMERICAL TIME STEP INTEGRATION ALONG ONE OF THE ATTRACTORS MAY JUMP TO AND FOLLOW THE OTHER ATTRACTOR 6HIS LEADS TO AN ERRONEOUS NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM 6HE JUMP MAY BE THE RESULT OF A BAD CHOICE OF ERROR BOUNDS OR TIME STEPS BUT IT MAY ALSO BE THE RESULT OF THE USE OF THE WRONG NUMERICAL ROUTINE 9E SHALL DEMONSTRATE BELOW THE ERRONEOUS RESULT OF THE APPLICATION OF AN 'ULER EXPLICIT ROUTINE WHICH SHOULD 0'8'4 BE APPLIED TO NONLINEAR EQUATIONS TO A SIMPLE NONLINEAR INITIAL VALUE PROBLEM +N SECTION WE SOLVED

!

NUMERICALLY WITH TWO PAIRS OF INITIAL CONDITIONS 6WO DIERENT ATTRACTORS WERE FOUND AND THEY ARE SHOWN ON GURE

(&2/$ 6HE NUMERICAL SOLUTION TO THE INITIAL VALUE PROBLEM USING A 4UNGE-UTTA ROUTINE 6HE TRANSIENT APPROACHES THE CORRECT ATTRACTOR

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.ET US NOW CONSIDER EQUATION WITH THE INITIAL CONDITIONS !

!

9E RST SOLVE THE INITIAL VALUE PROBLEM WITH A 4UNGE-UTTA ROUTINE FROM /#2.' 6HE TRANSIENT APPROACHES THE CORRECT ATTRACTOR WITH SMALL AMPLITUDE THAT BELONGS TO THE INITIAL CONDITIONS 6HE RESULT IS SHOWN ON GURE 9E NEXT SOLVE THE SAME INITIAL VALUE PROBLEM WITH AN EXPLICIT 'ULER ROUTINE ALSO FROM /#2.' 6HE RESULT IS SHOWN ON GURE AND IT IS CLEARLY SEEN THAT 6*' 64#05+'06 +0 6*+5 %#5' #2241#%*'5 6*' 16*'4 #0& +0%144'%6 #664#%614 6HE EXPLICIT 'ULER ROUTINE IS VERY POPULAR BECAUSE IT WORKS FAST #S LONG AS NO OTHER SOLUTION OR MEASURED RESULTS FROM A TEST EXIST FOR COMPARISON THE 'ULER SOLUTION WILL BE ACCEPTED +T IS OUTRIGHT DANGEROUS 6HE EXPLICIT 'ULER ROUTINE SHOULD NEVER BE USED FOR THE NUMERICAL SOLUTION OF NONLINEAR DYNAMIC PROBLEMS

(&2/$ 6HE NUMERICAL SOLUTION TO THE INITIAL VALUE PROBLEM USING AN EXPLICIT 'ULER ROUTINE 6HE TRANSIENT APPROACHES 6*' 9410) #664#%614 6HIS SIMPLE EXAMPLE DEMONSTRATES THAT GREAT CARE MUST BE TAKEN WHEN A NUMERICAL SOLVER IS CHOSEN FOR A GIVEN NONLINEAR DYNAMIC PROBLEM )ARG AND &UKKIPATI DISCUSS AND DEMONSTRATE THE USE OF EXPLICIT AND IMPLICIT ROUTINES FOR THE NUMERICAL SOLUTION OF LINEAR AS WELL AS NONLINEAR DYNAMIC PROBLEMS 6HEY TEST A SELECTION OF ROUTINES ON BOTH LINEAR AND NONLINEAR DYNAMIC PROBLEMS 6HEY COMPARE %27 TIME STABILITY AND ARTICIAL DAMPING IN THEIR EXAMPLES +N AN ':2.+%+6 4176+0' THE NEXT STEP DEPENDS EXPLICITLY ON KNOWN QUANTITIES ONLY +N THE NONLINEAR EXAMPLE WITH THE ROLLING WHEEL SET )ARG AND &UKKIPATI P ND THAT THE EXPLICIT SOLVERS ARE EITHER UNSTABLE OR INTRODUCE ARTICIAL DAMPING +N +/2.+%+6 4176+0'5 THE DEPENDENCE IS IMPLICIT SO THE NEXT STEP HAS TO BE CALCULATED BY THE SOLUTION OF AN EQUATION 5INCE VEHICLE DYNAMIC PROBLEMS ARE NONLINEAR THE SAID EQUATION WILL ALSO BE NONLINEAR +T MEANS THAT A NONLINEAR EQUATION HAS TO BE SOLVED NUMERICALLY FOR EACH TIME STEP FOR EXAMPLE BY A 24'&+%614 %144'%614 /'6*1& 6HE EXPLICIT SOLVERS ARE INECIENT FOR THE ANALYSIS OF THE SOCALLED 56+ PROBLEMS )ENERALLY A PROBLEM IS SAID TO BE STI IF THE SOLUTION CONTAINS

H. True

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BOTH SLOW AND VERY FAST PROCESSES 6HE FAST PROCESSES DECAY VERY FAST WITH TIME SO A VERY SHORT STEP LENGTH IS NEEDED TO KEEP THE EXPLICIT SOLVERS FROM BLOWING UP IN THE NUMERICAL SOLUTION (URTHERMORE THE EXPLICIT SOLVERS OFTEN BECOME UNSTABLE WHEN THEY ARE APPLIED TO STI PROBLEMS +N ORDER TO MAINTAIN STABILITY THE STEP SIZE AGAIN NEEDS TO BE VERY SMALL 8EHICLE DYNAMIC PROBLEMS AS WELL AS THE NUMERICAL SIMULATIONS OF REAL WORLD TESTS ARE USUALLY STI PROBLEMS 6HEREFORE THE APPLICATION OF IMPLICIT ROUTINES IS RECOMMENDED FOR THE NUMERICAL SOLUTION OF THESE INITIAL VALUE PROBLEMS 4ILL HAS APPLIED AN IMPLICIT 'ULER ROUTINE TO VEHICLE DYNAMIC PROBLEMS WITH SUCCESS 6RUE AND HIS COWORKERS HAVE MOST OFTEN USED THE .51&# ROUTINE FOR THE CAL CULATION OF BIFURCATION DIAGRAMS ALSO SUCCESSFULLY .51&# CHANGES SOLUTION STRATEGY AUTOMATICALLY DEPENDING ON WHETHER THE PROBLEM IS STI OR NOT 6HE 4UNGE-UTTA SOLVER RK HAS BEEN RECOMMENDED FOR YEARS AS THE MOST EECTIVE ROUTINE FOR THE PURPOSE OF SHADOWING CHAOTIC ATTRACTORS BUT IT IS AN EXPLICIT ROUTINE AND IT THEREFORE CANNOT BE REC OMMENDED FOR THE NUMERICAL ANALYSIS OF VEHICLE DYNAMIC PROBLEMS 6HERE EXIST HOWEVER ALSO IMPLICIT 4UNGE-UTTA SOLVERS FOR STI PROBLEMS #LL THESE 4UNGE-UTTA SOLVERS ARE A PART OF /#6.#$ /OST RECENTLY THE ROUTINE 5&+4- BY 0RSETT AND 6HOMSEN BASED ON THE THEORY FROM #LEXANDER HAS BEEN APPLIED BY *OMANN TO THE NUMERICAL ANALYSIS OF TWOAXLE FREIGHT WAGONS A STI PROBLEM WITH ROLLING AND SLIDING DRY FRICTION AND BUMPER STOPS 5&+4- IS A RECENT IMPLICIT ROUTINE WHICH CAN BE DOWNLOADED FROM THE WEB (OR A DETAILED DESCRIPTION OF TIMESTEPPING EXPLICIT AND IMPLICIT NUMERICAL ROUTINES AND RECOMMENDATIONS FOR THEIR APPLICATIONS THE INTERESTED READER IS REFERRED TO \ THE LITERATURE .AMBERT AND 1DERLIND 6HE TIMESTEPPING NUMERICAL METHODS ARE BY FAR THE MOST POPULAR NUMERICAL METHODS FOR THE SOLUTION OF VEHICLE DYNAMIC PROBLEMS BECAUSE THEY LEND THEMSELVES NATURALLY TO THE SOLUTION OF INITIAL VALUE PROBLEMS +T SHOULD BE MENTIONED HOWEVER THAT STEADY STATE SOLUTIONS HAVE BEEN FOUND BY NUMERICAL SOLUTION OF THE RELATED OPTIMIZATION PROBLEM +T IS CLAIMED ORAL INFORMATION THAT THE STABLE ATTRACTORS IN A BIFURCATION DIAGRAM FOR A VEHICLE CAN BE COMPUTED FASTER IN THAT WAY 6HE ENGINEER WITHOUT A GOOD BACKGROUND IN NUMERICAL ANALYSIS FACES A PROBLEM WHEN HE MUST CHOOSE A DIERENTIAL EQUATION SOLVER &'SOLVER #LTHOUGH A NUMERICAL SOLVER MAY DELIVER A SOLUTION WHEN IT IS APPLIED TO A GIVEN DYNAMIC PROBLEM IT CANNOT BE TAKEN FOR GRANTED THAT THE SOLUTION IS THE CORRECT ONE +T IS STRONGLY RECOMMENDED THAT THE ENGINEER APPLIES TWO DIERENT AND ACKNOWLEDGED ROUTINES TO THE SAME PROBLEM AT LEAST INITIALLY +F THE RESULT OF THE CALCULATIONS USING ONE OF THE SOLVERS DIERS CONSIDERABLY FROM THE RESULTS WHICH THE OTHER SOLVER YIELDS HE MUST CONCLUDE THAT AT LEAST ONE OF THE SOLVERS IS BAD IN HIS CASE 1N THE OTHER HAND HE CANNOT BE SURE THAT THE RESULTS ARE CORRECT EVEN IF THE TWO RESULTS AGREE WITH EACH OTHER

'$ 2*$/(" ) ,)21(,+ "/,00

(0",+1(+2(16

9HEN A VEHICLE DYNAMIC PROBLEM IS NOT SMOOTH THE DYNAMIC PROBLEM MUST BE SUP PLEMENTED WITH FUNCTIONS WHICH DENE THE LOCATION OF THE DISCONTINUITIES IN THE STATE SPACE ! I ! / J ! 0 WHERE / IS THE NUMBER OF DISCONTINUITIES AND 0 IS THE DIMENSION OF THE STATE SPACE 6HE FUNCTIONS ARE *;2'4 574(#%'5 IN THE STATE SPACE SEE CHAPTER $Y THE WORD DISCONTINUITIES IS UNDERSTOOD NOT ONLY DISCONTINUITIES

118


IN THE FUNCTIONS THEMSELVES ZEROORDER DISCONTINUITIES BUT ALSO DISCONTINUITIES IN THE DERIVATIVES UP TO SOME ORDER HIGHERORDER DISCONTINUITIES 6HE HYPER SURFACES ARE THE SOCALLED 59+6%*+0) $170+'5 #S AN EXAMPLE WE CONSIDER NOW A DYNAMIC PROBLEM WITH ONE SWITCHING BOUNDARY ! 6HE ANALYSIS IS EASILY GENERALIZED TO PROBLEMS WITH MORE SWITCHING BOUNDARIES AS LONG AS WE CAN DEAL WITH THEM ONE AT A TIME +N THE TWO DOMAINS DIVIDED BY THE SWITCHING BOUNDARY THE PROBLEM IS SMOOTH AND THERE WE SOLVE THE INITIAL VALUE PROBLEMS IN THE USUAL MANNER 9E ASSUME THAT ON ONE SIDE OF ! AND ON THE OTHER SIDE #LL WE IN ADDITION NEED TO CONSIDER IS THE REGION CLOSE TO THE POINT 2 WHERE THE SOLUTION CROSSES FROM ONE DOMAIN TO THE OTHER 6HE POINT 2 IS CALLED THE 64#05+6+10 21+06 9HEN WE INTEGRATE OUR DYNAMIC PROBLEM AS AN INITIAL VALUE PROBLEM WE MUST STOP THE INTEGRATION IN THE RST DOMAIN AT THE SWITCHING BOUNDARY 6HE SWITCHING BOUNDARY IS ONLY DENED IN TERMS OF THE &'2'0&'06 8#4+#$.'5 9E MUST THEREFORE CONTINUOUSLY CHECK IN OUR SOLUTION PROCEDURE AT WHICH TIME THE TRAJECTORY IS WITHIN A GIVEN VERY SMALL DISTANCE FROM ! OR HAS ALREADY CROSSED THE SWITCHING BOUNDARY +T IS THEN NECESSARY TO REFORMULATE THE SOLUTION PROCEDURE TO DETERMINE THE TIME 6 ITERATIVELY AT WHICH THE TRAJECTORY HITS OR TOUCHES THE SWITCHING BOUNDARY 6 CAN ONLY BE FOUND NUMERICALLY WITHIN A CERTAIN TOLERANCE BUT IT SHOULD BE DETERMINED AS ACCURATELY AS POSSIBLE 6HE TRAJECTORY DYNAMIC PROBLEM IS THEN INTEGRATED UP TO THE TIME 6 AND STOPPED 9E HAVE NOW FOUND THE TRANSITION POINT 2 +T IS DENED BY THE VALUES 6 J ! 0 +N THE MATHEMATICAL FORMULATION OF THE NONSMOOTH DYNAMIC PROBLEM IT MUST BE DENED WHAT WILL HAPPEN AT THE SWITCHING BOUNDARY 6HE COLLISION WITH THE SWITCHING BOUNDARY AND WHAT HAPPENS THERE IS CALLED AN '8'06 #S AN EXAMPLE OF AN EVENT WE MENTION A JUMP V IN THE RST DERIVATIVE OF ONE OF THE STATE VARIABLES SAY AT TIME 6 9E THEN FORMULATE A NEW INITIAL VALUE PROBLEM AT THE TIME 6 USING THE END STATE VARIABLES FOUND AT THE TRANSITION POINT 2 ADJUSTED BY THE LAW OF THE EVENT AS THE NEW INITIAL VALUES FOR THE INTEGRATION +N OUR EXAMPLE WE TAKE ALL OUR END STATE VARIABLES AT THE TIME 6 AND ONLY ADD THE GIVEN JUMP V TO THE DERIVATIVE OF AT THE TIME 6 6HE CONTINUATION OF OUR DYNAMIC PROBLEM IS THEN GIVEN BY THE NEW INITIAL VALUES AT THE TIME 6 COMBINED WITH THE SMOOTH DYNAMIC FORMULATION OF OUR PROBLEM IN THE 4'.'8#06 &1/#+0 +N OUR EXAMPLE THE RELEVANT DOMAIN WILL BE ON THE OTHER SIDE OF THE SWITCHING BOUNDARY THE SECOND DOMAIN 6HEN THE INTEGRATION CONTINUES IN THAT DOMAIN #S AN EXAMPLE WE INVESTIGATE THE DYNAMICS OF A WHEEL SET USED ON A RAILWAY WAGON AND PRESENTED IN )ARG AND &UKKIPATI P 6HE SYSTEM IS SHOWN ON GURE WHERE THE WHEEL SET IS TRAVELLING ALONG AN IDEAL TRACK 6HE NONLINEAR EQUATIONS OF MOTION FOR THE LATERAL DISPLACEMENT AND THE YAW ANGLE AND THE VALUES OF THE DIERENT PHYSICAL PARAMETERS CAN BE FOUND IN )ARG AND &UKKIPATI SECTION 6HE VARIATION OF THE DIERENCE IN THE ROLLING RADII OF THE WHEEL SET VERSUS THE LATERAL DISPLACEMENT IS SHOWN ON GURE

H. True

119

(&2/$ 6HE WHEEL SET MODEL

(&2/$ 6HE VARIATION OF DIERENT VARIABLES WITH LATERAL DISPLACEMENTS

(OR DIERENT ENTRIES WE HAVE THE FOLLOWING VALUES OF THE CONSTANTS AND THE DISTANCE ! IN THE EXPERIMENTS VAR

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(&2/$ 9HEEL SET DYNAMICS SOLUTION FOR 8 ! KMH 9HEN WE SOLVE THE PROBLEM FOR AN INITIAL LATERAL DISPLACEMENT ! ! WITH A SPEED OF KMH WE OBTAIN THE SOLUTION ON GURE WHERE THE DAMPING IS STRONG AND ONLY ONE PASSAGE OF THE DISCONTINUITY OCCURS *ERE THE SOLUTION WITH CONSTANT STEP SIZE CALLED 10'4- AND THE SOLUTION THAT TRACKS DOWN THE DISCONTINUITY CALLED )'4- ARE CLOSE 9HEN THE VELOCITY IS INCREASED TO NEAR THE *OPF BIFURCATION POINT OF 6*+5 274'.; 6*' 14'6+%#. 241$.'/ WHERE MORE PASSAGES OF DISCONTINUITIES OCCUR WE ND VERY DIERENT SOLUTIONS 6HE CONSTANT STEP SIZE SOLUTION IS MUCH LESS ACCURATE THAN THE ONE WHERE THE DISCONTINUITIES ARE TRACKED AS SEEN ON GURE 6HIS LEADS TO DIERENT INTERPRETATIONS OF THE DYNAMIC PROPERTIES +T MAY BE CRITICAL SINCE THIS CASE IS NEAR THE POINT WHERE THE STATIONARY SOLUTION BECOMES UNSTABLE .ET US SUMMARIZE OUR EXPERIENCE GAINED FROM DIERENT NUMERICAL TREATMENTS OF DIS CONTINUITIES IN VEHICLE DYNAMIC PROBLEMS 6HE SIMPLEST WAY TO HANDLE THE DISCONTINUITIES IS TO +)014' 6*' &+5%106+07+6+'5 #0& +06')4#6' #%4155 6*'/ #5 +( 6*'; &1 016 ':+56 1NE HOPES THAT THE SMOOTHING PROPERTY OF THE NUMERICAL ROUTINE WILL AUTOMATICALLY TAKE CARE OF THE PROBLEM WITH THE DISCONTINUITY +F THE DISCONTINUITY IS IN THE SECOND OR HIGHER DERIVATIVES OF THE FUNCTIONS IN THE DYNAMIC PROBLEM THIS STRATEGY MAY WORK WELL 6HE GENERAL ATTITUDE IS THAT AS LONG AS THE NUMERICAL INTEGRATION DELIVERS A SOLUTION THEN THE SOLUTION IS THE CORRECT ONE 6HIS IS RARELY TRUE AS WE JUST DEMONSTRATED IN THE LAST EXAM PLE # SOLUTION FOUND IN THIS WAY MAY EVEN BE QUALITATIVELY WRONG # CHAOTIC DYNAMIC BEHAVIOUR MAY BE TOTALLY MISSED 6HIS WAS ILLUSTRATED BY :IA AND 6RUE .AST BUT NOT LEAST THE NUMERICAL SOLVER WILL CONSUME A LOT OF %27TIME WITH THE INTEGRATION ACROSS THE DISCONTINUITY AND WE DO NOT KNOW THE RESULTING NUMERICAL APPROXIMATION OF

H. True

121

(&2/$ 9HEEL SET DYNAMICS SOLUTION FOR 8 ! KMH # 274'.; 6*'14'6+%#. 241$.'/

THE SMOOTHING FUNCTION 6HE PROBLEM WITH THE RESULTING SMOOTHING FUNCTION LEADS TO THE SECOND METHOD OF HANDLING THE DISCONTINUITIES *' 5/116*+0) (70%6+105 #4' 24'5%4+$'& +0 '#%* #0& '8'4; %#5' 6HEN AT LEAST WE KNOW HOW THE NUMERICAL ROUTINE HANDLES THE DISCONTINUITY AND THE %27TIME WILL BE REDUCED 6HE SMOOTHING FUNCTION MUST MODEL THE PHYSICAL REALITY IN THE BEST POSSIBLE WAY /ANY PEOPLE CLAIM THAT THIS METHOD IS THE OPTIMAL ONE BECAUSE IT MODELS THE PHYSICAL REALITY CLOSELY 6HE PROBLEM WITH THE METHOD IS HOWEVER THAT A GOOD MATHEMATICAL MODEL IS NOT ALWAYS KNOWN +N THE ARTICLES BY 6RUE AND #SMUND AND 6RUE AND 6RZEPACZ THEIR OWN SMOOTH MODEL OF STICK SLIP DRY FRICTION IN PLANE CONTACT BETWEEN STEEL OR CAST IRON SURFACES WAS APPLIED 6HE RESULTS OF THE NUMERICAL DYNAMIC ANALYSIS CONTAIN A LOT OF CHAOTIC DYNAMICS AS COULD BE EXPECTED FROM THE NATURE OF THE PROBLEM # COMPARISON WAS HOWEVER MADE BETWEEN AN IDEALLY KINEMATICAL MODEL AND A DYNAMIC MODEL OF AN '.#56+% +/2#%6 9HEN THE MOVING PART HAD ENTERED THE PART IN REST THE VELOCITY OF THE MOVING PART CHANGED SIGN IN THE KINEMATICAL MODEL +N THE DYNAMIC MODEL THE LAW OF ELASTIC DEFORMATION WAS APPLIED 6RUE AND 6RZEPACZ USED A CONSTANT AND VERY SMALL STEPLENGTH IN THE INTEGRATIONS

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OF THE DYNAMIC MODEL 6HE APPLICATION STOPPED WHEN THE TWO PARTS WERE SEPARATED AGAIN 1NLY THE DYNAMIC MODEL IS ABLE TO YIELD AN ESTIMATE OF THE CONTACT FORCES +N THE DYNAMIC MODEL THE %27TIME USED FOR INTEGRATION ACROSS THE IMPACT EXPLODED 6HE RESULTS DIERED AS COULD BE EXPECTED BUT BOTH RESULTS LIE ON THE SAME CHAOTIC ATTRACTOR +T IS INTERESTING TO NOTE HOWEVER THAT THE DYNAMICS OF THE SYSTEM IS THE SAME CHAOTIC ONE IN THE TWO CASES 6HE SITUATION WAS DISCUSSED IN CHAPTER WITH RELATION TO THE MODELLING OF THE IMPACT 6HE ROBUST DYNAMICS IS CLOSELY CONNECTED WITH THE CHAOTIC DYNAMICS IN THIS CASE 6HE ROBUSTNESS IS HOWEVER NOT GUARANTEED +N OTHER PROBLEMS THE DYNAMICS OF THE TWO CASES MAY DIER 6HE THIRD AND CORRECT WAY OF HANDLING THE DISCONTINUITIES NUMERICALLY IS TO 64#%&190 6*' 64#05+6+10 21+06 #5 #%%74#6'.; #5 2155+$.' 6HE TRANSITION POINT IS THE POINT IN THE STATE SPACE WHERE THE TRAJECTORY HITS THE SWITCHING BOUNDARY *'0 5612 6*' 6+/' +06')4#6+10 1( 6*' &;0#/+% 241$.'/ #6 6*#6 21+06 #22.; 6*' .#9 1( 64#05+6+10 #0& %106+07' 6*' 6+/' +06')4#6+10 1( 6*' &;0#/+% 241$.'/ 10 6*' 4'.'8#06 5+&' 1( 6*' 59+6%*+0) $170& #4; 6HIS IS THE SAFEST AND MOST ACCURATE METHOD TO DEAL WITH NONSMOOTH PROBLEMS 6HE EXTRA %27TIME SPENT IN THE DETERMINATION OF THE TRANSITION POINT IS MUCH LESS THAN THE TIME NOT SPENT FOR THE INTEGRATION OF THE DYNAMIC PROBLEM WITH VERY SMALL TIME STEPS ALONG STEEP GRADIENTS ACROSS THE DISCONTINUITY +T IS THE LOGICAL NUMERICAL APPLICATION OF THE RESULTS PRESENTED IN SECTION 6HE METHOD WAS USED WITH SUCCESS BY *OMANN TO INVESTIGATE THE DICULT PROBLEM OF THE DYNAMICS OF TWOAXLE FREIGHT WAGONS *IS MODELS CONTAIN BOTH ROLLING AND SLIDING FRICTION IN THE SUSPENSION AS WELL AS POSSIBLE IMPACTS BETWEEN THE AXLE BOXES AND THEIR GUIDANCE 6HE LACK OF PROPER CONSIDERATION OF THE DISCONTINUITIES IN THE NUMERICAL ANALYSIS LEADS IN GENERAL TO AN EVEN QUALITATIVELY ERRONEOUS RESULT OF THE CALCULATIONS +T IS IMPORTANT TO IDENTIFY COEXISTING ATTRACTORS IN NONLINEAR DYNAMIC SYSTEMS 6HE NECESSITY IS OBVIOUS WHEN THE ATTRACTORS ARE CHARACTERIZED BY DIERENT AMPLITUDES +T MAY SEEM LESS OBVIOUS IF THE ATTRACTORS HAVE ALMOST THE SAME AMPLITUDES AND ARE ONLY QUALITATIVELY DIERENT +F HOWEVER LIKE IN 6RUE AND #SMUND 6RUE AND 6RZEPACZ AND :IA AND 6RUE ONE OF THE COEXISTING ATTRACTORS IS A CHAOTIC ATTRACTOR THEN GREAT CAUTION MUST BE EXERCISED 9E REFER TO THE DISCUSSION OF CHAOTIC TRANSIENTS AT THE END OF CHAPTER 6HE EXISTENCE OF A COEXISTING CHAOTIC ATTRACTOR WILL OFTEN GIVE RISE TO VIOLENTLY OSCILLATING TRANSIENT MOTIONS WITH DANGEROUSLY HIGH AMPLITUDES 6HIS HAPPENS FOR TRANSIENTS APPROACHING THE CHAOTIC ATTRACTOR AS WELL AS FOR TRANSIENTS APPROACHING A NONCHAOTIC ATTRACTOR +T IS THEREFORE VERY IMPORTANT TO IDENTIFY ALL THE CHAOTIC ATTRACTORS IN A NONLINEAR DYNAMIC PROBLEM 6HE POSSIBLE LARGE AMPLITUDE TRANSIENTS MAY LEAD TO ACCIDENTS IN THE PHYSICAL WORLD

4 6 1, ",*-21$ 01 !)$ !/ +"'$0 ,% 3$'(")$ #6+ *(" -/,!)$*0 +2*$/ (" ))6

+N ORDER TO ND THE STABLE BRANCHES THE ATTRACTORS OF A VEHICLE DYNAMIC PROBLEM USE 2#6* (1..19+0) +0 56#6' 2#4#/'6'4 52#%' 9E SHALL ASSUME THAT 6*' 64#%- +5 +&'#. #0& (4'' 1( &+5674$#0%'5 4EMEMBER THAT WE WANT TO ND THE EIGENDYNAMICS OF THE VEHICLE 9E HAVE GIVEN A STABLE STEADY STATE SOLUTION TO OUR DYNAMIC PROBLEM FOR A CERTAIN VALUE OF THE CONTROL PARAMETER WHICH IN OUR CASE IS THE SPEED 8 SEE GURE 9E

H. True

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CAN THEN ND THE VECTOR SOLUTION AT A TIME AND STORE THE STATE VECTOR +T IS CALLED THE END VALUE 6HE PATH FOLLOWING IS PERFORMED STEPWISE +N EACH STEP WE INCREASE OR DECREASE 8 WITH A SMALL INCREMENT AND CALCULATE THE STEADY STATE SOLUTION OF THE RESULTING DYNAMIC PROBLEM USING THE END VALUE OF THE PRECEDING STEP AS THE INITIAL VALUE FOR THE NEW STEP IN 8 9E TAKE AS MANY STEPS AS NEEDED 6HE PROCESS WILL END WHEN THE TRANSIENT IN THE INTEGRATION APPROACHES ANOTHER STABLE STEADY STATE SOLUTION FOR A CERTAIN VALUE OF THE PARAMETER 8 IS CLOSE TO A BIFURCATION POINT THAT HAS JUST BEEN PASSED AND WHICH CAN NOW BE DETERMINED BY INTERPOLATION OF THE PARAMETER 8 OR BY THE 0EWTON4APHSON METHOD IN ORDER TO ND THE BIFURCATION POINT +N THIS WAY A SOLUTION OF THE DYNAMIC PROBLEM CAN BE TRACED OR (1..19'& AND BIFURCATION POINTS FOUND IN THE PARAMETERSTATE SPACE #T THE BIFURCATION POINTS WE TURN OUR ATTENTION TO THE NEW STABLE BRANCHES AND FOLLOW THEM IN DEPENDENCE ON 8 FOR BOTH GROWING AND FALLING VALUES #PPLY AS THE INITIAL CONDITION THE END VALUES OF THE NEW STEADY STATE SOLUTION +T IS HIGHLY RECOMMENDABLE TO CALCULATE THE EIGENVALUES OF THE DYNAMIC PROBLEM LINEARIZED AROUND THE STABLE STEADY STATE SOLUTION AND ND THEIR CHANGE WITH THE PARAMETER ACROSS THE BIFURCATION POINT ASSUMING THAT THE ,ACOBIAN EXISTS 6HE CALCULATION IS A VALUABLE VERICATION OF THE PROPERTIES OF THE FOUND BIFURCATION +N A BIFURCATION OF A PERIODIC SOLUTION FROM A STABLE STATIONARY SOLUTION TWO COMPLEX CONJUGATE EIGENVALUES SHOULD CROSS THE IMAGINARY AXIS AT THE BIFURCATION POINT FROM THE NEGATIVE TO THE POSITIVE REAL HALF PLANE 6HE SUCCESS OF THE METHOD IS BASED ON THE ASSUMPTION THAT THE STABLE BRANCHES CAN BE FOUND AS BIFURCATIONS FROM AT LEAST ONE STABLE STEADY STATE SOLUTION BRANCH OF THE PROBLEM THAT IS KNOWN A PRIORI +T MUST BE EMPHASIZED THAT IT IS NOT GUARANTEED THAT ALL THE STABLE BRANCHES CAN BE FOUND IN THIS WAY IN ALL DYNAMIC PROBLEMS +N VEHICLE DYNAMICS THE AIM IS TO DESIGN A VEHICLE THAT RUNS SMOOTHLY ALONG A GIVEN TRACK 9E SHALL THEREFORE MAKE THE BASIC ASSUMPTION THAT THE VEHICLE DYNAMIC MODEL TO BE INVESTIGATED HAS A KNOWN STEADY STATE SOLUTION IN A CERTAIN SPEED INTERVAL IN AN APPROPRIATE MOVING COORDINATE SYSTEM +N VEHICLE DYNAMICS THE SPEED 8 IS AN APPROPRIATE CONTROL PARAMETER AND IN MOST CASES THE APPROPRIATE COORDINATE SYSTEM IS ONE THAT MOVES WITH THE CONSTANT SPEED 8 ALONG THE TRACK OF THE VEHICLE # DYNAMIC MODEL OF A GOOD VEHICLE WILL MOST OFTEN HAVE A TRIVIAL STABLE SOLUTION FOR SUCIENTLY SMALL VALUES OF THE SPEED 8 IE 5 ! +T CAN BE USED AS THE INITIAL SOLUTION 6HE PATH FOLLOWING PROCEDURE CAN BE ACCELERATED AND AUTOMATED BY APPLICATION OF THE 4#/2+0) /'6*1& +N THE DYNAMIC SYSTEM THE CONTROL PARAMETER 8 IS REPLACED BY A MONOTONICALLY SLOWLY GROWING OR DECAYING FUNCTION OF TIME 8 +N THE RAMPING METHOD WE START THE INTEGRATION OF THE DYNAMIC PROBLEM AS USUAL WITH THE KNOWN END VALUE AND INTEGRATE UP TO AN APPROPRIATE END TIME 6 SO THAT =886> COVERS THE DESIGNED SPEED RANGE OF THE VEHICLE 4EPEAT THE PROCEDURE AS LONG AS NECESSARY TO DETERMINE ALL THE ATTRACTORS 6HE BIFURCATION DIAGRAM WILL IN PRACTICE BE QUITE WELL APPROXIMATED BY THE RAMPING SOLUTION BUT THE METHOD OVERSHOOTS THE BIFURCATION POINTS AND ONE OR MORE OF CLOSELY SITUATED BIFURCATION POINTS MAY BE MISSED 6HE BIFURCATION POINTS MUST AFTERWARDS BE RECALCULATED WITH GREATER ACCURACY 6HE SMALLEST PARAMETER VALUE FOR WHICH THERE EXISTS ANOTHER STABLE STEADY STATE SOLUTION ATTRACTOR TO OUR DYNAMIC PROBLEM IN ADDITION TO THE STATIONARY ONE IS THE %4+6+%#. 52''& OF THE VEHICLE 6HE PATHFOLLOWING METHOD HAS BEEN USED SUCCESSFULLY NOT

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ONLY BY 6RUE AND HIS COWORKERS +T IS DESCRIBED AND DISCUSSED BY 6RUE IN SEVERAL PAPERS 6RUE A AND ITS RESULTS HAVE BEEN VERIED IN ROAD TESTS 6HERE EXIST GOOD COMMERCIAL VEHICLE DYNAMIC SIMULATION ROUTINES 9HEN A POTENTIAL USER MUST SELECT A PROGRAM HE MUST TEST HIS REQUIREMENTS OF THE PROGRAM AGAINST THE PERFORMANCE OF THE PROGRAM AND AGAINST THE REQUIREMENTS LISTED IN THIS SECTION 6HE PRO GRAMS MUST CONTAIN AT LEAST TWO RECOGNIZED IMPLICIT SOLVERS FOR THE SOLUTION OF NONLINEAR DYNAMIC PROBLEMS AND THE USER MUST BE ABLE TO CONTROL THE NUMERICAL PARAMETERS SUCH AS STEP LENGTHS CONSTANT OR VARYING SIZE AND ERROR BOUNDS 'XPLICIT 'ULER ROUTINES ARE OF COURSE BANNED FROM USE

+F NOT TREATED PROPERLY THE INTERPRETATION OF THE DYNAMIC PROPERTIES FROM THE NUMERICAL RESULTS MAY BE VERY DIERENT FROM THE PROPERTIES OF THE REAL SYSTEM 9E RECOMMEND WHEN THE NUMERICAL TOOLS ARE SELECTED THAT A SIMILAR CARE IS SPENT AS THE CARE TAKEN OF THE MODELLING +T IS OF THE OUTMOST IMPORTANCE THAT PROPERTIES THAT INUENCE THE DYNAMIC BEHAVIOUR SUCH AS THE DISCONTINUITIES ARE PROPERLY ACCOUNTED FOR

*! )% !" ,()# ,$#!( &UE TO THE NONLINEAR COUPLING WITH THE SUPPORTING GROUND IT IS BASICALLY IMPOSSIBLE TO SPLIT THE DYNAMICS OF THE TOTAL VEHICLEGROUND SYSTEM INTO EECTS CAUSED BY THE PROPERTIES OF THE VEHICLE AND THOSE OF THE SUPPORT 6HE CONTACT FORCES BETWEEN THE RAILS AND WHEELS ARE IMPORTANT UNKNOWNS IN THE SIMULATIONS NOT ONLY BECAUSE THEY ACT AS AN INPUT TO THE DYNAMIC VEHICLE MODEL 6HEY ARE IMPORTANT FOR THE DESIGN OF THE VEHICLE AND THE DETERIORATION OF THE STRUCTURE OF THE SUPPORT 9E ARE HOWEVER LUCKY IN THE SENSE THAT THE DYNAMIC INUENCE OF THE VEHICLE ON THE GROUND ACTS ON A MUCH LARGER TIMESCALE THAN THE INSTANTANEOUS ACTION THE VEHICLE IS EXPOSED TO FROM THE GROUND 6HE FEEDBACK CAN THEREFORE BE NEGLECTED IN THE SHORTTERM MODELLING OF THE VEHICLE DYNAMICS +T IS HOWEVER POSSIBLE TO MAKE DEDUCTIONS AND ACHIEVE A BETTER UNDERSTANDING OF THE DYNAMICS OF THE TOTAL SYSTEM IF THE DYNAMICS OF SOME OF THE ELEMENTS IN THE SYSTEM IS KNOWN 6HE DYNAMIC FEATURES OF A VEHICLE MODEL SHOULD THEREFORE BE INVESTIGATED RST 6HE INVESTIGATION OF THE FULL NONLINEAR VEHICLE DYNAMIC MODEL STARTS WITH A NUMERICAL CALCULATION OF THE BIFURCATION DIAGRAMS FOR THE VEHICLE RUNNING ON AN IDEAL STRAIGHT TRACK SECTION 6HE CALCULATIONS MUST BE PERFORMED FOR ALL RELEVANT CONTACT GEOMETRIES AND FOR A RELEVANT SELECTION OF ADHESION COECIENTS 6HE SPEED IS CHOSEN AS THE CON TROL PARAMETER WHILE ALL OTHER PARAMETERS ARE KEPT CONSTANT 6HE RELEVANT ATTRACTORS IN THE PARAMETERSTATE SPACE MUST BE FOUND AND THE CRITICAL SPEED DETERMINED +F NECES SARY THE CALCULATION OF THE BIFURCATION DIAGRAMS CAN BE REPEATED WITH OTHER PARAMETER COMBINATIONS 0EXT THE INVESTIGATION CONTINUES WITH A SET OF NUMERICAL CALCULATIONS OF THE BIFURCATION DIAGRAMS FOR QUASISTATIONARY CURVING IN ARCS WITH SELECTED RADII #GAIN THE SPEED IS CHOSEN AS THE CONTROL PARAMETER # LOWER CRITICAL SPEED WILL OFTEN BE FOUND IN CURVES WITH A LARGE RADIUS THAN THE ONE FOUND BY DRIVING ON TANGENT TRACK 9HEN THE ATTRACTORS ARE KNOWN WE ARE READY TO MAKE NUMERICAL SIMULATIONS OF THE VEHICLE DYNAMICS ON REALISTIC TRACKS TRANSITION CURVES TURNOUTS AND IN OTHER SITUATIONS WHERE THE TRANSIENT RESPONSE IS IMPORTANT 6HESE SIMULATIONS SERVE ALSO AS A BASIS FOR COMPARISONS WITH MEASURED RESULTS FROM ROAD TESTS AND AS A VERICATION OF THE DYNAMIC

H. True

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MODEL 6HIS MAY GIVE RISE TO A GREAT VARIETY OF NONLINEAR MOTIONS +N A MASTER THESIS 'NGBO %HRISTIANSEN INVESTIGATED THE DYNAMICS OF THE %OOPERRIDER BOGIE ON A SINUOUS TRACK &EPENDING ON THE PHASE DIERENCE BETWEEN AND THE AMPLITUDE OF THE SINE FUNCTIONS OF THE RIGHT AND LEFT RAIL 'NGBO %HRISTIANSEN FOUND MANY OF THE NONLINEAR PHENOMENA IN THE BOOKS BUT ONLY WHEN THE SPEED WAS ABOVE THE CRITICAL SPEED 6HE DYNAMICS ABOVE THE CRITICAL SPEED WAS OBVIOUSLY A RESULT OF THE MODE INTERACTIONS BETWEEN THE HUNTING MOTION OF THE VEHICLE AND THE EXCITATIONS FROM THE TRACK 5INCE MOST RAILWAY FREIGHT VEHICLES TODAY OFTEN RUN AT SUPERCRITICAL SPEEDS SPECIAL CARE MUST BE TAKEN WHEN THEIR DYNAMICS IS MODELLED AND INVESTIGATED +N EVERY STEP OF THE INVESTIGATION IT IS IMPORTANT TO PAY ATTENTION TO THE NUMERICS #FTER HAVING SPENT SO MUCH EORT ON THE CONSTRUCTION OF THE THEORETICAL MODEL MANY INVESTIGATORS SEEM TO BELIEVE THAT THE JOB IS DONE /ORE EORT IS NEEDED AT THAT POINT THAN THAT SPENT BY TAKING JUST ANY NUMERICAL SOLVER THAT CAN BE FOUND ON THE WEB OR THAT IS INCORPORATED IN A COMMERCIAL SIMULATION PROGRAM OFTEN WITHOUT ANY INFORMA TION ABOUT THE TYPE OF SOLVER AND ITS PARAMETERS 9E RECOMMEND WHEN THE NUMERICAL TOOLS ARE SELECTED THAT A SIMILAR CARE IS SPENT AS THE CARE TAKEN OF THE MODELLING +T IS VERY IMPORTANT THAT PROPERTIES SUCH AS THE DISCONTINUITIES WHICH INUENCE THE DYNAMIC BEHAVIOUR ARE PROPERLY ACCOUNTED FOR SEE SECTION +T IS OF COURSE PERMITTED TO MAKE APPROXIMATIONS OF SOLUTIONS TO NONLINEAR SYSTEMS FOR INSTANCE BY LINEARIZATION OF NONLINEAR CHARACTERISTICS OR TERMS IN THE DYNAMIC SYSTEM +F THE TOTAL DYNAMIC SYSTEM SHALL BE LINEARIZED IT IS IMPORTANT THAT THE OPERATOR KNOWS WHICH STABLE SOLUTION IE STABLE BRANCH HE WILL LINEARIZE AROUND *E MUST ALSO INVESTIGATE WHETHER A LINEARIZATION IS PERMITTED SEE SECTION 6HE ,ACOBIAN ! = > ! 0 MUST EXIST AND THE REST TERM MUST SATISFY 5 5 FOR 5 IF THE TRIVIAL SOLUTION IS CONSIDERED +T IS THEREFORE IMPORTANT TO KNOW THE FULL NONLINEAR OPERATOR AS WELL AS ITS BIFURCATION DIAGRAM RST 6HE LINEARIZED OPERATOR CAN YIELD VALUABLE INFORMATION ABOUT CERTAIN PROPERTIES OF THE VEHICLE MODEL 6HE EIGENVAL UES WILL FOR INSTANCE INFORM ABOUT CHARACTERISTIC FREQUENCIES AND THEIR ATTENUATION AND POTENTIAL RESONANCES IN THE VEHICLE DYNAMIC SYSTEM 9HEN THE ANALYSES OF THE NONLINEAR DYNAMICS OF VEHICLE SYSTEMS ARE PERFORMED WITH ACCURATE THEORETICAL MULTIBODY SYSTEM MODELS THAT CONSIST OF ELEMENTS WITH WELL DENED AND VERIED DYNAMICS WITH DUE REGARD TO THE NONLINEAR NATURE OF THE SYSTEMS ON MODERN COMPUTER SYSTEMS BY WELL EDUCATED PERSONNEL USING SOFTWARE THAT IS CAREFULLY SELECTED WITH REGARD TO THE NATURE OF THE SPECIC DYNAMIC PROBLEM THEN THE RESULTS WILL BE SO ACCURATE THAT SIMULATIONS IN FUTURE CAN BE USED AS A DOCU MENTATION FOR CERTAIN VEHICLE PERFORMANCE REQUIREMENTS AND EVEN AS A PART OF THE SAFETY SPECICATIONS 6HIS WILL GIVE THE MANUFACTURERS THE NATIONAL BOARDS OF SAFETY AND LARGE USER ORGANISATIONS LIKE THE RAILWAY COMPANIES A STRONG AND RELIABLE TOOL FOR THE DESIGN AND TESTING OF VEHICLE SYSTEMS UNDER REALISTIC OR EVEN UNREALISTIC OPERATING CONDITIONS 6HE MODELS MAY ALSO BE VALUABLE TOOLS FOR INVESTIGATIONS OF ACCIDENTS 6HE ADVANTAGES OF SUCH THEORETICAL ANALYSES ARE LOWER COSTS COMPARED WITH ALTERNATIVE METHODS EXIBILITY

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WITH RESPECT TO DESIGN PARAMETERS AND OPERATING CONDITIONS AND IN MANY CASES A FASTER PRODUCTION OF RESULTS

!"!%'& , 4 #LEXANDER &ESIGN AND IMPLEMENTATION OF &+4- INTEGRATORS FOR STI SYSTEMS 22.+'& 7/'4+%#. #6*'/#6+%5 [ , $IRKEDAL 0IELSEN '9 '8'.12/'065 +0 6*' *'14; 1( !*''. #+. 106#%6 '%*#0+%5 2H& THESIS +// 6HE 6ECHNICAL 7NIVERSITY OF &ENMARK # %HUDZIKIEWICZ '.'%6'& .'/'065 1( 6*' 106#%6 41$.'/5 0'%'55#4; (14 +08'56+)#6+0) 6*' #+. '*+%.' ;56'/ ;0#/+%5 PAGES [ #DVANCED 4AILWAY 8EHICLE 5YSTEM &YNAMICS 9YDAWNICTWA 0AUKOWO6ECHNICZNE 9ARSAW 2OLAND 0- %OOPERRIDER 6HE HUNTING BEHAVIOR OF CONVENTIONAL RAILWAY TRUCKS 0 )+0''4+0) #0& 0&7564; [ 5 &AMME 7 0ACKENHORST # 9ETZEL AND $ 9

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