Multibody Dynamics. Education/Advanced Courses. Spring MULTIBODY DYNAMICS (FLERKROPPSDYNAMIK) (FMEN02, 7.5p)

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1 Multibody Dynamics Spring 28 Education/Advancd Courss MULTIBODY DYNAMICS (FLERKROPPSDYNAMIK) (FMEN2, 7.5p)

2 Cours coordinator Prof. Aylin Ahadi, Mchanics, LTH Prof. Emritus Pr Lidström 2

3 COURSE REGISTRATION Multibody Dynamics (FMEN2) Spring 28 pnr ftrnamn förnamn mail Abramowitz Ryan Adibi Pardis Backstam Alxandr Brggrn Jnny Brglund Darll Olivia Dalklint Anna Ek Hofmann Ludvig Fostr John Stwart Hllholm Linna Hodali Cabrra Flip Javir Hultgrn Viktor Itka Rita Jönsson Gustaf Kingsly Albrt Lloriux Virgini Lindqvist Max Lindström Malin Nybrg Johanns Ong Samul Prsson Gustav Sandll Karolina Sarajärvi Marko Schulthis Robin Stnson Dnnis Svanbrg Gustaf 3

4 FUNDAMENTALS OF MULTIBODY DYNAMICS LECTURE NOTES Multibody systm (MSC ADAMS ) Pr Lidström Kristina Nilsson Division of Mchanics, Lund Univrsity 4

5 Password: ulr Lonard Eulr

6 COURSE BOOK RESERVATION (HARD COPY) Fundamntals of Multibody Dynamics, Lctur Nots SEK 35 Nam Abramowitz Ryan Adibi Pardis Backstam Alxandr Brggrn Jnny Brglund Darll Olivia Dalklint Anna Ek Hofmann Ludvig Fostr John Stwart Hllholm Linna Hodali Cabrra Flip Javir Hultgrn Viktor Itka Rita Jönsson Gustaf Kingsly Albrt Lloriux Virgini Lindqvist Max Lindström Malin Nybrg Johanns Ong Samul Prsson Gustav Sandll Karolina Sarajärvi Schulthis Stnson Svanbrg Marko Robin Dnnis Gustaf Signatur 6

7 Cours program Cours litratur: Lidström P., Nilsson K.: Fundamntals of Multibody Dynamics, Lctur Nots (LN). Div. of Mchanics, LTH, 27. Th Lctur Nots will b on th Cours wb sit. Hand out matrial: Solutions to slctd Exrciss, Examination tasks (Assignmnts), Projct spcification. Exampls of prvious writtn tsts. Tachr: Prof. Aylin Ahadi (Lcturs, Exrciss, Cours coordination) Phon: , mail: aylin.ahadi@mk.lth.s Schdul (w. -7): Lcturs: Monday 3-5 Tusday 3-5 Thursday - 2, room M:IP2 Exrciss: Wdnsday 8 -, room M:IP2 7

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10 Cours objctivs and contnts Cours objctivs: Th objctiv of this cours is to prsnt th basic thortical knowldg of th Foundations of Multibody Dynamics with applications to machin and structural dynamics. Th cours givs a mchanical background for applications in, for instanc, control thory and vhicl dynamics. Cours contnts: ) Topics prsntd at lcturs and in th Lctur Nots. 2) Exrciss. 3) Examination tasks 4) Application projct Th scop of th cours is dfind by th curriculum abov and th lctur nots (Fundamntals of Multibody Dynamics, Lctur Nots). Th taching consists of Lcturs and Exrciss: Lcturs: Lcturs will prsnt th topics of th cours in accordanc with th curriculum prsntd abov. Exrciss: Rcommndd xrciss workd out by th studnt will srv as a prparation for th Examination tasks. Solutions to slctd problms will b distributd.

11 Examination

12 CONTENTS. INTRODUCTION 2. BASIC NOTATIONS 2 3. PARTICLE DYNAMICS 3 3. On particl systm Many particl systm RIGID BODY KINEMATICS Th rigid body transplacmnt Th Eulr thorm Mathmatical rprsntations of rotations Vlocity and acclration THE EQUATIONS OF MOTION Th Nwton and Eulr quations of motion Balanc laws of momntum and momnt of momntum Th rlativ momnt of momntum Powr and nrgy 6. RIGID BODY DYNAMICS 4 6. Th quations of motion for th rigid body Th inrtia tnsor Fixd axis rotation and baring ractions Th Eulr quations for a rigid body Powr, kintic nrgy and stability Solutions to th quations of motion: An introduction to th cas of 7. THE DEFORMABLE BODY Kinmatics Equations of motion Powr and nrgy Th lastic body THE PRINCIPLE OF VIRTUAL POWER Th principl of virtual powr in continuum mchanics 29 2

13 . THE MULTIBODY Multibody systms Dgrs of frdom and coordinats Multibody kinmatics Lagrang s quations Constitutiv assumptions and gnralizd intrnal forcs Extrnal forcs Th intraction btwn parts Th Lagrangian and th Powr thorm 295. CONSTRAINTS 3. Constraint conditions 3.2 Lagrang s quations with constraint conditions 35.3 Th Powr Thorm 323. COORDINATE REPRESENTATIONS 327. Coordinat rprsntations Linar coordinats 33.3 Floating fram of rfrnc 359 APPENDICES A A. Matrics A A.2 Vctor spacs A6 A.3 Linar mappings A2 A.4 Th Euclidan point spac A37 A.5 Drivativs of matrics A45 A.6 Th Frobnius thorm A54 A.7 Analysis on Euclidan spac A56 SOME FORMULAS IN VECTOR AND MATRIX ALGEBRA EXERCISES 3

14 Mchanics in prspctiv Quantum Nano Micro-mso - Macro-Continuum lngthscal Å nm µm mm 4

15 Prrquicits Multibody Dynamics Synthsis 3D Analysis Linar algbra Continuum mchanics Mchanics (Basic cours Finit lmnt mthod Matlab Mathmatics Mchanics Numrical analysis 5

16 Contnt Multibody Dynamics Rigid bodis Flxibl bodis Coordinats Constraints Lagrang s quations A multibody systm is a mchanical systm consisting of a numbr of intrconnctd componnts, or parts, prforming motions that may involv larg translations and rotations as wll as small displacmnts such as vibrations. Intrconnctions btwn th componnts ar of vital importanc. Thy will introduc constraints on th rlativ motion btwn componnts and in this way limit th possibl motions which a multibody systm may undrtak. 6

17 Cours objctivs Primary objctivs ar to obtain a thorough undrstanding of rigid body dynamics Lagrangian tchniqus for multibody systms containing constraint conditions a good undrstanding of th dynamics multibodis consisting of coupld rigid bodis som undrstanding of th dynamics of multibodis containing flxibl structurs 7

18 an ability to prform an analysis of a multibody systm and to prsnt th rsult in a writtn rport rad tchnical and scintific rports and articls on multibody dynamics som knowldg of industrial applications of multibody dynamics computr softwars for multibody problms 8

19 Industrial applications Arospac Automotiv Multibody systm dynamics (MBS dynamics) is motivatd by an incrasing nd for analysis, simulations and assssmnts of th bhaviour of machin systms during th product dsign procss. Mchanism Robotics 9

20 Commrcial MBS softwars Adams Dads Simpac Ansys Dymola/Modlinc Simulink/Matlab RcurDyn Adams Studnt Edition Multibody Dynamics Simulation 2

21 Multibody systm Classical stam ngin quippd with a slidr crank mchanism Slidr crank mchanism convrts rciprocating translational motion of th slidr into rotational motion of th crank or th othr way around. 2

22 Multibody systm Conncting rod Rvolut joint Flywhl Translatonal joint Slidr Rvolut joint Fundamnt Crank Figur.2 Stam ngin with slidr crank mchanism. Th crank is connctd to th slidr via a rod which prforms a translational as wll as rotational motion. Th connction btwn th slidr and th rod is maintaind through a so-calld rvolut joint and a similar arrangmnt conncts th rod and th crank. 22

23 Multibody systm A schmatic (topological) pictur of th slidr crank mchanism. Th mchanism consists of thr bodis; th slidr, th crank and th rod. Ths ar connctd by rvolut joints which allow thm to rotat rlativ on anothr in plan motion. It is practical to introduc a fourth part, th ground, which is assumd to b rsting in th fram of rfrnc. 23

24 Joints in thr dimntions Th rvolut joint works lik a hing and allows two connctd parts to rotat rlativ to on anothr around an axis, fixd in both parts. A prismatic or translational joint allows two connctd parts to translat rlativ to on anothr along an axis, fixd in both parts. Th position of th slidr can b masurd by th singl coordinat. W say that th prismatic joint has on dgr of frdom. 24

25 Configuration coordinats Rigid part in plan motion, 3DOF Configuration coordinats: θ, x, C y C 25

26 Multibody systm Guidr = Ground Slidr Rod A θ B ϕ Crank O Ground B ϕ C y C x C A O g x B Figur.3 Topology and coordinats of slidr crank mchanism. # Parts: 4 Configuration coordinats: x, θϕ,, x, y B C C 26

27 Grublr formula Two dimnsional m f= 3N ( ) ri i= Numbr of dgrs of frdom (DOF): f Numbr of parts: N (including th ground) Numbr of joints: m Numbr of DOF rmovd by joint i: r i Rvolut joint: ri = 2 Translational joint: ri = 2 27

28 Joints in thr dimntions ri = 5 ri = 5 m f= 6N ( ) ri i= 28

29 Dgrs of frdom and constraints B C Ground A g θ A θ Ground D B ϕ a) b) Figur.5 a) Thr-bar mchanism b) Th doubl pndulum. 29

30 m Grublr formula ( ) i, f= 3N r f i= Slidr crank mchanism: N = 4, m= 4, ri = 2 f= 34 ( ) 4 2= C Thr bar mchanism: N = m= 4, ri = 2 B f= 34 ( ) 4 2= A θ Ground D Doubl pndulum: N = 3, m 2, = i r = 2 Ground A θ g f= 33 ( ) 2 2= 2 B ϕ 3

31 Biomchanics A simpl multibody modl of a humanoid in plan locomotion N =, m, = i Ground Figur.6 Human locomotion. r = 2 f = 3( ) 2 = 3 2 = 3

32 Constraints Configuration coordinats: x, θϕ,, x, y B C C If th numbr of configuration coordinats is qual to th numbr of dgrs of frdom of th systm thn w say that w us a minimal st of coordinats. Solutions to ths quations rprsnt th motion of th multibody systm. Howvr, using this approach will hid important information on th dynamics of th systm. For instanc, intrnal forcs in th systm, such as forcs apparing in th joints, will not b accssibl using a minimal st of coordinats. 32

33 Constraints In ordr to b abl to calculat ths forcs on has to incras th numbr of configuration coordinats and along with this introduc constraint conditions rlating ths coordinats in accordanc with th proprtis of th joints btwn th parts. Configuration coordinats: x, θϕ,, x, y B C C Constraint at A: Constraint at B: L Rcosθ + cosϕ xc = 2 L Rsinθ sinϕ yc 2 = L xb cosϕ xc = 2 L sinϕ yc = 2 33

34 Multibody dynamics - th analysis of a mchanical systm rsults in a solution of a st of ordinary diffrntial quations To st up th diffrntial quations w nd a gomtrical dscription of th MBS and its parts including its mass distribution. (In th cas of flxibl parts w also nd to constitut th lastic matrial proprtis of ths parts.) W hav to analyz th connctions btwn parts in trms of thir mchanical proprtis. Proprtis of th connctions in th shap of joints will b rprsntd by constraint conditions, i.. quations rlating th configuration coordinats and thir tim drivativs. Finally th intraction btwn th MBS and its nvironmnt has to b dfind and givn a mathmatical rprsntation. Basd on th fundamntal principls of mchanics; Eulr s laws and th Principl of virtual powr w may thn formulat th diffrntial quations govrning th motion of th MBS. Th solution of ths quations will, in gnral, call for numrical mthods. 34

35 Multibody dynamics softwar Adams Figur.7 Exampls of MBS modls in ADAMS. 35

36 Th rigid body -dos not chang its shap during th motion ω B t c g P p cp Figur.8 Th rigid body. v c This modl concpt may b considrd to b th most important for MBS and many MBS ar qual to systms of coupld rigid bodis. Th gnral motion of a rigid body thus has six dgrs of frdom. vp = vc + ω pcp, P ap = ac + α pcp + ω ( ω pcp ) Th quations of motion F = acm Mc = Icω + ω Iω c I c is th inrtia tnsor of th body with rspct to its cntr of mass 36

37 Th flxibl body Figur.9 Th flxibl body. Th lastic nrgy U = 2 2 ( (tr ) ) dv( X ) 2 λ E + µ E B Grn-St.Vnant strain tnsor and Lam-moduli 37

38 Th Multibody kinmatics Conncting rod Rvolut joint Flywhl Configuration coordinats: Slidr Rvolut joint Fundamnt Crank 2 n q = q, q,..., q xprssd in trms of ths coordinats and thir tim drivativs Kintic nrgy: T = Tqq (, ) Potntial nrgy: U = Uq ( ) Equations of motion: (Lagrang s quations) d T T U ( ) + k,,..., k k k Q = k = n dt q q q Q k = Q( qq, ) k Gnralizd forc rprsnts xtrnal forcs on th systm and intrnal contact forcs btwn parts 38

39 Equations of motion undr constraints Lagrangian: L= T U Nonconsrvativ forcs: non non Q = Q (, tqq, ) k k non-consrvativ forcs including, for instanc, friction. Constraint quations: m< n n k g + g q ν ν =, ν =,..., m gνk = gνk(, tq) k k= Equations of motion: m d L L non ν ( ) Qk k,,..., k k λ gν = k = n dt q q ν = n k g + g kq ν ν =, ν =,..., m k= m kc Q = λ ν g, k =,..., n k ν = ν k raction forcs du to th kinmatical constraints Constraint forcs: ν ν λ = λ ( t), ν =,..., m 39

40 Chaptr 3: Particl Dynamics is lft for slf study and rcapitulation of basic concpt i mchanics. m v F You may try to solv som of th xrciss but don t spnd to much tim on this introductory chaptr! 4

41 Euclidan spac xyz,,... E Points in Euclidan spac y y x= u V Vctors in Translation vctor spac = x+ u V = ( 2 3) dim( ) = 3 basis a a= a+ 2a2+ 3a3= 2 3 a2 = a a 3 ( ) [ ] 4

42 Cartsian coordinat systm o E Fixd point, = ( 2 3) Fixd basis x= o+ r = o+ x + x + x = o+ r [ ] = o+ [ x] ox ox o E Fixd point, = ( 2 3) f f f f Fixd basis x= o + f [ x] f o = o+ [ o ] x= o+ [ o ] + f[ x] = o+ [ x] f[ x] = ([ x] [ o ] ) f f 3 3 f = B, B R, dtb 42

43 Cartsian coordinat systm Bx [ ] = ([ x] [ o ] ) Bx [ ] = [ x] [ o ] f f Transformation of coordinats for points [ x] = B ([ x] [ o] ) f [ x] = B([ x] [ o] ) f f 43

44 What about vctors? a V a= a [ ], a = f[ a], f = B f a = f[ a] = B[ a] = [ a] [ a] = B[ a] f f f [ a] = B[ a], [ a] = B [ a] f f 44

45 What about tnsors? A End( V ) A = [ A ], Af f [ A], f = B = f [ A] = B [ A] f B [ A] = B[ A] B T B = f f 45

46 Orthonormal basis ( ) = 2 3 i= j = δ = i j i j ij = = = 2 3 T 2 ( 2 3) T = I

47 Orthonormal bass = ( ), 2 3 ( ) f = f f f 2 3 f = B T BB= I, dt B= ± 3 3 Orthonormal matrix 47

48 Th tnsor product Notation: ab, V, a b End( V ) Dfinition: ( a bu ) = ab ( u) 48

49 Th vctor product Notation: ab, V, a b V a b b θ a a b= absinθ Notation: a V, End( V), a Skw( V ) Dfinition: ( a u ) = a u 49

50 Th dtrminant dt Q V ( Qa, Qb, Qc) = = dt[ Q] V ( abc,, ) ON-basis: ( ) = 2 3 5

51 Chaptr 4 :Rigid body kinmatics Rigidity condition: p ( t AB ) = constant, A, B 5

52 4. Rigid body transplacmnt χ(r A ) Position vctors pa = χ( ra), A χ(r A ) - mapping calld transplacmnt W choos a matrial point A as a rfrnc point, calld rduction point. 52

53 Rigid body transplacmnt Fix point A rduction point Transplacmnt: pa = χ( ra), A Rigidity condition: χ( r ) χ( r ) = r r,, B A B A AB Introduc a mapping dfind by: R ( a) : = χ( r + a) χ( r ), a V (*) A A A Th mapping has th following proprtis: RA( ) = RA( a) RA( b) = a b, ab, V RA( a) = a, a V Isomtric mapping th lnght of a vctor dos not chang! 53

54 Rigid body transplacmnt Isomtric mapping is linar: Linarity: R ( αa+ βb) = αr ( a) + βr ( b), ab, V, αβ, A A A R Notation for linar mapping: (w drop th paranthss) R ( a) = Ra A A Isomtry implis orthonormality. R is an orthonormal transformation: T T T RR= R R = V, R = R A A A A A A dt R = ± ddd R A 2 = two possibilitis A Th angl btwn th vctors dos not chang. Both lnght and angl ar prsrvd! RAaRB b = a.b 54

55 Rigid body transplacmnt Start with: From (*) follows: a = r r P A Rfrnc placmnt χ( r + a) = χ( r) + Ra, a V A A A χ( r ) = χ( r ) + R ( r r ) P A A P A Prsnt placmnt p p = R ( r r ) P A A P A p AP = Rr A AP 55

56 Rigid body transplacmnt p R r R SO( V ) AB = A AB A Rotation tnsor 56

57 What now, if w chang th rduction point from A to B? b= r r B Changing th rduction point A χ( r ) = χ( r ) + R ( r r ) P A A P A χ( r ) = χ( r ) + R ( r r ) P B B P B What is th rlationship btwn R and? A R B R a = χ( r + a) χ( r ) = χ( r + a+ b) χ( r + b) = χ( r + a+ b) χ( r ) B B B A A A A ( χ( r + b) χ( r)) = R( a+ b) Rb= Ra+ Rb Rb= Ra A A A A A A A A W may thus conclud that RA = RB = R, AB, Th orthogonal transformation rprsnting a transplacmnt of a rigid body is indpndnt of th rduction point! 57

58 Rigid body transplacmnt χ( r + a) = χ( r) + Ra, a V A A A χ( r + a) = χ( r) + Ra, a V B B B Dos th rotation tnsor dpnd on th rduction point? R B = R A? No! RB = RA = R! 58

59 Rigid body transplacmnt ur ( ) = p r = χ( r) r A A A A A Th displacmnt 59

60 Rigid body transplacmnt u = u + ( R r ) P A AP Thorm 4. A rigid body transplacmnt is uniquly dtrmind by th displacmnt of (an arbitrary) rduction point A; u A and an orthonormal transformation R SO( V ) which is indpndnt of th rduction point. 6

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62 Summary Box 4.: Rigid body transplacmnt A: rduction point u A : displacmnt of th rducion point A R: orthogonal transformation with dt R= u= ur ( ) = u + ( R )( r r ) A p= χ() r = χ( r) + Rr ( r ) A A A 62

63 Th translation A translation of a body is a transplacmnt whr all displacmnts ar qual, irrspctiv of th matrial point, i.. u = u = u,, A B AB What dos th associatd orthonormal tnsor look lik? u = u + ( R r ) P A AP ( R a ) = R= 63

64 Th translation u = u = u,,, A B AB 64

65 Th rotation Considr a rotation of th body around a fixd axis in spac through th point Rotation axis: ( A, n) i = 3 n introduc a Right-handd Ortho-Normal basis (a RON-basis ) i = ( i i i ) 2 3 Dcomposition: r = r + nnr ( ) AB AB AB Thn w writ r = AB i x + i2 x2 65

66 Th rotation g B g B p AB x 2 i 2 i r AB ϕ A i g = n A g 3 3 = i3 = n x 2 r = r + nnr ( ) AB AB AB r = i x + i x AB 2 2 Figur 4.9 Rotation around a fixd axis. Th vctor corrsponding to r AB in th prsnt placmnt is: p = p + nn ( p ) AB AB AB p = x + x AB 2 2 r = p = x + x + ( nr ) AB AB 2 AB 66

67 Th rotation = ( ), = n is a RON-basis. This basis is rlatd to i by g B g B p AB x 2 i 2 i r AB ϕ A i g = n A g 3 3 = i3 = n x 2 Figur 4.9 Rotation around a fixd axis. = icosϕ+ i2sinϕ 2 = i( sin ϕ) + i2cos ϕ, ϕ < 2π 3 = i3 = n This can b xprssd in matrix format. 67

68 Th rotation matrix cosϕ sinϕ ( 2 3) = ( i i2 i3) sinϕ cos ϕ, [ ] = i R i [ R] i cosϕ sinϕ = sinϕ cosϕ Th matrix rprsntation of th rotation R is calld th canonical rprsnataion. T T [ ] [ ] = [ ] [ ] = [ ], R R R R [ R] i i i i dt = i 68

69 Th rotation tnsor Now lt R b th tnsor dfind by: Ri = Ri = = 3 = 3 = Ri i n Ri = = i[ R] i p = p + n( n r ) = Ri x + Ri x + Rn( n r ) = R( i x + i x + n( n r )) = Rr AB AB AB 2 2 AB 2 2 AB AB p AB = Rr AB which mans that th rotation is a rigid transplacmnt 69

70 Summary Box 4.3: Rotation ( A, n): rotation axis A: rduction point u = : displacmnt of th rduction point A A cosϕ sinϕ R: orthonormal transformation, [ R] = sinϕ cosϕ i ϕ: rotation angl ϕ < 2π i = ( i i i ): rfrntial bas, i3= n=

71 Th Eulr thorm Thorm 4.2 (Th Eulr thorm, 775) Evry rigid transplacmnt with a fixd point is qual to a rotation around an axis through th fixd point. 7

72 i = ( i i i ), i = n Th canonical rprsntation Th transplacmnt is givn by: pab = RrAB, B W now try to find out if thr ar any points C with th proprty: pac = RrAC = rac i is vctor prpndicular to n If this is tru thn thr ar matrial vctors unaffctd by th transplacmnt. This brings us to th ignvalu problm for th tnsor R Rn = λn 72

73 Th charactristic quation, ignvalus [ ] [ ] p ( λ) = dt( λ R) = dt( λ R ) = R i λ cosϕ sinϕ 2 2 dt sinϕ λ cos ϕ = ( λ )(( λ cos ϕ) + sin ϕ) λ ± i p ( λ) = λ =, λ = ϕ R 23, 73

74 i = ( i i i ), i = n Th canonical rprsntation Rn = λn Eign valu problm: find a matrial points C such that r AC is an ignvctor to R with th corrsponding ignvalu. λ =, i 3 = n Ri [ ] = i R i i is vctor prpndicular to n [ R] i cosϕ sinϕ = sinϕ cosϕ cosϕ sinϕ i T R= i[ R] i = ( i i2 i3) sinϕ cosϕ i2 = i i3 74

75 Th canonical rprsntation cosϕ sinϕ i = ( i i2 i3) sinϕ cosϕ i2 = i3 i ( icos ϕ+ i2sin ϕ i( sin ϕ)+ i2cos ϕ i3) i2 = i3 ( i cos ϕ+ i sin ϕ) i +( i ( sin ϕ)+ i cos ϕ) i + i i ) = Equivalnt to th following tnsor-product rprsntation of R R= i i + ( i i + i i )cos ϕ+ ( i i i i )sinϕ n n+ cos ϕ( n n) + sinϕn 75

76 Summary Box 4.5: Rigid transplacmnt A: r A u A : r : p : R : : rduction point position vctor of th rduction point in th rfrnc placmnt displacmnt of th rducion point position vctor of a matrial point in th rfrnc placmnt position vctor of a matrial point in th prsnt placmnt rotation tnsor p= χ() r = ra + u ( ) { A + Rr ra translation rotation 76

77 Scrw rprsntation of th transplacmnt u A Sinc th translation changs with th rduction point A and th rotation R is indpndnt of this choic w may ask if thr is a spcific rduction point S such that th displacmnt is paralll to th rotation axis. u S L S r p= r + ns + Rr ( r) S Scrw lin { S rs ra ras n σ, σ } = = + + T ( R ) ua =, ϕ 2 ( cos ϕ) AS S Thorm 4.3 (Chasls Thorm, 83) Evry transplacmnt of a rigid body can b ralizd by a rotation about an axis combind with a translation of minimum magnitud paralll to that axis. Th rduction point for this so-calld scrw transplacmnt is locatd on th scrw lin givn in (4.28). 77

78 Th spcial orthogonal group Th st of all orthonormal tnsors on V forms an algbraic group, th orthonormal group, Bing a group mans that T T { Q Q Q QQ } O ( V ) = = = Q, Q2 O( V) QQ 2 O( V) O( V ) Q = Q O( V ) T Howvr, in gnral w hav QQ 2 QQ 2 and w say that th opration of combining orthonormal tnsors is non-commutativ. Th st of all rotations forms a subgroup in O( V ) calld th spcial orthonormal group { R R= } SO( V) = O( V ) dt 78

79 Non-commutativity of rotations Combining rotations is thus a non-commutativ opration as can b sn from th figur blow. Th spcial orthonormal group has th mathmatical structur of a so-calld Li-group. 79

80 Chaptr 4.3 Th main rprsntation thorm Thorm 4. To vry rotation tnsor R SO( V ) thr xists a unit vctor n and an angl ϕ so that R= R( n, ϕ) = n n+ cos ϕ( n n) + sinϕn This rprsntation is mainly uniqu. If R= thn n may b arbitrarily chosn, whil ϕ = ( modulo 2π). If R thn n is uniquly dtrmind, apart from a factor ±. For a givn n th angl ϕ is uniquly dtrmind ( modulo 2π ). For a dfinition of th tnsor a End( V ) s Appndix A

81 Th main rprsntation thorm using componnts R= R( n, ϕ) = n n+ cos ϕ( n n) + sinϕn n = ( 2 3) n= [ n] [ n] = n 2 n 3 n nn nn 2 nn 3 T n n = n n = n2 n n2 n3 = nn 2 nn 2 2 nn 2 3 n 3 nn 3 nn 3 2 nn 3 3 [ ] [ ] [ ] ( ) n3 n2 = = n n n2 n [ n ] [ n] [ ] 3 82

82 Th main rprsntation thorm using componnts R= n n+ cos ϕ( n n) + sinϕn [ ] = ( 2 3) n n [ n] = n = n2 n 3 [ R] [ n n] cos ϕ[ ( n n) ] sinϕ[ n ] = + + = T T [ n] [ n] + cos ϕ( [ ] [ n] [ n] ) + sinϕ[ n ] 83

83 How to calculat th rotation axis n and th rotation angl φ from th rotation tnsor R R= n n+ cos ϕ( n n) + sinϕn R n, ϕ? b = ( b b b ) 2 3 arbitrary RON-basis Eignvalu problm [ ] [ ] [ ] [ ] Rn = n ( R ) n = ( R ) n = b b [ R] b R R2 R3 = R2 R22 R23 R3 R32 R 33 R 3 3 [ n] b n = n2 n 3 R 3 84

84 Th ignvalu problm R λ R2 R3 n R R λ R n = R3 R32 R33 λ n 3 R λ R2 R3 pr( λ) = dt R2 R22 λ R23 R3 R32 R33 λ ± i p ( λ) = λ =, λ = ϕ R 23, cos λ 2 + λ 3 2 3, sin λ ϕ = ϕ = λ 2 2i 85

85 Th ignvalu problm λ = R R2 R3 n n R R R n = n R3 R32 R33 n 3 n 3 n [ R] cosϕ sinϕ = sinϕ cosϕ trr = R + R + R = + 2cosϕ trr R + R22 + R33 arccos( ) = arccos( ) 2 2 ϕ = R + R22 + R33 2π arccos( ) 2 86

86 Coordinat rprsntations of SO( V ) Eulr angls Bryant angls Eulr paramtrs Quatrnions 87

87 Eulr angls: ( ψθφ,, ) Rfrnc placmnt Prsnt placmnt [ ] o R R o o = = 88

88 Factorization: ( ψθφ,, ) R 3 ( ψ ) R ( θ ) R 3 ( φ) f = 3 3 g 3 θ f 3 g 2 = g g 2 f 2 f 2 ψ f 2 g = f g φ R= R ( φ) R ( θ) R ( ψ) 3 3 Evry rotation may b writtn as a product of thr simpl rotations. 89

89 Eulr angls: First rotation o W start th factorization by introducing th first rotation R3( ψ) : 3ψ which brings th o bas to th bas f = ( f f2 f ) o o 3 by rotating around th axis with dirction 3 angl ψ. ψ f = 3 3 ψ f 2 f 2 = ( ) 2 3 f = ( f f f ) 2 3 [ ψ ] o o f = R3( ψ) = R3( ) o, cosψ sinψ ( ψ) o = sinψ cosψ [ R ] 3 9

90 Eulr angls: Scond rotation θ Nxt w tak th rotation R( θ) : fθ which brings f to th RON-basis g = ( g g2 g3) by a rotation with th angl θ around an axis with th dirction f, th so-calld intrmdiat axis. g 3 θ f 3 g 2 f 2 f = ( f f f ) 2 3 g = f g = ( g g g ) 2 3 [ θ ] g = R( θ) f = f R( ), f ( θ) = cosθ sinθ f sinθ cosθ [ R ] 9

91 Eulr angls: Third rotation φ Finally w tak th rotation R3( φ) : g3φ which brings g to th final basis by a rotation th angl φ around an axis with dirction g3. = g g 2 g = ( g g g ) 2 3 φ = ( ) 2 3 g R 3( φ) g g [ R 3( φ) ], [ R ] = = g 3 cosφ sinφ ( φ) = sinφ cosφ g 92

92 Eulr angls: All rotations in combination R-substitution yilds: o f = R3( ψ) = R3( ) g = R( θ) f = f R( ) f = R3( φ) g = g[ R3( φ) ] g o[ ψ ] [ θ ] o [ ( φ) ] [ ( θ) ] [ ( φ) ] [ ( ψ) ] [ ( θ) ] [ ( φ) ] o o = R = g R3 = f R R3 = R g f g 3 R R f 3 g o [ R] o [ ] o [ ] [ ] o o o = R = R3( φ) R( θ) R3( ψ) = R3( ψ) R( θ) R3( φ) f g 93

93 Eulr angls: Th rotation o ψ θ φ f g [ R] o [ R ( ψ) ] o [ R ( θ) ] [ R ( φ) ] = = 3 f 3 g cosψ sinψ cosφ sinφ sinψ cosψ cosθ sinθ sinφ cosφ = sinθ cosθ cosψ cosφ sinψ cosθsinφ cosψ sinφ sinψ cosθcosφ sinψ sinθ sinψ cosφ+ cosψ cosθsinφ sinψ sinφ+ cosψ cosθcosφ cosψ sinθ sinθsinφ sinθcosφ cosθ 94

94 Eulr angls f Thorm 4.2 A rotation R is dtrmind by a st of thr ral coordinats, calld th Eulr 3 angls: ( ψ, θ, φ) R, ψ < 2π, θ π, φ < 2π. Convrsly ths angls ar ssntially dtrmind by R. In fact th angl θ [, π] θ ], π[ thn also ψφ, [ 2, π[ is uniquly dtrmind by R. If ar uniquly dtrmind by R. Whn θ = only ψ + φ is uniquly dtrmind and whn θ = π only ψ φ. 95

95 Eulr angls: Singularity Spcial cas: θ =, π [ R] o cosψ cosφ± sinψ sinφ cosψ sinφ± sinψ cosφ = sinψ cosφ± cosψ sinφ sinψ sinφ± cosψ cosφ = ± cos( ψ ± φ) sin( ψ ± φ) R R2 R3 sin( ψ ± φ) cos( ψ ± φ) = R2 R22 R23 R3 R32 R ± 33 Only ψ ± φ is dtrmind. Singular! 96

96 Eulr angls: Singularity [ R] o R R2 R3 = R2 R22 R23 = R3 R32 R 33 cosψ cosφ sinψ cosθsinφ cosψ sinφ sinψ cosθcosφ sinψ sinθ sinψ cosφ+ cosψ cosθsinφ sinψ sinφ+ cosψ cosθcosφ cosψ sinθ sinθsinφ sinθcosφ cosθ 2 cos θ = R, sinθ = cos θ 33 R23 R3 cos ψ =, sinψ = sinθ sinθ R32 R3 cos φ =, sinφ = sinθ sinθ θ =, π singular Ths formulas show that numrical difficultis ar to b xpctd for valus of θ clos to nπ, n=,,... 97

97 Th rotation angl trr = + 2cos ϕ = ( + cos θ) cos( φ+ ψ) + cosθ cosϕ = ( + cos θ)cos( φ+ ψ) + cosθ 2 ψ φ θ 98 Figur 4.5 Th gyroscop.

98 Chaptr Eulr paramtrs SSSSS ffff ttt fffffffffff rrrrrrrrrrrrrr: R= n n+ cos ϕ( n n) + sinϕn = + ( cos ϕ)( n n ) + sinϕn us of th trigonomtric idntitis ϕ cosϕ ϕ ϕ sin 2 = sinϕ = 2sin cos ϕ ϕ ϕ R= + 2sin n ( n ) + 2sin cos n IIIIIIIII ϕ ϕ = cos, = nsin

99 Eulr paramtrs tttt ww haaa R= + 2( )( ) + 2 With a spcific RON basis in which b = ( b b b ), RON-basis, 2 3 n= b + b + b n 2n, 2 3n3 n = ϕ ϕ = cos, = nsin 2 2 Thn w hav th following rlations: = b+ b22+ b33 ϕ ϕ ϕ ϕ = cos, = nsin, 2 = n2sin, 3 = n3sin n.n = = (,, 2, 3) R, = S = { = R } 3 S 3 R 4 This constraint on th Eulr paramtrs rprsnts th unit sphr Th paramtrs = (,, 2, 3) ar known as th Eulr paramtrs.

100 [ R() ] How to calculat matrix lmnts from Eulr paramtrs To indicat th Eulr paramtrs associatd with a rotation w writ R= R( ) = R(, ) = + 2( ) + 2, = (, ) S = b 2 3 b = ( b b b ), RON-basis, = ( + ) 2 ( 2 3) 2 ( 3+ 2) ( 2+ 3) 2 ( + 2) 2 ( 2 3 ) ( 3 2) 2 ( 2 3+ ) 2 ( + 3)

101 How to calculat Eulr paramtrs from matrix lmnts [ R] o R R R R R R R R R 2 3 = trr + R + R22 + R33 + = = Rii trr + 2Rii R R22 R33 i = =, i= 23,, No singularity! (In contrast to th Eulr angls) Thr is a 2 - corrspondnc btwn Eulr paramtrs and rotation matrics! 2

102 How to calculat Eulr paramtrs from matrix lmnts, condinud = (,,, ) (,,, ) R + R + R =± = R R , 2 R3 R3 R2 R2 =, 3 = 4 4 3

103 How to calculat Eulr paramtrs from matrix lmnts R + R + R =± = 4, 2 = R2 + R2 4, 3 = R3 + R3 423 = R32 + R23 + R R22 R33 =, =± 4 2 = R + R 4 2 2, 3 = R + R

104 Spcial cass: How to calculat Eulr paramtrs from matrix lmnts + R22 R R33 =, =, 2 =± 4 3 = R + R R33 R R22 =, =, 2 =, 3 =± 4 5

105 Chaptr Th composition of rotations Lt R and R 2 b two rotations. Th composition of R and R 2 is givn by R = RR 2 and this rprsnts anothr rotation. W hav th following Proposition 4. If R = R(,, ), R2 = R2( 2,, 2) and R= RR 2 thn R= R(, ) whr =, = 2, + 2, + 2 (4.8), 2, 2 Corollary 4. Two rotations commut if and only if thir rotation axs ar paralll, i.. RR = RR P

106 Chaptr Th xponntial map Rotation Skw-sym. Tnsor R: nϕ SO( V ) N = n Skw( V ) Expanding in Maclaurin sris: 2 3 ϕ 2 ϕ 3 R= + ϕn + N + N +... = xp( ϕn) = xp( nϕ ) 2! 3! Th xponntial map: 2 3 xp( A) : = + A+ A + A +..., A End( V ) 2! 3! 7

107 Chaptr 4.4 Tim-dpndnt transplacmnt, Motion vlocitiy and acclration rrrrr tttttttttttttt pp = χ( rp,) t = r A + ua() t + R()( t rp ra) = pa() t + R()( t rp ra), t = p A () t Th rigid motion is a st of rigid transplacmnts paramtrizd by tim t 8

108 Rigid body vlocity Spin tnsor: W = Angular vlocity: (axial vctor of ) W RR T T ω = ax( RR ) anti-symmtric tnsor, sinc: W T = ( RR T ) T = RR T = RR T = W RR = RR + RR = RR = RR T T T T T Rigid body vlocity fild: vp = va + ω pap, P 9

109 Rigid body rotation T W = RR R = WR = ( ω R ) = ω R R = ω R R( ) = R R= R( t) = (xp( ω( s) ds )) R t ω= ω() t t = R= R() t Thus, if th angular vlocity is known and th rotation tnsor is known at tim thn th rotation tnsor : is known.

110 Rigid body acclration ap = aa + α pap + ω ( ω pap ), P It is mor difficult to graphically illustrat th acclration fild than th vlocity fild. W look at th spcial cas whn th body rotats around an axis with fixd dirction α ω ω= nω, α = nω, n = Tangntial acclration Cntriptal acclration ω ( ω p ) AP g β g P p AP α p AP Th tangntial and normal parts of th rlativ acclration: α pap = ω n pap = ω pap sin β = ω d Bt A g a A 2 2 AP AP AP d ω ( ω p ) = ω ω p = ω p sin β = ω

111 4.4.2 Eulr-Poisson vlocity formula for moving bass Starting from a fixd RON-bas In th rfrnc placmnt = ( ) R 3 ω 2 = ( ) RON-basis following th rotation of th rigid body = R, = R, = R [ R] = [ R] 2

112 Eulr-Poisson vlocity formula for moving bass 3 R 3 ω 2 2 3

113 Eulr-Poisson vlocity formula for moving vctors and tnsors a = a() t = i ta() t 3 i= i ( ) 3 i i= a = a = a i rl a = ω a+ a rl 3 i i, j= A= A() t = () t () t A () t j ij ( ) 3 i i, j= A = () t () t A () t = A j ij rl A = ω A Aω ( ) + A rl 4

114 Angular acclration moving vctor a = ω a+ a rl 3 ω 3 ω= iωi i= rl rl ω= ω ω+ ω = ω 2 = rl 3 = i i= ω ω ω i 5

115 Angular vlocity and Eulr angls 6

116 Combining rotations Th addition of angular vlocitis, vrsion considr two rotation tnsors R R = t () R = R t 2 2 () th combind rotation tnsor R = RR 2 W introduc th angular vlocitis ω T = ax( RR ) ω T = ax( RR ) ( T ω = ax RR ) ω= ω2 + Rω 2 Addition rul not usd that oftn 7

117 Combining rotations Th addition of angular vlocitis, vrsion 2 3 R f 3 ω f f 2 R 2 g 3 ω g 2 f f = R g = 2 R f g ω gf g 2 R = RR g = R 2 T ω = ω = ax( R R ) f T ω = ax(( R ) R ) gf 2 f 2 T ω= ω = ax( RR ) g 8

118 Combining rotations Th addition of angular vlocitis, vrsion 2 3 R f 3 ω f f 2 R 2 g 3 ω g 2 f g ω gf g 2 ω = ω + ω g gf f Addition rul 2 9

119 2

120 2

121 Eulr angls, first rotation T ω = ω = ax( R R ) f [ ψ ] o o f = R3( ψ) = R3( ) o, cosψ sinψ ( ψ) o = sinψ cosψ [ R ] 3 By us of Lmma 4. 22

122 Eulr angls,scond rotation [ θ ] g = R( θ) f = f R( ), f ( θ) = cosθ sinθ f sinθ cosθ [ R ] 23

123 Eulr angls, third rotation [ φ ] = R3( φ) g = g R3( ), g cosφ sinφ ( φ) = sinφ cosφ g [ R ] 3 24

124 25

125 Exrcis :2 Givn a rotation tnsor R= R ( ψθφ,, ) whr ψθφ,, ar Eulr angls. Calculat th R (,, ) and R( 5, 35, 6 ). matrics [ ] 32

126 Exrcis :2 Th rotation matrix xprssd in Eulr angls: R( ψ, θ, φ ) := cos( ψ) sin( ψ) sin( ψ) cos( ψ) cos( θ) sin( θ) sin( θ) cos( θ) cos( φ ) sin( φ ) sin( φ ) cos( φ ) Convrsion from dgrs to radians: R d ( ψ, θ, φ ) := R ψ π, θ π, φ π Calculation of th rotation matrix: R d ( 5, 35, 6) = Chcking th rotation matrix: R d ( 5, 35, 6) R d ( 5, 35, 6) T = R d ( 5, 35, 6) = Calculating th rotation of th first basis vctor: R d ( 5, 35, 6) =

127 Exrcis :4 Th mchanism to control th dploymnt of a spaccraft solar panl from position A to position B is to b dsignd. Dtrmin th transplacmnt, i.. th translation vctor and rotation tnsor R, which can achiv th rquird chang of placmnt. Th sid facing th positiv x-dirction in position A must fac th positiv z-dirction in position B. Calculat th rotation vctor n ϕ corrsponding to th rotation. (Mriam & Kraig 7/). 34

128 Exrcis :4 3 n ϕ