Chapter 3: Kinematics and Dynamics of Multibody Systems

Transcription

1 Chapter 3: Knematcs and Dynamcs of Multbody Systems 3. Fundamentals of Knematcs 3.2 Settng up the euatons of moton for multbody-systems II

2 3. Fundamentals of Knematcs 2 damper strut sprng chasss x z y steerng 4 rod Dsposer/te rod Wheel carrer transverse control arm

3 3. Fundamentals of Knematcs 3 opologcal basc prncples. Open knematcal chan ( tree-structure ) n B bodes (wthout reference/nertal body) n onts he route between two bodes s dstnct each body has exactly one predecessor Hence t apples: n n B Each ont has a dedcated body and connects the body wth ts predecessor n B n 7

4 3. Fundamentals of Knematcs 4 opologcal basc prncples 2. Closed knematcal chan ( loop-structure ) n B bodes (wthout reference/nertal body) n onts each loop reures one extra ont Hence : n n B +n L n B 0 n 3 n L 3

5 3. Fundamentals of Knematcs 5 Basc notaton and coherences f - degree of freedom of ont f - Sum of all degree of freedom of onts f L - degree of freedom of loop Degree of freedom (Laufgrad) durng spatal moton (rübler; Kutzbach) f n 6 nb (6 f ) n L n n B f n f f 6 n L 6 n L

6 Prncples of assemblng knematcal chans wth closed loops 3. Fundamentals of Knematcs 6. sparse method (cuttng off at all bodes) prncple: constrants: dsconnectng all onts concurrence of all ont parameters 2. vector-loop method (dsconnectng loops) prncple: constrants: dsconnectng only one ont per loop closure condtons of the loops 3. topologcal methods (loops as transference elements) prncple: constrants: knematcal loops are prepared as knematcal transfer elements ( knematcal transformators) - transfer euatons n the loop (nonlnear and local) - coupler euatons between the loops (lnear and global)

7 3. Fundamental Procedures of Knematcs II Illustraton wth the example of a double wshbone suspenson (front wheel suspenson): 7 double wshbone wheel suspenson: σ + γ γ Rgd body model P S β s R L L 2 S S solated DOF R wheel axle r s β S Mult-body system 5 bodes 7 onts 2 knematcal loops System degrees of freedom compresson (angle β) steerng ( dsplacement s) solated degree of freedom gnored

8 3. Fundamentals of Knematcs 8 System overvew n 7 n B 5 n L n -n B 2 4 f 4 7 f 7 L 2 B 5 6 f 6 3 B 4 f 6 n B n (6 f ) B 6 5 ( ) L 3 f f 5 3 R P S S f B 3 B 2 R S 2 f 2 3

9 3. Fundamentals of Knematcs 9. sparse method (cuttng of all bodes) Euaton of constrants e.g. for 5 : r 2 2 wth w + r ( w [ x y z ϕ ψ θ ] ) r r (w ) 2r B 4 4r ont 5 r 4 B 2 r 2

10 3. Fundamentals of Knematcs 0 Double wshbone wheel suspenson wth cut onts: a) sparse method (cuttng off all bodes) 5 bodes all onts dsconnected 7 onts assembly: each ont has (6 f ) constrants degrees of freedom of ont euatons of constrants are dentfed from the closure condtons for poston and orentaton: r ( r ( ) )!! r ( ( r ) )

11 2. vector-loop Methods (dsconnectng loops) he relatve moton n the onts s descrbed through 3. Fundamentals of Knematcs f n g f Jont varables β, s β 2 a 3 B β 6,, β 7, β 8 Jont 5 Closure Euatons a e.g. for 2 : a + a2( β) a3( β2 ) + a4( β6, β7, β8 ) β B 3 a 2 a 4 B 2 Jont 2

12 Double Wshbone Suspenson wth dsonted loops 3. Fundamentals of Knematcs 2 vector-loop Method (dsconnectng loops) 5 Bodes Number of ndependant loops: n 7 Jonts L n -n B Jonts 2 and 5 are dsconnected Assembly: each Jont has (6 f ) Constrants Degrees of freedom of Jont Constrant euatons from closure condtons for poston and orentaton: a! 0! Ι Identtymatrx

13 3. Fundamentals of Knematcs 3 3. topologsche Methode (Loops are consdered as tranmsson elements) Am: to calculate the moton of every body of the system usng the nput parameters (generalsed coordnates) β 2 L 2 2 s s β 2,, β 3, (β 4 ) β β (, 2 ),...4 s s ( 2 ), β 6,, β 7, β 8 β 9,, β 0, β L ββ β 3,, β 4, (β 5 )

14 3. Fundamentals of Knematcs Double Wshbone Suspenson wth transmsson elements 4 topologsche Method (loops as transmsson elements) 5 Bodes Number of ndependant loops: 7 Jonts n L n -n B Assembly: each loop s a knematc transmsson element (knematc ransformator) Constrant euatons: - local: n general 6 non-lnear contrant euatons per loop - global: lnear couplng euatons between the loops; Number depends on the order of the couplng between the loops

15 3. Fundamentals of Knematcs 5 β 2 loop f 8, n L f L f -6n L Inputs: ββ, β 6 β 6,, β 7, β 8 Outputs: β 2, β 3,, β 4, (β 5 ), β 7, β 8 L β 2 β B β 8 β B 3 β 7 β 3,, β 4, (β 5 ) redundent Degree of freedom β 3 (β 5 ) β 4 β 6 B 2

16 3. Fundamentals of Knematcs 6 β 2 Loop f g 8, n L f L f g -6n L β 6,, β 7, β 8 Inputs: ββ, β 6 Outputs: β 2, β 3,, β 4, (β 5 ), β 7, β 8 L β Knematc ransformer β β 6 L fl 2 β 3 β 4 β 7 β 8 β 2 β 3,, β 4, (β 5 ) redundant Degree of freedom

17 7 3. Fundamentals of Knematcs s β 2,, β 3, (β 4 ) Loop 2 f, n L f L2 f -6n L -6 5 β 2 L 2 Inputs: β 2, ss, β 7, β 8, (β 4 ) Outputs: β 6, β 9, β 0, β, β 2, β 3 β 9,, β 0, β β 6,, β 7, β 8 β 2 2 s B β 8 β 3 B 5 β 7 (β 4 ) β 6 β 2 B 2 B β 4 β 9 β 0

18 β 2 s L 2 β 2,, β 3, (β 4 ) 3. Fundamentals of Knematcs Loop 2 f, n l f L2 f -6n l -6 5 Inputs: β 2, ss, β 7, β 8, (β 4 ) Outputs: β 6, β 9, β 0, β, β 2, β 3 8 β 6,, β 7, β 8 β 9,, β 0, β Knematc ransformer s β 9 β β 2 0 β L β 7 2 β β 8 2 (β f L2 4(5) β 4 ) 3 β 6

19 3. Fundamentals of Knematcs opologcal descrpton of the double wshbone suspenson 9 B β 2 2 s B 5 B 2 L β 8 Β 3, (β 4 ) β 7 β B 3 β 6 β 2 (β 5 ), β 4 β B 4 β 3 β 9 β 0 L 2

20 3. Fundamentals of Knematcs 20 L + L 2 β 2 ββ β 3 L β 4 fl 2 β 3 β 4 β 9 β 0 β β 2 β 3 β 6 β 7 β 8 + s β 9 β β 2 0 β L β 7 2 β β 8 2 (β f L2 4(5) β 4 ) 3 β 6 L fl 2 ββ β 6 β 2 β 7 β 8 ss L 2 f L2 4(5) (β 4 )

21 3. Fundamentals of Knematcs 2 Knematc Net (wthout solated degree of fredom) 2 s solated DoF (gnored) β β 3 β 4 β 9 β 0 β β 2 β 3 2 knematc Loops wth (2+4) β 6 loop degrees of freedom 6 β 2 L fl 2 β 7 L 2-4 Couplng euatons between β 8 f L2 4(5) the loops -4 System degrees of freedom 2 ββ ss (β 4 )

22 Vehcle as a normal Mult-body system 3.2 Euatons of moton for Mult-Body systems 22 Is modelled as a complex mult-body system wth knematc loops, consstng of rgd bodes. o begn wth elastc characterstcs are modelled as concentrated elastctes. In the case of holonomc Constrants: Normal/Usual Mult-body systems he euatons of moton n mnmal coordnates for a system wth f degrees of freedom In the general form: M () & + b (,) & Q (,,t) & M b Q (f x ) - Vector of general coordnates (f x f) - Mass matrx (symmetrc) (f x ) - Vector of general centrfugal and gyro Forces (f x ) - Vector of general appled forces

23 3.2 Euatons of moton for Mult-Body systems Specaltes n Mult-loop Mult-body systems and Mechansms 23 Because of the knematc loops, there are comparatvely less degrees of freedom n a System wth more number of bodes and constrants. Settng up complex euatons of moton: - manually (very strenuous!) - numercally and/or usng symbols wth the help of computers

24 3.2 Euatons of moton for Mult-Body systems 24 Example: he Fve-pont Wheel Suspenson (Spatal) Euatons of moton for the sprng compresson ( f Degree of freedom multlnk rear suspenson Damler-Benz

25 Example: he Fve-pont Wheel Suspenson (Spatal) 3.2 Euatons of moton for Mult-Body systems 25 f D o F Euatons of moton n the form s Sprngr m (s)s & + b(s,s) & Q(s,s,t) & Steerng gude Usng symbols wth Programmsystem NEWEUL Steerng arm (blocked) Wheel carrer

26 3.2 Euatons of moton for Mult-Body systems 26 Fve-pont wheel suspenson: suspensonmovement f... m (s)s + b(s,s). Q (s,s,t) wheel suspenson (cont.) m(s). b(s,s). b(s,s) Q (s,s,t).

27 3.2 Euatons of moton for Mult-Body systems Methods to set up the euatons of moton for mult-body systems and mechansms Holonom Systems consdered as normal Mult-body systems 27 n Bodes r Constrants f 6 n - r Degrees of freedom Prncple of lnear momentum and Prncple of conservaton of angular momentum (Newton-Euler-Euaton) Number of euatons of moton: 6 n > f ( f) Lagrange Euatons of frst order (Method ) Number of euatons of moton : 6 n > f Lagrange Euatons of scond order (Method 2) Number of euatons of moton : f d`alembert s Prncple (Method 3) 6 n (Lagrange Multplcators) Number of euatons of moton : f (Mnmum coordnates)

28 3.2 Euatons of moton for Mult-Body systems Methods to set up the euatons of moton. Method: Based on the Fundamental euatons of dynamcs N ( F m a ) δr ( F - Appled forces) 0 Wll be then appled n the LARANE euatons frst order for pont masses: ven s a system wth 28 N Mass ponts m, r, g geometrc Constrants f α (t ; r,,r N ) 0 ; α,,g k knematc Constrants N φβ l β ( t; r,..., rn ) v + dβ (t; r,..., rn ) β,,k 0 ; Hence the System has f 3 N - g - k Degrees of freedom

29 By applyng the LARANE Multplcators λ α, µ β We get the LARANE Euatons of frst order: 3.2 Euatons of moton for Mult-Body systems 29 m f α a F + (t; r,..., rn ) 0 g f k α λα + µ β α r β l β ;,,N ; α,,g (DAE Dfferental Algebrac Euatons) N φβ l β v + dβ 0 ; β,,k (3N+g+k) Euatons for the (3N+g+k) Unknowns x,y,z, λ α, µ β 3N g k Advantages:- applcable for holonomc and non-holonomc systems - Euatons can be easly set up - Reacton forces can be calculated drectly Dsadvantages:- more euatons than degrees of freedom - euatons are numercally unstable Euatons for rgd body systems are analogous!

30 3.2 Euatons of moton for Mult-Body systems Method: By ntroducng f ndependant general coordnates (correspondng to the number of degrees of freedom), 2,, f One can obtan from the fundamental dynamc euatons he LARANE euatons of second order for holonom Systems: d dt ( ) & Q ;,..., f

31 3.2 Euatons of moton for Mult-Body systems 3. Method: Euatons of moton obtaned from the d`alembert s Prncple wth the help of knematc Dfferentals: Systematc procedure to solve the constrant euatons makng use of the solved loop knematcs. Identfyng the ndependant loops - ntroducton of natural coordnates β,s - settng up local, non-lnear constrant euatons - local solutons from the outputs Keep at hand the knematc ransformator 2. Defnng the loop network - settng up the lnear couplng euatons knematc Network 3. Choosng the structural nputs Optmsng the soluton flow 3... relatve Knematcs β. β.. β

32 Forward knematcs (recursve) 3.2 Euatons of moton for Mult-Body systems 32 Absolute coordnates, Body B w Poston: [ x, y, z, ϕ, ψ θ ], Jont Coordnate β w w ( w, w 2,, w, β, t) Velocty: B B w& w& ( w,...., w ; w&,...., w& ; β, ; t Acceleraton: β & ) It represents here translaton (s) and rotaton (β) w& w&& ( w,...., w ; w&,...., w& ; w&&,...., w& ; β, β &, β & ; t ) β β & β & absolute Knematcs w w& w& &

33 3.2 Euatons of moton for Mult-Body systems Knematcs of two bodes oned together wth a ont Parameters of Moton of B are known Parameters of moton of B J are reured Elementary standard ont 33 Rotatonal transton from B to B β ω α ω + ω α + ω ω + α v a ω r α v r a α ω r B ranslatory transton from B to B B r r r v a r + r v a + ω + α r r + v + 2ω v + ω (ω r ) + a Known Unknown Only one dmensonal onts are allowed. In the case of mult dmensonal onts, they have to be represented wth sngle dmensonal onts wth vrtual bodes nbetween them.

34 34 Used representaons r, r v, v a, a ω, ω α, α Poston vector to the reference/consdered pont, Absolute velocty of the reference/consdered pont, Absolute acceleraton of the reference/consdered pont, Absolute angular velocty of the body Absolute angular acceleraton of the body, r connectng vector between the reference/consdered ponts, v, a velocty / Acceleraton of B relatve to B, ω, α Angular velocty and angular acceleraton of B relatve B.

35 Euatons of moton for Mult-Body systems Rotatonal ont θ s β s B B B B e e s&e ranslatory ont β & e,, e e ) r ( r a r v r r r β α β ω α α ω ω ω ω + α ω + && & se a se v 0 0 a a v v r r r && & α ω +

36 3.2 Euatons of moton for Mult-Body systems 36 ranslaton v v + ω r + v v + ω (r + r ' ) + v ' v + ω (r + r ' ) + se & a a + α r + 2ω v + ω ( ω r ) + a a + α (r + r ' ) + 2ω v ' + ω ( ω (r + r ' )) + a ' a + α (r + r ' ) + 2ω se & + ω ( ω (r + r )) + && se ' Rotaton ω ω + ω ω + ω ' ω + β& e ω α + ω ω + α α + ω ω ' + α ' α + ω β& e + βe &&

37 37 Used representaton r r r ' v a ω α se & && se β& e β&& e Vektor from reference pont B to the reference pont B Vektor from reference pont B to the ont pont, Vektor vom ont pont to the reference pont B, Velocty of relatve to ( translatory Jont), Acceleraton of relatve to ( translatory ont), Angular velocty of relatve to ( rotatonal ont), Angular acceleraton of relatve to ( rotatonal ont).

38 38 x x x x x z y x z y x e e e e e e (3.23) (3.24) (3.25) ` ` (3.26) x y x z y z ` u u ) cos ( 0 u u u 0 u u u 0 sn cos ) u, ( ϕ + ϕ + ϕι ϕ (3.27) ϕ ϕ ϕ ϕ cos sn 0 sn cos ` (3.28)

39 3.2 Euatons of moton for Mult-Body systems 39 Summary of Knematcs eneralsed coordnates natural/ontcoordnates global coordnates & & & relatve Knematcs global Knematcs β β& β & absolut Knematcs w w& w& &

40 3.2 Euatons of moton for Mult-Body systems Euatons of moton derved from the d`alember Prncple 40 Reacton forces on B Reacton moments on B n B { m & F ) δs + ( θ ω& + ω θ ω ) δφ } ( s& s s 0 Vrtual work done by the Reacton forces and moments on B m, θ s : Mass und nertal tensor (relatve to the center of mass) of the body & s& ω F, ω&, δ s, δφ : acceleraton of the centre of mass of the body : angular velocty and acceleraton : resultng appled forces and moments : vrtual translaton or rotatonal orsche, bzw. rotatorsche dsplacements

41 4 3.2 Euatons of moton for Mult-Body systems Prncple of conservaton of angular momentum Prncple of lnear momentum S S R R R F m a m J F m s F F m s F F F m s + + && && && && S S S R S S R R S S a J ω + ω θ + θ θ ω θ + ω ω θ + ω θ + ω ω θ φ φ & & & & D Alembert s Prncple vrtual work of the Reacton forces vanshes ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 a J J F m a m J J J F J F s F s F s ) F (s A B B B B B B n S S S S S S n R R S n R R n R R n R R n R R ω + ω θ + θ θ + + δ + δ φ + δ δ φ + δ δφ + δ φ + δ δ φ φ φ φ && &&

42 3.2 Euatons of moton for Mult-Body systems 42 One reures the transfoematon of δ [ ] M& + b δ Q δ ndependant Euatons of moton M & & + b Q : [f x ] - Vector of generalsed coordnates (f number of degrees of freedom) M : [f x f ] - System Mass matrx (symmetrc, regular) b : [f x ] - Vector of generalsed gyro and centrfugal forces Q : [f x ] - Vector of generalsed appled forces Introducng the knematc relatons o δ δ o s φ o J o o o o s δ & s J s & ; & + bs ; J s o o o Jφδ ; o ω& Jφ & & + bφ ;.. o Jφ o s o φ Jacob- Matrces

43 Elements of the euaton of moton 3.2 Euatons of moton for Mult-Body systems 43 - Mass matrx n B o o o o o M { J s J m + θ } s J φ s J φ - eneralsed gyro forces n B o o b { s s m + [ θ + θ ] J o a } o o o o o ω ω J φ a s φ s - eneralsed appled forces n B o o o o Q { J s F + J φ }

44 3.2 Euatons of moton for Mult-Body systems Partal dfferentaton of the absolute values wth respect to the generalsed coordnates 44 Motvaton: v s s s s& ( ) & & + L+ & { { f { J s Column Column f f V s already known through the Knematcs dependant on o o s J s o ; J φ (Jacob-Matrces) o φ o a s f f k 2 o s k & & k o aφ f f k 2 o φ k & & k

45 Euatons of moton for Mult-Body systems Knematc Dfferentals φ ϕ θ ψ s z y x w w J w. Dfferentaton to determne: w w w w & & f & f. 0 0 ~ () Formal: ϕ ϕ φ f f s w x x J J J L M O M L knematc: lobal Knematc (no dfferentaton) w w&.. & & & f. 0 0 ~ ()

46 3.2 Euatons of moton for Mult-Body systems 46 knematc Dfferentals w ~ () w& ~ () & w& w& else & k 0 th Column J{ w } ~ () w& Seperated nto translaton and rotaton: s ~ () s& φ ~ () ω

47 3.2 Euatons of moton for Mult-Body systems Dfferentaton to determne: aw k 2 w k & & k Formal: ~ w w& & & & + 0 aw Knematc: & ~ & & 0 lobal Knematc (wthout dfferentaton) w w& ~ w& &

48 3.2 Euatons of moton for Mult-Body systems Knematc Dfferentals w w& ~ w & ~ w& a && 0 Seperated nto RANSLAION and ROAION a a s φ ~ && s ~ ω&

49 49 Euatons of moton wth knematc Dfferentals 3.2 Euatons of moton for Mult-Body systems Newton-Euler-euaton n B { m & F ) + ( θ ω& + θ ω ) } ( s& s δφ 0 δ s ω s! Dfferental relatonshps δs f ~ () f ~ s& δ & s& s& ().. ~ ; + & s& δφ f ~ () ω δ ; ω& f ~ ω () & ~ & + ω& M & & + b Q Dfferental euatons of moton of mnmal order

50 3.2 Euatons of moton for Mult-Body systems 50 coeffcents M,k n B { () (k) ~ () (k) + ~ ( )} m ~ s& ~ s& ω θ s ω b n B ~ ~ { m + ω [ θ ω + ω θ ω ]} s& ~ () & s& ~ () s s Q n B { s& () F + } ~ ω ()

51 3.2 Euatons of moton for Mult-Body systems Euatons of Moton of Complex Multbody System usng Knematcal Dfferentals 5 Input: Mass and Inerta m, Appled Forces and orues F, State-Varables ( t ) (t),..., θ s [ (t)] ; & ( t ) [ & (t),..., (t)] f & f k & k & && 0 0 k lobal Knematcs lobal Knematcs lobal Knematcs (Poston) (Velocty) (Acceleraton) s ~ s& ~ ω ~ ~ & s& ω& Euatons of Moton (Mnmal Order) M & & + b Q

52 3.2 Euatons of moton for Mult-Body systems 52 Double wshbone suspenson: σ + γ γ Rgd body model: P β s S R L L 2 S S r s Mult-body system: 5 bodes 7 onts 2 knematc loops R S Wheel axle 2 observable system degrees Of freedom: β - compresson (angle ) -steerng(dsplacement s) Lagrange. order Lagrange 2. order Knematc Dfferentals Dynamc und knematc Euatons are set up and worked wth n parallel. Descrpton results n Body coordnates and/or Relatve coordnates. Only dynamc euatons. Knematc euatons wll be set up Earler und ncorporated nto the Dynamc euatons. Descrpton results n Mnmal coordnates (Relatve coordnates n the most cases): Applcable only for holonomc Systems. Usng d`alembert s Prncple ranston over to dynamc Euatons n Mnmal coordnates;. e. Settng up the knematc euatons beforehand n Mnmal coordnates und ts Incorporaton nto the dynamc Euatons. & & & β Relatv- β & knematcs β & Absolute knematcs lobal Knematcs w w& w& &

53 Fundamental problems of the Dynamcs. Drect Problems gven: forces ^ generalsed forces Q [Q,, Q f ] 3.2 Euatons of moton for Mult-Body systems reured: moton ^ generalsed acceleratons & [ &&,..., & f ] and generalsed coordnates respectvely [,, f ] as a soluton of the Dfferental euaton M.... Mf && ḅ M Mf.... Mff && f bf Q f Q... reured gven Non-lnear Problem 53

54 3.2 Euatons of moton for Mult-Body systems Inverse Problem gven: moton ^ generalsed coordnates [,, f ] reured: loads ^ generalsed forces Q [Q,, Q f ] M.... M Mf M M f.... ff && && f ḅ b... Lnear Problem 3. Reactve forces (Zwangskräfte) gven: loads and moton reured: reactve forces f f gven reured Prncple of lnear momentum and prncple of conservaton of angular momentum (Newton Euler) Q Q...