Basic concept of dynamics 3 (Dynamics of a rigid body)

Transcription

1 Vehicle Dynaics (Lecture 3-3) Basic concept of dynaics 3 (Dynaics of a rigid body) Oct. 1, 2015 김성수

2 Vehicle Dynaics Model q How to describe vehicle otion? Need Reference fraes and Coordinate systes 2

3 Equations of otion of a rigid body q Three different type of Equations of Motion l Newton-Euler Equations of otion l Virtual work for of Newton-Euler Equations of otion l Lagrange Equations of otion

4 Newton-Euler Equations of otion (2D) Equations of Translational Motion d ( ivi ) F + f = = a dt i i i i å å å r = å iri v = å i v i a = å a F + f = a i i i i i i = å i åf = a

5 Newton-Euler Equations of otion (2D) r = r i P Y ( M ) = r F + r f = r a p i i i i i r i a = a + α r + ω ( ω r) i p Y r p X ( M ) = [ r a + r ( α r) + r ( ω ( ω r))] p i i p ( M ) k = {( xi + yj) [( a ) i + ( a ) j] p i i p x p y + ( xi + yj) [ ak ( xi + yj)]} r i 2 2 ( M p ) ik = i[ - y( ap ) x + x( ap ) y + a x + a y ] 2 ( M p ) i = i[ - y( ap ) x + x( ap ) y + ar ] k r p X

6 Newton-Euler Equations of otion (2D) Y r i r p X I p å = ò 2 ( M p ) i = i[ - y( ap ) x + x( ap ) y + ar ] 2 ( M p ) i = d[ - y( ap ) x + x( ap ) y + ar ] å ò ò ò ò 2 ( M p ) i = d[ - y( ap ) x + x( ap ) y + ar ] = r 2 ( yd)( ap ) x ( xd)( ap ) y ( r d) i x i + y j = ( xd) i + ( yd) j = ò r d = x i + y j M = - y( a ) + x( a ) + I a 2 ( r d) p p x p y p ò = ò d where If point P = point å M = ò I a a

7 Newton-Euler Equations of otion (2D) åf = a å å F F = ( a ) x x = ( a ) y y å M = I a I ( r 2 d) What is? = ò

8 Newton-Euler Equations of otion (2D) å M = I a åf = a I = ò 2 ( r d) Moent of inertia is a easure of the resistance of a body to angular acceleration in the sae way that ass is a easure of the body s resistance to linear acceleration I = ò 2 r d 2 I r rdv = ò I = rò 2 r dv

9 Newton-Euler Equations of otion (2D) u Parallel axis theore I = r d = [( d + x ') + y ' ] d = ( ' + ' ) + 2 ' x y d d x d d d = r d 2 dx ' d d d = I + d ò 2 ò ò ò ò ò ò ò u Radius of yration I Moent of inertia about the z axis passing through the ass center I k = k 2 I =

10 Newton-Euler Equations of otion (2D) å å å M = - y( a ) + x( a ) + I a p p x p y p I = I + r = I + ( x + y ) p M y a x a I x y 2 2 p = - ( p ) x + ( p ) y + a + ( + ) = y[ -( a ) + ya] + x[( a ) + xa] + I a p x p y = y - a - xw + x a + yw + I a 2 2 [ ( ) x ] [( ) y ] = - y( a ) + x( a ) + I x y 2 a = a + r - r P a w a M = - y( a ) + x( a ) + I a = ( M ) p x y k p a å The oents of the external forces about P are equivalent to the su of the kinetic oents of the coponents of a about P plus the kinetic oent of I a.

11 Newton-Euler Equations of otion (2D) å å F F = ( a ) x x = ( a ) y y å M = I a or å M = å ( M ) p k p

12 Newton-Euler Equations of otion (2D) q A single pendulu exaple = Free body diagra Kinetic diagra

13 Newton-Euler Equations of otion (2D) q A single pendulu exaple Free body diagra Kinetic diagra =

14 Equations of otion of a rigid body (3D) q Virtual work for of Newton-Euler Equations of Motion Newton s equations of otion for differential ass d(p) is ò p && r d(p) - F(P)d(P) -[ f (P,R)d(R)]d(P) = 0 where F(P) is external force per unit ass at point P and f(p,r) is internal force per unit asses at point P and R

15 Equations of otion of a rigid body Variational equations of otion (Virtual work for of equations of otion) ò ò ò ò δr && r d(p) - δr F(P)d(P) - δr f (P,R)d(R)d(P) = 0 PT P PT PT For rigid body, work due to internal forces is workless, thus PT ò ò δr f (P,R)d(R)d(P) = 0 Variational equations of otion for a rigid body becoes ò && ò PT P PT δr r d(p) - δ r F(P)d(P) = 0 which ust holds for all virtual displaceents rigid body p δr that can occur in a

16 Equations of otion of a rigid body To represent equations of otion in ters of Cartesian coordinates selected for a rigid body, virtual displaceents of point P can be represents in ters of virtual displaceents or the origin of the body reference frae and virtual rotation P p p δr = δr + Aδπ% s = δr - As% δπ Siilarly, the acceleration of point P ay be written as && r = && r + Aω% & ' s + Aω & % ' s = && r - As% ω & ' + Aω% ' ω% ' s p P P P P Substituting above equations into equations of otion yields ò PT P PT δr r d(p) -ò δ r F(P)d(P) T T p T p p = ( δr + δ π s% A )(r && - As% ω & ' + Aω% ω% s ) d(p) ò - ò && T T p T ( δr + δ π s% A )F(P)d(P) = 0

17 Equations of otion of a rigid body T d( P) + T P d( P) - T δr && r δr (Aω& % + Aω% ω % ) s δr F(P) d( P) ò ò ò ò ò ò ò T P - d π ò s% F (P) d( P) = 0 T P T T P P T P P + d π s d( P) A && r + d π s% ω& % s d( P) + d π s% ω% ω% s d( P) ( d p ) s P ω%& s P d( p) = - s% P s% P ( ) ω& ò º J ω& ò P P J º - s s% d(p) P 2 P 2 P P P P é( y ) + ( z ) - x y - x z ù ê ú = ò ê- x y x + z - y z úd P ê P P P P P 2 P 2 -x z - y z ( x ) + ( y ) ú ë û P P P 2 P 2 P P ( ) ( ) ( ) ò P P P P s% ω% ω% s = -s% ω% s% ω = -ω% s% s% ω - ω s s% ω - s ω ω% s P P = -ω% s% s% ω ( d p ) ò P P P P s% ω% ω% s d( p) = ω% - s% s% ( ) ω = ω% J ω P P PT P P T P % T % ab % + ab = ba % + ba T

18 Equations of otion of a rigid body Rearranging equations T T δ r { r && + ω& - ( F - d) } + δπ ( && r + J ω & + ω% J ω - n ) = 0 rw T rw where æ é rw ù é && ( - ö T T r ù é F d) ù [δ r,δ π ] ç ê - 0 T ú ê ú ê ú = è ë rw J û ëω& û ën - ω% J ω û ø ò ò ò p p p ò rw ò ò c º I, F º F, n º s% A F, p T d(p) (P)d(P) (P)d(P) J º - s% s% d(p), º - A s% d(p)= - A( ρ% ' ), p d = Aω% ' ω% ' s ' d(p)= Aω% ' ω% '( ρ' ) c Above equations of otion are general equations for a single rigid body

19 Equations of otion of a rigid body If the body reference frae is located at the center of ass, then soe ters disappear. where æ é rw ù é && ( - ö T T r ù é F d) ù [δ r,δ π ] ç ê - 0 T ú ê ú ê ú = è ë rw J û ëω& û ën - ω% J ω û ø º I, F º F, n º s% A F, p T d(p) (P)d(P) c (P)d(P) J º - s% s% d(p), º - A s% d(p)= - A( ρ% ' ), c ò ò ò p p p ò rw ò ò c p d = Aω% ' ω% ' s ' d(p)= Aω% ' ω% '( ρ' ) Equations of otion for a single rigid body using the centroidal body fixed reference frae is as follows æ é 0 ù é && ö T T r ù é F ù [δ r,δ π ] ç ê ú ê ú ê ú = è ë Jc û ëω& û ënc - ω% Jcω û ø dr d which ust hold for all virtual displaceents and virtual rotation π of the centroidal body-fixed frae x - y - z c

20 Equations of otion of a rigid body For a rigid body that oves in space freely under the action of the resultant external force and torque n r and d π' arbitrary, so by the orthogonality theore, F,d r && = F J ω & = n - ω% J ω c c c Newton Equation Euler Equation The equations of otion are valid only if the body-fixed centroidal. Orthogonality theore; A vector b = [ b1, b2,..., b ] T n is the zero vector if and only if T s b = 0 For all T s = [ s, s,..., s ] in R 1 2 n n x - y - z frae is in n-diensional space n R Proof of the orthogonality theore is in the reference: E.J. Haug, Interediate Dynaics, Prentice-Hall, pp. 107.

21 Rigid Body Equations of Motion in Orientation eneralized Coordinate (1) Mixed Kinetic-kineatic equations r && = F J ω & = n - ω% J ω c c c 1 T p & = ω 2 T p p -1 = 0 In this forulation, Euler paraeters are used as orientation generalized coordinates, however in velocity and acceleration, ω and ω& are used as generalized coordinates. Since the angular velocity integrable paraeters. ω is not (2) Equations of otion in Euler paraeters Angular velocity and angular acceleration relationships are as follows; δπ = 2δp ω & = 2p&& T T δ r ( r && - F ) + δπ (J ω & + ω% J ω - n ) = 0 c c ω = 2p& Substituting above relationships into the variational for of Euler Equations

22 Rigid Body Equations of Motion in Orientation eneralized Coordinate T A T T T T A δr [r && - F ]+ δp [4 J p&& - 8& J p & - 2 n ] = 0 dr c which ust hold for all and dp that satisfy kineatic constraints and the Euler paraeter noralization constraint of T δp p = 0 Using the Lagrange ultiplier theore to introduce an Euler paraeter noralization Lagrange ultiplier b such that δr [r && - F ]+ δp [4 J p&& - 8& J p & - 2 n + βp] = T A T T T T A c c dr dp which ust hold for all and, if there is no constraint acting on the rigid body. For single floating body, there is no kineatic constraint except Euler paraeter noralization constraint. Therefore and dp are arbitrary. T T A T c c dr By the orthogonality theore, the Newton-Euler equations in Euler paraeters are A r && = F 4 J p && = 2 n + 8& J p & - βp c with noralization constraints T p p -1 = 0 0

23 Rigid Body Equations of Motion in Orientation eneralized Coordinate In this forulation, Euler paraeters are used as orientation generalized coordinates, also in velocity and acceleration, tie derivative of Euler paraeters are used. Lagrange Multiplier Theore: Let b be avector in R n and let A be an x n atrix, if for all s in R n such that T s b = 0 As = 0 then, there exists a vector lada in R, called Lagrange ultiplier, such that T T T T T s b s A λ s b A λ for all s in Rn. Due to the orthogonality theore, this is equivalent to If A has full row rank, then lada is unique. + = ( + ) = 0 b T + A λ = 0 Proof of the Lagrange Multiplier theore is in the reference: E.J. Haug, Interediate Dynaics, Prentice-Hall, pp. 121.

24 Virtual work for of Equations of otion (2D) q A single pendulu exaple

25 Virtual work for of Equations of otion (2D) q A single pendulu exaple

26 Virtual work for of Equations of otion (2D) q A single pendulu exaple

27 Virtual work for of Equations of otion (2D) q A single pendulu exaple

28 Virtual work for of Equations of otion (2D) q A single pendulu exaple with transforation using independent coordinate

29 Virtual work for of Equations of otion (2D)

30 Equations of otion of a rigid body (3D) q Lagrange Equations with

31 Equations of otion of a rigid body (3D) q Lagrange Equations

32 Equations of otion of a rigid body (3D) q Lagrange Equations

33 Lagrange Equations of otion (2D) q A single pendulu exaple

34 Lagrange Equations of otion (2D) q A single pendulu exaple

35 HoeWork for Equations of otion (2D) (1) Derive EQM for a cart-pendulu syste (2) Derive EQM for a quarter car odel Using any of three ethods