Rigid Body Dynamics ~ f = p h g

Rigid Body Dynamics ~ f = p h g
AOE 5204 Definition of rigid body. Linear momentum and extension of Newton’s second law. Angular momentum and development of Euler’s law. Euler’s equations, moments of inertia. Coupling between rotational and translational motion. Special cases: axisymmetric torque-free; asymmetric torque-free; Lagrange top. Recall the fundamental dynamics equations We need to develop these equations for finite, rigid, bodies ~ f = d t p g h

Rigid Body Definition AOE 5204 Particle model is an idealization that is useful for describing vehicle motion for many applications; e.g., Aircraft performance Spacecraft orbits Only translational motion is of interest Rigid body model is an idealization that is useful for describing more complicated vehicle motions; e.g., Aircraft roll, pitch, and yaw Spacecraft pointing Marine vessel motion Both translational and rotational motion are of interest A rigid body is an idealized model of a solid body of finite dimension in which deformation is neglected. Alternatively, the distance between any two points in the body remains constant in any motion. The motion of a rigid body is described by the position and velocity of any point in the body with respect to an inertial origin, and the orientation and angular velocity of a body frame with respect to an inertial frame

Rigid Body Models ~ f = p h g
AOE 5204 Defining rigid body models takes two distinct approaches in the literature, both of which arrive at the basic equations of motion One approach: begin with a system of n particles and take the limit as n  to arrive at a continuous body Newton’s Second Law naturally extends to the system of particles and the resulting limit of a continuous body Euler’s Law results from application of Newton’s Second Law Second approach: begin with a continuous body and assert that Newton’s Second Law and Euler’s Law are independent principles that apply to finite bodies, whether rigid or deformable ~ f = d t p g h

~ v ~ r Particle Model ^ i ^ i ^ i m i i i 3 2 1 M o d e l t h r i g b
AOE 5204 ~ v i M o d e l t h r i g b y a s m f n p c , = 1 ; T v ~ u X D m i ^ i 3 ~ r i ^ i 2 ^ i 1 i s a n e r t l o g F f m

~ v ~ r Particle Model (2) ^ i ^ i ^ i m m i i ~ f ~ f ~ f e x t r n a
AOE 5204 ~ v i T h e s y t m l i n a r o u _ ~ p = X 1 v f + x S N w c d L ( b ) . ~ f j i m i m j ~ f i j ^ i 3 ~ f e x t r n a l i ~ r i ^ i 2 ^ i 1

~ v ~ r Particle Model (3) ^ i ^ i ^ i m i i i 3 2 1 T h e a n g u l r
AOE 5204 ~ v i T h e a n g u l r m o t f s y p i c _ ~ = X 1 d [ v ] + x ; m i ^ i 3 ~ r i ^ i 2 ^ i 1

~ v ~ r Continuous Model ^ i ^ i ^ i d m B 3 2 1 M o d e l t h r i g b
AOE 5204 ~ v M o d e l t h r i g b y a s c n u m , B N w S E L T ~ p = Z v q - x _ f d m B ^ i 3 ~ r ^ i 2 ^ i 1

~ v ~ r Continuous Model (2) ^ i ^ i ^ i d m B 3 2 1 W h a t d o e s i
AOE 5204 ~ v W h a t d o e s i n g r l ~ p = R B v m ? I - u f c b y ; Z ( ) V 1 x 2 3 T q : , . A j z k d m B ^ i 3 ~ r ^ i 2 ^ i 1

Continuous Model (3) a c o b ^ b ^ b ^ b F o r e x a m p l , t h s i n
AOE 5204 F o r e x a m p l , t h s i n g v b d y f ( ~ ; ) w c . W u = Z B V 3 2 1 M z - k a ^ b 3 c ^ b 2 o ^ b 1 b T h e r c t a n g u l p i s m d o f , b . N y w

Continuous Model (4) a c o b ~ r ^ b ^ b ^ b d m T h e ¯ r s t m o n f
AOE 5204 T h e r s t m o n f i a b u , ~ c g l v d y p - . w = Z B V [ 1 2 3 ] a ^ b 3 d m c ~ r o ^ b 2 o ^ b 1 b T h e r s t m o n f i a c b u d , p w v

Continuous Model (5) o ~ r ~ r ~ r ^ b ^ b ^ b d m c T h e m a s c n t
AOE 5204 T h e m a s c n t r , i d p o b u w f ~ v . = Z B V + ( ) ^ b 3 ~ r o d m ~ r c ^ b 2 o ^ b 1 ~ r o c c P u t i n g a l o e h r , w b c = 1 2 [ ] T s v y -

Continuous Model (6) T h e l i n a r m o t u s d y b ¯ ~ p = Z v w f -
AOE 5204 T h e l i n a r m o t u s d y b ~ p = Z B v w f - c V g , k O + W h a t i s e v l o c y ~ f d r - n m ? T b w : g p , O u ! = +

Continuous Model (7) T h u s t e l i n a r m o ~ p = Z ³ v + ! £ ´ d c
AOE 5204 T h u s t e l i n a r m o ~ p = Z B v O + ! b d c y f g - , w I f o i s t h e m a c n r , l u p y ~ = v O

Continuous Model (8) L i n e a r m o t u p c l : ~ = v + ! £ f _ H w d
AOE 5204 L i n e a r m o t u p c l : ~ = v O + ! b f _ H w d x s h q g ? R F ^ T A p l y i n g t h e f o r m u a ~ d v = _ , w b + ! c s x ( ) 1 T q k -

Continuous Model (9) N o t e h a ~ ! i s g n r l y b d f m q u , p - ;
AOE 5204 N o t e h a ~ ! i s g n r l y b d f m q u , p - ; . = ( _ ) R T c : v T h e t r a n s l i o q u f m v c : ~ p = + ! _ x - , d g 1 W b w ?

~ v ~ r ~ r Continuous Model (10) F B d m o o i T h e r a t w o f m s
AOE 5204 T h e r a t w o f m s n g u l i c : - , d ~ H b = Z B v p y . U T h e q u a n t i y ~ v d m s f c o - r l , b k g p w . ~ v B d m ~ r o o ~ r F i

~ v _ ~ r ~ r ~ r Continuous Model (11) F B d m o o o i T h e a n g u
AOE 5204 T h e a n g u l r m o t , ~ b i s = Z B _ d w p v c f y . T h e q u a n t i y _ ~ r o d m s - f l , b k g c p w . ~ v B _ ~ r o d m ~ r o o ~ r F i

Continuous Model (12) W e w i l o r k t h a n g u m , ~ b : = Z £ _ d
AOE 5204 W e w i l o r k t h a n g u m , ~ b : = Z B _ d T v c y s ! . [ ] E x p ( ) ~ h o = Z B [ r ( ! ) ] d m R e w i t s n g a W v c f b , p l u - . T

Continuous Model (13) I n t r o d u c e a w m h i l b j , y s ~ 1 - ¯
AOE 5204 I n t r o d u c e a w m h i l b j , y s ~ 1 - p v : = U g ! ( ) 2 C x Z B V f ^ T T h e a n g u l r v o c i t y s d p - f m , ~ = Z B ( ) 2 1 ! j : I

Continuous Model (14) W e h a v c o m p u t d b y i n g r l s - ~ , w
AOE 5204 W e h a v c o m p u t d b y i n g r l s - ~ , w x f : = T ^ S I 2 1 T h e a n g u l r m o t i s ~ = I ! w Z B 2 1 d H c ? k f y v : b - j p .

Continuous Model (15) ~ I = Z h r 1 ¡ i d m · n ^ b o ¸ £ ¤ R e c a l
AOE 5204 ~ I = Z B h r 2 1 i d m T n ^ b o R e c a l t , k p u f s w g T a k i n g t h e d o p r u c s v ^ b ~ I = Z B 1 m w y f , x l F :

Continuous Model (16) a c o b ~ r ^ b ^ b ^ b d m T h e m a t r i x v
AOE 5204 T h e m a t r i x v s o n f - c p u d l I b = Z B 1 V k y w 2 4 + 3 5 q g [ ] E a ^ b 3 d m c ~ r o ^ b 2 o ^ b 1 b F o r e x a m p l , I 1 = b c 3 2 + Y u s h d t y q i n g - .

Continuous Model (16) a c o b ~ r ^ b ^ b ^ b d m o C a r y i n g o u
AOE 5204 a ^ b 3 d m c ~ r o ^ b 2 o ^ b 1 b C a r y i n g o u t h e s l d I b = m 2 4 1 3 ( + c ) 5 N p , T . A w v f x ^ W

Inertia Tensor E x p r e s a n g u l m o t v c - , i y d f F : h = ~ ¢
AOE 5204 E x p r e s a n g u l m o t v c - , i y d f F b : h = ~ ^ ! I T C q N w . A n g u l a r m o e t b : ~ h = Z B _ d I ! w ( ) 2 1 i T s p c , k .

Example Inertia Matrix
AOE 5204 a ^ b 3 d m c ~ r o ^ b 2 o ^ b 1 b C a r y i n g o u t h e s l d I b = m 2 4 1 3 ( + c ) 5 N p , T . A w v f x ^ W

Parallel Axis Theorem c o ~ r ^ b ^ b ^ b S u p o s e w k n t h m f i
AOE 5204 S u p o s e w k n t h m f i r a x b c , d y - l g T I W ^ b 3 ~ r c o c o ^ b 2 ^ b 1 T e c h n i a l y , t w o f r m s u v d g - p ; x

Parallel Axis Theorem (2)
AOE 5204 M o m e n t f i r a x b u c : I = Z B T 1 d W w s + h E C y - p , k g T h e i n t g r a l s x p o m f b u ( d F ) . P : I = c + 1 A v y E S w - q Q D ?

Application of Parallel Axis Theorem
AOE 5204 L o k u p t h e m n s f i r a c y l d , w g v x b C I = 2 4 ( 3 + ) 1 5 P T ; : ^ b 1 ^ b 1 A c y l i n d r a s p e f t h - g u o . T = 4 m , 1 : 5 k ^ b 2 3 I Note typographical error in posted version

Application of Parallel Axis Theorem (2)
AOE 5204 F o r t h e p a n l I c = m d i g 2 4 ( b + ) 1 3 5 x s : T [ ] ~ r c o o c ^ b 1 ^ b 1 C o m p u t e h n f i r a c s b d y l A P T ~ = ( 2 + ) ^

AOE 5204 F o r t h e p a n l ( w i s m c f y d ) : I = g 2 4 b + 1 3 5 T u ^ x . W ? , - N v ! S U q ~ r c o o c ^ b 1 ^ b 1 T h e m o n t f i r a p l b u c y d s g , - P A

AOE 5204 S o m e b s r v a t i n W p l d P A T h g I u c y f H w ? , x - = 1 ~ r o c o c ^ b 1 ^ b 1 N e x t s p : C a l c u h m o n f i r b d y

Change of Vector Basis Theorem
AOE 5204 S u p o s e w k n t h m f i r a x b , d y - c F T I l W ; v g ^ a 3 ^ b 2 ^ b 3 o ^ a 2 ^ a 1 ^ b 1 T h e m o n t f i r a s , ~ I c l b j u w x p g d - y v

Change of Vector Basis Theorem (2)
AOE 5204 W e h a v I = Z B r T 1 d m l t i o n s p b w F x R , u : ? S g f T h e i n t g r a l s o b v u y m f x p d F , w c - R I = q N

Parallel Axis Theorem and Change of Vector Basis Theorem
AOE 5204 ^ b 3 ^ a 3 ^ b 2 ~ r c o ^ b 3 c o ^ b 2 o ^ a 2 ^ b 1 ^ a 1 ^ b 1 P a r l e x i s t h o m : I = c + T 1 A n v y C h a n g e o f v c t r b s i m : R I = U d p u l x

Principal Axes T h e m o s t i p r a n l c f C g V B x d b y S v ² : ~
AOE 5204 T h e m o s t i p r a n l c f C g V B x d b y S v : ~ u I = E F a c t : A l e i g n v u s o f r , y m . d h C T x p b -

Principal Axes (2) C o m p u t e h i g n v a l s d - c r f I , x F T b
AOE 5204 C o m p u t e h i g n v a l s d - c r f I , x F T b = w k ( ; ) I n M a t l b , f o r e x m p w c - u h i g s y [ V D ] = ( ) d v S : 1 ; 2 3 W

Principal Axes (3) I = % a f b r i c t e d x m p l 1 . 9 3 7 6 8 2 4
AOE 5204 I = % a f b r i c t e d x m p l 1 . 9 3 7 6 8 2 4 > [ V , D ] g ( ) 5 - * n s > V * D a n s = . 3 9 7 2 5 1 6 - 8 4 ' I [ x ] X d i g R b F t h e p r c l f m

A Comprehensive Example
AOE 5204 A s p a c e r f t i o m d 3 g b : y l n , . T h F w ^ x ' 2 1 k / 5 4 u = 8 7 ; 9

Comprehensive Example (2)
AOE 5204 S t e p s o l u i n : F r a c h b d y , w I f - m x ( ) A d I o p a n r t c y l b i m C u e s f - , x h g A detailed solution with Matlab code is posted on the Handouts page

Angular Momentum Principle
AOE 5204 O b s e r v a t i o n : I f = c , h ~ _ g l w - p Ä T m u B a c k t o h e d n i ~ = Z r _ m D : Ä R l O ; - f g )

Angular Momentum Principle (2)
AOE 5204 A n g u l a r m o e t p i c : ~ h = I ! b _ H w d x s q f ? R v F ^ T + N y E x p r e s i n g t h w o q u a F b v c = I ! _ + T l y , m f d

Rotational Equations of Motion
AOE 5204 R e c a l p r v i o u s y d k n - m t q _ = 1 2 + 4 T ! Q ( ) S h I f , b g W e c a n u m r i l y t g h q o s f d R - p : , v ~ = ( ; _ ! ) T N w 3 1 x F b

Coupled Equations of Motion
AOE 5204 T h e c o m p l t s f u d r a n i q - g b y : ! = I 1 v ( ; R ) _ + Q T h e r s t w o q u a i n p v d l c f m - y N 1 3 , x 2 R

Euler’s Equations A c o m n a p r h i s t u b - e = I ! g l q _ ( ) ¡
AOE 5204 A c o m n a p r h i s t u b - e = I ! g l q _ ( ) + 1 T x f E d , ; w v F o r a p i n c l f m e , w t I = 2 4 1 3 5 d g [ ] T h v s u - k ( . ) x y :

Euler’s Equations (2) A s u m i n g a p r c l f e , E - q t o y b w h
AOE 5204 A s u m i n g a p r c l f e , E - q t o y b w h _ ! 1 = I 2 3 + G v d ( ) k R A s p r e v i o u l y n t d , h q c a - g w m k . S f b : x ( I 1 = 2 )

Axisymmetric, Torque-Free Rigid Body
AOE 5204 B e g i n w t h _ ! 1 = I 2 3 + S A a d C S e t i n g = _ ! 1 A C 2 3 I r a ) ( T h o w q u s b c m :

Axisymmetric, Torque-Free Rigid Body (2)
AOE 5204 S i n c e t h s y m o p l , w a v u r g x D ^ = ( A C ) : _ ! 1 2 d b - f Ä + q T h e a n g u l r v o c i t y b s m x , ! 3 = d w p f : _ 1 A C 2 S q - ) (

AOE 5204 H a v e ! 3 = D n d ^ ( A C ) r i Ä 1 + 2 T h s o l u t q c b w ' p - . C a n l s o b e w r i t ! 1 = ^ + ' h p c 2 m - u d . D f x W y >

AOE 5204 W e h a v i n t g r d k c s q u o f l m x y b w z : ! 1 = ^ + ' 2 3 A K n o w i g t h e a u l r v c y s f m , k q ( _ p ) A H 4 I ! d - x b Y

AOE 5204 W i t h e s o l u n ! 1 = a ^ + ' 2 c 3 w g r k m , x p d - E f Á Ã F o r t h i s e f E u l a n g , p ! = ( ) _ d c 1 Á Ã 2 3 + A y ^ ' S q m b

AOE 5204 C o m b i n g t h e k c s l u a d O D E v : ^ + ' = _ Á Ã w r > , y f p ( ) x A < 2 F A s u m e ~ h ( c o n t ) i a l g d w ^ 3 , = [ ! 1 2 C ] T p b S v f r : +

AOE 5204 R e c a l t h s i n ( + x ) = o I f 1 r u , _ Á 2 T d Ã > . m b F i r s t w o k n e m a c q u : ^ + ' = _ Á Ã D v d b y T h , ( 1 ) 2 I p A C

AOE 5204 Axisymmetric, Torque-Free Rigid Body (8) Summary S o l u t i n f E e r ' s q a : ! 1 = ^ + 2 c 3 w h ( ) > A C S o l u t i n f k e m a c s q : _ Á = Ã ^ 1 C p A 2 + P r v N g d w h b y - j x

Euler’s Equations Recap
AOE 5204 A s u m i n g a p r c l f e , E - q t o y b w h _ ! 1 = I 2 3 + G v d ( ) k R A s p r e v i o u l y n t d , h q c a - g w m k . S f b : x ( I 1 = 2 )

Asymmetric, Torque-Free Rigid Body
AOE 5204 g = , w h i c ) ~ o n s t : _ ! 1 I 2 3 O r m a x f T e l k y Z B d E D v p B e g i n w t h _ ! 1 = I 2 3 + U l k a x s y m r c , q u o p d

Asymmetric, Torque-Free Rigid Body (2)
AOE 5204 I n s c a l r f o m , t h e w v d q u i T 2 : = 1 ! + 3 S - ( k g ) ; ' / y E x W b ~ ? _ ! = I 1 T 2 S h o w t a i s c n : d ( ) +

AOE 5204 W e h a v 2 T = I 1 ! + 3 A s u m , w i t o l f g n r - y > d < W e c a n s o l v f r ! 2 1 d 3 i t m , T h I u b - q _ = ( ; ) g k w p .

AOE 5204 T h e s o l u t i n f r ! 2 = a ( ; k ) v J c b p m [ 1 ] . S d , : W In Matlab, elliptic functions are computed using the function call [sn,cn,dn]=ellipj(tau,m) where m = k2

AOE 5204 W e h a v t g n r l s o u i . A y p c ? _ ! 1 = I 2 3 S q d f w j ( ; ) z , T b x W e w a n t o d r m i h s b l - y f p C c : P g ( . , I 1 > 2 3 ) k ; S ! = [ ] T

AOE 5204 O n e w a y t o i v s g b l r z u h m q . H - ! = [ ] T B , A d p x f + ! 1 = + 2 3 S u b s t i e h x p r o n E l ' q a _ I ( ) d g c f j m

AOE 5204 _ ! 1 = I 2 3 T h e s q u a t i o n r l , c - p d y Y m w E ' f x g b O ( 6 ) D i e r n t a _ ! 1 d s u b 3 o Ä + ( I 2 ) = C l y h c - , q f m x k p g > w ; <

Rigid Body Dynamics ~ f = p h g

Similar presentations

Presentation on theme: "Rigid Body Dynamics ~ f = p h g"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Rigid Body Dynamics ~ f = p h g

Similar presentations

Presentation on theme: "Rigid Body Dynamics ~ f = p h g"— Presentation transcript:

Similar presentations

About project

Feedback