Coordinate representations The MultiBody · 2014. 3. 4. · 2014-03-04 1 The MultiBody Coordinate...

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The MultiBodyCoordinate representations

sum res kcQ Q Q

T sum2 1 n

0 m 1

q M Q 0g gq 0

( , ) ,n n2 2M M t q

res cif i ric ec bQ Q Q Q Q Q

( , , )sum sum 1 nQ Q t q q

( , ) ,m 10 0g g t q ( , ) m ng g t q

kc TQ g

det( ) ,2M 0 rank g m

... 5

1

...

3

4

2

( , ) n n2 2M M t q ( , , )sum sum 1 nQ Q t q q

ComplexityHighLow

( , ) n n2 2M M t q ( , , )sum sum 1 nQ Q t q q

Coordinate representations

The complexity of and as functions will depend on the specific q-coordinate system used. There will, in general, be a pay-off between and concerning complexity.

2M sumQ ( , , )t q q

2M sumQ

ComplexityHighLow

q q ˆ( , )q q t qCoordinate change:

( , ,..., )1 2 n nq q q q

( , ) ( , ( ); ), , qx X t t q t X X t T 0B

( , , ; ) ,n

kq 0 k

k 1t q q X q

x x b b ( , ; )( , ; ) q

k k k

t q Xt q X

q

b b

0B

X

tB

x

1q

( )q q t

- : nq space

nq

( , ; )q t q


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The system of configuration coordinates is said to be regular for the multibody if, forall fixed ( , )t q ,

, , , ,...,n

q k k kk

k 1w 0 X w w 0 k 1 n

q

0B

This is equivalent to the statement that, for all fixed ( , )t q , the functions

( , ; ), , ,...,q qk k t q X X k 1 n

q q

0B

are linearly independent vector fields on 0B . Furthermore, the system of configurationcoordinates is said to be regular for the part if, for all fixed, ( , )t q

, , , ,...,n

q k k k0k

k 1

w 0 X w w 0 k 1 nq

B

Coordinate representation. Rigid part

( ) RON-basis fixed in inertial frameo o o o1 2 3e e e e

, , , , , 1 2 3 4 5 6q x q y q z q q q

( , ; ) ( , ; ), , ,q q o o o

0 1 1 2 2 3 31

t q X t q Xt q

b 0 b e b e b e

( , ) ( , ; ) ( , , )( )o 1 o 2 o 3 4 5 6q O 1 2 3 cx X t t q X x q q q q q q X X e e e R

( ), ( ), ( )4 c 5 c 6 c4 5 6X X X X X Xq q q

R R Rb b b

( , , ) ( )4 5 6q q q SO R R V


o

4q

e

R

oo oT4 4q q

e

RR e e ,oo oT

eR e R e

sin cos cos cos sin sin sin cos cos cos cos sincos cos sin cos sin cos sin sin cos cos sin sin

sin sin cos sin cos

4 6 4 5 6 4 6 4 5 6 4 5

4 6 4 5 6 4 6 4 5 4 5

6 5 6 5 5

q q q q q q q q q q q qq q q q q q q q q q q

q q q q q

Using Euler angles the q-cordinate system defined by:

is regular.

( , ) ( , ; ) ( , , )( )o 1 o 2 o 3 4 5 6q O 1 2 3 cx X t t q X x q q q q q q X X e e e R


,( , , , )

3o o ij

0 1 2 3 i ji j 1

e e e e R

R e e

( , , , )T 4 10 1 2 3e e e e e

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )

2 20 1 1 2 0 3 1 3 0 2

ij 2 21 2 0 3 0 2 2 3 0 1

2 21 3 0 2 2 3 0 1 0 3

2 e e 1 2 e e e e 2 e e e eR 2 e e e e 2 e e 1 2 e e e e

2 e e e e 2 e e e e 2 e e 1

2 2 2 20 1 2 3e e e e 1

Euler parameters:

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Coordinate representation

( , ) ( , ( ); ) ( , ( )) ( )m

iq 0 i

i 1x X t t q t X x t q t X

α a

( , ), ( ), , ,...,in m

kk k ik

k 0 i 1

t qx q X k 0 1 nq

αb b a

( ) , ,..., , i i X i 1 m X a a 0V B

: ( )i End α V

( , ) ( , ; ) ( , ( )) ( )m

iq i

i 1X t t q X t q t X

F F α a

( , ) ( , ) , : , ,...,i i i it q t q i 1 m α α 1 Special case:

( , ) ( ), , ,...,in

k iki 1

t q X k 0 1 nq

b a

Shape functions, linearly independent

( ; ) ( )n

kq 0 k

k 1x q X x X q

a

( , ) , ,...,i i it q q i 1 n

Linear coordinates

( ), ,..., , i i X i 1 n X a a 0B

( )n

kk

k 1X q

x a

( , ) ( , ; ) ( )n

kq k

k 1X t t q X X q

F F a

Transplacement

,

( , , ) ( )3

4 5 6 o o ij o ij oTi j

i j 1

q q q q q

R e e e e

,( , ; ) ( )

3o 1 o 2 o 3 o o ij

q O 1 2 3 i j ci j 1

x t q X x q q q X X q

e e e e e

, , , ,i ijq q i 1 2 3

Linear coordinates for rigid parts

Configuration coordinates (12 in number):

( )( )ij ij T3 3q q I Constraints (6 in number):

( ; ) ( ; )q qx q X X q X u

( ; ) ( )n

kq id 0 k id

k 1

q X x X q X

aidq q

( ; ) ( )( ) ( )n

k kq k id id

k 1

q X X q q q q

u u a a

1 2 na a a a

Linear coordinates

Displacement:

0 1T T 0 Scleronomous coordinate systems:

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Linear coordinates

Proposition 11.2

T2

1T T q M q2

(11.34)

where ( )T

0M dv X a a0B

(11.35)

and

1 2 1 2 1 n

2 1 2 2 2 nT

n 1 n 2 n n

a a a a a aa a a a a a

a a

a a a a a a

The constant matrix M is positive definite.

Linear coordinates

Proposition 11.3 Using linear coordinates the complementary inertia force for isidentically zero, i.e. cif

1 nQ 0 .

Tk k

d T T q Mdt q q

Theorem 11.1 Using linear coordinates the equations of motion for the multibody may be written

T i c b T

1 n

0 m 1

q M Q Q Q g 0g gq 0

Linear coordinates

( )o o o o1 2 3 Oe e e e

1k

oT 3 1k k 2k

3k

aaa

ea e a ( ) ( ) 0

ok kX X

ea e a

o Aa e11 12 1n

3 n21 22 2n

31 32 3n

a a aA a a a

a a a

( )n n

T o T o T oT o T n n

1

A A A A A A

a a e e e e

( )T0M A A dv X

0B

3oi

i 1 i

aa eX

, 3

oO i i i

i 1X X

e X X

Linear coordinatesDeformation measures

( )o o o o1 2 3 Oe e e e

Corollary 11.4 The deformation gradient and the left Cauchy-Green strain tensor are givenby

,

3o o

q i j q iji 1

F

F e e and ,,

3o o

q i j q iji j 1

C

C e e

where

,

nkki

q ijk 1 j

aF q

X and ,

,

( )n

k l T Tk lq ij

k l 1 i j i j

C q q q q

a a a aX X XX

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, 3

oO i i i

i 1X X

e X X , where 1

2 i i1 i2 in

3

AA A A a a a

A

3o o

i ii 1

A A

a e e

Corollary 11.5

,,

( ; ) ( , )3

o oq q i j q ij

i j 1q X C q X

C C e e

where , , ( , ) ( )T

q q ijij ijC C q X q X q

and the so-called Cauchy-Green matrix is defined by

( ) ( )( ) ( )3

T n nk kij ij

k 1 i j

A X A XX

X X

The Cauchy-Green matrix has the properties T

ij ji , ( )Tid ij id ijq X q


Furthermore, for the Green- St.Venant tensor

,,

( ; ) ( )3

o oq q i j q ij

i j 1q X E X

E E e e

where

, ( ) ( ( ) )Tq ij ij ij

1E X q X q2


Linear coordinatesSelection of global shape functions

( , ) ( , , , )3

i i 1 2 3i 1

X t p t

p e X X X

( ), ,..., , i i X i 1 n X a a 0B

( ; ) ( , ),q Ox q X x X t p

, , ,( , , , ) ( ) , ( )31 2

1 2 3 1 2 3 1 2 33

jj 1

i 1 2 3 i 1 2 3 i i

0 k

ap t a t a t

X X X X X X

Shape functions:

,( , ) ( , , , ) ( , , , ) ( ) 31 2

1 2 33

jj 1

3

m i i 1 2 3 i 1 2 3 i 1 2 3i 1

0 m

X t p t p t a t

P p e X X X X X X X X X

dim( ) ( )( )( )m1 m 1 m 2 m 32

Plinear space,mP

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,p p ,, ,...,iZ i 1 n 0B ,

( ), , , , ki i k k pq Z i 1 2 3 Z e p

( ) , , , , , kij i k j k pq Z i j 1 2 3 Z e p e

2Z

0B

1Z

3Z

4Z

5Z 6Z

8Z 7Z

( )ik ik Xa a

,

( , ) ( , , , ) ( ) ( ) ( ), k p k p

3 3 30 k ki i 1 2 3 i ik ij ijk

i 1 i 1 Z i j 1 ZX t p t q t X q X X

p e a a 0BX X X

( )ijk ijk Xa a

Nodal points:

Configuration coordinates:

Shape functions:


,8n

dim( )mm 3 60 P

dim( )mm 1 12 P

p

,p

,4n

, , ,p 1 2 3 4Z Z Z Z

,p

Elasticity

Proposition 11.4 The second Piola-Kirchhoff stress tensor S for an isotropic linear elasticmaterial, with Lamé moduli and , is given by

,,

( ; ) ( ; )3

o oq i j q ij

i j 1

q X S q X

S S e e

where

, , ( ; ) ( ( ) ) ( ( ) )3

T Tq ij q ij kk ij ij ij

k 1

S S q X q X q 1 q X q2

Elasticity

Theorem 11.2 The strain energy density of an isotropic linear elastic material, with Lamémoduli and , is given by

,

( ; ) ( ( ( ( ) )) ( ( ) ) )3 3

T 2 T 2e e ii ij ij

1 1 i j 1

1u u q X q X q 1 q X q4 2

Total elastic energy:0

( ) ( ; ) ( )e eU U q u q X dv X B

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ElasticityProposition 11.5

i i i i Te1 2 n

UQ Q Q Q q Kq

where the stiffness matrix, n nK , is defined by ( ) ( ) ( ) ( ) ( )1 2 3K K q 2 k q k q 2 k q and the matrices , and n n

1 2 3k k k are defined by

( ) ( ( ) ) ( ) ( )3

T1 ii ii

i 1

1k q q X q 1 2 X dv X4

0B

( ) ( )( ) ( )0

3T T

2 jj kk kkj k

1k q q q 1 dv X4

B

( ) ( ) ( )3

T3 jk jk

j k

1k q q q sym dv X4

0B

Theorem 11.3 (The equations of motion)

( )T T T

1 n

0 m 1

q M q K q Q g 0g gq 0

Elasticity

( ) ( ),n nK K q Sym

ric ec bQ Q Q Q

( )id n nK q 0

m ng 0

( ) n 1Mq K q q 0

1 nQ 0 ,0 m 1g 0 No constraints:

ElasticityLinearization

( )( ) ( ) (( ) ( ))( ) ( )tidid id id id id id

K qK q q K q q q K q q q o q qq

( ) idq q t q Equilibrium solution:

( )(( ) )( ) ( ) ( ) ( )tidid id id id id id

K q q q q o q q K q q o q qq

( )( )tidid id

K qK qq

,idM z K z 0 idz q q

The Euler-Bernoulli Beamin plane motion

( )1 2 3 Oi i i

, A 1 00 LX X X i0C E X X

( , ) ( , ) ( , ) , 1 1 2 2X t u X t u X t u X u u i i i 0C

1 2i i i1

2

uu

u

Ron-basis fixed in inertial frame:

Beam center line in reference placement:

Center line displacement:

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1i

2i

X

( , )X tu

3 i

x

A

B

0C

tC

0A O

, ( , )t Xx x X X t u 0C CE

Beam center line in present placement:

( )0 0a area A

( , ) ( )idX t q q u a

( )1 2 6 Tq q q q

Linear coordinates:

Configuration coordinates:

Shape functions:


( ) ( )1 2 1X Ha i X , ( ) ( )2 2 2X Ha i X

( ) ( )3 2 3X Ha i X , ( ) ( )4 2 4X Ha i X 00 L X

( ) ( )5 1 1X ha i X , )( ) (6 1 2X ha i X

Hermite polynomials

( ) ( ) ( )2 31

0 0

H 1 3 2L L

X X

X , ( ) ( ) ( )2 32

0 0 0

H 2L L L

X X X

X

( ) ( ) ( )2 33

0 0

H 3 2L L

X X

X , ( ) ( ) ( )2 34

0 0

HL L

X X

X

Third order polynomials

( ), ( ) , ( ), ( )1 1 2 2 3 3 4 4id id 0 id 0 id 0 0q q p 0 q q p 0 L q q p L q q p L L

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 3 3 4 4id 1 id 2 id 3 id 4p p q q H q q H q q H q q H X X X X X

( ) , 2 30 1 2 3 0p p a a a a 0 L X X X X X

A B

1q 3q

2q 5q 4q 6q

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( )10

h 1L

X

X ( )20

hL

X

X

( ), ( )5 5 5 5id id 0q q p 0 q q p L

( ) ( ) ( ) ( ) ( )5 5 6 6id 1 id 2p p q q h q q h X X X

, ( ) 0 1 0p p b b 0 L X X X

First order polynomials

1 2 3 4 5id id id id id

id O 1 6id 0

q q q q q 0q X X

q L

a i X

Transplacement of beam center line

( ; ) ( )q id Oq X X X q q X q u a a

( ) ( )( ) ( ) ( ) ( )

1 21

1 2 3 42

0 0 0 0 h hAA

H H H H 0 0A

X X

X X X X

Aa i

( ; )q Oq X X Aq i

Rigid transplacement of beam center line

( , ) cos sin , 1 2 0X t 0 L

i i XX


( ) cos5 6

0 0

1 1q qL L

,

( ( ( ) ) ( ( ) ) ( ( ) )1 2 2 2 3 2

0 0 0 0 0 0 0 0 0

6 1 6q q 1 4 3 qL L L L L L L L L

X X X X X X

( ( ) ) sin4 2

0 0 0

1q 2 3L L L

X X

cossin

sin

cossin

6 50 3 1

01 2 3 42 4

01 2 3 46 5

020

q q Lq q L

3q 2q 3q q 0q q L

2q q 2q q 0q q L

q L

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( ) ( ) ( ) cos5 6 5 6 5 51 2

0

q h q h q q q qL

X

X X X

( ) ( ) ( ) ( )1 2 3 4 1 21 2 3 4

0

q H q H q H q H q qL

X

X X X X

cos sincos )( ; ) ( )

sin cossin

5 5

q O O O1 1

q qq X X Aq X X

0q q

i i i

XX

X

Velocity of beam center line

( , )X t Aq u u i

( )3 3

2 2 o T o T T oT o T T

I

Aq Aq q A Aq q A Aq

x u u u e e e e

TA A

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

21 1 2 1 3 1 4

22 1 2 2 3 2 4

23 1 3 2 3 3 4

24 1 4 2 4 3 4

21 1 2

22 1 2

H H H H H H H 0 0H H H H H H H 0 0H H H H H H H 0 0H H H H H H H 0 0

0 0 0 0 h h h0 0 0 0 h h h

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X

X X X

The kinetic energy

( ) ( ) ( ( ))2 T T T T Tk 0 0 0

1 1 1 1E dv X q A Aq dv X q A A dv X q q Mq2 2 2 2

x 0 0 0B B B

( )0 0L L

T T T0 0

0 0 00 0

m mM A A dv X A A a d A Ada L L

X X0B

The mass matrix:

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

0 0 0 0

0 0 0 0

0 0 0 0

0

L L L L2

1 1 2 1 3 1 40 0 0 0

L L L L2

2 1 2 2 3 2 40 0 0 0L L L L

23 1 3 2 3 3 4

0 0 0 0 0L

4 10

H d H H d H H d H H d

H H d H d H H d H H d

mH H d H H d H d H H dL

H H d

X X X X X X X X X X X


X X X X X X X X X X

X X X

X

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

0 0 0

0 0

0 0

L L L2

4 2 4 3 40 0 0

L L2

1 1 20 0

L L2

2 1 20 0

H H d H H d H d

0 0 0 00 0 0 0

0 00 0 156 22 54 13 0 00 0 22 4 13 3 0 00 0 54 13 156 22 0 0m

13 3420h d h h d

h h d h d

X X X X X X X

X X X X X

X X X X X

X

22 4 0 00 0 0 0 140 700 0 0 0 70 140

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The displacement gradient

( ) ( )

( ) ( ) ( ) ( )

1

1 2

2 1 2 3 4

A0 0 0 0 h hA

A H H H H 0 0

X XXX X X X X

X

( )idA q q

u iX X

( )10

1hL

X , ( )20

1hL

X

( ) ( ( ) )21

0 0 0

6HL L L

X X

X , ( ) ( ( ) )22

0 0 0

1H 1 4 3L L L

X X

X

( ) ( ( ) )23

0 0 0

6HL L L

X X

X , ( ) ( ( ) )24

0 0 0

1H 2 3L L L

X X

X

( ) ( )2

2 21 2e 0 0 2

u uu Ea EI

X X

The specific elastic energy

( )1 1id

u A q q

X X( )

2 22 2

id2 2

u A q q

X X

( ( )) ( ) ( ( )) ( )2 2

T T1 1 2 2E 0 id id 0 id id2 2

A A A Au Ea q q q q EI q q q q

X X X X

( ) ( ( ) ( ) )( ) ( ) ( )2 2

T T T T1 1 2 2id 0 0 id id id2 2

A A A Aq q Ea EI q q q q k q q

X X X X

The specific stiffness matrix

( ) ( )2 2

T T1 1 2 20 0 2 2

A A A Ak Ea EI

X X X X

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

(

20 1 0 1 2 0 1 3 0 1 4

20 2 1 0 2 0 2 3 0 2 4

20 3 1 0 3 2 0 3 0 3 4

0 4

EI H EI H H EI H H EI H H 0 0

EI H H EI H EI H H EI H H 0 0

EI H H EI H H EI H EI H H 0 0

EI H

X X X X X X X

X X X X X X

X X X X X X X

X

) ( ) ( ) ( ) ( ) ( ) ( )21 0 4 2 0 4 3 0 4

0 02 20 0

0 02 20 0

H EI H H EI H H EI H 0 0Ea Ea0 0 0 0L LEa Ea0 0 0 0L L

X X X X X X X

( ) ( ) ( ) ( )Te E id id

1 1U u dv X q q k q q dv X2 2

0 0BB

( ) ( ( ))( ) ( ) ( )T Tid id id id

1 1q q kdv X q q q q K q q2 2

0B

The elastic energy

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12

The stiffness matrix

( )0L

00

K kdv X a kd X0B

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 0 0 0

0 0 0 0

0 0

L L L L2

1 1 2 1 3 1 40 0 0 0

L L L L2

2 1 2 2 3 2 40 0 0 0L L

20 3 1 3 2 3

0 0

H d H H d H H d H H d

H H d H d H H d H H d

EI H H d H H d H



X X X X X X X ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

0 0

0 0 0 0

L L

3 40 0

L L L L2

4 1 4 2 4 3 40 0 0 0

d H H d

H H d H H d H H d H d

0 0 0 00 0 0 0

X X X X


0 0

0 0

0L L 3 2 2

0 0 0 0 0 0 02 2

0 0 0 00 0 0 0L L 2 2

0 0 0 00 02 2

0 00 0 0 00 0

0 0 12 6 12 6 0 00 0 6 4 6 2 0 00 0 12 6 12 6 0 00 0 6 2 6 4 0 0EIEa Ea1 1 L a L a Ld d 0 0 0 0

EI L EI L I I

a L a LEa Ea1 1 0 0 0 0d d I IEI L EI L

X X

X X

The potential energy in the gravitational field

( ) ( ( ))5 6 1 2 3 40 0 0Oc 1 2

L L L11 q q q q q q2 2 6 6

p i i

( ( ))1 2 3 40 0g Oc

mgL L1V m q q q q2 6 6

p g

( )2 g g i

Lagrangian for the beam

e gL T V T U V

( ) ( ) ( ( ))T T 1 2 3 40 0id id

mgL L1 1 1q Mq q q K q q q q q q2 2 2 6 6

156 22 54 13 0 022 4 13 3 0 054 13 156 22 0 0mM13 3 22 4 0 04200 0 0 0 140 700 0 0 0 70 140

03 2 20 0 0 0 0

0 02 2

0 0 0 0

0 0

12 6 12 6 0 06 4 6 2 0 012 6 12 6 0 06 2 6 4 0 0EIK

L a L a L0 0 0 0I I

a L a L0 0 0 0I I

2i

1i 3 i

i

1f

2f

A

B

u

Ox

OAp

Floating frame of reference

( )1 2 3 Af f f( )1 2 3 Oi i iFixed frame: Floating frame:

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13

The kinetic energy

( )T2 2

2 21 24 4

4 3q q1 m mT3 42 420 2q q

x x

( (sin cos ) (cos sin )) ( )22 4 2 4

T 20 01 2 A

mL mLq q q q q M q2 6 6 6

x x

( sin cos )( ) ( ) ( ) ( )2

2 4 2 4 2 2 2 2 40 0 01 2 1 2

mL mL mLm 3 m 3q q q q q q12 30 2 2 6 30 2

x x x x

( (sin cos ) (cos sin )T2 2 2 4 2 4

2 01 24 4

1 1q q mLm q q q q105 140

1 12 2 6 6q q140 105

x x

( sin cos )( )2 41 2

m q q12

x x

T2 2

e 34 40

4 2q q1 EIU2 42 Lq q

The elastic energy

The potential energy in the gravitational field

( sin cos ( ))2 40 0g Oc 2

L LV m q q gm2 12

p g x

MBD-project

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MBD-projectIn all Tasks below parts and are assumed to be rigid and the motion of these parts isassumed to be plane. The RON-basis ( )i j k is fixed to the machine foundation which in itsturn is assumed to be fixed in an inertial frame. Task 1: Part is assumed to be rigid.

a) Introduce a set of configuration coordinates for the slider and crank mechanism thatwill make it possible to calculate the reactions at the joints at A and B.

b) Formulate, using the coordinates defined above, appropriate constraint conditions. c) Formulate the equations of motion for the slider and crank mechanism using Lagrange

- d’Alemberts equations with constraints. The numerical solution of these equations(using, for instance, Matlab) is not required.

d) Construct an ADAMS-model (Model_1) for the slider and crank mechanism. Build astarting configuration as in Figure 1 above where f 130cm .

e) Simulate the motion of the multibody. Required results:

1) The equations of motion according to a) – c) above. 2) The ADAMS model: Give a verbal description and a picture of Model_1. 3) Plot the force ( )F F t during the time interval 0 t 5s . 4) Plot the angular velocity of the part during the time interval 0 t 5s . 5) Plot the reaction forces and moments at the joint at A during the time interval 0 t 5s .

MBD-project

MBD-project MBD-project

Task 2: Part is an elastic beam.

a) Construct an ADAMS-model (Model_2) for the slider and crank mechanism. Use forthe modeling of the beam: Flexible bodies→Discrete Flexible Link with 21 elements.

b) Simulate the motion of the multibody. Required results:

1) The ADAMS model: Give a verbal description and a picture of Model_2. 2) Plot the angular velocity of the part during the time interval 0 t 5s . 3) Plot the reaction forces and moments at the joint A during the time interval 0 t 5s .

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MBD-project MBD-projectDefinition of the external force: The velocity of the slider is denoted B Bxv i and the external force is given by ( )F F i where

if ( ) ,

if 0 B 5

B 0B

F x 0F F x F 10 N

0 x 0

MBD-project

Initial condition: The motion of the multibody is starting from rest in the placement shown in Figure 1 where f 130cm .

MBD-project

Project report Produce a short report where:

1) the equations of motion for the slider-crank mechanism are derived and presented. Donot use numerical values here but use the parameter names given in the projectspecification ( I , 0L , m ...etc.) or parameters introduced and defined by yourself.

2) the Required results defined above are presented as ADAMS-plots. 3) a discussion of the results is accounted for where the following questions are

answered: Does the flexibility of the crank, introduced in Model_2, influence the results

in any way? Will the motion of the Slider and crank mechanism reach a “steady state” and

if so why does this happen?

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MBD-project Short ADAMS manual

for the Multibody Dynamics Project

Slider and Crank Mechanism

1. Starting ADAMS

Open ADAMS-VIEW: Program→MSC.Software→MSC.ADAMS.2012→Aview Name and save your model in a catalogue where you can find it!

Choose “Units: MKS”

From ”View”-menu choose: Toolbox and Tool bar From “Settings” choose: “Units: Angle: Radians”

From “Settings”-menu” choose the following “Working Grid…”

X Y

Size (2.5m) (1.0m)

Spacing (2cm) (2cm)

Zoom in the entire grid of the work area.

2. Constructing the model: Task 1

Coordinate representations The MultiBody · 2014. 3. 4. · 2014-03-04 1 The MultiBody Coordinate...

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Transcript of Coordinate representations The MultiBody · 2014. 3. 4. · 2014-03-04 1 The MultiBody Coordinate...