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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
Coordinate representations
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Coordinate representations
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
Coordinate representation. Rigid part
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
Coordinate representation. Rigid part
,( , , , )
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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Linear coordinatesDeformation measures
, 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
Linear coordinatesDeformation measures
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 coordinatesDeformation measures
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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Linear coordinatesSelection of global shape functions
,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:
Linear coordinatesSelection of global 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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The Euler-Bernoulli Beamin plane motion
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:
The Euler-Bernoulli Beamin plane motion
( ) ( )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
Rigid transplacement of beam center line
( ) 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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Rigid transplacement of beam center line
( ) ( ) ( ) 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 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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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 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
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
X X X X X X X 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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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 .
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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