On The Dynamics of Multibody Systems With Springs: Abstract
On The Dynamics of Multibody Systems With Springs: Abstract
On The Dynamics of Multibody Systems With Springs: Abstract
A14_316
I. Problem formulation
z1
z2
C1
y1
O1
x2
C2
O2
[ ]
[ Q ] = [[1 ]T {u } {u } {u }] ,
[Q] = [1 ]T [ Q ] .
x1
O0
X
Fig. 1. Solid rigid linked by springs.
(2)
(3)
{1 } = [X O1 YO1 Z O1 ]T ,
{1 } = [ 1 1 1 ]T ,
T
{q1 } = [{1 }T {1 }T ] ,
J y1z1 , J z1x1 ;
m1
{m1 } = 0
0
0
{S1 } = m1 z C1
y
C1
{J O }
1
{ }
{ } { }
y2
s_doru@yahoo.com, doru.stanescu@upit.ro
stelian_mihalcea_07@yahoo.com
1
Jx
1
= J y1x1
J
z1x1
0
m1
0
z C1
0
xC1
J x1 y1
J y1
J z1 y1
m1
0
0
(4)
(5)
(6)
(7)
y C1
x C1 ,
0
(8)
J x1z1
J y1z1 ,
J z1
(9)
13th World Congress in Mechanism and Machine Science, Guanajuato, Mxico, 19-25 June, 2011
[M ] = [Q] [m[S][]A]
q1
[A ][S]T [Q]
,
[Q]T [J O1 ][Q]
= Fq1 ,
dt q& 1 q1
AB =
(10)
(12)
0 zB yB
{rB } = [x B y B z B ] , [rA ] = z B 0 x B ,(25)
y B x B
0
it results the generalized forces column matrix [7]:
X B X A
[I ]
k ( AB l 0 )
{F} =
[Q]T [r ][A ]T YB Y A , (26)
AB
A
1
Z B Z A
in which
T
T T
=
q& 1 q& 1
T T
=
q1 q 1
T
L
q 6
{Fq } = [Fq
1
Fq
T
,
q& 6
L Fq
(14)
],
T
(15)
= M q 1 {q& 1 } ,
q& 1
T
q1
(13)
[ ]
[ ] {q& }
{ } [ ]
1
{ } { }
1
X A X O1
Y = Y + [A ]{r }
1
A ,
A O1
Z A Z O
1
(16)
Mq
Mq
1
1
{q& 1 } L {q& 1 }T
{q& 1 }T
X O1
1
and if we call
~
& {q& } + T ,
Fq = M
q1
1
1
q1
it results the Lagrange equations [7]
~
&& 1 } = Fq + Fq
M q {q
[ ]
where
[ ]
1
=
2
X A )2 + (Y B Y A )2 + (Z B Z A )2 . (22)
(11)
{ }
(X B
A14_316
(27)
X B X O2
Y = Y + [A ]{r }
(28)
2
B .
B O2
Z B
Z
O2
In the case of n springs, the equation (26) is rewritten as
n
k (A B l )
{F} = i i i 0 i
Ai Bi
i =1
(17)
X X A .
i
[I]
Bi
T
T Y Bi Y Ai .
[Q] r Ai [A1 ]
Z Bi Z Ai
(29)
[ ]
(18)
(19)
Z'
Z
C
uC O
mg
Y'
O'
A'
A
X'
u
k
O'0
O0
Base
B'
13th World Congress in Mechanism and Machine Science, Guanajuato, Mxico, 19-25 June, 2011
matrix;
1) ,
2) ,
3) , [K
K (AB
K (AB
K (AB
AB ] , [K C ] , [K ] , the
rigidity matrices given by [3]
s
1)
K (AB
= k 1 {U}{U}T ,
L
[ ] [ ] [ ]
[ ]
]T ,
2
A
A]
[K (AB3) ] = ks2 [[00]] [r ][u][0+] [u][r ] ,
A
A
(1)
( 2)
( 3)
[K AB ] = [K AB ] + [K AB ] + [K AB ] ,
[0]
[K C ] = mg [0]
,
2 [0] [rC ][u C ] + [u C ][rC ]
( 2)
AB
b c d e f ]T ,
X Y Z ]T ,
[K ] = [K C ] + [K Ai Bi ] ;
(38)
(39)
(40)
(41)
i =1
J X , J Y , J Z , J XY , J XZ , J YZ , the inertial
moments of the rigid body;
[J O ] , [M ] , the matrix of the inertial moments,
respectively the inertial matrix
J X J XY J XZ
[J O ] = J YX J Y J YZ ,
J ZX J ZY
J Z
(42)
{U} = [a
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[M ] = m[I ] m[rC ] ;
m[rC ] [J O ]
(33)
,(34)
(35)
0 c b
0 Z A YA
{u} = c 0 a , {r A } = Z A 0 X A ; (36)
b a 0
Y A X A
0
C , the weight centre of the rigid body;
m , the mass of the rigid body;
{u C } , the unit vector of the weight force at
equilibrium;
X C , YC , Z C , the OC vector projections onto the
OX , OY , OZ axes;
OX , OY , OZ axes;
{U} , {} , {rB } , the matrices given by [3]
3
13th World Congress in Mechanism and Machine Science, Guanajuato, Mxico, 19-25 June, 2011
{U} = [a
{} = [ X Y
~
~T
b c d ~
e f ,
(43)
Z X Y Z ]T ,
(44)
0 Z B YB
{rB } = Z B 0 X B ;
(45)
YB X B
0
~
~ ( 2) , ~
, K
K AB , K , the matrices given by
AB
[ ][ ][ ][]
[K~ ] = k 1 Ls {U}{U~ } ,
~ (1)
K
AB
(1)
AB
,
B ]
A
A
~ (1)
~ ( 2)
= K AB + K AB ,
[K~ ] [ ] [ ]
[K~ ] = [K ] ;
AB
(46)
(47)
Ai Bi
(49)
A' B' =
(A' B')2
=
(Lu AB )2
[AB ( A B )]2
(48)
i =1
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(51)
V AB = s 2 2 s (u AB ) + 1 (u AB )2
L
2
(52)
s
2
+ ( AB ) ,
L
k s {U} + mg{U
i i
} = {0} .
(63)
i =1
1
(53)
( OA ) .
2
Further on, taking into account that the AB vectors
partial derivatives with respect to the arguments X , Y ,
Z , X , Y , Z , are equal to the A vectors partial
derivatives with respect to the same parameters and if we
call {FeAB } the column matrix of the excitation by
components
A
A
ks
s
,
B
+ 1 (u B ) u
(54)
L
L
q i
q i
A = + OA +
{F } + {G} + {F } = {0} ,
n
j =1
AjB j
(64)
{}
{}
[ ]
13th World Congress in Mechanism and Machine Science, Guanajuato, Mxico, 19-25 June, 2011
VII. Application
0 2l
0 2l
0 0
~ 1)
K (AB
,
0 0
0 0
0 4l 2
(70)
0
0 7l
1 0 0
0 1 0
0
0
5l
7l
5l
0
k0 0 1
~ 2)
=
K (AB
,
2
2
7l
5l
0
40 0 l
0 0 3l 21l 2 15l 2
0
0
0 22l 2
l 3l 0
and, in the same way, for the CD spring the successive
expressions were obtained
{U} = 2 [ 1 1 0 0 0 2l ]T ,
2
0 0 l
~
2
T
[ 1 1 0 0 0 2l ] , [rC ] = 0 0 3l ,
U =
2
l 3l 0 (71)
3l
3l
2l
[ ]
B
k
A
X
2l
l
mg
~
k
3l
E
3l
mg
k 2
l
2
(67)
s
1
= .
L
4
For the AB spring the following relations were
reached
2
[1 1 0 0 0 2l ]T ,
2
0 0 l
~
2
T
U =
[1 1 0 0 0 2l ] , [r A ] = 0 0 3l ,
2
l 3l 0 (68)
{U} =
0 7l
1 0 0 0
0 1 0 0
0 5l
5l
0
k 0 0 1 7l
~ ( 2)
=
K CD
.
2
2
5l
0
4 0 0 l 7l
0 0 3l 21l 2 15l 2 0
0 22l 2
l 3l 0 0
The EF spring elongation being null, it results
{U} = U~ = [0 1 0 0 0 0]T
{}
(72)
(73)
[ ]
0 0 7l
0 0 5l ,
7l 5l 0
7l 2 5l 2
0
2
2
[rA ][rB ] = 21l 15l
0 ,
0
22l 2
0
0
0
0
0
0
0
[ ]
[rB ] =
0
0
0
0
0
0
0 7l
0
[rD ] = 0 0 5l ,
7l 5l 0
7l 2 5l 2
0
2
2
[rC ][rD ] = 21l 15l
0 ,
0
0
22l 2
1 1 0 0 0 2l
1 1 0 0 0 2l
~ (1)
3k 0 0 0 0 0 0
K CD =
,
8 0 0 0 0 0 0
0 0 0 0 0 0
2
2l 2l 0 0 0 4l
Fig. 3. Application.
s =
1
1
0
0
0
2l
{}
F
O0
1
1
3k 0
=
8 0
0
2l
[ ]
2l
A14_316
{}
(69)
(74)
13th World Congress in Mechanism and Machine Science, Guanajuato, Mxico, 19-25 June, 2011
[K~ ]
EF
[]
0
0
~ 0
= k
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
0
0
(75)
~
and the K matrix is
~
~ 1)
~ 2)
~ (1)
~ ( 2)
~
K = K (AB
+ K (AB
+ K CD
+ K CD
+ K EF , (76)
[] [
] [ ] [ ] [ ] [ ]
wherefrom
0 ~ 0 0
0 20l
5
4k
0 0
0
0
0 5 +
k
~
k 0
0
2 14l
0
0
K =
.
4 0
0
2l 14l 2 0
0
0
0 0 30l 2
0
0
4l
0
0 0
0 32l 2
[]
{} = [ 0
0 0 0 0 0 ]T cos t ,
resulting the excitation force
5 0 20l 0
0
~
k
{Fe } = K {} =
cos t
0
4
2
4l 0 + 32l 0
[]
A14_316
(77)
(78)
(79)
VIII. Conclusions
In our work we presented, in a specific multibody type
form, the dynamics of a rigid body linked with linear
elastic springs by another one, with imposed motion. We
obtained the equations of motion and we studied the case
of small oscillations around the equilibrium position. For
the theory described we completely solved a practical
application. The presented method can be used in most
general situations such as vehicle suspensions, seismic
excitations, etc.
Acknowledgement
The second authors contribution to this paper is based on
the European Program Dezvoltarea colilor doctorale
prin acordarea de burse tinerilor doctoranzi cu frecven
POSDRU/88/1.5/S/52826.
References
[1] Shabana, A., A., Dynamics of Multibody Systems, 3rd edition,
Cambridge University Press, New York, 2005.
[2] Amironache, F., Fundamentals of multibody Dynamics. Theory and
Applications, Birkhuser, Boston, Basel, Berlin, 2006,
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