Jan Cristian Grigore, Nicolae Pandrea
One considers that the rigid with constraints acted by the external for ces suported by the elatic bars embedded at the base. Under the in fluence of the external forces appear the connection forces and the displacements in the kinematic pairs. One deduces the mathematical model for the computation of the reactions and the displacements and one makes a numerical applications.
: rigid, constraints, elastic, joint bars
In the paper
[a], using the plückerian co ordinates and the relative displace ments method [8], it elaborated the computation model of the efforts from elastic bars spatially arranged, bars that suspended a rigid with constraints without clear ances.In this paper it will customize the general model for the plane elastic bars system and it will be made an application for such a system.
! " #
One considers that the rigid body relative to the axis system O0XYZ(fig. 1), suspended by the elastic bars
A
iB
i embedded in thepointsA ,
iB
i and one consi ders that the rigid has constraints without clearances.An example of such constraint is made (fig. 1), when the point
D
of the rigid remains located on the fixed rigid surface equationg
(
X
,
Y
,
Y
)
=
0
ANALELE UNIVERSITĂłII
“EFTIMIE MURGU” RE IłA
$ Using the general notations [1]
[ ]
K
i the stiffness matrix of the barA
iB
i relative to the reference systemOXYZ
[ ]
K
the general stiffness matrix of the elastic bars system[ ]
∑
[ ]
=
=
ni i
k
k
1
, (1)
[ ]
S
the possible displacements matrix from constraints[ ]
ξ
the column matrix of the scalar parameters of the displacements from con straints[ ]
U
the restrictions matrix from constraints[ ]
η
the column matrix of the scalar parameters of the restrictions from con straints[ ]
F
the column matrix of the components of the external forces torsor which act the rigidone obtains [1] the matrix equation of the elastic equilibrium
{ }
F
+
[ ]
U
{ }
η
=
[ ][ ]
K
S
{ }
ξ
(2) and taking into account the conditions[ ] [ ] [ ] [ ] [ ] [ ]
U
TS
=
0
;
S
TU
=
0
(3) from (2) one deduces the matrices of the scalar parameters{ }
ξ
=
[
[ ] [ ] [ ]
S
T⋅
K
⋅
S
]
−1⋅
[ ]
S
T⋅
{ }
F
(4)[ ] [ ] [ ]
[
U
T⋅
K
−1U
]
−1[ ] [ ]
U
Tk
−1{ }
F
−
=
η
(5){ }
F
i=
[ ] [ ][ ][ ]
T
i −1k
iS
ξ
,
i
=
1
,
2
,...
n
(6)% &
% '
In the case of the plan bars system (fig. 2)
$ !
using the notations
i i i
i
y
A
z
A
,
the inertial central principal axes of the normal section inA
i by the barA
iB
ii i
x
A
, the axis of the centres of gravity of the normal sections, sections considered identicali
l
, the lenght of the barA
iB
ii
A
~
, the area of the normal sectioni
I
, the inertial moment of the normal section relative to the axisA
iz
ii
E
, the Young modulusi i
k
k
1,
2 , the stiffness defined by the relations3 2
1
;
i i i i i
i i i
l
I
E
k
l
A
E
k
=
=
(7)[ ]
k
i , the local stiffness matrix of the barA
iB
i[ ]
=
2 2 2
2 2 1
4
6
0
6
12
0
0
0
i i i i
i i i i i
l
k
l
k
l
k
k
k
k
(8)i
(
X
i,
Y
i)
, the co ordinates of the pointA
i[ ]
T
i[ ]
T
i~
,
, the position matrices[ ]
[ ]
α
α
α
−
α
=
α
+
α
α
−
α
α
α
α
−
α
=
1
0
0
cos
sin
sin
cos
~
,
1
sin
cos
cos
sin
0
cos
sin
0
sin
cos
i i i
i i i
i
i i i i i i i i
i i
i i
i
X
Y
T
Y
X
Y
X
T
(9)
[ ]
K
i , the stiffness matrix of the barA
iB
i in theOXY
system[ ] [ ][ ]
[ ]
~
−1=
i i ii
T
k
T
K
(10)where
[ ]
[ ] [ ]
[ ]
T i i T ii
T
T
T
T
~
−1=
;
−1=
~
(11)one obtains the stiffness matrix of the elastic bars system
[ ]
∑
[ ]
=
=
ni i
K
K
1
(12)
% !
For the restrictions matrix computation one takes into account the relation
[ ] [ ][ ]
U
=
T
u
(13)where:
[ ]
u
, is the restrictions matrix in the loacal reference system[ ]
U
, is the restrictions matrix in the general reference systemOXY
[ ]
T
, the position forces matrix of the local system relative to general refe rence system.If the local reference system has the axes parallel to the axes
OX ,
OY
and the origin has the co ordinates(
X ,
Y
)
, the position matrix is[ ]
−
=
1
0
1
0
0
0
1
X
Y
For the bearing constraint of the rigid on the fixed rigid equation curve
(
X
,
Y
)
=
0
g
in pointD
(
X
D,
Y
D)
, denoting withg ,
Xg
y the partial derivatives calculated in the pointD
, there are obtained the expressions{ }
;
{ }
;
0
* ** * *
*
−
=
=
X D Y D
Y X Y
X
g
Y
g
X
g
g
U
g
g
u
(15)where
2 2 * 2 2 *
;
X X
Y Y
Y X
X X
g
g
g
g
g
g
g
g
+
=
+
=
(16)and for the fixed point constraint (joint) in
D
(
X
D,
Y
D)
, there are obtained the expressions[ ]
[ ]
−
=
=
D
D
X
Y
U
u
0
1
0
1
;
0
0
1
0
0
1
(17)
% % '
For the matrix (small) displacements computation one takes into account the relation
[ ]
S
=
[ ]
T
~
[ ]
s
(18) where:[ ]
s
, is the displacements matrix in the local reference system[ ]
S
, is the displacements matrix in the general reference systemOXY
[ ]
T
~
, is the position matrix for displacements of the local system relative to the systemOXY
If the local reference system has its axes parallel to
OX ,
OY
, the origin has the co ordinates(
X ,
Y
)
the position matrix is[ ]
−
=
1
0
0
1
0
0
1
~
X
Y
For the bearing constraint of the rigid on the fixed rigid curve equation
(
X
,
Y
)
=
0
g
in pointD
(
X
D,
Y
D)
,there are obtained the expressions{ }
{ }
;
1
0
;
1
0
0
0
* * *
*
−
−
=
−
=
X DD Y X
Y
X
g
Y
g
S
g
g
s
(20)and for the fixed point constraint (joint) in
D
(
X
D,
Y
D)
there are obtained the ex pressions{ }
{ }
−
=
=
1
;
1
0
0
D D
X
Y
S
s
(21)( )
One considers a rigid straight bar
A
1A
2 from fig. 3, of lenght2
b
propped in1
A
on the rigid equation straight lineX
cos
β
−
Y
sin
β
+
b
sin
β
=
0
The rigid bar
A
1A
2 embedded bars inB
i,
i
=
1
,
2
at the base and in2
,
1
,
i
=
A
i at the rigid bar.If the force
F
acts upon the rigid bar on the axis directionOY
let us de terminate the displacementsξ
1,
ξ
2 the reactionN
from the pointA
1 and the efforts from bars.$ % $ (
Choosing the reference system from fig.3 it results the equalities
β
−
=
β
=
=
α
=
α
=
=
=
=
=
=
sin
;
cos
;
0
;
;
* *
2 1 3 2 22 21 1
12 11
Y
X
g
g
l
EI
k
k
k
l
EA
k
k
k
[ ]
( )
( )
−
−
=
1
0
0
0
1
0
1
0
1
4
6
0
6
12
0
0
0
1
0
1
0
1
0
0
0
1
2 2 2 2 1b
kl
l
k
l
k
k
k
b
k
i i i (23)[ ] [ ] [ ]
+
⋅
=
+
=
2 1 2 2 2 2 1 2 14
6
0
6
12
0
0
0
2
b
k
kl
l
k
l
k
k
k
K
K
K
(24){ }
[ ]
β
β
=
β
−
β
−
β
=
1
0
0
cos
sin
;
cos
sin
cos
b
S
y
U
(25)(
β
+
β
−
β
β
)
+
β
+
β
=
cos
sin
3
cos
6
sin
4
cos
12
sin
2 2 2 2 2 1 2 2 2 2 2 2 2 1bl
b
l
k
k
l
k
b
k
D
(26)(
)
β
+
β
−
+
β
=
ξ
ξ
cos
6
sin
2
4
2
cos
2 1 2 1 2 2 2 1l
k
b
k
b
k
l
k
D
F
(27)(
)
[
+
β
−
β
]
=
η
=
4
2sin
6
2cos
1 2 2
1
k
l
k
b
k
lb
D
Fk
N
(28){ }
(
)
(
)
{ }
(
)
(
)
ξ
+
+
β
ξ
ξ
+
β
ξ
ξ
−
β
ξ
=
ξ
+
+
β
ξ
ξ
+
β
ξ
β
ξ
+
β
ξ
=
2 2 1 2 2 1 2 2 2 1 2 2 1 1 2 2 2 1 2 2 1 2 2 2 1 2 2 1 1 14
cos
6
6
cos
12
sin
;
4
cos
6
6
cos
12
cos
sin
b
k
l
k
l
k
l
k
k
b
k
F
b
k
l
k
l
k
l
k
k
k
F
(29) 2 1 2 2 2 04
6
b
k
l
k
lb
k
tg
+
=
β
(30)If
0
<
β
<
β
0 the reactionN
is negative, ifβ
=
β
0 the reactionN
is null and ifβ
>
β
0the reactionN
is positive.In a numerical application if one considers that the elastic bars with
m
8
.
0
;
m
N
10
2
⋅
11⋅
2=
=
−l
E
, have a circular sectiond
=
16
mm
, thatβ
=
10
,N
5000
=
F
and the rigid bar isb
=
0
.
4
m
, there are obtained the results:rad
09
.
7
rad,
27
.
3
N,
28802
,
rad
10
5
.
0
4 1 20
=
⋅
=
ξ
=
ξ
=
{ }
{ }
−
⋅
⋅
=
⋅
⋅
=
rad
83
.
2
0
N
26
.
2
10
5
;
rad
836
.
2
0
N
4
.
3
10
5
3 2 31
F
F
This work was supported by CNCSIS – UEFISCSU, project number PN II RU 683/2010
[1] Grigore, J. C., Pandrea, N., Calculul liniar elastic al barelor care sus pendă un solid rigid cu constrângeri fără jocuri (în curs de publicare), The 3rd International Conference on International Conference″Computational Mechanics and Virtual Engineering″COMEC 2010, 29 – 30 OCTOBER 2009, Brasov, Romania
[2] Grigore, J. C., Pandrea, N., Calculul liniar elastic al barelor care sus pendă un solid rigid cu constrângeri cu jocuri (în curs de publicare), The 3rd International Conference on International Conference″Computational Mechanics and Virtual Engineering″COMEC 2010, 29 – 30 OCTOBER 2009, Brasov, Romania
[3] Buzdugan Gh.,RezistenŃa materialelor, Editura Tehnică BucureGti, 1980 [4] Courbon , J.,Calculul des structures, Dunora, Paris, 1972
[5] Dimentberg, F.,Teoriza vintov i ee prilozheniya, Nauka, Moskva, 1978 [6] Grigore, J. C., Dinamica mecanismelor cu jocuri, Editura UniversităŃii
din PiteGti, 2010
[7] Pandrea, N.,Metoda deplasărilor relative în calculul sistemelor elastice. Buletinul Institutului de ÎnvăŃământ Superior, PiteGti, 1978
[8] Pandrea, N., Elemente de mecanica solidelor în coordonate plück eriene, Editura Academiei Române, BucureGti 2000
[9] Pandrea, M., Calculul liniar elastic în coordinate plückeriene, Editura UniversităŃii, PiteGti, 2006
[10] Voinea, R., Pandrea, N., ContribuŃii la o teorie matematică generală a cuplelor cinematice, IFTOMM Inz. Symp.,Vol B, Bucharest 1973
Addresses:
• Lecturer PhD.Eng. Grigore Jan Cristian, University of PiteGti, Depart ment of Applied Mechanics, str. Târgul din Vale, nr. 1, 110040, PiteGti
jan_grigore@yahoo.com
