MODELING OF THE DAMPED OSCILLATIONS OF THE VISCOUS BEAMS STRUCTURES WITH SWIVEL JOINTS FOR HARMONIC MODE

Text-only Preview

IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
MODELING OF THE DAMPED OSCILLATIONS OF THE VISCOUS
BEAMS STRUCTURES WITH SWIVEL JOINTS FOR HARMONIC MODE
M. Dawoua Kaoutoing1, G.E Ntamack2, K. Mansouri3, T.Beda4, S. Charif D'Ouazzane5
1, 2, 4 Groupe de Mecanique et des Materiaux, GMM, Departement de Physique, Faculte des Sciences, Universite de
Ngaoundere, Cameroun, [email protected], [email protected], [email protected]
3 Laboratoire des Signaux, Systemes Distribues et Intelligence Artificielle (LSSDIA) ENSET de Mohammedia, Universite
Hassan II Mohammedia Casablanca, Maroc, [email protected]
5Laboratoire de Mecanique, Thermique et Materiaux, LMTM, Ecole Nationale de l'Industrie Minerale, ENIM, B.P. 753
Rabat, Maroc, [email protected]
Abstract
Mechanic studies realized on the two dimensional beams structures with swivel joints show that in statics, the vertical displacement is
continuous, but the rotation is discontinuous at the node where there is a swivel joint. Moreover, in dynamics, many authors do not
usually take into account the friction effect, modeling of these structures. We propose in this paper, a modeling of the beams structures
with swivel joints which integrates viscosity effects in dynamics. Hence this work we will present the formulation of motion equations
of such structures and the modal analysis method which is used to solve these equations.
Keywords: Beams, Swivel joint, Viscosity, Vibration, Modal Method.
---------------------------------------------------------------------***----------------------------------------------------------------------
1. INTRODUCTION
consider that one of the nodes is embedded in the beam and
the other is a steering joint [1]. As an example let us consider
Swivel joint is a spherical mechanical piece used as
the structure of the following figure with a swivel joint at node
articulation in the framework, which allows turning over in all
2.
directions [1]. The swivel joint does not transmit moment. Its
action is reduced to a force passing through its center [2, 3].
ur
The work carried out on the framework in beams with swivel
Y
joints indicates that, in statics there is continuity of the arrow,
p
-
but a discontinuity of rotation to the swivel node [4, 5.6]. In
dynamics, frictions are often neglected during the evaluation
of the degrees of freedom of the structures containing swivel
joints. In this work, we propose a technique of calculation
X
which helps to evaluate the vertical displacement and rotation,
taking into account the frictions in the calculation of the
degrees of freedom of the structures in beams with swivel
joint in dynamics. The evaluation of these degrees of freedom
L
L
is based on the setting in equation of these structures in
dynamics and given their solutions by the modal method.
1
3
2
This paper is organized in the following ways: in the first part,
we present the model which enables us to establish the motion
equations of such structures. This step is followed by the
presentation of the solution of these equations by the modal
Figure 1: Structure in beams with a swivel joint
method of analysis. The last part of this work is related to the
analysis and the discussion of results.
The node with swivel joint is modeled by two nodes, a node
kneecap and an embedded node as shown in figure 2:
2. THEORETICAL MODEL
When a swivel joint is inserted between two beams, the node
that makes connection between the two points, we can
__________________________________________________________________________________________
Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org
151

IJRET: International Journal of Research in Engineering and Technology
ISSN: 2319-1163
ur
It is a system of uncoupled equations where each i mode is put
in the following form [4]:
- P
f (t)
2
&& + 2
i
& + =
i
i
i
i
i
i
mi
(3)
2
4
Equation (3) has as the following solution:
1
3

H () ( )

1
( )1
1
T
jt
q

=
x x
F e
1
2
0

-
m
1
1


Figure 2: Modeling the swivel joints, node 2 with swivel

H ()

joint, node 4 with embedded node
(2) (2
2
)T
jt
q

=
x
x
F e
2

2
0

m

2
2
Study of the swivel joint's problem consists of determining the
q(t) =



value of displacements and rotations of node 2 and 4. In statics
H ( ) (3) (3
3
)T
jt
q =
x x
F e
several works are related to the evaluation of these degrees of
3
2
0

m

3
3
freedom [5, 6]. In these references vertical displacements of


nodes 2 and 4 are identical. The rotation of node with swivel

H () (4) (4
4
)T
jt
q =
x
x
F e
joint, node 2 is locally evaluated by solving the elementary
4
2
0

m



system of statics equations. But to determine the rotation of
4
4
(4)
the embedded node 4, it is initially necessary to make the
assembly of the global stiffness matrix of all the frameworks,
Where:
by taking into account the disturbance of the elements with
(i)
x
are the eigenvectors;
swivel joint nodes. In this work, we will propose a technique
i
to evaluate the same degrees of freedom, in dynamics and
i are the self throb;
introducing viscosity for the swivel joint nodes .The motion
equations of a framework in dynamics with external forces can
H i amplification dynamic factor of.
be formulated as following:
To calculate the rotation of node 2, we have to solve the
Mu&& + Cu& + Ku = f (t)
(1)
elementary system (1), by writing in the member of the
f t =
2 ( )
0
elementary force
. But to calculate the rotation of
In equation (1),
node 4, it is initially necessary to assemble all global matrices
M, is the mass matrix;
of the structure. In the continuation of this work, we present
C, is the damping matrix;
the solutions obtained in the evaluation of the rotation of the
K, the stiffness matrix;
node kneecap 2 on the case of figure 1 structure.
F(t), is the external disturbance.
3. SIMULATIONS AND ANALYSIS OF RESULTS
In general to solve the system of equations (1), inverse
methods are used, which consists of going from physical space
For simulations, we will consider identical beams of constant
to space modes, to find the solution in modal space and to
cross-section S, quadratic moment IZ, density and Young's
come back to physical space [3, 6, 7]. In the case of
modulus E. The selected beams have the length L, of type IPN
framework with viscosity, by considering structure of figure 1
of iron with the following mechanical characteristics:
example f, by using modal method analysis, we obtain a
system in the following form:
4
E=210000
MPa,
I =77,67 cm
S=7,57
cm2,
z
m
0 0 0
&
&
2
m 0
0
0
& k
0 0 0
f
()t
3
= 7850Kg / m [8].
1
1
1 11
1
1
1
1




0 m 0 0
&
&
0 2m
0
0
& 0 k 0 0


f
()t
2
2
2 2 2
2 2
2
+

+

=
2
The useful part, after taking into account the boundary
0
0 m 0& 0
0 2m
0 & 0
0 k 0


f()t conditions, the motions equation of the structure in figure 1,
3
3
3 3 3
3
3
3







3
subjected to harmonic excitations is in the form:
0 0 0 m
&
&
0
0
0 2m
& 0 0 0 k




f
()t

4
4
4 4 4
4
4
4
4
(2)
__________________________________________________________________________________________
Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org
152

IJRET: International Journal of Research in Engineering and Technology
ISSN: 2319-1163
S

L 280 0
v& C 0
v& EI6 0
v s
P in
t



2
z
f
2

+ 1

2 +

&
&
3


2

= 0

16800 8L

C
L
L &
2
0 2

2
0 82

2
0
(5)
The solution of this system is:
12



-
P
t
1
0
(
v )
t v()
t v ()
t
(
v )
t e ( c
Aos t
s
Bin )
t


= + = =
+
+
t

-
h
p
a
a
K (
f
1
-

) (+2) sin( )
2
2

(6)
With:
Figure 3: Graphs rotation and vertical displacement of node 2
of damping structure against time


f
=
1

When taking, v = 0.5m , v&
1ms-
=
, = 0.5rad and
0
0
0
&
-1
=1 rad s under harmonic forces of vibration amplitude
0
and:
P
=
0=10N and in the case of weak oscillations
(
0.5 ), we
(
-
obtain the solution represented in figure 4:
t )= Ae
t (t + )
Where:
=
2
1
f
-
The amplitude and the phase are:

2
&





2
0 +

0
A = 0 +








&
0 +
(
0 )


= arctan



0


Figure 4: Graph of rotation of the node 2 of viscous swivel
The self throbs of the structure are:
joint under harmonic force against time
EI
When we plotted the rotation curves as a function of time, in
=
EI
6
z
et = 40, 98
z
1
figure 3, for t varying from 0 to 40 s per step of 0.01 s, we
4
sL
2
4
sL
observed the attenuation of rotation, which characterizes the
presence of damping. Beyond 0.12 s, rotation stops probably
1
because of viscosity. To look further into this phenomenon, it
When taking v = 0.5m , v&
1ms-
=
, = 0.5rad , and
0
0
0
is necessary to make several tests while varying the damping
&
-1
=1 rad s we obtain in the case as of free vibrations the
ratio.
0
solution represented on figure 3:
On figure 4, t varies from 0 to 40 s and we did not observe the
disappearance of signal. This is due to the presence of the
external forces which are supposed to be harmonic. With this
assumption rotation is maintained during the vibration of the
__________________________________________________________________________________________
Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org
153

IJRET: International Journal of Research in Engineering and Technology
ISSN: 2319-1163
framework. But that in the case of figure 3 (damped free
oscillations) or the case of figure 4 (quenched forced
oscillations) a major analysis of these swivel joints requires
the comparison between the rotation of the swivel node and
the node embedded in order to better understand the influence
of these connections in the structures. This is the direction in
which we will pursue our research in this field.
CONCLUSIONS
The work aims at proposing a method of modeling beams
structures with swivel joints by taking into account the
frictions in dynamics. The Lagrange's method allowed us to
establish the motion equations of frameworks. The technical
modal analysis permitted to solve the system of motion
equations obtained and the cancellation of the transmission of
moments in swivel nodes to the embedded node enabled.
Graphs of the results give the opportunity to see the behavior
of the deadened and free forced structures. But the completion
of these swivel joints study requires investigation of several
comparison and damping ratios between the behavior of the
swivel nodes and the embedded nodes at the node where there
is a swivel joint.
REFERENCES
[1]
J.L. Batoz et G.Dhatt, Modelisation des structures par
elements finis. Hermes Volume 2: poutres et plaques
(1990).
[2]
L. R. Rakotomanana, Elements de dynamique des
solides et structures deformables, Universite de
Rennes 1, (2006).
[3]
M. J.Turner, R. W. Clough, H. C. Marlin, and L. J.
Topp, Stiffness and deflection analysis of complex
structures. J. Aero. Sci., vol.23, (1956), PP 805-823.
[4]
J.F. Imbert, Analyse des structures par elements finis.
Ecole nationale superieure de l'aeronautique et de
l'espace. 3eme edition, Cepadues edition 111, rue
Nicolas - Vauquelin 31100 Toulouse.
[5]
H. Bouabid, S. Charif d'Ouazzane, O. Fassi-Fehri et K.
Zine-dine A, Representation of swivel joints in
computing tridimensional structures 3eme Congres de
Mecanique, Tetouan, (1997).
[6]
G.E. Ntamack, M. Dawoua Kaoutoing, T. Beda, S.
Charif D'Ouazzane. Modeling of swivel joint in two
dimensional beams frameworks. Int. J. Sc. and Tech.
3, 1, (2013), 21-25.
[7]
R. J. Guyan, Reduction of stiffness and Mass
Matrices , AIAA, 3; 80, (1965).
[8]
Kerguignas,
La
methode
des
deplacements:
application a la resolution des structures planes a noeuds
rigides, EMI-RABAT, (1982).
[9]
A. Bennani, V. Blanchot, G. Lhermet, M. Massenzio, S.
Ronel, Dimensionnement des structures, (2007).
__________________________________________________________________________________________
Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org
154