Modeling of the damped oscillations of the viscous beams structures with swivel joints for harmonic mode

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org 151
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 Mécanique et des Matériaux, GMM, Département de Physique, Faculté des Sciences, Université de
Ngaounderé, Cameroun, dawouakaoutoingmaxime@yahoo.fr, guyedgar@yahoo.fr, tbeda@yahoo.fr
3
Laboratoire des Signaux, Systèmes Distribués et Intelligence Artificielle (LSSDIA) ENSET de Mohammedia, Université
Hassan II Mohammédia Casablanca, Maroc, khmansouri@hotmail.fr
5
Laboratoire de Mécanique, Thermique et Matériaux, LMTM, Ecole Nationale de l’Industrie Minérale, ENIM, B.P. 753
Rabat, Maroc, charif.enim@hotmail.com
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
Swivel joint is a spherical mechanical piece used as
articulation in the framework, which allows turning over in all
directions [1]. The swivel joint does not transmit moment. Its
action is reduced to a force passing through its center [2, 3].
The work carried out on the framework in beams with swivel
joints indicates that, in statics there is continuity of the arrow,
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
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
is based on the setting in equation of these structures in
dynamics and given their solutions by the modal method.
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
method of analysis. The last part of this work is related to the
analysis and the discussion of results.
2. THEORETICAL MODEL
When a swivel joint is inserted between two beams, the node
that makes connection between the two points, we can
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
the structure of the following figure with a swivel joint at node
2.
Figure 1: Structure in beams with a swivel joint
The node with swivel joint is modeled by two nodes, a node
kneecap and an embedded node as shown in figure 2:
L L
321
X
Y
p


IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Figure 2: Modeling the swivel joints, node 2 with swivel
joint, node 4 with embedded node
Study of the swivel joint’s problem consists of determining the
value of displacements and rotations of node 2 and 4. In statics
several works are related to the evaluation of these degrees of
freedom [5, 6]. In these references vertical displacements of
nodes 2 and 4 are identical. The rotation of node with swivel
joint, node 2 is locally evaluated by solving the elementary
system of statics equations. But to determine the rotation of
the embedded node 4, it is initially necessary to make the
assembly of the global stiffness matrix of all the frameworks,
by taking into account the disturbance of the elements with
swivel joint nodes. In this work, we will propose a technique
to evaluate the same degrees of freedom, in dynamics and
introducing viscosity for the swivel joint nodes .The motion
equations of a framework in dynamics with external forces can
be formulated as following:
( )Mu Cu Ku f t    (1)
In equation (1),
M, is the mass matrix;
C, is the damping matrix;
K, the stiffness matrix;
F(t), is the external disturbance.
In general to solve the system of equations (1), inverse
methods are used, which consists of going from physical space
to space modes, to find the solution in modal space and to
come back to physical space [3, 6, 7]. In the case of
framework with viscosity, by considering structure of figure 1
example f, by using modal method analysis, we obtain a
system in the following form:
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
0 0 0 2 0 0 0 0 0 0 (
0 0 0 0 2 0 0 0 0 0
0 0 0 0 0 2 0 0 0 0
0 0 0 0 0 0 2 0 0 0
m m k f
m m k
m m k
m m k
   
   
   
   
        
        
                 
          
                  
 
 
 
 
2
3
4
)
( )
( )
( )
t
f t
f t
f t
 
 
 
 
 
  
(2)
It is a system of uncoupled equations where each i mode is put
in the following form [4]:
2 ( )
2 i
i i i i i i
i
f t
m
      
(3)
Equation (3) has as the following solution:
   
   
   
   
1 11
1 02
1 1
2 22
2 02
2 2
3 33
3 02
3 3
4 44
4 02
4 4
( )
( )
( )
( )
( )
T j t
T j t
T j t
T j t
H
q x x F e
m
H
q x x F e
m
q t
H
q x x F e
m
H
q x x F e
m












 
 
 
 
 
 
  
 
 
 
 
   (4)
Where:
 i
x are the eigenvectors;
i
i are the self throb;
iH amplification dynamic factor of.
To calculate the rotation of node 2, we have to solve the
elementary system (1), by writing in the member of the
elementary force   02 tf . But to calculate the rotation of
node 4, it is initially necessary to assemble all global matrices
of the structure. In the continuation of this work, we present
the solutions obtained in the evaluation of the rotation of the
node kneecap 2 on the case of figure 1 structure.
3. SIMULATIONS AND ANALYSIS OF RESULTS
For simulations, we will consider identical beams of constant
cross-section S, quadratic moment IZ, density  and Young’s
modulus E. The selected beams have the length L, of type IPN
of iron with the following mechanical characteristics:
E=210000 MPa,
4
=77,67 cmzI S=7,57 cm2
,
3
/7850 mKg [8].
The useful part, after taking into account the boundary
conditions, the motions equation of the structure in figure 1,
subjected to harmonic excitations is in the form:
3
p

2 41
P


__________________________________________________________________________________________













































0
sin
80
06
0
0
80
0280
1680
0
2
2
23
2
2
2
1
2
2
2
tPv
LL
EIv
C
Cv
L
SL fz







(5)
The solution of this system is:
   
1
2
0
22
1
( ) ( ) ( ) ( ) ( cos sin ) sin( )
1 2
t
h p a a f
P
v t v t v t v t e A t B t t
K

   
 

 
       
   
(6)
With:



f

and:
 t =  

tAe t
Where:
 =
2
1  f
The amplitude and the phase are:
 
















 








 

0
00
2
002
0
arctan








A
The self throbs of the structure are:
1 4
6 zEI
sL


 et 2 4
40,98 zEI
sL



When taking 0 0.5v m ,
1
0 1v ms
 , 0 0.5rad  , and
-1
0 1 rad s  we obtain in the case as of free vibrations the
solution represented on figure 3:
Figure 3: Graphs rotation and vertical displacement of node 2
of damping structure against time
When taking, 0 0.5v m ,
1
0 1v ms
 , 0 0.5rad  and
-1
0 1 rad s  under harmonic forces of vibration amplitude
P0=10N and in the case of weak oscillations ( 0.5  ), we
obtain the solution represented in figure 4:
Figure 4: Graph of rotation of the node 2 of viscous swivel
joint under harmonic force against time
When we plotted the rotation curves as a function of time, in
figure 3, for t varying from 0 to 40 s per step of 0.01 s, we
observed the attenuation of rotation, which characterizes the
presence of damping. Beyond 0.12 s, rotation stops probably
because of viscosity. To look further into this phenomenon, it
is necessary to make several tests while varying the damping
ratio.
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

__________________________________________________________________________________________
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, «Modélisation des structures par
éléments finis. Hermes Volume 2: poutres et plaques»
(1990).
[2] L. R. Rakotomanana, «Eléments de dynamique des
solides et structures déformables», Université 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 éléments finis.
Ecole nationale supérieure de l’aéronautique et de
l’espace». 3ème édition, Cepaduès édition 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» 3ème
Congrès de
Mécanique, Tétouan, (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 méthode des déplacements:
application à la résolution des structures planes à nœuds
rigides», EMI-RABAT, (1982).
[9] A. Bennani, V. Blanchot, G. Lhermet, M. Massenzio, S.
Ronel, «Dimensionnement des structures», (2007).

Modeling of the damped oscillations of the viscous beams structures with swivel joints for harmonic mode

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Modeling of the damped oscillations of the viscous beams structures with swivel joints for harmonic mode