It is proposed to investigate in this paper the damped vibrations of an incompressible liquid contained in a deformable tank. A linearized formulation describing the small movements of the system is presented. At first, a diagonal damping is introduced in the reduced equations of the hydroelastic sloshing problem. We obtain a nonclassically damped coupled system with a damping matrix that is not symmetric. Then, by projecting the system onto its complex modes, the frequency and time responses for different type of loads are built. A numerical application is illustrated on a test case.

References

References
1.
Chiu
,
E.
,
Schotté
,
J.-S.
,
Farhat
,
C.
, and
Ohayon
,
R.
, 2011, “
Flutter Analysis of a Wing-Store Configuration With Slosh Effects
,”
Proceedings of the 30th International Forum of Aeroelasticity and Structural Dynamics
,
Paris
,
France
.
2.
Abramson
,
H. N.
, 1966, “The Dynamic Behaviour of Liquids in Moving Containers,” NASA Report No. SP-106.
3.
Moiseyev
,
N. N.
, and
Rumyantsev
,
V. V.
, 1968,
Dynamic Stability of Bodies Containing Fluid
, (
Springer-Verlag Applied Physics and Engineering
, Vol.
6
),
Springer-Verlag, Berlin
.
4.
Smith
,
C. C.
, 1948, “The Effect of Fuel Sloshing on the Lateral Stability of a Free-Flying Airplane Model,” Langley Memorial Aeronautical Laboratory, Technical Report No. NACA RL L8C16.
5.
Ibrahim
,
R. A.
, 2005,
Liquid Sloshing Dynamics: Theory and Applications
,
Cambridge University Press
,
Cambridge, England
.
6.
Faltinsen
,
O. M.
, and
Timokha
,
A. N.
, 2009,
Sloshing
,
Cambridge University Press
,
Cambridge, England
.
7.
El-Kamali
,
M.
, 2010, “Ballottement des Liquides Avec Tension Superficielle: Etudes Dynamique et Statique,” Ph.D. thesis, Conservatoire National des Arts et Métiers, Paris, France.
8.
El-Kamali
,
M.
,
Schotté
,
J.-S.
, and
Ohayon
,
R.
, 2010, “
Three Dimensional Modal Analysis of Sloshing Under Surface Tension
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
87
105
.
9.
Schotté
,
J.-S.
, 2001, “Influence de la Gravité sur les Vibrations Linéaires d’une Structure Élastique Contenant un Liquide Incompressible,” Ph.D. thesis, Conservatoire National des Arts et Métiers, Paris, France.
10.
Morand
,
H. J.-P.
, and
Ohayon
,
R.
, 1995,
Fluid-Structure Interaction: Applied Numerical Methods
,
Wiley
,
New York
.
11.
Schotté
,
J.-S.
, and
Ohayon
,
R.
, 2009, “
Various Modelling Levels to Represent Internal Liquids in the Vibratory Analysis of Complex Structures
,”
Comp. Methods Appl. Mech. Eng.
,
198
, pp.
1913
1925
.
12.
Schotté
,
J.-S.
, and
Ohayon
,
R.
, 2005, “
Incompressible Hydroelastic Vibrations: Finite Element Modelling of the Elastogravity Operator
,”
Comput. Struct.
,
83
, pp.
209
219
.
13.
Caughey
,
T.
, 1960, “
Classical Normal Modes in Damped Linear Systems
,”
J. Appl. Mech.
,
27
, pp.
269
271
.
14.
Henderson
,
D. M.
, and
Miles
,
J. W.
, 1994, “
Surface-Wave Damping in a Circular Cylinder With a Fixed Contact Line
,”
J. Fluid Mech.
,
275
, pp.
285
299
.
15.
Johnson
,
C. D.
,
Kienholz
,
D. A.
, and
Rogers
,
L. C.
, 2003, “
Finite Element Prediction of Damping in Beams With Constrained Viscoelastic Layers
,”
Shock Vib. Bull.
,
51
, pp.
71
82
.
16.
Tsai
,
M. H.
, and
Chang
,
K. C.
, 2000, “
A Study on Modal Strain Energy Method for Viscoelastically Damped Structures
,”
Proceedings of the Twelfth World Conference on Earthquake Engineering
,
Auckland, New Zealand
.
17.
Anquez
,
L.
, 1974, “
Etude de l’Amortissement des Vibrations Dans un Liquide Peu Visqueux
,”
Acta Astronaut.
,
1
, pp.
172
184
.
18.
Girault
,
V.
, and
Raviart
,
P.-A.
, 1981,
Finite Element Approximation of the Navier-Stokes Equations
,
Springer-Verlag
,
Berlin
.
19.
Wang
,
W.
,
Li
,
J.
, and
Wang
,
T.
, 2008, “
Modal Analysis of Liquid Sloshing With Different Contact Line Boundary Conditions Using FEM
,”
J. Sound Vib.
317
, pp.
739
759
.
20.
Kidambi
,
R.
, 2009, “
Capillary Damping of Inviscid Surface Waves in a Circular Cylinder
,”
J. Fluid Mech.
,
627
, pp.
323
340
.
21.
Martel
,
C.
,
Nicolás
,
J. A.
, and
Vega
,
J. M.
, 1998, “
Surface-Wave Damping in a Brimful Circular Cylinder
,”
J. Fluid Mech.
,
360
, pp.
213
228
.
22.
Miles
,
J. W.
, and
Henderson
,
D. M.
, 1998, “
A Note on Interior vs. Boundary-Layer Damping of Surface Waves in a Circular Cylinder
,”
J. Fluid Mech.
,
364
, pp.
319
323
.
23.
Case
,
K. M.
, and
Parkinson
,
W. C.
, 1957, “
Damping of Surface Waves in an Incompressible Liquid
,”
J. Fluid Mech.
,
2
, pp.
172
184
.
24.
Tisseur
,
F.
, and
Meerbergen
,
K.
, 2001, “
The Quadratic Eigenvalue Problem
,”
SIAM Rev.
,
43
, pp.
235
286
.
25.
Saad
,
Y.
, 1992,
Numerical Methods for Large Eigenvalue Problems
,
Halsted
,
New York
.
26.
Gohberg
,
I.
,
Lancaster
,
P.
, and
Rodman
,
L.
, 1982,
Matrix Polynomials
,
Academic
,
New York
.
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