Aeroelasticity and Structural Dynamics
Hydroelasticity and sloshing of tanks filled with liquid
Influence of incompressible internal fluids on the vibrations of structures
As a general rule, the models for analyzing the vibratory and aeroelastic behavior of aeronautic vehicles are based on a modal representation of the structure. The purpose of the research that we have carried out in the last few years has been to improve that modal representation by taking into account the influence of internal liquids (fuels) in the models. Whereas the traditional industrial approach is to represent the internal liquids as added masses distributed on the walls of the tanks, here we interested in a method that can take into account (at the first order) the liquid's internal dynamics and the effects of sloshing. In 1992, Morand & Ohayon proposed a hydroelastic model, called "with gravity", the specific feature of which was that it did not ignore the effects of gravity, unlike a traditional hydroelastic models, and which could represent, using a linear model, all of the couplings between the movements of the fluid's free surface and the movements of the whole or the deformations of the tank.
Firstly, we validated the different aspects of this model:
- In the first place, using the experimental measurements recorded by X.-J. Chai in 1996 on a tank filled with water, parallelepipedal in shape, considered to be fixed and rigid, at the bottom of which a flexible plate was fixed. The frequencies of the first antisymmetric coupled modes obtained by the hydroelastic fluid model with gravity are very close to the frequencies measured (figure 1).
Figure 1a: Display of the 3 first antisymmetric modes of Chai's experimental tank.
Calculated natural frequency 1.58 Hz (exp. measurement: 1.56 Hz)
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Figure 1b: calculated natural frequency 2.42 Hz (exp. measurement: 2.39 Hz)
Figure 1c: calculated natural frequency 2.98 Hz (exp. measurement: 2.95 Hz)
Figure 2: Parameters of the Kreis and Klein benchmark
Figure 2b: calculated natural frequency: 0.91 Hz (Kreis & Klein value: 0.99 Hz)
- The lack of experimental results available in the literature led us to carry out a campaign of experiments. A first series of experiments was done on a polyurethane tank in the shape of a parallelepiped partially filled with water (figure 3). The system's very low natural frequencies revealed sloshing modes that correspond to the theoretical values and the modes calculated by finite elements (table 1).
Figure 3: experimental tank equipped with capacitive probes and a surface exciter
 |
Analytics
(Hz) |
Finite
elements (Hz) |
Exp
measures (Hz) |
Max
error
(%) |
| 1,15 |
1,126 |
1,156 |
3 |
| 1,693 |
1,683 |
1,697 |
1 |
| 2,077 |
2,073 |
2,1 |
1 |
Tab. 1: Comparison between theoritical, calculated and measured natural sloshings frequencies |
Secondly, our concern was to adapt the fluid-structure formulation presented with a view to its use in an industrial environment. In order to minimize the effort of implementation in a standard structural finite element code and the additional computing time due to the introduction of internal fluids, we developed reduced models from the complete formulation. To do this, we considered two approaches:
- The first dates from the 50s and consists of representing the fluid's dynamics by a set of spring-mass or pendulum type equivalent mechanical systems. As the traditional method for determining the configuration of these systems is based on the knowledge of an analytical solution of the liquid's sloshing equation, only tanks with basic geometries can be dealt with. By using our finite elements modeling of the liquid, we have extended this method to three dimensional tanks of any shape.
- The second aims to construct a reduced model by projecting complete fluid-structure equations on an appropriate modal base (for example, the base of the fluid's sloshing modes).
These two approaches were validated and compared on different test cases (Table 2).
|
Natural freq.
(Hz) |
Complete
model
(hydro. with g) |
Weights-springs |
Modal
reduction |
| 1st bending |
0,152 |
0,135 |
0,152 |
| 1st torsion |
0,594
1,47
1,72
... |
0,546
1,43
1,71
... |
0,594
1,47
1,72
... |
| 2nd bending |
0,689
1,54
1,80
... |
0,668
1,52
1,77
... |
0,689
1,54
1,80
... |
| 3rd bending |
3,45 |
3,45 |
3,45 |
| 1st tank yaw |
5,99 |
6,02 |
5,99 |
|
Comparison between the weights-springs model and the reduced model
by modal projection and the complete model
(at the bottom: illustration of the 1st mode of 2nd bending)
The representation of the internal fluid by equivalent mechanical systems was applied to a compartmentalized tank of an aircraft tail plane (figures 4 and 5) in the context of a collaboration with Airbus France. A significant impact of the sloshing of the fuel at some of the aircraft's natural frequencies was found.
Figure 4: tail plane tank of a four engine aircraft
Figure 5: division into compartments and filling of the tank
As regards applications of this work to the field of aeroelasticity, a feasibility study was carried out with Dassault Aviation on a simplified model of a military aircraft wing fitted with external tanks (Fig. 6 and 7). The influence of the sloshing in the tanks on the aircraft's stability in flight was observed in a qualitative way.
Figure 6: modeling of the sloshing of fuel in an auxiliary tank
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Figure 7: example of a hydroelastic mode
