A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system

Authors

Flávio Luiz Cardoso-Ribeiro, Denis Matignon, Valérie Pommier-Budinger

Abstract

This work is motivated by an aeronautical issue: the fuel sloshing in the tank coupled with very flexible wings. Vibrations due to these coupled phenomena can lead to problems like reduced passenger comfort and maneuverability, and even unstable behavior. Port-Hamiltonian systems (pHs) provide a unified framework for the description of multi-domain, complex physical systems and a modular approach for the interconnection of subsystems. In this work, pHs models are proposed for the equations of liquid sloshing in moving containers and for the structural equations of beams with piezoelectric actuators. The interconnection ports are used to couple the sloshing dynamics in the moving tank to the motion the beam. This coupling leads to an infinite-dimensional model of the system in the pHs form. A finite-dimensional approximation is obtained by using a geometric pseudo-spectral method that preserves the pHs structure at the discrete level. Experimental tests on a structure made of a beam and a tank were carried out to validate the finite-dimensional model of liquid sloshing in moving containers. Finally, the pHs model proves useful to design an active control law for the reduction of sloshing phenomena.

Keywords

Port-Hamiltonian systems; Shallow-water sloshing; Moving container; Dynamic coupling; Interconnection of systems; Active control

Citation

• Journal: Journal of Fluids and Structures
• Year: 2017
• Volume: 69
• Issue:
• Pages: 402–427
• Publisher: Elsevier BV
• DOI: 10.1016/j.jfluidstructs.2016.12.007

BibTeX

@article{Cardoso_Ribeiro_2017,
  title={{A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system}},
  volume={69},
  ISSN={0889-9746},
  DOI={10.1016/j.jfluidstructs.2016.12.007},
  journal={Journal of Fluids and Structures},
  publisher={Elsevier BV},
  author={Cardoso-Ribeiro, Flávio Luiz and Matignon, Denis and Pommier-Budinger, Valérie},
  year={2017},
  pages={402--427}
}

Download the bib file

References

• Abzug, (2002)
• Alemi Ardakani, H. A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing. Journal of Fluids and Structures vol. 65 30–43 (2016) – 10.1016/j.jfluidstructs.2016.03.013
• ALEMI ARDAKANI, H. & BRIDGES, T. J. Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. European Journal of Applied Mathematics vol. 21 479–517 (2010) – 10.1017/s0956792510000197
• ARDAKANI, H. A. & BRIDGES, T. J. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions. Journal of Fluid Mechanics vol. 667 474–519 (2011) – 10.1017/s0022112010004477
• Alemi Ardakani, H. & Bridges, T. J. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions. European Journal of Mechanics - B/Fluids vol. 31 30–43 (2012) – 10.1016/j.euromechflu.2011.08.004
• Ascher, (1998)
• Banks, (1996)
• Breedveld, P. C. Port-based modeling of mechatronic systems. Mathematics and Computers in Simulation vol. 66 99–128 (2004) – 10.1016/j.matcom.2003.11.002
• Cardoso-Ribeiro, F. L., Pommier-Budinger, V., Schotte, J.-S. & Arzelier, D. Modeling of a coupled fluid-structure system excited by piezoelectric actuators. 2014 IEEE/ASME International Conference on Advanced Intelligent Mechatronics 216–221 (2014) doi:10.1109/aim.2014.6878081 – 10.1109/aim.2014.6878081
• Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation.. IFAC-PapersOnLine vol. 49 290–297 (2016) – 10.1016/j.ifacol.2016.07.456
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Duindam, (2009)
• Farhat, C., Chiu, E. K., Amsallem, D., Schotté, J.-S. & Ohayon, R. Modeling of Fuel Sloshing and its Physical Effects on Flutter. AIAA Journal vol. 51 2252–2265 (2013) – 10.2514/1.j052299
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Graham, E. W. & Rodriquez, A. M. The Characteristics of Fuel Motion Which Affect Airplane Dynamics. http://dx.doi.org/10.21236/ADA073847 (1951) doi:10.21236/ada073847 – 10.21236/ada073847
• Hamroun, B., Dimofte, A., Lefèvre, L. & Mendes, E. Control by Interconnection and Energy-Shaping Methods of Port Hamiltonian Models. Application to the Shallow Water Equations. European Journal of Control vol. 16 545–563 (2010) – 10.3166/ejc.16.545-563
• Hamroun, B., Lefevre, L. & Mendes, E. Port-based modelling and geometric reduction for open channel irrigation systems. 2007 46th IEEE Conference on Decision and Control 1578–1583 (2007) doi:10.1109/cdc.2007.4434237 – 10.1109/cdc.2007.4434237
• Hodges, (2011)
• Kunkel, Differential-algebraic equations: analysis and numerical solution. Eur. Math. Soc. (2006)
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Macchelli, A. & Melchiorri, C. Control by interconnection of mixed port Hamiltonian systems. IEEE Transactions on Automatic Control vol. 50 1839–1844 (2005) – 10.1109/tac.2005.858656
• Macchelli, A., Melchiorri, C. & Stramigioli, S. Port-Based Modeling and Simulation of Mechanical Systems With Rigid and Flexible Links. IEEE Transactions on Robotics vol. 25 1016–1029 (2009) – 10.1109/tro.2009.2026504
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Multi-variable port Hamiltonian model of piezoelectric material. 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566) vol. 1 897–902 – 10.1109/iros.2004.1389466
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
• Morris, K. & Ozer, A. O. Strong stabilization of piezoelectric beams with magnetic effects. 52nd IEEE Conference on Decision and Control 3014–3019 (2013) doi:10.1109/cdc.2013.6760341 – 10.1109/cdc.2013.6760341
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Olver, (1993)
• Paynter, (1961)
• Petit, N. & Rouchon, P. Dynamics and solutions to some control problems for water-tank systems. IEEE Transactions on Automatic Control vol. 47 594–609 (2002) – 10.1109/9.995037
• Prieur, C. & de Halleux, J. Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations. Systems & Control Letters vol. 52 167–178 (2004) – 10.1016/j.sysconle.2003.11.008
• Robu, B., Baudouin, L., Prieur, C. & Arzelier, D. Simultaneous $H_\infty$ Vibration Control of Fluid/Plate System via Reduced-Order Controller. IEEE Transactions on Control Systems Technology vol. 20 700–711 (2012) – 10.1109/tcst.2011.2144984
• Rodriguez, H., van der Schaft, A. J. & Ortega, R. On stabilization of nonlinear distributed parameter port-controlled Hamiltonian systems via energy shaping. Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228) vol. 1 131–136 – 10.1109/.2001.980086
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Schaft, A. J. Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems. Advanced Dynamics and Control of Structures and Machines 127–167 (2004) doi:10.1007/978-3-7091-2774-2_9 – 10.1007/978-3-7091-2774-2_9
• Surveys in Differential-Algebraic Equations I. (Springer Berlin Heidelberg, 2013). doi:10.1007/978-3-642-34928-7 – 10.1007/978-3-642-34928-7
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• Seslija, M., Scherpen, J. M. A. & van der Schaft, A. Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems. Automatica vol. 50 369–377 (2014) – 10.1016/j.automatica.2013.11.020
• Sidi, M. J. Spacecraft Dynamics and Control. (1997) doi:10.1017/cbo9780511815652 – 10.1017/cbo9780511815652
• Trefethen, Spectral Methods in MATLAB. Soc. Ind. Appl. Math. (2000)
• Voß, T. & Scherpen, J. M. A. Port-Hamiltonian Modeling of a Nonlinear Timoshenko Beam with Piezo Actuation. SIAM Journal on Control and Optimization vol. 52 493–519 (2014) – 10.1137/090774598
• Voss, T., Scherpen, J. M. A. & Onck, P. R. Modeling for control of an inflatable space reflector, the nonlinear 1-D case. 2008 47th IEEE Conference on Decision and Control 1777–1782 (2008) doi:10.1109/cdc.2008.4739177 – 10.1109/cdc.2008.4739177
• Voss, T., Scherpen, J. M. A. & Onck, P. R. Modeling for control of an inflatable space reflector, the nonlinear 1-D case. 2008 47th IEEE Conference on Decision and Control 1777–1782 (2008) doi:10.1109/cdc.2008.4739177 – 10.1109/cdc.2008.4739177