September  2013, 33(9): 4017-4040. doi: 10.3934/dcds.2013.33.4017

The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

1. 

Department de Matemática Aplicada I, Universitat Politecnica de Catalunya, Barcelona, E-08028, Spain

2. 

Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, Rio Hondo 1, Mexico City, 01000, Mexico

3. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  January 2012 Revised  December 2012 Published  March 2013

We consider the motion of a planar rigid body in a potential two-dimensional flow with a circulation and subject to a certain nonholonomic constraint. This model can be related to the design of underwater vehicles.
    The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.
Citation: Yuri N. Fedorov, Luis C. García-Naranjo, Joris Vankerschaver. The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017
References:
[1]

P. Appell, "Traite de Mechanique Rationelle," Vol. 2, Gauthier-Villars, Paris, 1953. Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616.  Google Scholar

[3]

A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. and Chaot. Dyn., 8 (2003), 449-462.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaev, On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 16 (2006), 013118, 7 pp.  Google Scholar

[5]

T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlin. Sci., 21 (2011), 325-385.  Google Scholar

[6]

T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws, IEEE Trans. Automat. Control, 53 (2008), 339-349.  Google Scholar

[7]

S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it, in "The Selected Works on Wing Theory of Sergei A. Chaplygin," Translated from the 1933 Russian original by M. A. Garbell, (1956), 42-72. Google Scholar

[8]

S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem, (in Russian) Math. Sbornik, 28 (1911), 303-314; English translation in Reg. Chaot. Dyn., 13 (2008), 369-376.  Google Scholar

[9]

F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 051, 12 pp.  Google Scholar

[10]

M. J. Field, Equivariant dynamical systems, Trans. Am. Math. Soc., 259 (1980), 185-205.  Google Scholar

[11]

Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface, (Russian) Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 54 (1987), 67-75; Eng. transl. in Mechanics of Solids, 22 (1988), 65-73. Google Scholar

[12]

Yu. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259.  Google Scholar

[13]

Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A, 43 (2010), 434013, 18 pp.  Google Scholar

[14]

L. C. García-Naranjo and J. Vankerschaver, Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation,, preprint, ().   Google Scholar

[15]

I. S. Gradshteyn and I. M. Ryzhik, "Tables of Integrals, Series, and Products," $7^{th}$ edition, Elsevier/Academic Press, Amsterdam, 2007.  Google Scholar

[16]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515.  Google Scholar

[17]

E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid J. Nonlinear Sci., 15 (2005), 255-289.  Google Scholar

[18]

S. Kelly and R. Hukkeri, Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift, IEEE Transactions on Robotics, 22 (2006), 1254-1264. Google Scholar

[19]

G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik," Teubner, Leipzig, 1877. Google Scholar

[20]

H. Lamb, "Hydrodynamics," $6^{th}$ edition, Dover, NY, 1932. Google Scholar

[21]

N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica J. IFAC, 33 (1997), 331-346.  Google Scholar

[22]

T. Levi-Civita and U. Amaldi, "Lezioni di Meccanica Razionale. Vol. 2. Dinamica dei Sistemi con un Numero Finito di Gradi di Liberta," (Italian) Nuova ed. N. Zanichelli, Bologna, 1951, 1952. Google Scholar

[23]

L. Milne-Thomson, "Theoretical Hydrodynamics," $5^{th}$ edition, revised and enlarged edition, MacMillan and Co. Ltd., London, 1968. Google Scholar

[24]

P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001.  Google Scholar

[25]

S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid, Reg. and Chaot. Dyn., 6 (2001), 33-38.  Google Scholar

[26]

R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints, in "Proceedings of the First International Symposium on Impact and Friction of Solids, Structures, and Intelligent Machines" (ed. A. Guran), World Scientific, Singapore, (2000), 113-116. Google Scholar

[27]

J. Roenby and H. Aref, Chaos in body-vortex interactions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871-1891.  Google Scholar

[28]

P. G. Saffman, "Vortex Dynamics," Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. Google Scholar

[29]

B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227. doi: 10.1063/1.1445183.  Google Scholar

[30]

B. N. Shashikanth, A. Sheshmani, S. D. Kelly and W. Mingjun, Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings, J. Math. Fluid. Mech., 12 (2010), 335-353. doi: 10.1007/s00021-008-0291-0.  Google Scholar

[31]

J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223.  Google Scholar

[32]

J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 609-629. doi: 10.1134/S1560354710040143.  Google Scholar

[33]

D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.  Google Scholar

show all references

References:
[1]

P. Appell, "Traite de Mechanique Rationelle," Vol. 2, Gauthier-Villars, Paris, 1953. Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616.  Google Scholar

[3]

A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Reg. and Chaot. Dyn., 8 (2003), 449-462.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaev, On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 16 (2006), 013118, 7 pp.  Google Scholar

[5]

T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlin. Sci., 21 (2011), 325-385.  Google Scholar

[6]

T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws, IEEE Trans. Automat. Control, 53 (2008), 339-349.  Google Scholar

[7]

S. A. Chaplygin, On the effect of a plane-parallel air flow on a cylindrical wing moving in it, in "The Selected Works on Wing Theory of Sergei A. Chaplygin," Translated from the 1933 Russian original by M. A. Garbell, (1956), 42-72. Google Scholar

[8]

S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier theorem, (in Russian) Math. Sbornik, 28 (1911), 303-314; English translation in Reg. Chaot. Dyn., 13 (2008), 369-376.  Google Scholar

[9]

F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 051, 12 pp.  Google Scholar

[10]

M. J. Field, Equivariant dynamical systems, Trans. Am. Math. Soc., 259 (1980), 185-205.  Google Scholar

[11]

Yu. N. Fedorov, Rolling of a disc over an absolutely rough surface, (Russian) Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 54 (1987), 67-75; Eng. transl. in Mechanics of Solids, 22 (1988), 65-73. Google Scholar

[12]

Yu. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259.  Google Scholar

[13]

Yu. N. Fedorov and L. C. García-Naranjo, The hydrodynamic Chaplygin sleigh, J. Phys. A, 43 (2010), 434013, 18 pp.  Google Scholar

[14]

L. C. García-Naranjo and J. Vankerschaver, Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation,, preprint, ().   Google Scholar

[15]

I. S. Gradshteyn and I. M. Ryzhik, "Tables of Integrals, Series, and Products," $7^{th}$ edition, Elsevier/Academic Press, Amsterdam, 2007.  Google Scholar

[16]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515.  Google Scholar

[17]

E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid J. Nonlinear Sci., 15 (2005), 255-289.  Google Scholar

[18]

S. Kelly and R. Hukkeri, Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift, IEEE Transactions on Robotics, 22 (2006), 1254-1264. Google Scholar

[19]

G. R. Kirchhoff, "Vorlesunger Über Mathematische Physik, Band I, Mechanik," Teubner, Leipzig, 1877. Google Scholar

[20]

H. Lamb, "Hydrodynamics," $6^{th}$ edition, Dover, NY, 1932. Google Scholar

[21]

N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica J. IFAC, 33 (1997), 331-346.  Google Scholar

[22]

T. Levi-Civita and U. Amaldi, "Lezioni di Meccanica Razionale. Vol. 2. Dinamica dei Sistemi con un Numero Finito di Gradi di Liberta," (Italian) Nuova ed. N. Zanichelli, Bologna, 1951, 1952. Google Scholar

[23]

L. Milne-Thomson, "Theoretical Hydrodynamics," $5^{th}$ edition, revised and enlarged edition, MacMillan and Co. Ltd., London, 1968. Google Scholar

[24]

P. K. Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001.  Google Scholar

[25]

S. M. Ramodanov, Motion of a circular cylinder and a vortex in an ideal fluid, Reg. and Chaot. Dyn., 6 (2001), 33-38.  Google Scholar

[26]

R. H. Rand and D. V. Ramani, Relaxing nonholonomic constraints, in "Proceedings of the First International Symposium on Impact and Friction of Solids, Structures, and Intelligent Machines" (ed. A. Guran), World Scientific, Singapore, (2000), 113-116. Google Scholar

[27]

J. Roenby and H. Aref, Chaos in body-vortex interactions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1871-1891.  Google Scholar

[28]

P. G. Saffman, "Vortex Dynamics," Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. Google Scholar

[29]

B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly, The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227. doi: 10.1063/1.1445183.  Google Scholar

[30]

B. N. Shashikanth, A. Sheshmani, S. D. Kelly and W. Mingjun, Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings, J. Math. Fluid. Mech., 12 (2010), 335-353. doi: 10.1007/s00021-008-0291-0.  Google Scholar

[31]

J. Vankerschaver, E. Kanso and J. E. Marsden, The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266. doi: 10.3934/jgm.2009.1.223.  Google Scholar

[32]

J. Vankerschaver, E. Kanso and J. E. Marsden, The dynamics of a rigid body in potential flow with circulation, Reg. Chaot. Dyn., 15 (2010), 609-629. doi: 10.1134/S1560354710040143.  Google Scholar

[33]

D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.  Google Scholar

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