Hamiltonian structures and stability for rigid bodies with flexible attachments

  • P. S. Krishnaprasad
  • J. E. Marsden
Article

Abstract

The dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy-Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrations of the string, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.

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References

  1. R. Abraham & J. Marsden [1978]. Foundations of Mechanics, Second Edition, Addison-Wesley.Google Scholar
  2. G. Alvarez-Sanchez [1986]. Control of Hamiltonian systems with symmetry, Thesis, Univ. of Calif., Berkeley.Google Scholar
  3. S. Antman [1972]. The theory of rods, Handbuch der Physik VI, C. Truesdell, ed., Springer-Verlag, 641–703.Google Scholar
  4. S. Antman [1974]. Kirchhoff's problem for nonlinearly elastic rods, Quart. J. Appl. Math. 32, 221–240.Google Scholar
  5. S. Antman & J. Kenny [1981]. Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity, Arch. Rational Mech. An. 76, 289–354.Google Scholar
  6. S. Antman & A. Nachman [1980]. Large buckled states of rotating rods, Nonlinear Analysis TMA 4, 303–327.Google Scholar
  7. V. Arnold [1966a]. Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits, Ann. Inst. Fourier 16, 319–361.Google Scholar
  8. V. Arnold [1966b]. An a priori estimate in the theory of hydrodynamic stability, Izv. Vyssh. Uchebn. Zved. Mat. 54, 3–5, English Transl.: Am. Math. Soc. Transl. 1979 [1969], 267–269.Google Scholar
  9. V. Arnold [1978]. Mathematical Methods of Classical Mechanics, Springer-Verlag.Google Scholar
  10. J. Ball & J. Marsden [1984]. Quasiconvexity at the boundary, positivity of the second variation and elastic stability, Arch. Rational Mech. An. 86, 251–277.Google Scholar
  11. J. Baillieul & M. Levi [1983]. Dynamics of rotating flexible structures, Proc. IEEE Conf. CDC, San Antonio. TP2, 808–813.Google Scholar
  12. J. Baillieul [1983]. Modeling and control of flexible and articulated spacecraft, Proc. CISS, Johns Hopkins University, 95–102.Google Scholar
  13. T. Benjamin [1972]. The stability of solitary waves, Proc. Roy. Soc. London 328 A, 153–183.Google Scholar
  14. J. Bona [1975]. On the stability theory of solitary waves, Proc. Roy. Soc. London 344 A, 363–374.Google Scholar
  15. F. Bretherton & D. Haidvogel [1965]. Two-dimensional turbulence above topography, J. Fluid Mech. 78, 129–154.Google Scholar
  16. V. Guillemin & S. Sternberg [1984]. Symplectic Techniques in Physics, Cambridge University Press.Google Scholar
  17. D. Holm, B. Kupershmidt, & C. Levermore [1984], Canonical maps between Poisson brackets in Eulerian and Lagrangian descriptions of continuum mechanics, Phys. Lett. 98 A, 389–395.Google Scholar
  18. D. Holm, J. Marsden, & T. Ratiu [1985]. Nonlinear stability of the Kelvin-Stuart cats eye solutions, Proc. AMS-SIAM Summer Seminar, Lectures in Applied Math. AMS vol. 123, 171–186.Google Scholar
  19. D. Holm, J. Marsden, T. Ratiu & A. Weinstein [1983]. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Physics Letters 98 A, 15–21.Google Scholar
  20. D. Holm, J. Marsden, T. Ratiu & A. Weinstein [1984]. Stability of rigid body motion using the energy-Casimir method. Cont. Math. AMS 28, 15–23.Google Scholar
  21. D. Holm, J. Marsden, T. Ratiu & A. Weinstein [1985]. Nonlinear stability of fluid and plasma equilibria, Physics Reports 123, 1–116.Google Scholar
  22. P. Holmes & J. Marsden [1983], Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32, 273–310.Google Scholar
  23. C. Hubert [1981]. The attitude dynamics of dynamics explorer A, Attitude Control Analysis, RCA Astro-Electronics, AAS 81-123.Google Scholar
  24. T. Kane & D. Levinson [1980]. Formulation of the equations of motion for complex spacecraft, J. Guidance and Control 3, No. 2, 99–112.Google Scholar
  25. T. Kane, P. Likins & D. Levinson [1983]. Spacecraft Dynamics, New York: McGraw-Hill.Google Scholar
  26. P. Krishnaprasad & C. Berenstein [1984]. On the equilibria of rigid spacecraft with rotors, Systems and Control Letters 4, 157–163.Google Scholar
  27. P. Krishnaprasad [1983]. Lie-Poisson structures and dual-spin spacecraft, Proc. 22nd IEEE Conf. on Decision and Control, IEEE, New York, 814–824.Google Scholar
  28. P. Krishnaprasad [1985]. Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonlinear Analysis: TMA vol. 9, No. 10, 1011–1035.Google Scholar
  29. C. Leith [1984]. Minimum Enstrophy Vortex, AIP Proc. 106, 159–168.Google Scholar
  30. D. Lewis, J. Marsden, R. Montgomery & T. Ratiu [1986]. The Hamiltonian structure for dynamic free boundary problems. Physica 18 D, 391–404.Google Scholar
  31. P. Likins [1974]. Analytical dynamics and nonrigid spacecraft simulation, JPL Technical Report, TR 32-1593.Google Scholar
  32. E. Magrab [1979]. Vibrations of Elastic Structural Members, Sijthoff & Nordhoff, Netherlands.Google Scholar
  33. J. Marsden & T. J. R. Hughes [1983]. Mathematical Foundations of Elasticity, Prentice-Hall.Google Scholar
  34. J. Marsden & T. Ratiu [1986]. Reduction of Poisson Manifolds, Lett. Math. Phys. 11, 161–170.Google Scholar
  35. J. Marsden, T. Ratiu & A. Weinstein [1984a]. Semi direct products and reduction in mechanics, Trans. Am. Math. Soc. 281, 147–177.Google Scholar
  36. J. Marsden, T. Ratiu & A. Weinstein [1984b]. Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, Cont. Math. AMS 28, 55–100.Google Scholar
  37. J. Marsden & A. Weinstein [1974]. Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121–130.Google Scholar
  38. J. Marsden, A. Weinstein, T. Ratiu, R. Schmid & R. G. Spencer [1983]. Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-ISIMM Symposium on “Modern Developments in Analytical Mechanics,” Torino (June 7–11, 1982), Atti della Academia delle Scienze di Torino 117, 289–340.Google Scholar
  39. L. Meirovitch [1974], Bounds on the extension of antennas for stable spinning satellites, J. Spacecraft and Rockets, March, 202–204.Google Scholar
  40. L. Meirovitch & J. Juang [1974]. Dynamics of a Gravity-Gradient Stabilized Flexible Spacecraft, NASA Contractor Report: NASA CR-2456.Google Scholar
  41. R. Montgomery, J. Marsden & T. Ratiu [1984]. Gauged Lie-Poisson Structures, Cont. Math. AMS, 28, 101–114.Google Scholar
  42. P. Morrison [1986]. A paradigm for joined Hamiltonian and dissipative systems, Physica 18 D, 410–419.Google Scholar
  43. A. Nachman [1985]. Buckling and vibration of a rotating beam (preprint).Google Scholar
  44. E. Reissner [1973]. On a one-dimensional large-displacement finite-strain beam theory, Studies in Appl. Math. 52, 87–95.Google Scholar
  45. E. Reissner [1981]. On finite deformation of space-curved beams, ZAMP 32, 734–744.Google Scholar
  46. J. C. Simo [1985]. Finite strain beam formulattion : I, to appear in Comp. Meth. App. Mech. Engg. Google Scholar

Copyright information

© Springer-Verlag GmbH & Co. KG 1987

Authors and Affiliations

  • P. S. Krishnaprasad
    • 1
  • J. E. Marsden
    • 2
  1. 1.Department of Electrical EngineeringUniversity of MarylandCollege Park
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeley

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