### Abstract

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out fer perturbations to these relative equilibria, to confirm their stability properties.

Original language | English (US) |
---|---|

Pages (from-to) | 303-343 |

Number of pages | 41 |

Journal | Dynamical Systems |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications

### Cite this

*Dynamical Systems*,

*19*(4), 303-343. https://doi.org/10.1080/14689360412331309160

**Dynamics of multibody systems in planar motion in a central gravitational field.** / Sanyal, Amit; Bloch, Anthony; Mcclamroch, N. Harris.

Research output: Contribution to journal › Article

*Dynamical Systems*, vol. 19, no. 4, pp. 303-343. https://doi.org/10.1080/14689360412331309160

}

TY - JOUR

T1 - Dynamics of multibody systems in planar motion in a central gravitational field

AU - Sanyal, Amit

AU - Bloch, Anthony

AU - Mcclamroch, N. Harris

PY - 2004/12/1

Y1 - 2004/12/1

N2 - Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out fer perturbations to these relative equilibria, to confirm their stability properties.

AB - Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out fer perturbations to these relative equilibria, to confirm their stability properties.

UR - http://www.scopus.com/inward/record.url?scp=11144297251&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11144297251&partnerID=8YFLogxK

U2 - 10.1080/14689360412331309160

DO - 10.1080/14689360412331309160

M3 - Article

AN - SCOPUS:11144297251

VL - 19

SP - 303

EP - 343

JO - Dynamical Systems

JF - Dynamical Systems

SN - 1468-9367

IS - 4

ER -