Plannar motion and stability for a rigid body with a flexible attachment in a field of central gravitational force
一类刚-梁耦合系统平面稳态运动的稳定性

Planar motion for a rigid body attached with an elastic beam in a field of central gravitational force was investigated, and both of orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by Lagrangian equation. The system has a first integral which indicates the conservation of angular momentum of the system. In the case of conservative system, the system has a class of relative equilibria which correspond to stationary motions of the system. Each stationary motion is defined by steady rotation of the system at an angular velocity equal to the orbital velocity, with the beam being in deformed state. By taking the energy-momentum functional as the Lyapunov function, the sufficient conditions for the stability of the relative equilibria were given. In the case of the energy dissipation, if the dissipation comes from the material viscous-damping of the beam, such energy dissipation do not change the relative equilibria of the sysstem, so the sufficient conditions for the stability of the stationary motions are the same.