Bifurcation and Stability Analysis of the Hamiltonian–Casimir Model of Liquid Sloshing

Motion stability of a spacecraft is discussed. A canonical Hamiltonian model for liquid sloshing is presented for a moving rigid body. An equivalent mechanical pendulum model is used to represent the fuel slosh inside the container. In this model sloshing is represented by the moving mass, the rest of the mass of the spacecraft is assumed to be stationary. The spacecraft structure is considered to be an elliptical rigid shape and the steady rotation along the x-axis is taken as the major-axis rotation. Motion stability for the present model is analyzed using the Lyapunov theory with Casimir energy functions. Conditions for stability and instability are derived for a steady principal axis rotation of the rigid body. Simulation results are presented to distribute the region into stable and unstable regions. Besides this, the nonlinear behavior of the system is analyzed under the influence of an external force acting periodically. Chaos is observed through a bifurcation diagram. The time history map and phase portrait are also presented to analyze the nonlinear behavior of the system.