An Investigation of Spinup Dynamics of Axial Gyrostats Using Elliptic Integrals and the Method of Averaging
Abstract
A gyrostat is a rigid body containing an internal source of angular momentum which does not alter the geometry of the system. Because of their relative simplicity, gyrostats are frequently used to model dual-spin spacecraft. An axial gyrostat is composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor aligned with a principal axis of the platform. Rotation of the rotor relative to the platform provides a source of internal angular momentum, and does not affect the moment of inertia tensor of the gyrostat. In this thesis, we consider the dynamics of axial gyrostats in the absence of energy dissipation and external torques. Spinup of the rotor is effected by a small constant internal axial torque, ga. The dynamics are described by four first-order ordinary differential equations, which admit exact solutions in terms of elliptic functions for ga = 0. Application of the method of averaging reduces the problem to a single first-order differential equation, which is studied analytically and numerically. This single equation accurately describes most spinup trajectories. However, when spinup trajectories cross the instantaneous separatrices of the ga = 0 system, the Sigma(1) frequency of the unaveraged system vanishes. The averaged equations are not valid near these separatrix crossings, and we develop an approximate equation valid in a neighborhood of the separatrix, which we use to connect the averaged solutions across the separatrix. When applied iteratively in a neighborhood of the separatrix, the approximate equation agrees qualitatively with the exact solution.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1991
- Accession Number
- ADA249728
Entities
People
- Christopher D. Hall
Organizations
- Air Force Institute of Technology