Abstract
Incorporating both viscous friction torque and external torque, a mathematical model of a gyrostat system consisting of three rotors and a fixed outer frame is configured. This model is transformed into a Kolmogorov-type system for force analysis. The force field in the gyrostat system includes four different torques—inertial, internal, dissipative, and external. Correspondingly, four different energies are identified and the interconversion of energies is analyzed. The Casimir power being equivalent to the error power between the supplied power and the dissipative power is found and used to analyze the mechanism underlying the different dynamical behaviors. A four-wing chaotic attractor is found when the gyrostat is subject to a combination of inertial, internal, and dissipative torques as well as the addition of an external torque. The bifurcation of the Casimir power and leaps in Casimir energy level are used as indicators of the different definitive dynamics, which is demonstrated to be identical to that appearing in the state bifurcation and the Lyapunov exponent spectrum.
Highlights
The dynamics of the gyrostat model have been extensively studied over the past two centuries, and they are important for applications in the gyrostat dynamics of satellites, spacecraft, aircraft, and robots
The earlier achievements concerning the equations of motion for a gyrostat and the analytical solutions of the equations governing the gyrostat dynamics were proposed by Wittenberg [1]
The Casimir power is equivalent to the error power between the supplied power and the dissipative power
Summary
The dynamics of the gyrostat model have been extensively studied over the past two centuries, and they are important for applications in the gyrostat dynamics of satellites, spacecraft, aircraft, and robots. The Melnikov equation provides the necessary conditions for the existence of chaotic motion [15, 16] These features of the model cannot identify the mechanism or explain the cause of the dynamic behavior generated and the energy exchange corresponding to the different force fields in the gyrostat system. If a system is transformed into the Kolmogorov model, its underlying mechanism is able to be uncovered by establishing the link between force (torque) and definitive behavior.
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