A novel approach to solving Euler’s nonlinear equations for a 3DOF dynamical motion of a rigid body under gyrostatic and constant torques

• Published in: Journal of low frequency noise, vibration, and active control Vol. 44; no. 1; pp. 111 - 129
• Main Authors: TS, Amer, AH, Elneklawy, HF, El-Kafly
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.03.2025; Sage Publications Ltd; SAGE Publishing
• Subjects: Angular velocity; Decoupling; Degrees of freedom; Exact solutions; Liapunov functions; Mechanical systems; Nonlinear differential equations; Nonlinear equations; Rigid structures; Series expansion; Symmetrical bodies; Torque; Gyrostatic motion; fixed points; constant torques; rigid body; novel solutions; nonlinear dynamics
• ISSN: 1461-3484; 2048-4046
• DOI: 10.1177/14613484241293859

This research focuses on approaching a new technique to solve the rotary movement of a three degrees-of-freedom (DOF) semi-symmetric rigid body (RB) affected by a gyrostatic torque (GT) and under the influence of the RB’s constant-fixed torques (RBCFTs). The governing nonlinear differential equations (NDEs) for the RB are derived from the famous Euler’s equations, along with another system to calculate Euler’s angles, which gives an accurate solution for the RB’s position during its motion. In the case of a semi-symmetric body about its minor axis, novel analytical solutions for the RB’s angular velocities are presented in addition to the solutions of Euler’s angles. Our procedure to obtain these solutions arises through decoupling the governing NDEs into two coupled DEs, and the averaged time parameter is used. General solutions of the later DEs are presented in view of series expansions. The impact of the internal and external torques on the body’s motion is graphically simulated, which gives an overview of the periodicity of the derived solution and the effect of various values of these torques on the solution. Moreover, these simulations present other prospects for the body’s stability by analyzing them using the famous Lyapunov function. The importance of this study arises from its wide applications in our lives in many fields, such as mathematics and physics. It also holds immense potential for enhancing mechanical systems, elucidating celestial motion, and enhancing spacecraft efficiency.