article · Journal of low frequency noise, vibration and active control
This research provides a three-dimensional mathematical model describing the rotational dynamics of an asymmetric charged rigid body. The system experiences a combination of a gyrostatic moment, an electromagnetic force field, time-varying body-fixed torques, and a constant axial torque. Euler equations of motion are formulated to evaluate the attitude dynamics of bodies that are nearly symmetrical. From these equations, new analytical solutions for the angular velocities are established by expressing the time-varying torques through integrals that are evaluated in closed form. The investigation specifically examines scenarios involving a constant spin-axis torque alongside transverse torques defined as polynomial functions of time. While exact solutions are derived for axisymmetric bodies, approximate analytical solutions are generated for Eulerian angles under small-angle assumptions. Motion stability and parameter contributions are verified through computer-generated phase plane diagrams and graphical simulations.
Understanding how charged rotating objects move under magnetic forces and varied torques is essential for controlling complex physical systems. By providing precise mathematical solutions for these motions, this work assists engineers and scientists in predicting dynamic behaviours, analysing stability, and designing better controls for complex mechanical and aerospace systems without relying entirely on resource-intensive numerical simulations.
This work represents early-stage theoretical modelling and simulation. The mathematical frameworks could potentially be utilised by aerospace engineers and mechanical systems designers seeking to optimise spacecraft performance, stabilise rotating machinery, or model celestial mechanics. However, because the study is confined to mathematical derivations and computer simulations, practical implementation in commercial systems remains at a foundational stage and requires real-world experimental testing.
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The 3D modeling analysis for the rotary motion of an asymmetric rigid body (RB) that gains a charge is presented. Under the effect of a gyrostatic moment (GM), an electromagnetic force field (EFF), time-varying body-fixed torques (TVBFTs), and constant axial torque (CAT), Euler’s equation of motion (EOM) is derived to describe the body’s EOM. The process is to derive the analytic solutions for the general attitude motion of the RB that is nearly symmetrical; therefore, a novel analytical solution for the angular velocities of the body has been approached. These new solutions are obtained by considering torques that vary over time and expressing them as integrals. Additionally, a novel closed-form evaluation mechanism for these integrals is offered. Specifically, the case of a constant torque around the spin axis and transverse torques represented by polynomial functions of time is explored. When dealing with an axisymmetric RB subject to a CAT, the solutions obtained from Euler’s EOM are exact. However, it is important to note that novel analytic solutions for the Eulerian angles are approximations, as they rely on the assumption of small angles. Nonetheless, these approximations have broad applicability to a wide range of practical problems. The method’s precision is demonstrated through the graphical simulation of the proposed solutions. Additionally, a computer program is utilized to create diagrams and phase plane curves, highlighting the contribution of various body parameters to the motion. These plots depict the contributions of various values regarding GM, charge, and CAT. Motion stability is also examined through phase diagrams. In addition to presenting novel solutions and outcomes for the problem, this study plays a vital role in multiple scientific and engineering fields as it has the potential to optimize mechanical systems, explain celestial motion, and improve spacecraft performance.
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DOI: 10.1177/14613484241276381
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