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This study focuses on determining the required minimum-time (MT) for the spatial motion of a free rigid body (RB) experiencing gyrostatic moments (GMs) and viscous friction. The study assumes that the body’s center of mass coincides with the original point of two Cartesian systems of coordinates. An optimal control law for slow motion is established, and the corresponding time and phase pathways are analyzed. The innovative results are presented for two new cases through various graphs highlighting the positive effects of the GMs. A comparison is achieved between the obtained results and previous outcomes that did not consider gyrostatic moments, showing remarkable consistency with slight deviations that are discussed. The practical applications of this study, which limits itself to using gyroscopic theory to maintain the stability and balance of vehicles in which gyroscopes are used, as well as figuring out the trajectory of aircraft and marine vehicles, are what make it noteworthy.

This work focuses on the dynamical rotatory motion of an asymmetric rigid body (RB) subjected to a constant body-fixed torque and a vector of a gyrostatic moment (GM). The motion is considered in the absence of the first two components of the GM. The novelty of the current work is the conversion of the stability analysis of this body from a two dimensional phase plane to three ones phase space. Specifically, two scenarios of stationary torques are considered, the first one is directed on the minor or major axis, while the latter is presumed to act on the middle axis. Therefore, three dimensional phase space routes are generated. In both scenarios some novel analytical and simulation outcomes are provided regarding equilibrium manifolds, periodic solutions or non-periodic ones, separator surfaces, as well as the extreme periodic solutions. The significance of this work is due to its great applications, especially those that use the gyro’s theory.

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange's scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body's movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler-Poisson are solved through the utilization of Poincaré's small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body's position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems., (© 2024. The Author(s).)

This paper studies the shortest time of slowing rotation of a free dynamically asymmetric rigid body (RB), analogous to Euler's case. This body is influenced by a rotatory moment of a tiny control torque with closer coefficients but not equal, a gyrostatic moment (GM) due to the presence of three rotors, and in the presence of a modest slowing viscous friction torque. Therefore, this problem can be regarded as a semi-optimal one. The controlling optimal decelerating law for the rotation of the body is constructed. The trajectories that are quasi-stationary are examined. The obtained new results are displayed to identify the positive impact of the GM. The dimensionless form of the regulating system of motion is obtained. The functions of kinetic energy and angular momentum besides the square module are drawn for various values of the GM's projections on the body's principal axes of inertia. The effect of control torques on the body's motion is investigated in a case of small perturbation, and the achieved results are compared with the unperturbed one. For the case of a lack of GM, the comparison between our results and those of the prior ones reveals a high degree of consistency, in which the deviations between them are examined. As a result, these outcomes generalized those that were acquired in previous studies. The significance of this research stems from its practical applications, particularly in the applications of gyroscopic theory to maintain the stability and determine the orientation of aircraft and undersea vehicles., (© 2022. The Author(s).)