Sufficiently small rotations of Lagrange's gyro

In this study, the motion of Lagrange's gyro about its fixed point in the presence of a perturbed torque, a gyroscopic torque, and a varied restoring one is searched. We assume sufficiently small angular velocity components in the direction of the principal axes that differ from the dynamical symmetry one and a restoring torque that is considered to be greater than the perturbing one. In this manner, we replace the familiar small parameter that was used in previous works with a large one. In such cases, the gyro equations for motion (EOM) are formulated in the form of a two-degrees-of-freedom (DOF) autonomous system. We average the obtained system to get periodic solutions and motion's geometric interpretation of the problem using the large parameter. The regular precession and the pure rotation of the motion are obtained. A numerical study is evaluated for asserted the used techniques and showed the influence of the changing parameters of motion on the gyro behavior. The trajectories of the motions and their stabilities are discussed and analyzed. The novelty of this work lies in how to adapt the method of large parameter (MLP) to solve the rigid body problem, especially since it has been assumed initially that its angular velocity or its initial energy are very small. MSC (2000): 70E20, 70E17, 70E15, 70E05

Automated summary

This study investigates the motion of Lagrange's gyro around its fixed point under the influence of perturbed torque, gyroscopic torque, and varied restoring torque. The gyro equations for motion are formulated as a two-degrees-of-freedom autonomous system, assuming small angular velocity components and a greater restoring torque compared to the perturbing torque. The study explores periodic solutions and geometric interpretation of the motion using a large parameter. Numerical analysis is conducted to evaluate the techniques used and examine the impact of changing motion parameters on gyro behavior.