Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

In this paper, the motion of a rigid body in a singular case of the natural frequency ( ω = 1 / 3 ) is considered. This case of singularity appears in the previous works due to the existence of the term ω 2 − 1 / 9 in the denominator of the obtained solutions. For this reason, we solve the problem from the beginning. We assume that the body rotates about its fixed point in a Newtonian force field and construct the equations of the motion for this case when ω = 1 / 3 . We use a new procedure for solving this problem from the beginning using a large parameter ε that depends on a sufficiently small angular velocity component r o . Applying this procedure, we derive the periodic solutions of the problem and investigate the geometric interpretation of motion. The obtained analytical solutions graphically are presented using programmed data. Using the fourth-order Runge-Kutta method, we find the numerical solutions for this case aimed at determining the errors between both obtained solutions.

The Stability of Certain Motion of a Charged Gyrostat in Newtonian Force Field

This work aims to study the stability of certain motions of a rigid body rotating about its fixed point and carrying a rotor that rotates with constant angular velocity about an axis parallel to one of the principal axes. This motion is presumed to take place due to the combined influence of the magnetic field and the Newtonian force field. The equations of motion are deduced, and moreover, they are expressed as a Lie–Poisson Hamilton system. The permanent rotations are calculated and interpreted mechanically. The sufficient conditions for instability are presented employing the linear approximation method. The energy-Casimir method is applied to gain sufficient conditions for stability. The regions of linear stability and Lyapunov stability are illustrated graphically for certain values of the parameters.

The Slow Spinning Motion of a Rigid Body in Newtonian Field and External Torque

In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r 0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r 0 ⟶ 0 , define a large parameter proportional to 1 / r 0 , and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.

The Motion of a Rigid Body in the Presence of a Gyrostatic : الحركات المغزلية السريعة لجسم متماسك في مجال قوة نيوتوني في حالة وجود العزم

In this study، the rotational motion of a rigid body about a fixed point in the Newtonian force field with a gyrostatic momentum about the z-axis is considered. The equations of motion and their first integrals are obtained and reduced to a quasi-linear autonomous system with two degrees of freedom with one first integral. Poincare's small parameter method is applied to investigate the analytical peri­odic solutions of the equations of motion of the body with one point fixed، rapidly spinning about one of the principal axes of the ellipsoid of inertia. A geometric interpretation of motion is given by using Euler's angles to describe the orientation of the body at any instant of time.

On the motion of a gyro in the presence of a Newtonian force field and applied moments

This work focuses on the motion of a dynamical model that consists of a symmetric rigid body (gyro) that rotates about a fixed point similar to Lagrange’s gyroscope. This body is acted upon by external forces represented by a Newtonian force field, gyro torques about the principal axes of inertia of the gyro and perturbing moments acting on the same axes. Assuming that, the gyro initially has a high angular velocity about the dynamic axis of symmetry. The averaging technique is used to obtain a more appropriate averaging system for the governing system of equations of motion in terms of a small parameter. Therefore, the analytical solutions of this system for two applications, depending on different forms of perturbing moments, are presented. These solutions are represented graphically to clarify the effectiveness of the different parameters of the body on the motion.

On the dynamical motion of a gyro in the presence of external forces

This work shed light on the motion of a symmetric rigid body (gyro) about one of its principal axes in the presence of a Newtonian force field besides a gyro moment in which its second component equals null. It is assumed that the body center of mass is shifted slightly relative to the dynamic symmetry axis. The governing equations of motion are investigated taking into account some initial conditions. The desired solutions of these equations are achieved in framework of the small parameter method. The periodic solutions for the case of irrational frequencies are investigated. Euler’s angles have been used to interpret the motion at any time. The geometrical representations of the obtained solutions and the phase plane schemas of these solutions are announced during several plots. Discussion of the results is presented to reinforce the importance of the considered gyro moment and the Newtonian force field. The significance of this problem is due to the framework of its several applications in different industries such as airplanes, submarines, compasses, spaceships, and guided missiles.

On Permanent Rotations of a System of Two Coupled Gyrostats in a Central Newtonian Force Field

This paper studies the problem of the motion of a chain of two gyrostats coupled by an ideal spherical joint. The chain moves about a fixed point in a central Newtonian force field. Under the assumption that the gyrostatic moment of each gyrostat is constant relative to its carrier, the paper establishes and analyzes the conditions for existence of the chain’s permanent rotations about a vertical axis. For a case when each gyrostat has the mass distribution analogous to the one of a Lagrange gyroscope, the paper derives and analyzes the necessary conditions for stability of the permanent rotations. The findings of the paper extend corresponding results in the dynamics of a single gyrostat to a case of the multibody chain as well as generalize some of the known properties of permanent rotations in the many-body dynamics.

Precessional Motions of a Chain of Coupled Gyrostats in a Central Newtonian Force Field

This paper studies the problem of the motion of a classical model of modern analytical multibody dynamics — a chain of gyrostats coupled by ideal spherical joints. The chain moves about a fixed point in a central Newtonian force field. The paper develops the equations of chain’s motion and then establishes and analyzes the conditions for existence of some classes of precessional motions of the chain, under the assumption that the barycenter of each gyrostat is located on the line connecting the points where it is attached to other gyrostats. The findings of the paper extend corresponding results in the dynamics of a single gyrostat to a case of the multibody chain as well as generalize some of the known properties of precessional motions in the dynamics of many bodies.