In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector \(\vec{\ell}\) whose components are li (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter e (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincare small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.

On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector

In this article, we study the three-dimensional rotational motion of a symmetric rigid body (gyro) about a fixed point under the influence of a gyrostatic moment vector about the principal axes of inertia according to Kovalevskaya’s conditions. The six nonlinear differential equations of motion beside their first integrals are reduced to two differential equations from second order composed an autonomous system with one first integral. It is taken into consideration that the gyro spins with high angular velocity about one of the orthogonal principal axes. The small parameter method is used to obtain the analytical solutions of the governing system. The attained solutions are represented graphically through several plots concerning with the different values of the gyrostatic moment vector to evaluate the performance of these values on the gyro’s motion. Euler’s angles are estimated to describe the motion at any instant. The fourth-order Runge–Kutta algorithms through Matlab packages are used to obtain the numerical solutions for the mentioned problem and to represent these solutions graphically. A comparison between the analytical and the numerical solutions shows high consistency between them which indicates the powerful of the presented analytical technique. The phase plane plots are presented in order to give a full description about the regular motion of the gyro. The considered subject has a great interest in many applications such as submarines, satellites and aircrafts.

The Rotational Motion of a Symmetric Rigid Body Similar to Kovalevskaya’s Case

In this paper, the problem ofthe existence ofperiodic solutions of motion ofa gyrostatfixed at one point under the action ofa central Newtonian force field, and a gyrostatic momentum [i (i : 1,2,3; l1 : 42 = 0, {/3 ab O) similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter 8 (Elfimov, 1978). The equations of motion will be studied under certain initial conditions ofmotion. The Poincaré small parameter method (Malkin, 1959; Nayy‘eh, 1973) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are geometrically interpreted to give the forms ofEuler angles.

L. Sperling

Papers

On the Existence of Periodic Solutions of a Gyrostat Similar to Lagrange’s Gyroscope