There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Numerical methods for Engineers and Scientists

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Introduction To Perturbation Techniques

Synchronous states of slowly rotating pendula

Numerical methods for partial differential equations

The rotational motion of a heavy solid about a fixed point in the presence of a gyrostatic moment vector is investigated in this paper. It is supposed that the body is rapidly spinning about the major or the minor principle axis of the ellipsoid of inertia. The Krylov-Bogoliubov-Mitropolski technique is modified to obtain the periodic solutions of the equations of motion of the body with nonzero basic amplitude. These solutions are performed by computer codes to get their graphical representations. The result of this study was compared with similar previous works.

Application of the Krylov-Bogoliubov-Mitropolski Technique for a Rotating Heavy Solid under the Influence of a Gyrostatic Moment

In der vorliegenden Arbeit wird die Bewegung eines festen Korpers um einen festen Punkt in einem zentralen Newtonschen Kraftfeld betrachtet. Es wird angenommen, das das Massenzentrum des Korpers nicht notwendig mit dem festen Punkt zusammenfallt und das dessen Tragheitsellipsoid beliebig ist. In die Betrachtung wird mit einbezogen, das der Korper eine hinreichend grose Anfangswinkelgeschwindigkeit (r0) um die kleine oder die grose Hauptachse des Tragheitsellipsoids hat und das der Parameter (l/r0) klein ist. Die Bewegungsgleichungen und deren verfugbare erste Integrale werden auf ein quasilineares autonomes System mit zwei Freiheitsgraden und mit einem ersten Integral reduziert. Es werden die Methode von Poincare [1] und deren Modifikationen [2] und [3] angewendet, um periodische Losungen fur das erhaltene autonome System in dem Falle zu konstruieren, wenn zwei Frequenzen des erzeugenden Systems verschieden, aber kommensurabel sind (mit Ausnahme von w = 1/2, 1, 2). Wir beschranken uns darauf, solche Losungen und die Ausdrucke fur die Eulerschen Winkel fur den Grenzfall γ ≈ 0 zu bestimmen. Schlieslich wird ein Runge-Kutta-Verfahren der Ordnung 4 [4] angewendet, um die numerischen Losungen des autonomen Systems zu untersuchen. Danach werden graphische Darstellungen sowohl fur die numerische als auch fur die analytische Losung gewonnen. Ein Vergleich zwischen ihnen zeigt, das die Ergebnisse weitgehend zusammenfallen und die Abweichungen sehr klein sind.



In the present paper the motion of a rigid body about a fixed point in a central Newtonian force field is considered. It is assumed, that the center of mass of the body is not necessarily coinciding with the fixed point and its ellipsoid of inertia is arbitrary. It is taken into consideration, that the body has a sufficiently large initial angular velocity (r0) about the minor or the major principal axis of the ellipsoid of inertia and that the parameter (1/r0) is small. The equations of motion and their available first integrals are reduced to a quasilinear autonomous system of two degrees of freedom with one first integral. The method of Poincare [1] and its modifications [2] and [3] are applied to construct periodic solutions for the autonomous system obtained in the case when the two frequencies of the generating system are distinct but commensurable (except ω = 1/2, 1, 2). We restrict ourselves to determine such solutions and the expressions of the Eulerian angles for the limiting case γ ≈ £ 0. At the end, a fourth order Runge-Kutta method [4] is applied to investigate the numerical solutions of the autonomous system. Then, the graphical representations for both the numerical and the analytical solutions are obtained. A comparison between them shows that the results coincide and the deviations are very small.

Limiting case for the motion of a rigid body about a fixed point in the Newtonian force field

In the present study, the motion of a pendulum on an ellipse is considered. The supported point of this pendulum moves on an elliptic path while the end point moves with arbitrary angular displacements. Applying Lagrange's equation, the equation of motion is obtained in terms of a small parameter e. This equation represents a quasilinear system of second order which can be solved in terms of a generalized coordinate φ.

On the Motion of the Pendulum on an Ellipse

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

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Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

Poincare's small parameter method and the Krylov-Bogoliubov asymptotic method are among the number of basic methods used for the study of nonlinear oscillations. Poincare's method was developed in conformity with stationary (periodic) oscillations [1], although it may be extended to nonstationary oscillations [2]. The Krylov-Bogoliubov method may be used, first of all, for a study of nonstationary oscillations, but it is, of course, completely applicable to periodic oscillations [3]. In the present paper, the method of Krylov-Bogoliubov-Mitropolski [4] is modified to investigate the periodic solutions for the equations of motion of a heavy solid, with one fixed point, rapidly spinning about the major or the minor axis of the ellipsoid of inertia.

A. I. Ismail

Papers

Application of the Krylov-Bogoliubov-Mitropolski Technique for a Rotating Heavy Solid under the Influence of a Gyrostatic Moment

Limiting case for the motion of a rigid body about a fixed point in the Newtonian force field

On the Motion of the Pendulum on an Ellipse

Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

On the application of Krylov-Bogoliubov-Mitropolski technique for treating the motion about a fixed point of a fast spinning heavy solid