الفهرس Only 14 pages are availabe for public view
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Abstract
The objective of this thesis is to perform an exhaustive analysis of the pendulum -like motion of a rigid body carrying a rotor. By pendulum like--motions of the rigid body about a fixed point we mean motions of the body about a horizontal axis which preserves a fixed position in space. Those motions are possible only around a principal axis of inertia of the body at the fixed point while the centre of mass lies in the plane of the other two principal axes. The thesis consists a brief historical introduction, four chapters and a list of references.
Chapter one:
We describe the derivation of the equation in the variation about the trajectory of the motion by linearizing the orbital second order differential equation which introduced by H. Yehia. Following, we write the variational equation for each of the two types of motion (rotation and vibration) in various forms, as a linear equation with rational coefficients (Fuchs equation) and as equation with periodic coefficients in elliptic, trigonometric and Hill forms for a symmetric and dynamically symmetric body.
Chapter two:
We give necessary information from the theory of stability of differential equations. We display several periodic differential equations such as Ince’s equation, Lame’s equation, Lame wave equation, in a manner which indicates their relation to one another and at the same time their status in the general scheme of differential equations.
Chapter three:
We consider the stability of plane vibration motion for dynamically symmetric body in presence of gyrostatic moment. The equation in the variation for pendulum vibrations take the form of a deformation of lame’s equation
(1)
in which depend on the moments of inertia, on the gyrostatic moment of the rotor and ( the modulus of the elliptic function) depends on the total energy of the motion. The determination of the zones of stability and instability of plane motion reduces to finding conditions on those parameters for existence of primitive periodic solutions (with periods ). In fact, reversing the sign of together with a shift in the argument leaves equation (1) invariant. This means that the picture of stability zones is symmetric with respect to the sign of the gyrostatic moment. A theory of this equation is developed analogous to that of Ince for lame’s equation. The zones of stability are illustrated in a graphics form by plotting surfaces separating them in the three dimensional space of parameters. To complete the picture of zones of stability, we perform a numerical study of the stability of the variational equation and scan the domain of possible values of parameters. Finally, we display some figures that show complete agreement between analytical and numerical results. While the equation in the variation for pendulum rotations take the form of a deformation of lame’s equation
(2)
in which depend on the moments of inertia, on the gyrostatic moment of the rotor and ( the modulus of the elliptic function) depends on the total energy of the motion. The determination of the zones of stability and instability of plane motion reduces to finding conditions on those parameters for existence of primitive periodic solutions (with periods ). In fact, equation (2) is not symmetric with respect to the sign of the gyrostatic moment. We found some difficulties to find the analytical solution of this equation due to the existence of the term .
Chapter four:
We use the variational equation in the form of Hill equation. Then we study the stability problem of the small oscillations of asymmetric body when its centre of mass is on the plane x-y and obtained the surface bounding the stability zone.