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Regul. Chaotic Dyn.:

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Regul. Chaotic Dyn., 2013, том 18, выпуск 5, страницы 539–552 (Mi rcd138)  

Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)

On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor

Hamad M. Yehiaa, E. G. El-Hadidyb

a Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
b Department of Mathematics, Faculty of Science, Damietta University, Damietta, Egypt

Аннотация: One of the most notable effects in mechanics is the stabilization of the unstable upper equilibrium position of a symmetric body fixed from one point on its axis of symmetry, either by giving the body a suitable angular velocity or by adding a suitably spinned rotor along its axis. This effect is widely used in technology and in space dynamics.
The aim of the present article is to explore the effect of the presence of a rotor on a simple periodic motion of the rigid body and its motion as a physical pendulum.
The equation in the variation for pendulum vibrations takes the form

$$\frac{d^2\gamma_3}{du^2}+\alpha[\alpha \nu^2+\frac{1}{2}+\rho^2 - (\alpha+1)\nu^2sn^2u+2\nu \rho \sqrt{\alpha}cnu ]\gamma_3=0,$$

in which $\alpha$ depends on the moments of inertia, $\rho$ on the gyrostatic momentum of the rotor and $\nu$ (the modulus of the elliptic function) depends on the total energy of the motion. This equation, which reduces to Lame’s equation when $\rho=0$, has not been studied to any extent in the literature. The determination of the zones of stability and instability of plane motion reduces to finding conditions for the existence of primitive periodic solutions (with periods $4K(\nu)$, $8K(\nu)$) with those parameters. Complete analysis of primitive periodic solutions of this equation is performed analogously to that of Ince for Lame’s equation. Zones of stability and instability are determined analytically and illustrated in a graphical form by plotting surfaces separating them in the three-dimensional space of parameters. The problem is also solved numerically in certain regions of the parameter space, and results are compared to analytical ones.

Ключевые слова: stability, pendulum-like motions, planar motions, periodic differential equation, Hill’s equation, Lame’s equation


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Тип публикации: Статья
MSC: 70E50, 70H14, 70J25
Поступила в редакцию: 05.04.2013
Принята в печать:09.09.2013
Язык публикации: английский

Образец цитирования: Hamad M. Yehia, E. G. El-Hadidy, “On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor”, Regul. Chaotic Dyn., 18:5 (2013), 539–552

Цитирование в формате AMSBIB
\by Hamad M. Yehia, E. G. El-Hadidy
\paper On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 5
\pages 539--552

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    Citing articles on Google Scholar: Russian citations, English citations
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    Эта публикация цитируется в следующих статьяx:
    1. H. M. Yehia, E. Saleh, S. F. Megahid, “New solutions of classical problems in rigid body dynamics”, Mech. Res. Commun., 69 (2015), 40–44  crossref  isi  scopus
    2. H. M. Yehia, S. Z. Hassan, M. E. Shaheen, “On the orbital stability of the motion of a rigid body in the case of Bobylev–Steklov”, Nonlinear Dyn., 80:3 (2015), 1173–1185  crossref  mathscinet  isi  scopus
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