On the application of KBM method for the 3-D motion of asymmetric rigid body

Amer, T. S.; Abady, I. M.

doi:10.1007/s11071-017-3537-7

Cited by 31 publications

(15 citation statements)

References 20 publications

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“…is technique considered through the weak nonlinear equation of the second order in terms of the small parameter was defined in the problem [7][8][9][10][11]. Here, we consider applying the large parameter ε.…”

Section: E Krylov-bogoliubov Techniquementioning

confidence: 99%

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On New Modifications of Some Perturbation Procedures

Ismail

2021

Discrete Dynamics in Nature and Society

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In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.

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Section: E Krylov-bogoliubov Techniquementioning

confidence: 99%

“…is technique is devoted to finding approximated periodic solutions using the small parameter as in [7][8][9][10][11]. In our work, we modify the approximated expansions using the large parameter to become…”

Section: The Krylov-bogoliubov-mitropolski Techniquementioning

confidence: 99%

On New Modifications of Some Perturbation Procedures

Ismail

2021

Discrete Dynamics in Nature and Society

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“…The importance of this component lies in the fact that the solutions that have been reached do not contain any singular points. It has contributed to achieve a new frequency called the Amer’s frequency 26 , 27 , 36 , which in turn showed that there are no singularities in the solutions of RB’s motion at all.…”

Section: Introductionmentioning

confidence: 99%

Studying the influence of external moment and force on a disc’s motion

Amer

Elkafly

2022

Sci Rep

Self Cite

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In this work, the influence of a gyrostatic moment vector (GMV) and the Newtonian field (NF) on the rotatory motion of a restricted rigid body (RB) according to disc case around a fixed point is examined. The basic equation of the body motion is used to get the regulating motion’s system as well as the three available independent first integrals. The system’s six equations and these integrals were reduced to two equations of a quasi-linear two-degrees-of-freedom autonomous system and one first integral. The disc has been presumed to be quickly rotating around one of the ellipsoid of inertia's main axis. Poincaré’s method of small parameter (PMSP) is applied to acquire the periodic solutions of the controlling system of the body’s motion. Euler's angles are utilized to characterize the body’s configuration at any instant in which it is graphed, as well as the obtained solutions to explore the good action of the body’s parameters on its motion. The phase plane graphs of these solutions are presented to examine their stabilities. The relevance of this work may be traced to its wide range of applications in fields as diverse as physics, engineering, and life sciences, including assembly and machine design.

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“…This problem was generalized in [17] when the body moves under the presence of two components of the constant gyrostatic moment vector while the attained solutions don't have any singular points at all, due to that the authors used another frequency different from the used frequency in [15,16] by a small quantity depends on the third component of the gyrostatic moment vector. Recently, KBM method is utilized in [18] to the solutions of the equations of motion of a rigid body in a general case, i.e. without any restrictions on the locations of the body center of mass and one the values of principal moments of inertia.…”

Section: Introductionmentioning

confidence: 99%

On the Spinning Motion of a Disc under the Influence a Gyrostatic Moment

Bek

Amer

Gamiel

2021

Springer Proceedings in Mathematics &Amp; Statistics

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This work outlines the motion of a disc about one of its fixed point different from its center of mass in the presence of a constant gyrostatic moment about the principal axes of inertia. The governing system of motion consists of six nonlinear differential equations and their first integrals are reduced to another quasilinear autonomous one of 2DOF besides one first integral. Initially, it is hypothesized that the body is rapidly spun about one of its principal axes. The method of small parameter of Poincaré is used to achieve the desired approximate solutions of the equations of motion. Euler's angles are used to interpret the motion of the body at any blink. The numerical solutions of autonomous system are investigated using the fourth order Runge-Kutta algorithms (RKA). The comparison between both two solutions reveals that the numerical solutions are in well agreement with the approximate ones and the deviation between them is very slightly. The importance of this work is focused on its great applications in many fields such as in engineering, physics and industrial applications for example ships stabilizers, racing cars, pointing devices for computer, satellites and like.

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On the application of KBM method for the 3-D motion of asymmetric rigid body

Cited by 31 publications

References 20 publications

On New Modifications of Some Perturbation Procedures

On New Modifications of Some Perturbation Procedures

Studying the influence of external moment and force on a disc’s motion

On the Spinning Motion of a Disc under the Influence a Gyrostatic Moment

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