Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

Journal of Low Frequency Noise, Vibration and Active Control

Amer

2023

Self Cite

In this study, the motion of Lagrange’s gyro about its fixed point in the presence of a perturbed torque, a gyroscopic torque, and a varied restoring one is searched. We assume sufficiently small angular velocity components in the direction of the principal axes that differ from the dynamical symmetry one and a restoring torque that is considered to be greater than the perturbing one. In this manner, we replace the familiar small parameter that was used in previous works with a large one. In such cases, the gyro equations for motion (EOM) are formulated in the form of a two-degrees-of-freedom (DOF) autonomous system. We average the obtained system to get periodic solutions and motion’s geometric interpretation of the problem using the large parameter. The regular precession and the pure rotation of the motion are obtained. A numerical study is evaluated for asserted the used techniques and showed the influence of the changing parameters of motion on the gyro behavior. The trajectories of the motions and their stabilities are discussed and analyzed. The novelty of this work lies in how to adapt the method of large parameter (MLP) to solve the rigid body problem, especially since it has been assumed initially that its angular velocity or its initial energy are very small. MSC (2000): 70E20, 70E17, 70E15, 70E05

Section: Description Of the Problemmentioning

confidence: 99%

Sufficiently small rotations of Lagrange’s gyro

Journal of Low Frequency Noise, Vibration and Active Control

Amer

2023

Self Cite

“…The only road to exiting from this complicated problem is by achieving a large parameter and using it through the large parameter technique to solve the problem. 1,28,50,52 The motion equations system is reduced, applying the first integrals, to a quasilinear independent one of 2 freedom degrees and 1 first integral. The analytic periodic solutions of the problem system are integrated and analyzed dynamically in presence of the first integrals of the motion.…”

Section: Introductionmentioning

confidence: 99%

“…This work presents the application of the large parameter technique [50][51][52] for considering the disc problem motion under new initial conditions, in a new domain, and in presence of the gyro torque around one of the principal axes of the body inertia ellipsoid. Initially, we give the body a sufficiently low angular speed component about one of these axes (weak oscillation of the disc).…”

Section: Introductionmentioning

confidence: 99%

A slow rotary dynamic motion of a disc under the influence of torque with the concept of a large parameter

Journal of Low Frequency Noise, Vibration and Active Control

Abdulaziz

2022

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A rotary motion is studied, which is used as a paradigm in both mathematics and dynamics. In this motion, the body moves around its axis. Important examples of this motion are the earth, wheels for cars or bicycles, cooling fans, airplanes, radars, and motion of the discs. In this article, we consider one of these motions such as the slow rotational motion of a disc of high natural frequency around a fixed point different from its center of mass. Suppose a constant gyrostatic couple acts on the body about the axis of symmetry. Let us consider the axis of symmetry coincide with one of the main axes of inertia. The controlled nonlinear independent system of equations of motion is obtained in the form of six nonlinear differential equations in the presence of three independent first integrals. This system is reduced to an alternative independent quasi-linear system consisting of two differential equations with only one integral. At first, when the body rotates slowly with a small angular speed about the axis of the symmetry, we achieve a large parameter for the motion which is proportional to the inverse of the angular velocity component. Then, we use the large parameter technique to obtain approximate and high-frequency analytic solutions for this motion. This technique enables us to introduce new conditions of the motion, save a lot of energy required to move the disc, and obtain new approximated solutions in a new domain of the disc parameters. We present a digital example of the problem to clarify the inferences of the motion depending on the parameters of the body. On the other hand, we use the Runge–Kutta algorithmic method to obtain the numerical solutions corresponding to the approximate solutions of the considered independent system. We investigate the inaccuracies and errors of analytic and numerical solutions by comparing them. This problem offers many applications in gyroscope dynamics, industrial engineering, astrophysics, navigation, satellites, and space engineering.

“…Moreover, the method of averaging is used in many studies, e.g., [15][16][17][18][19][20] to obtain the averaging system of the governing one for di erent cases of the rotational motion of asymmetric RB such as in a uniform eld of force, in the existence of gyro moment, in an NFF, and in the presence of a point charge on the axis of dynamic symmetry. Recently, in [21][22][23][24][25], the large parameter method is utilized to obtain the approximate solutions of this problem under certain initial conditions. e numerical results of the ywheel motion are found in [26] when the body is in uenced by NFF in addition to the gyro moment vector, while the vibrating motion of the RB is investigated numerically in [27] for the position of relative equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Existential Properties of Algebraic Integrals of a Rigid Body

Amer

2022

Advances in Astronomy

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In this article, we consider kinematical considerations of a rigid body rotating around a given fixed point in a Newtonian force field exerted by an attractive center with a rotating couple about their principal axes of inertia. The kinematic equations and their well-known three elementary integrals of the problem are introduced. The existence properties of the algebraic integrals are considered. Besides, we search as a special case of the fourth algebraic integral for the problem of the rigid body’s motion around a fixed point under the action of a Newtonian force field with an orbiting couple. Lagrange’s case and Kovalevskaya’s one are obtained. The large parameter is used for satisfying the existing conditions of the algebraic integrals. The comparison between the obtained results and the previous ones is arising. The numerical solutions of the regulating system of motion are obtained utilizing the fourth-order Runge-Kutta method and are plotted in some figures to illustrate the positive impact of the imposed forces and torques on the behavior of the body at any time.