Substantial condition for the fourth first integral of the rigid body problem

Amer, T. S.; Amer, W. S.

doi:10.1177/1081286517716733

Cited by 20 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These formulas depend on four arbitrary constants θ 0 , ψ 0 , ϕ 0 , and r 0 , where r 0 is sufficiently small. The obtained analytical periodic solutions are considered a generalization of the ones obtained in [13,14].…”

Section: Geometric Interpretation Of the Motionmentioning

confidence: 99%

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

Ismail

2020

International Journal of Aerospace Engineering

View full text Add to dashboard Cite

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

show abstract

Section: Geometric Interpretation Of the Motionmentioning

confidence: 99%

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

Ismail

2020

International Journal of Aerospace Engineering

View full text Add to dashboard Cite

show abstract

“…The exact solutions of these equations demand another adoption fourth first integral. With the exception of a few particular circumstances including Euler–Poinsot, Lagrange–Poisson, and Kovalevskaya, numerous trials have been carried out to find this integral in its entire generality 2 . A large number of integrable cases for the RB’s motion were obtained in 3 – 5 .…”

Section: Introductionmentioning

confidence: 99%

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

Amer

Elkafly

2022

Sci Rep

View full text Add to dashboard Cite

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.

show abstract

“…In 2 , the classical Kovalevskaya top is generalized to a heavy motion of a gyrostat when the gyrostatic moment (GM) is applied and a generalization of Burn’s problem is found. Whereas, the possibility of obtaining the fourth-first integral for the RB motion through a simple procedure is investigated in 3 . The novel integrable case for the dynamics of the RB problem, which generalizes the prior cases, is discussed in 4 .…”

Section: Introductionmentioning

confidence: 99%

Modeling a semi-optimal deceleration of a rigid body rotational motion in a resisting medium

El-Sabaa

Amer

Sallam

et al. 2022

Sci Rep

View full text Add to dashboard Cite

This paper studies the shortest time of slowing rotation of a free dynamically asymmetric rigid body (RB), analogous to Euler’s case. This body is influenced by a rotatory moment of a tiny control torque with closer coefficients but not equal, a gyrostatic moment (GM) due to the presence of three rotors, and in the presence of a modest slowing viscous friction torque. Therefore, this problem can be regarded as a semi-optimal one. The controlling optimal decelerating law for the rotation of the body is constructed. The trajectories that are quasi-stationary are examined. The obtained new results are displayed to identify the positive impact of the GM. The dimensionless form of the regulating system of motion is obtained. The functions of kinetic energy and angular momentum besides the square module are drawn for various values of the GM’s projections on the body’s principal axes of inertia. The effect of control torques on the body's motion is investigated in a case of small perturbation, and the achieved results are compared with the unperturbed one. For the case of a lack of GM, the comparison between our results and those of the prior ones reveals a high degree of consistency, in which the deviations between them are examined. As a result, these outcomes generalized those that were acquired in previous studies. The significance of this research stems from its practical applications, particularly in the applications of gyroscopic theory to maintain the stability and determine the orientation of aircraft and undersea vehicles.

show abstract

Substantial condition for the fourth first integral of the rigid body problem

Cited by 20 publications

References 21 publications

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

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

Modeling a semi-optimal deceleration of a rigid body rotational motion in a resisting medium

Contact Info

Product

Resources

About