scholarly journals TRANSLATIONAL-ROTATIONAL MOTION OF A NONSTATIONARY AXISYMMETRIC BODY

2021 ◽  
Vol 2 (336) ◽  
pp. 131-137
Author(s):  
M. Zh. Minglibayev ◽  
S. B. Bizhanova
Keyword(s):  
Rotational Motion ◽  
Evolution Equations ◽  
Second Harmonic ◽  
First Integral ◽  
Approximate Expression ◽  
Axisymmetric Body ◽  
Order System ◽  
Dynamical Structure ◽  
Symmetric Density ◽  
Principal Axes

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

scholarly journals INVESTIGATION OF THE EVOLUTION EQUATIONS OF THE AXISYMMETRIC BODY WITH

2020 ◽  
Vol 69 (1) ◽  
pp. 247-252
Author(s):  
M.Zh. Minglibayev ◽  
◽  
S.B. Bizhanova ◽  
Keyword(s):  
Coordinate System ◽  
Rotational Motion ◽  
Evolution Equations ◽  
Second Harmonic ◽  
Variable Mass ◽  
Dynamic Structure ◽  
Approximate Expression ◽  
Axisymmetric Body ◽  
Osculating Elements ◽  
Two Bodies

In this article we consider mutually gravitating non-stationary two bodies: first body is «central», it is a sphere with a spherical density distribution, the second body is «satellite», which has an axisymmetric dynamic structure and form. Newtonian force interaction is characterized by an approximate expression of the force function, which takes into account the second harmonic. The differential equation of translational-rotational motion of the axisymmetric body is derived with variable mass and variable size in a relative coordinate system. The axes of the own coordinate system for nonstationary two bodies coincides with the main axes of inertia and this position remains unchanged during evolution. The mass of bodies are varied isotropically in the different rates. The problem is investigated by methods perturbation theory. The equations of secular perturbations of translational-rotational motion of satellite are deduced in the analogues osculating elements Delaunay-Andoyer. The solutions of the differential equations of the perturbed motion are obtained by the numerical method and the graphs are constructed using the Wolfram Mathematica package.

scholarly journals Angular velocity of rotation of extended bodies in general relativity

1996 ◽  
Vol 172 ◽  
pp. 309-320
Author(s):  
S.A. Klioner
Keyword(s):  
General Relativity ◽  
Angular Velocity ◽  
Rigid Body ◽  
Rotational Motion ◽  
The Body ◽  
Astronomical Observations ◽  
Newtonian Approximation ◽  
Accurate Definition ◽  
Principal Axes ◽  
Definition Of

We consider rotational motion of an arbitrarily composed and shaped, deformable weakly self-gravitating body being a member of a system of N arbitrarily composed and shaped, deformable weakly self-gravitating bodies in the post-Newtonian approximation of general relativity. Considering importance of the notion of angular velocity of the body (Earth, pulsar) for adequate modelling of modern astronomical observations, we are aimed at introducing a post-Newtonian-accurate definition of angular velocity. Not attempting to introduce a relativistic notion of rigid body (which is well known to be ill-defined even at the first post-Newtonian approximation) we consider bodies to be deformable and introduce the post-Newtonian generalizations of the Tisserand axes and the principal axes of inertia.

scholarly journals On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

2021 ◽  
Vol 23 (2) ◽  
pp. 159-170
Author(s):  
Alexander A. Kosov ◽  
Eduard I. Semenov
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Exact Solutions ◽  
Rotational Motion ◽  
First Integral ◽  
System Of Differential Equations ◽  
Limit Values ◽  
Gyroscopic Forces ◽  
Parametric Families ◽  
Solutions Of Equations

Abstract. A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

scholarly journals Rotational Maneuvers of Copepod Nauplii at Low Reynolds Number

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 78
Author(s):  
Kacie T. M. Niimoto ◽  
Kyleigh J. Kuball ◽  
Lauren N. Block ◽  
Petra H. Lenz ◽  
Daisuke Takagi
Keyword(s):  
Reynolds Number ◽  
Rotational Motion ◽  
High Speed ◽  
The Body ◽  
Three Dimensions ◽  
Linear Path ◽  
Physical Mechanisms ◽  
High Speed Video ◽  
Video Observations ◽  
Principal Axes

Copepods are agile microcrustaceans that are capable of maneuvering freely in water. However, the physical mechanisms driving their rotational motion are not entirely clear in small larvae (nauplii). Here we report high-speed video observations of copepod nauplii performing acrobatic feats with three pairs of appendages. Our results show rotations about three principal axes of the body: yaw, roll, and pitch. The yaw rotation turns the body to one side and results in a circular swimming path. The roll rotation consists of the body spiraling around a nearly linear path, similar to an aileron roll of an airplane. We interpret the yaw and roll rotations to be facilitated by appendage pronation or supination. The pitch rotation consists of flipping on the spot in a maneuver that resembles a backflip somersault. The pitch rotation involved tail bending and was not observed in the earliest stages of nauplii. The maneuvering strategies adopted by plankton may inspire the design of microscopic robots, equipped with suitable controls for reorienting autonomously in three dimensions.

scholarly journals Rotations of small, inertialess triaxial ellipsoids in isotropic turbulence

2017 ◽  
Vol 821 ◽  
pp. 517-538 ◽  
Author(s):  
Nimish Pujara ◽  
Evan A. Variano
Keyword(s):  
Angular Velocity ◽  
Strain Rate ◽  
Rotational Motion ◽  
Isotropic Turbulence ◽  
Direct Numerical Simulations ◽  
Homogeneous Isotropic Turbulence ◽  
Particle Rotation ◽  
Principal Axes ◽  
Total Particle ◽  
Angular Velocities

The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The total particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results of Chevillard & Meneveau (J. Fluid Mech., vol. 737, 2013, pp. 571–596) who showed results of the rotation rate of the particle’s principal axes. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase as particles become elongated, regardless of whether they are axisymmetric. This trend is explained by considering the contributions of vorticity and strain rate to particle rotation. It is found that the majority of particle enstrophy is due to fluid vorticity. Strain-rate-induced rotations, which are sensitive to shape, are mostly cancelled by strain–vorticity interactions. The remainder of the strain-rate-induced rotations are responsible for weak variations in particle enstrophy. For particles of all shapes, the majority of the enstrophy is in rotations about the longest axis, which is due to alignment between the longest axis and fluid vorticity. The integral time scale for particle angular velocities about different axes reveals that rotations are most persistent about the longest axis, but that a full revolution is rare.

scholarly journals The Motion of a Rigid Body in the Presence of a Gyrostatic : الحركات المغزلية السريعة لجسم متماسك في مجال قوة نيوتوني في حالة وجود العزم

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Ghadir Ahmed Sahli
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
First Integral ◽  
Geometric Interpretation ◽  
The Body ◽  
Parameter Method ◽  
Principal Axes ◽  
Linear Autonomous System ◽  
Newtonian Force Field

In this study، the rotational motion of a rigid body about a fixed point in the Newtonian force field with a gyrostatic momentum  about the z-axis is considered. The equations of motion and their first integrals are obtained and reduced to a quasi-linear autonomous system with two degrees of freedom with one first integral. Poincare's small parameter method is applied to investigate the analytical peri­odic solutions of the equations of motion of the body with one point fixed، rapidly spinning about one of the principal axes of the ellipsoid of inertia. A geometric interpretation of motion is given by using Euler's angles to describe the orientation of the body at any instant of time.

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