Gyrostatic Moment Research Articles

Modeling analysis on the influence of the gyrostatic moment on the motion of a charged rigid body subjected to constant axial torque

Modeling analysis on the influence of the gyrostatic moment on the motion of a charged rigid body subjected to constant axial torque

The asymptotic solutions for the motion of a charged symmetric gyrostat in the irrational frequency case

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange’s scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body’s movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler–Poisson are solved through the utilization of Poincaré’s small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body’s position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems.

Evaluating the motion of a charged solid body having a globular cavity

The dynamical motion of a charged symmetrical solid body (SB) about its main symmetric axis is addressed in this work. The body is supposed to have a chamber with a globular shape, in which it contains a mobile mass and filled with a viscous liquid. This mass is assumed to be coupled via a double elastic band at a point that is situated along the dynamical symmetry axis and produces viscous friction. The body’s motion is influenced by a gyrostatic moment (GM), whose first two components on the body’s main axes are chosen to be zero. Additionally, the body is subject to the effects of an electromagnetic field, owing to a located point charge on the body’s dynamic symmetry axis. On the indicated motion, the combined impact of the viscous liquid and the mobile mass is investigated. The achieved outcomes show that, in the presence of internal dissipation, the system's motion tends towards a constant rotation around its axis at the point of highest inertia as time approaches infinity. To demonstrate how the body’s parameters affect the motion, the obtained outcomes are graphed. The method of fourth-order Runge-Kutta (4RKM) is employed to generate the governing system's numerical results, which are then displayed in additional diagrams. This work's relevance is concentrated on its considerable potential for usage in the submarine and gyroscope industries. The novelty of this work lies in the intricate interplay of rotational dynamics due to the GM, viscous liquid and electromagnetic effects to achieve new results, besides the gained numerical outcomes. Therefore, these results are considered novel and have significant theoretical and practical relevance. Moreover, it can be used to study the motion of celestial bodies, such as planets or moons, which might have internal cavities filled with fluids or other materials. The charged nature could be relevant for bodies with significant magnetic fields.

Analyzing the dynamics of a charged rotating rigid body under constant torques

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler’s equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler’s angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body’s parameters have on the motion’s behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body’s motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

Движение гиростата под действием следящего момента силы

The gyrostat moves around a fixed point under the forces influence in pseudo-Euclidean space with signature (3,1). The resulting moment of the external forces system and the gyrostatic moment are specified relative to the coordinate orthoreper, invariably associated with the gyrostat carrier, and are constant. Three types of gyrostat motion are given, in each of which the resulting force vector-moment is collinear to of the main axis of the gyrostat inertia, and the gyrostatic moment is orthogonal to this axis. The vector manifold of the gyrostat angular velocities and the field of its phase trajectories in the state space are found with the parametric integration method using. Integration of the gyrostat motion equations is carried out in quadratures and elliptic functions of time.

Движение гиростата вокруг центра инерции в полуевклидовом пространстве

The inertial motion of a gyrostat in a semi-Euclidean space with given index and defect is studied. A gyrostat with a constant gyrostatic moment moves so that its carrier rotates around a fixed center of inertia. Criteria for the existence of regular motions are obtained as a condition for the presence of axial structural-kinetic symmetry of the gyrostat. The properties of nutational, precessional, vibrational-rotational motions are studied and their description in configuration and phase spaces is given. The quadrature dependences of the gyrostat motion parameters in elliptic functions of time are determined. Parametric equations of hodographs of angular velocity and kinetic momentum vectors are found. The study was carried out for the case of the gyrostat angular momentum eigenvector.

Studying the influence of a gyrostatic moment on the motion of a charged rigid body containing a viscous incompressible liquid

The rotational motion of a charged rigid body (RB) is examined. The RB has a spherical cavity that contains an incompressible viscous liquid. The influence of a gyrostatic moment (GM), constant torques at the body-connected axes, and the action of the torque of a resistant force, due to the shape of the liquid, are considered. Assuming the liquid has a sufficiently high velocity, the Reynolds number does indeed have a small value. The regulating system of motion is derived in an appropriate formulation through Euler's equations of motion. The averaging method is used to approach a suitable form of the motion's governing system. In addition to using Taylor’s method to reach a solution for the averaged equations of motion of the RB, some initial conditions are considered to approach the required results. The asymptotic approach of the averaged system besides the numerical analysis enables us to obtain the appropriate results of the problem. To draw attention to the beneficial effects of the different values of the body’s parameter on the motion's behavior, these results are graphed through a computer program along with the associated phase plane curves. These diagrams illustrate the influence of several values respected to the GM, charge, body-constant torques, and resistive force torque. The stability of the RB's motion has also been discussed through the represented phase plane diagrams. These results are viewed as a generalization of prior ones, which have been reported for the scenario of an uncharged body or the absence case of the GM. The significance of the obtained results is due to its numerous real-world applications in life, such as for spaceships and wagons carrying liquid fuel.

Stability analysis of an acted asymmetric rigid body by a gyrostatic moment and a constant body-fixed torque

This work focuses on the dynamical rotatory motion of an asymmetric rigid body (RB) subjected to a constant body-fixed torque and a vector of a gyrostatic moment (GM). The motion is considered in the absence of the first two components of the GM. The novelty of the current work is the conversion of the stability analysis of this body from a two dimensional phase plane to three ones phase space. Specifically, two scenarios of stationary torques are considered, the first one is directed on the minor or major axis, while the latter is presumed to act on the middle axis. Therefore, three dimensional phase space routes are generated. In both scenarios some novel analytical and simulation outcomes are provided regarding equilibrium manifolds, periodic solutions or non-periodic ones, separator surfaces, as well as the extreme periodic solutions. The significance of this work is due to its great applications, especially those that use the gyro’s theory.

Linear Invariant Relations for the Equations of Motion of a Gyrostat with a Variable Gyrostatic Moment

Linear Invariant Relations for the Equations of Motion of a Gyrostat with a Variable Gyrostatic Moment

Simulation of a Subjected Rigid Body Motion to an External Force and Moment

PurposeThis work intends to investigate the rigid body’s motion around a specific fixed point (analogous to Lagrange’s scenario) in the presence of a gyrostatic moment (GM) besides the attraction of a Newtonian force field (NFF). This task is carried out by presuming that the body is quickly rotating about one of the major or minor principal axes of the inertia ellipsoid.MethodThe controlling system of six nonlinear differential equations (DEs) along with three first integrals is boiled down to an appropriate system of two DEs in addition to only one integral. Therefore, the analytic solutions of this system are obtained utilizing the approach of Poincaré small parameter (APSP).ResultsEuler's angles for the motion under investigation are derived to assess this motion at any instant of time. Additionally, phase plane graphs are displayed using computer codes to depict the stability behavior of the dynamical motion at any time.ConclusionThese achieved outcomes are thought of as a generalization of the ones that were found in some of previous works, in the absence of all applied forces and moments. This work presents a distinctive contribution in several crucial areas, particularly in engineering applications that have used the gyroscopic theory to determine the orientation and maintain the stability of various vehicles, such as spaceships, airplanes, submarines, and racing cars.

On a Class of Semi-Regular Gyrostat Precessions with Variable Gyrostatic Moment

On a Class of Semi-Regular Gyrostat Precessions with Variable Gyrostatic Moment

Studying the vibrational motion of a rotating symmetrically charged solid body subjected to external forces and moments

This paper studies the rotatory vibrating motion of a dynamically symmetric charged solid body (SB) connected with a spring around a fixed point as a novel model. The body is assumed to be influenced by a Newtonian force field (NFF) besides extrinsic moments like perturbing, restoring, and gyrostatic moment (GM). These moments are guided along the body’s principal axes of inertia. The body is assumed to have a high starting angular velocity in the direction of the dynamic symmetry axis, and the perturbing moments (PM) are thought to be less than the restoring moment. These assumptions provide us with a tiny parameter, to use the averaging technique (AT), and to acquire the averaging system of the regulating one. Several applications are introduced to achieve the solutions of this system separately. These solutions are graphed to reveal the positive influences of the body’s numerical values on the behavior of the body, taking into account the various impacts of the applied moments, the electromagnetic field, and the Newtonian one for all examined applications. The obtained results are considered a generalization of those which were obtained in related previous works. For the projections of the body angular velocity, the proportion of differences as a function of the investigated parameters varies between 17% and 35%, whereas the other parameters are varied between 0.0011% and 0.37%, respectively. Therefore, these outcomes are considered to be significant due to their applications in life specially that based on the theory of vibrating systems in physics and engineering fields.

On a Class of Semi-Regular Gyrostat Precessions with Variable Gyrostatic Moment

The article considers the problem of the motion of a gyrostat under the action of potential and gyroscopic forces in the case of a variable gyrostatic moment. The conditions for the existence of semi-regular precessions characterized by the constancy of their own rotation rate are studied. A new solution of the equations of the Kirchhoff–Poisson class is constructed, based on a special type of three invariant relations with respect to the main variables of these equations.

On a Class of Semi-Regular Gyrostat Precessions with Variable Gyrostatic Moment

The article considers the problem of the motion of a gyrostat under the action of potential and gyroscopic forces in the case of a variable gyrostatic moment. The conditions for the existence of semi-regular precessions characterized by the constancy of their own rotation rate are studied. A new solution of the equations of the Kirchhoff–Poisson class is constructed, based on a special type of three invariant relations with respect to the main variables of these equations.

Analysis of the Rotational Motion of a Solid Body in the Presence of External Moments

PurposeIn this work, the vibrating motion of a dynamically symmetric solid body with an elastic string, around a fixed point, as a new model, is investigated.HypothesesThe body spins as a result of external moments along the body’s principal axes, such as perturbing, restoring, and gyrostatic moments. It has been supposed that this body has a rapid angular velocity at the beginning of motion in the direction of its symmetry dynamic axis.MethodsThe averaging approach is employed to transform the governing system of motion into another appropriate averaging one to gain the asymptotic solutions of this system.ConclusionThese solutions are graphed and discussed in different plots depending on the numerical values of the body’s physical properties. The positive effects of the gyrostatic moment vector on the motion are examined in some cases. Furthermore, the obtained results generalize the previously related studies.ApplicationsThe importance of the investigated dynamical system is due to its applications in a variety of domains, including physics and engineering.

Precessions of a Gyrostat under the Action of Potential and Gyroscopic Forces in the Case of a Variable Gyrostatic Moment

Precessions of a Gyrostat under the Action of Potential and Gyroscopic Forces in the Case of a Variable Gyrostatic Moment

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

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.

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

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.

Использование системы двигателей-маховиков при реализации режима солнечной ориентации космического аппарата

The study is devoted to maintaining the solar orientation mode of the gyrostat spacecraft in the low earth orbit through a long period. The device shape is close to a cylinder with two fixed solar panels installed along the cylinder longitudinal axis symmetrically relative to it. In the solar orientation mode, the normal to the device solar arrays plane is invariably directed towards the Sun and the longitudinal axis oscillates relative to the orbit plane. To implement the specified spacecraft motion mode, a system of four flywheel engines is used, which rotation axes are directed in parallel to the side edges of the quadrangular pyramid. Position of the pyramid side edges relative to the coordinate system rigidly connected to the device is determined by two angles, which are the parameters of the flywheel engine system. The law of the gyrostatic moment control is considered ensuring attenuation of the perturbed spacecraft motion in the vicinity of its solar orientation and limiting accumulation of its intrinsic kinetic moment of the flywheel engines by controlling the device rotation angle around the normal to the light-sensitive side of the solar arrays. It is demonstrated that selection of certain parameters of the flywheel engines coordinate system would assist in implementing the solar orientation mode without unloading the gyrostatic moment during the entire flight time. Results of numerical simulation of the complete system of the spacecraft motion equations in its solar orientation mode are presented taking into account the influence of gravitational and aerodynamic moments, which confirms correctness of the selected parameter values.