Unperturbed Motion Research Articles

PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES

In the process of operation, the simplest elements (hereinafter elastic bodies) of machines and mechanisms under the influence of external and internal factors carry out complex oscillations ‒ a combination of longitudinal, bending and torsion combinations in various combinations. In general, mathematical models of the process of such complex phenomena in elastic bodies, even for one-dimensional calculation models, are boundary value problems for systems of partial differential equations. A two-dimensional mathematical model of oscillatory processes in a nonlinear elastic body is considered. A method of constructing an analytical solution of the corresponding boundary-value problems for nonlinear partial differential equations is proposed, which is based on the use of Ateba functions, the Van der Pol method, ideas of asymptotic integration, and the principle of single-frequency oscillations. For "undisturbed" analogues of the model equations, single-frequency solutions were obtained in an explicit form, and for "perturbed" ‒ analytical dependences of the basic parameters of the oscillation process on a small perturbation. The dependence of the main frequency of oscillations on the amplitude and non-linearity parameter of elastic properties in the case of single-frequency oscillations of "unperturbed motion" is established. An asymptotic approximation of the solution of the autonomous "perturbed" problem is constructed. Graphs of changes in amplitude and frequency of oscillations depending on the values of the system parameters are given.

Attitude Dynamics of Small Magnetic Axisymmetric Satellites in Near-Equatorial Low Earth Orbits/Very Low Earth Orbits

This study deals with the attitude motion of small magnetic axisymmetric satellites in near-equatorial circular low Earth orbits/very low Earth orbits subjected to the joint action of the environmental torques, namely, aerodynamic, gravitational, and magnetic. It is shown that if the satellite has only a longitudinal component of the intrinsic magnetic moment, it is convenient to separate the attitude motion into the unperturbed motion and the perturbations from the nonpotential components of the considered torques. The bifurcation analysis of the critical points of the unperturbed dynamic potential demonstrates the influence of the main system parameters on their existence and stability and provides an intuitive understanding of the possible regimes of attitude motion. Numerical simulations show that, in addition to regular attitude motion of the considered satellites, chaotic regimes are also possible.

Optimality principles of perturbed and unperturbed human squat motions using inverse optimal control

Optimality principles of perturbed and unperturbed human squat motions using inverse optimal control

Active space debris removal by ion multi-beam shepherd spacecraft

Contactless transportation of a passive object by an ion beam generated by an active spacecraft's thruster is a promising way of space debris removal. The study proposes a modification of the ion beam shepherd concept using an array of impulse transfer thrusters. The multi-beam scheme provides significant increase the generated ion force, which reduces a space debris object deorbit time, thereby reducing the probability of its collision with other orbital objects. The aim of the work is to study the possibility of using several ion beams for effective contactless transportation of a space debris object. Controlling the angular motion of the object permits orientation of the object to maximize the generated ion force. The article proposes a control law for impulse transfer thrusters, which is based on the calculation of the energy of the object's unperturbed motion, and two control schemes that implement this law: (1) relay control, which implies turning the thruster on and off, and (2) ion beam axis direction control for one of the thrusters. For a space debris object of a cylindrical shape, a comparison of the time and fuel costs required for deorbiting the object using one, two, and three impulse transfer thrusters is made. It is shown that the addition of thrusters significantly reduces the descent time, but has little effect on the mass of required fuel. For the case of two engines, the best angle of ion beams axes direction and the most preferable angular motion mode are determined. Numerical simulation of space debris removal is carried out for the case when the space debris object is in the least favorable mode of angular motion and the proposed control schemes are used. The ion beam direction control scheme showed better results than the relay scheme. The use of multiple ion beams opens up new possibilities for creating new ion beam shepherd spacecraft design and developing new control methods and laws.

Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

Calculation-empirical estimation of the vertical dynamic coefficient in the event of additional oscillations of the car

According to the results of derailment investigations, the ensuring of train traffic safety was found to be a paramount and urgent task. The growth of freight density and speed of movement on the railway network leads to a significant increase in the number of damaged wheels and rails and, as a result, to an increase in the level of force interaction during load transfer, which provokes the likelihood of a wagon derailment. The operation safety level of rolling stock on railways is determined mainly by the presence of a car derailment stability margin, which depends on the vertical dynamics and the coefficient kdv, which changes throughout the entire time of movement and, accordingly, on the technical condition of the car and the track as a whole. The vertical addition to each element of this “wheel-rail” system depends on many factors, therefore, the level of operational safety of the car, as a mechanical system, is determined mainly by its sprung and non-sprung car parts. The calculation schemes used to study the stability of unperturbed motion are non-linear systems. Non-linearities occur due to the gaps in axle boxes, side bearings and similar connections, non-linearity of the interaction forces of wheels with rails and, at present, also in some types of spring suspension. When moving in a curve, the cart is known to perform a complex movement. The rotation and transverse displacement of the bogie is prevented by frictional forces between the wheels and the rails. The bogie is affected by a part of the centrifugal force, unbalanced by the elevation of the outer rail and depending on the speed of movement, as well as by the guiding force from the side of the outer rail. It is known the process of a car movement results in the appearance of horizontal transverse (relative to the direction of movement) forces, which are determined by the level of interaction between the wheel flanges and the rails. The nature of the perturbed movement of the car leads to the appearance of horizontal transverse with respect to the axis of the path of movement. At relatively low speed, the forces that arise after wearing out of the gaps in the track and when the wheel flanges rest against the rail heads, at a certain level of irregularities, are not dangerous for the wheel-rail system. But with an increase in the speed of the car, these forces increase so that there is a threat to traffic safety as the likelihood of derailment increases.

Stability of unperturbed motion governed by the ternary differential system of Lyapunov-Darboux type with nonlinearities of degree four

For the ternary differential system of Lyapunov-Darboux type with nonlinearities of degree four, using the Lie algebra admitted by this system, was obtained the analytic first integral, determined the Lyapunov function and the conditions of stability of the unperturbed motion.

Сверхзвуковой флаттер удлиненной прямоугольной пластинки с одним свободным краем, растянутой по потоку газа

In the article, in a linear formulation, the influence of the initial stress state of an elongated rectangular elastic plate with one free edge, stretched along the gas flow, on the stability of the unperturbed motion of the "plate-flow" linear dynamic system is studied under the assumption that there are concentrated inertial masses on its free edge and moments. An analytical solution of the problem of stability of a dynamical system is obtained. The stabilizing effect of tensile forces on the system is shown.

On the exo-planet precession under torqes due to three celestial bodies with the evolution of the satellite's orbit

We investigate the non-resonant evolution of the axial tilt of hypothetical exo-Earth in the gravitational field of a star, planet's satellite (exo-Moon) and outer planet (exo-Jupiter). The exo-Earth is assumed to be rigid, axially symmetric ($A=B$) and almost spherical. We assume the orbits of the both exo-planets to be Keplerian ellipses with focus in the star, the orbit of exo-Moon to be an evolving Keplerian ellipse with slowly changing of ascending node longitude and periapsis argument. Assuming the frequencies of the unperturbed orbital elliptical motion to be of the order of unity, we obtain the canonical averaged equations describing the perturbed oscillations of the exo-Moon spin axis. These equations contain parameters changing slowly over time. Using the smallness of the planets' masses relative to the mass of the star, we have obtained simplified equations of oscillations of the exo-Earth spin axis by the small parameter method. Time integration of simplified equations gives the axial tilt of exo-Moon as a function of time. It is shown that the torques from the exo-Jupiter create a secular, long-period oscillation mode in axial tilt with a frequency equals to frequency of unperturbed spin axis precession of the exo-Earth. The impact of the exo-Moon on the evolution of the exo-Earth spin axis is that short-period harmonics appear in the oscillations of the axial tilt. The frequency of such oscillations is close to the precession frequency of the ascending node longitude of the exo-Moon orbit. We have calculated the evolution of exo-Earth axial tilt for two exo-planetary systems, i.e., for a system similar to the solar system, and for a planetary exo-system 7 Canis Majoris. The effect of destabilization (stabilization) of the exo-Earth tilt oscillations due to the torques exerted by exo-Moon and exo-Jupiter is described.

PROBLEMS OF EVOLUTION OF RIGID BODY MOTION SIMILAR TO LAGRANGE TOP

The problem of evolution of the rigid body rotations about a fixed point continues to attract the attention of researches. In many cases, the motion in the Lagrange case can be regarded as a generating motion of the rigid body. In this case the body is assumed to have a fixed point and to be in the gravitational field, with the center of mass of the body and the fixed point both lying on the dynamic symmetry axes of the body. A restoring torque, analogues to the moment of the gravity forces, is created by the aerodynamic forces acting on the body in the gas flow. Therefore, the motions, close to the Lagrange case, have been investigated in a number of works on the aircraft dynamics, where various perturbation torques were taken into account in addition to the restoring torque. Many works have studied the rotational motion of a heavy rigid body about a fixed point under the action of perturbation and restoring torques. The correction of the studied models is carried out by taking into account external and internal perturbation factors of various physical nature as well as relevant assumptions according to unperturbed motion. The results of reviewed works may be of interest to specialists in the field of rigid body dynamics, gyroscopy, and applications of asymptotic methods. The authors of this papers present a new approach for the investigation of perturbed motions of Lagrange top for perturbations which assumes averaging with respect to the phase of the nutation angle. Nonlinear equations of motions are simplified and solved explicitly, so that the description of motion is obtained. Asymptotic approach permits to obtain some qualitative results and to describe evolution of rigid body motion using simplified averaged equations. Thus it is possible to avoid numerical integration. The authors present a unified approach to the dynamics of angular motions of rigid bodies subject to perturbation torques of different physical nature. These papers contains both the basic foundations of the rigid body dynamics and the application of the asymptotic method of averaging. The approach based on the averaging procedure is applicable to rigid bodies closed to Lagrange gyroscope. The presented brief survey does not purport to be complete and can be expanded. However, it is clear from this survey that there is an literature on the dynamics of rigid body moving about a fixed point under the influence of perturbation torques of various physical nature. The research in this area is in connection with the problems of motion of flying vehicles, gyroscopes, and other objects of modern technology

Locally optimal control laws for Earth-bound solar sailing with atmospheric drag

Solar sailing is a spacecraft propulsion method relying solely on solar radiation pressure to provide thrust and is therefore propellantless by nature. Although it represents a practical and promising propulsion system particularly suited for heliocentric flight regimes, the majority of sailcraft missions flown to date have remained Earth-bound and more Earth-bound missions are scheduled for the near future. However, the fundamental dynamics and trajectory optimization of a solar sail around the Earth have only been investigated to a limited extent, often neglecting the effect of non-negligible perturbations in the dynamics and the optimal control problem. Among these perturbations are the effect of eclipses, non-spherical gravity, and aerodynamic drag. Their magnitude can be comparable to, or even exceed that of solar radiation pressure and their effect on the solar-sail dynamics should be investigated to ensure the sailcraft's transfer capabilities and controllability. This article does so by including these perturbations in the dynamics and by considering aerodynamic drag in the optimal control problem. Using this formulation, it is shown that the optimal control problem is independent of the solar-sail loading parameter and that, by solving it, locally optimal steering laws can be derived to effectively change individual orbital elements. These newly derived steering laws form an extension to the laws found by McInnes for unperturbed solar-sail Earth-bound motion. By accounting for the perturbations in the derivation of the steering laws, it is possible to characterize how the perturbations affect the solar-sail maneuvering capabilities. This is quantified based on the established increase of the targeted orbital element. Furthermore, a range of different starting orbits will be considered to analyze the effects of perturbations in different orbital regimes. As demonstration of the real need for this investigation, NASA's Advanced Composite Solar Sail System (ACS3) mission will be considered as real-case scenario. This mission is scheduled for launch in mid-2022 and may benefit from the steering laws derived in this article to prove the maneuverability of solar sails in Earth orbit.

Controllability and stability capacity of a roll pair motion research

The process of deriving the equation of the perturbed motion of a roll pair during the processing of leather material is discussed in the article, taking into account the influence of dynamic factors. It is shown that one of the reasons for the unstable stress state on the contact surfaces in the roller mechanism are dynamic factors arising from inaccuracies in the manufacture of its individual parts, assembly defects, and the occurrence of an oscillatory process in the roller mechanism, as well as due to the non-uniform thickness of the processed material at its gripping during starting and stopping the machine. Methods for determining optimal controls are shown; they provide asymptotic stability of the unperturbed motion of a roll pair and the torque applied to the upper roll as a function of generalized coordinates.It is shown that the width of the contact strip of the clamp, which depends on the radii of the rolls and the hardness of the coatings, has a significant effect on the efficiency of the rolls. The larger the shaft radius, the lower the actual pressure per unit contact area.
 It is shown that the squeezing efficiency increases with the improvement of the conditions for the removal of the squeezable liquid from the rolls (with their horizontal arrangement), with an increase in its temperature and a decrease in viscosity. Efficiency decreases with increasing material speed and thickness.
 It is shown that the width of the contact strip of the clamp, which depends on the radii of the shafts and the stiffness of the coatings, has a significant impact on the efficiency of the shafts. The larger the shaft radius, the lower the actual pressure per unit contact area

Lyapunov’s stability of the unperturbed motion governed bythe

There were obtained the conditions of stability after Lyapunov of the unper-turbed motion for the system S3 (1,3) in the non-critical case. It was constructed the Lyapunov series for the ternary differential system S3 (1,3) of Darboux type in the critical case and determined the conditions of stability of the unperturbed motion governed by this system.

The Effect of Atmospheric Drag Force on the Elements of Low Earth Orbital Satellites at Minimum Solar Activity

Researching and modeling perturbations is essential in astrodynamics because it gives information on the deviations from the satellite's normal, idealized, or unperturbed motion. Examined the impact of non-conservative atmospheric drag and orbital elements of low-earth-orbit satellites under low solar activity. The study is consisting of parts, the first looks at the effects of atmospheric drag on LEO satellites different area to mass ratios, and the second looks at different inclination values. Modeling the impacts of perturbation is included in each section, and the final portion determines the effects of atmospheric drag at various node values. The simulation was run using the Celestial Mechanics software system's SATORB module (Beutler, 2005), which solves the perturbation equations via numerical integration. The findings were examined using Matlab 2012. Conclusion that the impacts are stronger for retrograde orbits, which is due to the fact that the satellite moves in the opposite direction. The atmospheric drag effects for all orbital elements were increased by increasing the area to mass ratio. When the node value rises, the size parameter changes slightly, but the other orbital elements change. At varying inclinations, it is found that the changes in orbital elements due to atmospheric drug.

An analytic solution of full-sky spherical geometry for satellite relative motions

Herein, an exact and efficient analytic solution for an unperturbed satellite relative motion was developed using a direct geometrical approach. The derivation of the relative motion geometrically interpreted the projected Keplerian orbits of the satellites on a sphere (Earth and celestial spheres) using the solutions of full-sky spherical triangles. The results were basic and computationally faster than the vector and plane geometry solutions owing to the advantages of the full-sky spherical geometry. Accordingly, the validity of the proposed solution was evaluated by comparing it with other analytic relative motion theories in terms of modeling accuracy and efficiency. The modeling accuracy showed an equivalent performance with Vadali’s nonlinear unit sphere approach, which is essentially equal to the Yan–Alfriend nonlinear theory. Moreover, the efficiency was demonstrated by the lowest computational cost compared with other models. In conclusion, the proposed modeling approach illustrates a compact and efficient closed-form solution for satellite relative motion dynamics.

Reimagining full wave rf quasilinear theory in a tokamak

The velocity dependent resonant interaction of particles with applied radiofrequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto (J. Plasma Phys., vol. 86, 2020, 815860302). The behaviour is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). For a wave of parallel wave number ${k_{||}}$ interacting with weakly collisional plasma species of collision frequency $\nu$ and thermal speed ${v_{\textrm{th}}}$ , the effective collision frequency becomes of order $\nu {({k_{||}}{v_{th}}/\nu )^{2/3}} \gg \nu $ . The narrow boundary layers that arise because of the diffusive nature of the collisions allow a physically meaningful wave–particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behaviour of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave–particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.

Displacements of Points of a Viscoelastic Ball Caused by Tides

The derivation of equations of motion is considered in the problem of motion of two viscoelastic bodies in the central gravitational field of massive spherical rigid homogenous body with the help of the d’Alembert Lagrange variation principle. It was used the Kelvin– Voigt model of viscous forces and classical theory of small strains. The method for separation of motions was applied in order to get an approximate solution of quasi-static equation of the theory of elasticity on the unperturbed motion. There were obtained the displacements of each point of the bodies, caused by centrifugal, elastic and dissipative forces.

An Approach to Intelligent Control of Complex Industrial Processes: An Example of Ferrous Metal Industry

This paper proposes an original approach to the design of intelligent control systems for complex time-varying (nonstationary) objects in which industrial processes are repeatedly reproduced over a certain period of time. Such reproduction allows accumulating data and knowledge about control programs, conditions and results of their implementation, and the trajectories of controlled variables. The motion of a controlled object is decomposed into perturbed and unperturbed motions. The unperturbed programmed motion of the object is described using expert systems and the concept of case-based reasoning (CBR); the perturbed motion of the object, using artificial neural networks.

Influence of kinetic effects on long wavelength gradient-drift instability in high-frequency range in Hall thruster

A typical for Hall thruster discharges strong inhomogeneity of plasma parameters is a source of various types of instabilities. Some types of such instabilities (long high-frequency waves, in particular) are assumed to contribute to anomalous electron conductivity, which is essential to understand for the development of a reliable modelling tool for Hall thrusters. In this paper, the results of an analytical study of the influence of kinetic effects on dispersion properties of high-frequency perturbations in Hall thruster plasmas are presented, including the effect of the gradient magnetic electron drift on the long wavelength high-frequency instability. It was shown that when the long waves with phase velocity on the order of the unperturbed electron drift motion occur, the electron guiding center deviates significantly from the electric field equipotential line. The observed phenomenon leads to a self-sustained cycle of an intensive energy exchange between the waves and the electrons and a long-wavelength high-frequency instability inciting each other.

Chaotic motion around a black hole under minimal length effects

We use the Melnikov method to identify chaotic behavior in geodesic motion perturbed by the minimal length effects around a Schwarzschild black hole. Unlike the integrable unperturbed geodesic motion, our results show that the perturbed homoclinic orbit, which is a geodesic joining the unstable circular orbit to itself, becomes chaotic in the sense that Smale horseshoes chaotic structure is present in phase space.