Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

The energy-optimal motion of a vibration-driven robot in a resistive medium

The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. © Elsevier Ltd. All rights reserved.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

Optimal control of the motion of a multilink system in a resistive medium

The optimal control of the motion of a system consisting of a main body and one or two links joined to it by cylindrical joints in a resistive medium is investigated. The resistance force of the medium acting on the moving body is assumed to depend on their velocity. The control is accomplished through high-frequency angular oscillations of the links. The equations of motion are analysed, and the mean velocity of translational motion of the system is estimated under certain assumptions. Optimal control problems are formulated and solved, and the laws of control of the oscillations of the links for which the maximum mean velocity of motion is obtained are found as a result. The data obtained are in qualitative agreement with observations of the swimming of fish and animals. The results of this study can be used in developing mobile robots that move in a liquid.     

Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

Rotations of a near-symmetrical satellite in an elliptical orbit with Mercury-type resonance

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.     

Optimal control of the rectilinear motion of a two-body system in a resistive medium

The rectilinear motion of a two-body system is considered. One of the bodies (the main body) interacts with a resistive environment, while the other body (the internal body) interacts with the main body but does not interact with the environment. The force applied to the internal body leads to a reaction that acts on the main body and produces a change in its velocity, which causes a change in the resistance of the environment to the motion of the main body. Thus, by controlling the motion of the internal body, one can control the external force acting on the main body and, as a consequence, the motion of the entire system. A periodic motion of the internal body relative to the main body, which generates the motion of the main body with periodically changing velocity and the maximum displacement for the period, is constructed for a wide class of laws of resistance of the environment to the motion of the main body.The principle of motion considered is appropriate for mobile mini- and micro-robots. The body (housing) of such robots can be hemetically sealed and smooth, without protruding parts, which enables these robots to be used for the non-destructive inspection of miniature engineering structures such as thin pipe-lines, as well as in medicine. Problems of optimizing the control modes for such systems are of interest both to researchers in the field of optimal control and to specialists in applied mechanics and robotics.     

The effect of the elasticity of an orbital tether system on the oscillations of a satellite

An orbital tether system, including a satellite (a rigid body), an elastic ponderable tether and a terminal load, is investigated. A mathematical model is obtained using Lagrange's equation of the second kind, which enables the plane translational motion of the centres of mass of the elements of the system and the rotational motion of the satellite and the tether to be investigated. It is shown that the equations of motion for the new independent variable, that is, the true anomaly angle, obtained on the assumption that the motion of the centre of mass of the system is independent of the relative motion of its elements, are an extension of the known mathematical models. The effect of the elasticity of the tether on the angular oscillations of the tether and the satellite is investigated. The model constructed can be used both to analyse of the deployment of a tether system as well as to investigate of the combined behaviour of a satellite and a tether about the natural centres of mass.     

The quickest transfer of a spatial dynamic object into a circular domain

The spatial problem of the time-optimal transfer of a point mass by a limited force onto a terminal set in the form of a circle without fixing the final velocity is investigated. The optimal modes of motion are constructed and investigated for arbitrary initial values of the three-dimensional position and velocity vectors using the maximum principle. The governing relations are obtained in the form of fourth-order and eighth-order algebraic equations for the minimum time of motion, which enable the dependence on the initial data to be investigated constructively. The qualitative features of the solution due to a jump discontinuity in the minimum time of motion, which lead to jumps in the control vector, are established. The problem is solved approximately by perturbation methods for the cases of motion close to singular ones. A complete investigation of the control problem for the motion of an object in the plane of a circle and close to it is presented using an original numerical-analytical approach.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.     

The effect of external perturbing moments on the dynamics of a uniaxial single-flywheel attitude control system of a spacecraft PMM vol. 31, no. 6, 1967, pp. 1095–1103

Effective damping of relative craft oscillations in a single-flywheel uniaxial attitude control system whose purpose is to keep a spacecraft oriented in some direction in space, e.g. towards the Sun, can be achieved by nonrigid coupling of its body with the axis of rotation of the flywheel [1]. This can be accomplished by adequately restricting two of the degrees of freedom of the flywheel in a three-degree universal suspension by means of a proportional damper of any type.

In the present paper we shall consider the stability of free oscillations of a spacecraft equipped with a flywheel on a movable axis, as well as the motion of tlie oriented craft axis under the action of external perturbing moments and the perturbed motion of the craf itself relative to the oriented axis.     

Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses

The problem of the optimal control of a rigid body moving along a rough horizontal plane due to motion of two internal masses is solved. One of the masses moves horizontally parallel to the line of motion of the main body, while the other mass moves in the vertical direction. Such a mechanical system models a vibration-driven robot–a mobile device able to move in a resistive medium without special propellers (e.g., wheels, legs or caterpillars). Periodic motions are constructed for the internal masses to ensure velocity-periodic motion of the main body with maximum average velocity, provided that the period is fixed and the magnitudes of the accelerations of the internal masses relative to the main body do not exceed prescribed limits. Based on the optimal solution obtained for a fixed period without any constraints imposed on the amplitudes of vibration of the internal masses, a suboptimal solution that takes such constraints into account is constructed.     

The motion of a cylindrical body in a stratified fluid under the action of a radiation force

The free motion of a thin cylindrical body is investigated based on a previously derived expression for the radiation force acting on moving point sources in a stratified fluid. The fundamental equations of motion are derived, the limits of applicability of the approximation used are indicated and the results of calculations of typical trajectories of a body which begins to move with a specified velocity from a position of neutral buoyancy at an angle to the horizon are presented. Calculations of the trajectory of motion of a thin cylindrical body in a stratified fluid when the total radiation force is taken into account show that the effect of the lateral component of this force is considerable and leads not only to quantitative corrections but also to qualitative effects (for example, to an increase in the oscillations of the body and a change in its direction of motion). The results obtained pertain both to the motion of solids in fluids and to the translational motion of vortex dipoles in weakly stratified media.     

Optimal motion of a multilink system in a resistive medium

We consider the motion in a resistive medium of a mechanical system consisting of a main body and one or two links attached to it by means of cylindrical joints. The motion is controlled through high-frequency periodic oscillations of the links. For this system, an equation of motion is deduced and the average velocity of locomotion is estimated under certain assumptions. This velocity is positive if the angular velocity of diverting the attached links is less than the angular velocity of bringing them to the axis of the body. An optimal control problem of maximizing the average velocity is formulated and solved. An example is given.     

The dynamics of pyramidal bodies within the framework of the local interaction model

Two-dimensional inertial motion of pyramidal bodies in a medium is investigated, on the assumption that the force exerted by the medium on their surface is described by the local interaction model. Assuming unseparated flow around the bodies and small perturbations applied at the initial time to the parameters of rectilinear motion, an analytical solution is constructed of the problem of the two-dimensional motion of slender bodies with bases whose contour is a rhombus or a star consisting of four symmetrical cycles. It is shown that the solution provides the basis for a complete parameterc analysis of the dynamics of the body and for evaluating the forces and torques experienced by the body along its trajectory. A criterion for the stability of the body is found, using which, knowing the velocity, mass and position of the body's centre of gravity, one can determine the form of the perturbed motion of the pyramidal body. It is shown that the body shape is one of the most important factors affecting the stability of motion, and that, of all bodies with the same shape and position of the centre of mass, those with the least mass have the largest reserve of stability. The analytical results are confirmed by numerical solution of the Cauchy problem for the system of equations of motion obtained without the simplifying assumptions.     

Optimum velocity for a person moving under rain

The total rain received by a moving body has previously been modelled by defining a wetness function. Several cases such as one-dimensional motion of an inclined plane, two-dimensional motion of an inclined plane, motion with time-varying velocity, inclined rain for an inclined plane, rain on a cylindrical surface and three-dimensional motion of a convex body were treated in detail. One of the major conclusions was that for a fixed distance, assuming vertical rain, the body should travel as fast as possible since the total wetness decreases with increasing velocity. The wetness function was shown to decrease asymptotically to a constant value as the velocity increases and in the high-speed range, increasing the velocity does not decrease the wetness substantially. One might think that the excess amount of energy required for a higher speed does not compensate for the small fraction of decrease in wetness. In this work, criteria are developed for a critical speed (optimum speed), for which the wetness is small enough for a reasonable energy consumption. Three cases are investigated: (1) vertical rain; (2) rain inclined towards the body; (3) rain inclined away from the body. In the first two cases, there is no absolute minimum for the wetness function and the optimum velocity is determined by special criteria. The third case is somewhat different, however, and if the inclination angle is higher than a critical value, an absolute minimum for wetness is obtained and the optimum velocity for this case is defined to be the velocity corresponding to this absolute minimum. Therefore the definition of optimum velocity is qualitatively different from the first two cases.