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.     

Optimal control of vibrationally excited locomotion systems

Optimal controls are constructed for two types of mobile systems propelling themselves due to relative oscillatory motions of their parts. The system of the first type is modelled by a rigid body (main body) to which two links are attached by revolute joints. All three bodies interact with the environment with the forces depending on the velocity of motion of these bodies relative to the environment. The system is controlled by high-frequency periodic angular oscillations of the links relative to the main body. The system of the other type consists of two bodies, one of which (the main body) interacts with the environment and with the other body (internal body), which interacts with the main body but does not interact with the environment. The system is controlled by periodic oscillations of the internal body relative to the main body. For both systems, the motions with the main body moving along a horizontal straight line are considered. Optimal control laws that maximize the average velocity of the main body are found.     

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.     

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 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.     

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.     

Optimal Motion of a Two-Body System in a Resistive Medium

Locomotion of a mechanical system consisting of two rigid bodies, a main body and a tail, is considered. The system moves in a resistive fluid and is controlled by angular oscillations of the tail relative to the main body. The resistance force acting upon each body is assumed to be a quadratic function of its velocity. Under certain assumptions, a nonlinear equation is derived that describes the progressive motion of the system as a whole.     

Optimality conditions for the control of a double-membrane complex system by a modal decomposition technique

The optimal of damping out the oscillations of an elastically rectangular double-membrane system by means of point-wise actuators is solved analytically. The membrane is clamped along the boundaries. The motion of the system is initiated by given initial displacement and velocity conditions. The basic control problem is to minimize the deflection and the velocity of displacements at a specified time with the minimum expenditure of actuation energy. A quadratic performance functional is chosen as the cost functional which comprises the functionals of the deflection, velocity and the point-wise actuators. Necessary and sufficient conditions of optimality are investigated. The necessary conditions of optimality are obtained from a variational approach and formulated in the form of degenerate integrals which lead to explicit optimal control laws for the actuators. Numerical results are given for various problem parameters and the efficiency of the control mechanism is investigated.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Analysis and optimization of the motion of a body controlled by means of a movable internal mass

The controlled horizontal motion of a body in the presence of dry friction forces is investigated. Control is accomplished by means of a movable mass that can move within the body in a bounded range. Some simple modes of periodic relative motions of the movable mass, under which the entire system moves as a whole, are investigated. Constraints are imposed on the relative displacement, velocity and acceleration of the movable mass. The optimum parameters of this relative motion, under which the maximum mean velocity of the body is reached, are determined.     

Analysis and optimization of the rectilinear motion of a two-body system

The rectilinear motion of a system of two interacting bodies when there is a dry friction force acting on both of them is considered. It is assumed that the relative velocity of the bodies can vary practically instantaneously, while the distance between them has upper and lower limits. The periodic motion of the system as a whole is constructed, and the mean velocity of motion and the energy costs per unit of path are determined. The optimum values of the parameters for which the highest mean velocity is reached with the superimposed limitations are obtained.     

Conservative methods of controlling the rotation of a gyrostat

A method for shaping the control of the rotation of a gyrostat consisting of a rigid body, within which there are three rotors rotating about non-coplanar axes rigidly connected to the body, is discussed. The state of the system is defined by the position and angular velocity of rotation of the body, as well as by the angular velocities of the rotors. Control is achieved by torques applied to the rotors. The idea behind the proposed control method is to choose the controlling torques so that the angular velocities of rotation of the rotors are linear functions of the components of the angular velocity vector of the body. The linear dependence thus specified defines a 3 × 3 matrix, that is, a “controlled inertia tensor.” This matrix, which is specified by the parameters of the control selected, does not necessarily have the properties of an inertia tensor. As a result of such a choice of controls, the equations that define the variation of the angular velocity of the body are written in a form similar to Euler's dynamical equations. The system of equations obtained is used to formulate and solve problems of controlling the angular motion of a satellite in a circular orbit. The proposed method for constructing controlling actions enables both the Lagrangian structure of the equations of motion and the fundamental symmetries of the problem to be maintained. Expressions for the torques acting on the rotors and realizing the motion of the required classes are written in explicit form.     

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.     

UNSTEADY FREE CONVECTIVE MHD FLOW AND MASS TRANSFER THROUGH POROUS MEDIUM IN A ROTATING SYSTEM WITH FLUCTUATING HEAT SOURCE/SINK AND CHEMICAL REACTION

An investigation of unsteady MHD free convective flow and mass transfer during the motion of a viscous incompressible fluid through a porous medium in the presence of heat source, bounded by an infinite vertical porous surface, in a rotating system is presented. The porous plane surface and the porous medium are assumed to rotate in a solid body rotation. The vertical surface is subject to uniform constant suction perpendicular to it and the temperature at this surface fluctuates in time about a non-zero constant mean. Analytical expressions for the velocity, temperature and concentration fields are obtained using perturbation technique. Normal 0 false false false EN-US X-NONE X-NONE     

The unsteady motion of a viscous fluid between rotating parallel walls

The motions of a viscous incompressible fluid, rotating initially as a rigid body with constant angular velocity together with parallel walls which bound it, acted upon by suddenly starting longitudinal oscillations of one of the walls, are investigated. The walls make an arbitrary angle with the axis of rotation. In general, the solution is obtained in the form of the sum of an infinite series and is represented by an integral containing an elliptic function. A number of special cass of the motion of the wall is considered. The results obtained are used to investigate certain structures of the boundary layers on the walls.     

Control of an elastic manipulator arm using load position and velocity feedback

The rotation of an elastic manipulator arm about one of its ends in the horizontal plane is investigated. A load is attached to the other end. The motion is effected by an electric motor. The control is constructed in the form of linear feedback on the position of the load, its velocity, and the angular velocity of the arm. The stability of the control process is investigated. It is shown that when there are no viscous damping forces proportional to the angular velocity of the arm, load position and velocity feedback leads to undamped oscillations of the system and the desired equilibrium position is not stabilized. Asymptotic stability domains in the feedback coefficient space when viscous damping is present are constructed. Comparison shows these domains to be smaller than corresponding domains for a completely rigid body.     

Characteristic features of the high-velocity motion of bodies in dense media

The characteristic features of the high-velocity motion of conical and pyramidal bodies are investigated when the force acting on their surface is described by a local interaction model. It is assumed that the pressure on the body surface is represented by a binomial formula that is quadratic in the velocity. Three friction models are used to represent the tangential stresses: constant friction, friction that is proportional to the pressure and mixed friction. Analytical solutions of problems of the plane inertial motion of slender bodies with a base contour in the form of a circle, a rhombus or a star consisting of four cycles are constructed for an unseparated flow past the bodies and small perturbations imposed on the parameters of the linear motion at the initial instant of time. A criterion for the stability of the motion is found that enables the perturbed motion of the body to be determined when the medium parameters and the velocity, mass and shape of the body are known. The analytical results are validated by a numerical solution of the Cauchy problem for a system of equations of motion obtained without simplifying assumptions.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

On hydromagnetic rotating flow of an Oldroyd-B fluid near an oscillating plate

An initial value investigation is made of the motion of an incompressible, viscoelastic, conducting Oldroyd-B fluid bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with a constant angular velocity about an axis normal to the plate. The flow is generated from rest in the rotating viscoelastic system due to harmonic oscillations of a given frequency superimposed on the plate in presence of a transverse magnetic field. The exact solutions for the velocity field and the wall shear stress are obtained. The results are examined quantitatively for a particular case of an impulsively moved plate and the effects of various flow parameters on them are discussed. Many known results are found to emerge as limiting cases of the present analysis.     

The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point

The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application.