The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of the resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.     

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 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.     

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.     

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.     

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.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

Vibrations of a rotating oblate ellipsoidal shell

We consider the vibrations of an oblate ellipsoidal shell with a central rigid insertion rotating with a constant angular velocity about a vertical symmetry axis. The spin axis rotates uniformly. We construct partial differential equations describing the periodic elastic motion of the shell due to the translational, relative, and coriolis inertial forces. The displacement in the equatorial region of the shell is found as a function of the spin angular velocity for different values of the geometric parameters of the shell.Translated Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 81–84, 1989.     

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.     

周期载荷下超弹性圆柱壳的动力响应

研究了不可压超弹性圆柱壳在内表面周期载荷及突加常值载荷作用下的运动与破坏等动力响应问题．通过对所得描述圆柱壳内表面运动的非线性常微分方程解的数值计算和动力学定性分析,发现存在一个临界载荷;当突加常值载荷或周期载荷的平均载荷值小于这一临界值时,圆柱壳的运动随时间的演化是周期性的或拟周期性的非线性振动,而当其大于这一临界值时,圆柱壳将被破坏．另外,准静态问题的解可作为突加常值载荷作用下系统动力响应解的不动点,且不动点的性质与动力响应解及圆柱壳运动的性质有关．讨论了圆柱壳的厚度和载荷等参数对临界载荷值和圆柱壳运动特性的影响．     

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.     

Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid

We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.     

Drag force and virtual mass of a cylindrical porous shell in potential flow

We study the two-dimensional potential flow due to a circular cylinder in motion relative to an unbounded fluid. The cylinder consists of a thin, circular porous shell with fluid inside. The full nonlinear hydrodynamic problem is solved by Fourier expansion of Green's theorem. The truncated series is determined numerically by sampling points around the circle. A dimensionless shell parameter is introduced. For homogeneous porous shells, a maximal drag force occurs at the value 0.433 for the shell parameter, but the virtual mass is a monotonous function of the shell parameter. For an inhomogeneous shell, we have found a maximal value for the virtual mass which is 5% above the value for a rigid cylinder. Some of the results may be relevant to offshore engineering, especially in connection with porous coating of platform legs to reduce the total force.     

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 optimal periodic motions of a two-mass system in a resistant medium

The rectilinear motions of a two-mass system, consisting of a container and an internal mass, in a medium with resistance, are considered. The displacement of the system as a whole occurs due to periodic motion of the internal mass with respect to the container. The optimal periodic motions of the system, corresponding to the greatest velocity of displacement of the system as a whole, averaged over a period, are constructed and investigated using a simple mechanical model. Different laws of resistance of the medium, including linear and quadratic resistance, isotropic and anisotropic, and also a resistance in the form of dry-friction forces obeying Coulomb's law, are considered.     

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.     

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.     

一类非线性发展方程组的定解问题

将不可压缩的广义neo-Hookean材料组成的超弹性圆柱壳径向对称运动的数学模型归结为一类非线性发展方程组的初边值问题.利用材料的不可压缩条件和边界条件求得了描述圆柱壳内表面径向运动的二阶非线性常微分方程.给出了微分方程的周期解(即圆柱壳的内表面产生非线性周期振动)的存在条件,讨论了材料参数和结构参数对方程的周期解的影响,并给出了相应的数值模拟.     

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.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.