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 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.     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

The self-propulsion of a body with moving internal masses in a viscous fluid

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier — Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.     

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 stability boundaries of the steady motion of a satellite with a gyroscope

Parts of the asymptotic stability boundaries of the uniform motion of the centre of mass of a system of bodies consisting of an asymmetrical satellite with a three-axis gyroscope in a circular orbit are investigated by the second Lyapunov method. Terms of the Lyapunov function that are higher than the second order are enlisted for the investigation. The sign-definiteness criterion of inhomogeneous forms is employed for the corresponding function. Parts of the stability boundaries in which the steady motion investigated is asymptotically stable are established using the Lyapunov asymptotic stability theorem. Application of the Barbashin and Krasovskii theorems reveals parts of the stability boundaries in which the steady motion is unstable. It is established that the asymptotic stability of the steady motion investigated is solved by expanding the Lyapunov function to sixth-order terms.     

Dynamics of a Stewart platform

The kinematics and dynamics of a Stewart platform supported by six pneumocylinders are considered. The differential equations of motion are written, and the forces providing the fulfillment of the formulated law of motion are calculated. The inertia and weight of pneumocylinders are introduced into consideration to refine the equations of motion. The obtained equations are used to study the motion of a loaded Stewart platform, providing the stability of this motion by means of feedback control. Some numerical examples are given.     

Weak Convergence of a Numerical Method for a Stochastic Heat Equation

Weak convergence with respect to a space of twice continuously differentiable test functions is established for a discretisation of a heat equation with homogeneous Dirichlet boundary conditions in one dimension, forced by a space-time Brownian motion. The discretisation is based on finite differences in space and time, incorporating a spectral approximation in space to the Brownian motion.     

Phenomenological model of interaction of a plate with a flow

In the present paper, a finite-dimensional phenomenological model of unsteady interaction of a rigid plate with a flow is proposed. It is assumed that the plate performs translational motion across the flow. The internal dynamics of the flow is modeled by the attached second order dynamical system. It is shown that the model allows satisfactory agreement with experimental data. With the developed model an inverse problem of dynamics is examined for the situation where the plate performing uniform translational motion at some moment begins uniform deceleration and finally stops. It is shown that for sufficiently large values of the plate acceleration for some time range the flow does not resist the motion of the plate but “accelerates” it. It is shown also that the equations of motion in the context of the proposed model can be reduced to the integro-differential form, and comparison with the known model of S. M. Belotserkovsky is performed. The structural resemblance of the motion equations for a body in flow in both models is noted. The domain of applicability of the quasi-stationary model is examined. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 43–62, 2005.     

Understanding reversals of a rattleback

A counterintuitive unidirectional (say counterclockwise) motion of a toy rattleback takes place when it is started by tapping it at a long side or by spinning it slowly in the clockwise sense of rotation. We study the motion of a toy rattleback having an ellipsoidal-shaped bottom by using frictionless Newton equations of motion of a rigid body rolling without sliding in a plane. We simulate these equations for tapping and spinning initial conditions to see the contact trajectory, the force arm and the reaction force responsible for torque turning the rattleback in the counterclockwise sense of rotation. Long time behavior of such a rattleback is, however, quasi-periodic and a rattleback starting with small transversal oscillations turns in the clockwise direction.     

固壁近旁Stokes流中粘性液滴的运动和变形

发展了一种模拟固壁近旁轴对称Stokes流中粘性液滴的运动和变形及直接计算固壁上应力的边界积分方法．用此方法对不同的液滴-固壁初始相对间距、粘度比、表面张力和浮力联合参数以及环境流动参数情况进行了数值实验．数值结果显示,由于环境流动和浮力的作用,随着时间的推进,液滴在轴向压缩,在径向拉伸．当环境流动的作用弱于浮力作用时,随着时间的推移,液滴上升并向上弯,固壁上由液滴运动所引起的应力不断减小．当环境流动的作用强于浮力作用时,随着时间的推移,液滴变得越来越扁．在这种情形,当大初始间距时,壁面上的应力随液滴的演变而增大;当小初始间距时,由环境流动、浮力及壁面对流动的较强作用的联合影响,此应力随液滴的演变而减小．由于液滴运动所引起的壁面应力的有效作用仅限于对称轴附近的一个小范围内,且此范围随液滴与固壁的初始间距增大而增大．应力的大小随初始间距增大而大为减小．表面张力对液滴变形有阻止作用．液滴粘性会减小液滴的变形和位置迁移．     

Time-optimal steering of an object moving in a viscous medium to a desired phase state

The two-point problem of the time-optimal attainment of a desired phase state by a multidimensional dynamic object is investigated. The motion occurs in a viscous medium by means of a limited force. The open-loop and/or feedback control laws constructed by numerical-analytical methods for arbitrary initial data. An asymptotically approximate solution of the maximum principle boundary-value problem is presented for short and long time intervals. The singularities of the optimal trajectory are established for the initial and final parts of the motion. The solution obtained of the two-point problem of the optimal control of the motion of a dynamic object in a homogeneous viscous medium by means of a force of bounded modulus is compared with the known solutions in special formulations.     

On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

梢涡流场中气泡轨迹及形态数值模拟

针对气泡在舰船尾迹涡流场运动特性,根据其是否为尾涡所捕获,将数值模拟过程分为两个阶段:准球状运动阶段和非球状运动阶段．分别应用单向耦合质点粒子追踪法（PTM）和边界元法（BEM）模拟这两个阶段,将第1阶段结束的物理量作为第2阶段的初始条件,从而完成整个数值模拟过程．在已有数值研究结果和实验数据基础上,探讨空化发生条件,追踪尾迹空泡运动轨迹,模拟尾迹气泡的运动、变形、溃灭等,以及被尾涡捕获后的撕裂等运动特性,旨在为优化设计尾流场提供参考．     

Novel three-piston pump design for a slipper test rig

Slipper's micro motions including the squeezing motion, spinning motion, and tilting motion have a significant impact on its lubricating condition and dynamic behavior. However, few experimental studies are on these micro motions within a real axial piston pump, especially the slipper's spinning motion. The experimental investigations on the slipper in the past mainly focused on the parameters of the oil film such as pressure, thickness, and temperature. The sensors were often installed in the fixed swash plate when the cylinder block was chosen to rotate. Alternatively, the sensors were mounted in the fixed modified slipper when the swash plate rotated. The biggest challenge of the direct measurements of these micro motions is the space limitation for the sensor installation due to the compact structure of axial piston pumps as well as the slipper's macro motion. This paper presents a new three-piston pump for the slipper test rig which can provide enough installation space for the sensor. To realize the cylinder block balance, a hold-down plate is first introduced into this three-piston pump. In addition, a detailed set of relevant equations is derived to evaluate the functionality of the hold-down plate. Finally, the slipper's spinning motion was measured directly and continuously using this three-piston pump, which confirmed the capability of the slipper test rig.     

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.     

Feasible command strategies for the control of motion of a simple mobile robot

The concept of feasible command strategies is introduced and its applicability is demonstrated by solving a guidance and control problem. This problem concerns the motion of a system which is composed of a rolling disk and a controlled slender rod that is pivoted, through its endpoint, about the disk center. The motion of the disk-rod system is subjected to state and control constraints, and it serves as a model for the motion of a simple mobile robot. In addition, the concept of path controllability is introduced and a condition is derived for the system motion path controllability. The derivation of this condition enables one to design closed-loop control laws for the system motion.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Process of establishing a plane-wave system on ice cover over a dipole moving uniformly in an ideal fluid column

We consider a planar evolution problem for perturbations of the ice cover by a dipole starting its uniform rectilinear horizontal motion in a column of an initially stationary fluid. Using asymptotic Fourier analysis, we show that at supercritical velocities, waves of two types form on the water–ice interface. We describe the process of establishing these waves during the dipole motion. We assume that the fluid is ideal and incompressible and its motion is potential. The ice cover is modeled by the Kirchhoff–Love plate.     

Modelling and control of the motion of a disk rolling on a spherical dome

This work deals with the modelling and control of the motion of a disk rolling without slipping on a rigid spherical dome. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. First, a mathematical model of the motion of the disk rolling on the dome is derived. Then, by using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the spherical dome.