Example of motion of a cylindrical solid in a viscous liquid

The problem of motion of a pulsating solid (an infinitely long circular cylinder) in an oscillating viscous liquid in the presence (or absence) of an external stationary force is considered. The perturbation method is applied. It is found that the solution of the time-average motion of a body exists if and only if body pulsations, liquid vibrations, and external forces satisfy a certain relation. The presence of a plane analog of the phenomenon of predominantly unidirectional motion of a compressible solid in an oscillating liquid is established.     

Synthesis of a controller for stabilizing the motion of a rigid body about a fixed point

A method for the approximate design of an optimal controller for stabilizing the motion of a rigid body about a fixed point is considered. It is assumed that rigid body motion is nearly the motion in the classical Lagrange case. The method is based on the common use of the Bellman dynamic programming principle and the averagingmethod. The latter is used to solve theHamilton–Jacobi–Bellman equation approximately, which permits synthesizing the controller. The proposed method for controller design can be used in many problems close to the problem of motion of the Lagrange top (the motion of a rigid body in the atmosphere, the motion of a rigid body fastened to a cable in deployment of the orbital cable system, etc.).     

Accelerating motion of a sphere in a maxwell fluid

The translatory accelerating motion of a sphere due to an arbitrarily applied force in an unlimited Maxwell fluid is considered. The exact solutions for the velocity of the sphere for three particular types of accelerating motion are presented. The first is for a falling sphere; the second is for the decelerating motion of a sphere after the force which maintains the sphere with a constant velocity is removed; the third is for the motion of the sphere subjected to an impulsive force. The exact solutions are expressed in terms of real, regular, definite integrals which can be evaluated by numerical technique. Also presented are the asymptotic solutions for the velocity of the sphere in all three cases which are valid for small values of time.     

Stability in a case of motion of a paraboloid over a plane

We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

Motion of a Sphere in a Liquid Caused by Vibrations of Another Sphere

The motion of an absolutely rigid sphere in a nonuniformly vibrating, ideal, incompressible liquid is considered. The liquid vibrates under the action of an absolutely rigid sphere vibrating in a specified manner. Refined conditions are obtained under which the inclusion sphere recedes from the vibrating sphere, approaches it, and does not perform mean motion. It is found that under nonuniform vibrations of the liquid, the motion of the inclusion can depend on the geometrical parameters of the hydromechanical system.     

Features of Movement of a Rotational Body

Vertical motion of a rotational body in an air environment as a mechanical model of a rotochute is considered. It is assumed that, in the process of motion, the symmetry axis of the rotational body remains vertical and the rotational body itself rotates relative to this axis. The aerodynamic impact model is based on a quasistatic approach. Steady regimes of motion are identified, their stability is analyzed, and certain features of transition regimes are explored, including those related to the exchange between the energy of rotational motion and the energy of translational motion.     

一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉

通过理论分析和数值仿真,研究了一类二维冲击振动系统在一种强共振条件下的Hopf分叉与亚谐分叉。分析并证实了该类系统在此共振条件下可由稳定的周期1 1振动分叉为周期4 4振动或概周期振动,讨论了亚谐振动和概周期振动向混沌运动的演化过程。     

Motion of a sphere down an inclined plane in a viscous flow

The motion of a rigid particle near a wall in a fluid flow is an important element of particle transport by fluids. The aim of this study was to carry out an experimental and theoretical investigation of the gravity-induced motion of a rigid sphere in a viscous fluid in the presence of a transverse flow. The experimental study of this configuration is a way of understanding the specific features of the hydrodynamically constrained particle motion. It is established that the transverse motion of the fluid substantially increases the particle settling velocity, which grows with increase in the transverse flow velocity. This effect is most pronounced for small angles of inclination of the plane. The difference in the particle settling velocities in the presence and absence of the transverse flow could reach a factor of two.     

Motion of a liquid bounded by a flexible film

The planar motion of an ideal incompressible liquid bounded by a flexible inextensible film is examined. Some qualitative characteristics of this motion are noted, and the hydrodynamic impact phenomena that can arise are studied.     

Continuous Restricted Radial Motion of a Gas under the Action of a Piston

The possibility of continuous conjugation of the straightline radial motion of a gas sphere toward the center and away from the center with the motion where the gas in the entire sphere stops simultaneously is shown. The motion is described by an invariant submodel of rank 1. Time reflections allow one to construct a solution that describes a periodic continuous restricted motion of the gas sphere under the action of a piston.     

Integrability of the motion of a rolling disk of finite thickness on a rough plane

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometric functions. The integrability results are used to construct various bifurcation diagrams of the steady motion of the disk. The bifurcations of the steady motion of a disk on a rough plane complements the author's bifurcation analysis of the steady motion of the disk on a smooth plane ( [M. Batista, Steady motion of a rigid disk of finite thickness on a horizontal plane, Int. J. Non-Linear Mech. 41 (4) (2006) 605–621]).     

Small oscillations of a liquid in a partitioned cylindrical vessel

The equations of motion of a rigid body whose cavity is partially filled with an ideal fluid have been obtained in works of Moiseev [1, 2, 3], Okhotsimskii [4], Narimanov [5], and Rabinovich [6]. All the equation coefficients have been calculated for a cavity in the form of a circular cylinder or two concentric cylinders.The problem of fluid motion in a partitioned cylindrical cavity was considered by Rabinovich [7]. It was also considered by Bauer [8], who analyzed the particular case of vessel motion in the plane of one of the partitions.In the following we consider the two-dimensional motion of a cylinder with radial and annular baffles, and a definition is given of the velocity potential in the case of arbitrary positioning of the radial baffles with respect to the motion plane. Formulas are obtained for determining the parameters of a mechanical analog of the wave oscillations, which consists of two mathematical pendulum subsystems.     

一类刚-梁耦合系统平面稳态运动的稳定性

研究了中心力场中的一类刚-弹耦合系统的平面运动动力学，模型是带有一悬臂 梁的刚体. 综合考虑了系统轨道运动与姿态运动，在Lagrange力学体系下给出了系统的运 动方程，在保守系统和考虑梁的材料黏滞阻尼两种情况下，利用能量-动量方法给出了一类 相对平衡点稳定性的充分条件.     

Dynamics of a satellite carrying a point mass moving about it

We study the dynamics of a complex system consisting of a solid and a mass point moving according to a prescribed law along a curve rigidly fixed to the body. The motion occurs in a central Newtonian gravitational field. It is assumed that the orbit of the system center of mass is an ellipse of arbitrary eccentricity.We obtain equations that describe the motion of the carrier (satellite) about its center of mass. In the case of a circular orbit, we present conditions that should be imposed on the law of the relative motion of the mass point carried by the satellite so that the latter preserves a constant attitude with respect to the orbital coordinate system. In the case of a dynamically symmetric satellite, we consider the problem of existence of stationary and nearly stationary rotations for the case in which the carried point moves along the satellite symmetry axis.We consider several problems of dynamics of the satellite plane motion about its center of mass in an elliptic orbit of arbitrary eccentricity. In particular, we present the law of motion of the carried point in the case without eccentricity oscillations and study the stability of the satellite permanent attitude with respect to the orbital coordinate system.     

Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass

The motion of a flexible cantilever beam carrying a moving spring-mass system is investigated. The beam is assumed to be an Euler–Bernouli beam. The motion of the system is described by a set of two nonlinear coupled partial differential equations where the coupling terms have to be evaluated at the position of the mass. The nonlinearities arise due to the coupling between the mass and the beam. Due to the nonlinearities the system exhibits internal resonance which is investigated in this work. The equations of motion are solved numerically using the Rayleigh–Ritz method and an automatic ODE solver. An approximate solution using the perturbation method of multiple scales is also obtained.     

Accelerated supersonic motion of a plate with a finite angle of attack

The article considers the supersonic unsteady-state motion of a plate with a finite (but infinitely small) angle of attack with an associated shock wave in an ideal gas. The laws governing the change of the velocity and the angle of inclination of the plate are assumed to be arbitrary, and the Strouhal numbers are assumed to be small. Under the latter assumption and with Mach numbers in the perturbed region not too close to unity, the prehistory of the motion is sufficiently well described by the instantaneous values of the parameters of the unsteady state and their derivatives. This fact permits a considerable simplification of the equations describing the perturbed unsteady-state motion, and an analytical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 70–75, July–August, 1970.     

Bifurcation and chaos of a simple walking model driven by a rhythmic signal

The walk of animals is achieved by the interaction between the dynamics of their mechanical system and the central pattern generator (CPG). In this paper, we analyze dynamic properties of a simple walking model of a biped robot driven by a rhythmic signal from an oscillator. In particular, we examine the long-term global behavior and the bifurcation of the motion that leads to chaotic motion, depending on the model parameter values. The simple model consists of a hip and two legs connected at the hip through a rotational joint. The joint is driven by a rhythmic signal from an oscillator, which is an open loop. In order to analyze the bifurcation, we first obtained approximate solutions of the walking motion and then constructed discrete dynamics using the Poincaré map. As a result, we found that consecutive period-doubling bifurcations occur as the model parameter values change, and that the walking motion leads to chaotic motion over the critical value of the model parameters. Moreover, we approximately obtained the period-doubling solutions and the critical value by employing a Newton-Raphson method. Our analytical results were verified by the numerical simulations.