Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field

We present complete results devoted to the study of the equations of motion of a dynamically symmetric four-dimensional rigid body in a nonconservative force field. The form of the body is taken from the dynamics of real two- or three-dimensional rigid bodies interacting with a resisting medium according to the streamline flow around laws under which a non-conservative pair of forces acts on the body and forces the body center of mass to move rectilinearly and uniformly.     

Dynamical systems with variable dissipation: approaches, methods, and applications

This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can arouse the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under streamline flow-around conditions. The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense; i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci). The author obtains new families of phase portraits of systems with variable dissipation on lowerand higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness, He discovers new integrable cases of rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in an over-running medium flow.     

Auto-oscillations during deceleration of a rigid body in a resisting medium

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.     

Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field

This work is a relatively final result in studying the equations of motion of a dynamically symmetric, four-dimensional rigid body in a nonconservative force field in two logically possible cases of its tensor of inertia. The form of the force field considered is taken from the dynamics of real three-dimensional rigid bodies interacting with a medium.     

Simulation of the Spatial Action of a Medium on a Body of Conical Form

We consider amathematical model of the spatial action of a medium on the axisymmetric rigid body whose external surface has a part that is a circular cone.We present a complete system of equations of motion under the quasistationary conditions. The dynamical part forms an independent system of the sixth order in which the independent subsystems of lower order are distinguished. We study the problem of stability with respect to the part of variables of the key regime—the spatial rectilinear translational deceleration of the body. For a particular class of bodies, we show the inertial mass characteristics under which the key regime is stable. For a plane analog of the problem, we obtain a family of phase portraits in the space of quasivelocities.     

Asymptotic stability and associated problems of dynamics of falling rigid body

We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.      

On conditions of existence of permanent rotations of connected rigid bodies system about vertical vector

In contrast to the classical problem of motion of a heavy rigid body about a fixed point where the permanent rotations are well known and completely investigated [7, 3] as the most simple and good visually demonstrated type of motions, in multibody mechanics under an increasing of quantity of the system bodies, mechanical parameters and the order of differential motion equations the study of such motions is more complicated problem. The problem on permanent rotations of two connected rigid bodies under influence of gravity force was investigated in [2, 4]. In this paper a system consisting of arbitrary constant quantity, n, of heavy rigid bodies which are sequentially jointed in a chain is considered. The conditions of existence of motions when each body permanently rotates about the vertical vector are determined. These conditions are analyzed in a general case when the bodies angular velocities are different.     

Properties of solutions for the generalized Euler‐Poisson equations

The present paper deals with equations, which generalize the known Euler‐Poisson equations for the motion of a heavy rigid body about a fixed point. These equations arise in dynamics of systems of coupled rigid bodies. In these equations the generalized inertia tensor depends upon components of vertical vector, i.e. it is not constant. Our aim is to analyze Lyapunov stability of stationary solutions and orbital stability of periodic solutions of the equations under study.     

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion for a rigid body with an elastic beam in a field of central gravitational force was investigated, and both of the orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by the variational principle, and on view point of generalized Hamiltonian dynamics, the sufficient conditions for the stability of one class of relative equilibria were given by the energymomentum method.     

The Hess case in rigid-body dynamics

Generalizations of the Hess integral are presented for different forms of the equations of motions of rigid body. The general conditions for the existence of this integral, which is due to the presence of additional explicit symmetries of the equations of motion, are pointed out. Problems of reducing the order, of the explicit integration and the qualitative analysis of the motion of a rigid body subject to these conditions are considered. Analogues of Hess cases for a gyroscope in gimbals and the Chaplygin equations describing the fall of a rigid body in a fluid are indicated for the first time.     

On stability of certain key types of rigid body motion in a nonconservative field

In this activity the qualitative analysis of spatial problems of the real rigid body motions in a resistant medium is fulfilled. A nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack is constructed. An analysis of plane and spatial models (in the presence or absence of an additional tracking force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigid Body Dynamics Using Clifford Algebra

In this paper the dynamics of rigid bodies is recast into a Clifford algebra formalism. Specifically, the algebra Cℓ(0, 6, 2), is used and it is shown how velocities, momenta and inertias can be represented by elements of this algebra. The equations of motion for a rigid body are simply derived by differentiating the momentum of the body.     

The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler — Jacobi — Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.     

On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid

We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body. We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity.     

Brownian dynamics of rigid body by fluctuating hydrodynamic equations

In this work, Brownian dynamics of rigid body in an incompressible fluid with fluctuating hydrodynamic equations is presented. To demonstrate the Brownian motion of rigid body, fluctuating hydrodynamic equations have been coupled with equations of motion of rigid body. Thermal fluctuation is included in the fluid equations via random stress terms unlike the random terms in the conventional Brownian dynamics type approach. Calculation of random stress terms in the fluid is easier in comparison to the random terms in the particle motion. Direct numerical simulation for the Brownian motion of rigid body with a meshfree framework is analysed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Dynamics of a rigid body on rockers

We study the dynamics of a rigid body on rockers with a nonspherical contact surface. It is shown that in the case of small oscillations the equations of motion contain strong nonlinearity, which makes it possible to avoid resonance (nonperiodic) oscillations. We study free and forced oscillations under a harmonic force on the plane of whose parameters bifurcation curves are constructed separating the periodic and nonoscillatory processes. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 51–55.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

The equations of motion of a rigid body without parametrization of rotations

New dynamic equations are proposed for a rigid body, without using local parametrization of the rotation group to describe the rotational part of the motion. A simple system of differential-algebraic equations, well suited for constructing the equations of motion of articulated bodies, is obtained.     

The motion of a body supported at three points on a rough plane

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.