Heavy Rigid Body Research Articles

Dynamics of the suslov problem in a gravitational field: Reversal and strange attractors

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

New solutions of classical problems in rigid body dynamics

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.

On the general solution of the problem of the motion of a heavy rigid body in the Hess case

A solution of the Euler-Poisson equations in the Hess case is represented in the form of the family of its singular points together with the asymptotic behaviour of the solution at these points. A complete list of single-valued and finite-valued solutions in the Hess case is given. A representation for limiting periodic solutions is obtained and a precise condition for the existence of these solutions is found. Bibliography: 25 titles.

On the regular precession of an asymmetric rigid body acted upon by uniform gravity and magnetic fields

In 1947 Grioli discovered that an asymmetric heavy rigid body moving about a fixed point can perform a regular precession, which is the rotation of the body about an axis fixed in it, while that axis precesses with the same uniform angular velocity about a non-vertical axis fixed in space.In the present note, we show that a magnetized asymmetric rigid body moving about a fixed point while acted upon by uniform gravity and magnetic fields can perform a regular precession about a horizontal axis fixed in space orthogonal to the magnetic field. This motion does not contain Grioli's as a special case since the gravity and magnetic effects are coupled and can vanish only simultaneously.

On the orbital stability of the motion of a rigid body in the case of Bobylev–Steklov

The problem of motion of the heavy rigid body about a fixed point admits simple periodic solutions in few cases. Examples are the pendulum-like plane motions, Grioli’s case and Bobylev–Steklov case. Noting that only stable motions can be realized due to the inevitable deviations in the initial conditions and in the determination of the distribution of mass in the body, the study of stability acquires an increasing importance. The stability of plane motions was considered in several works. Grioli’s case was studied recently by Markeyev. The aim of the present work is to study stability in the linear approximation for Bobylev and Steklov’s case. The use of Euler–Poisson equations and their integrals for the study of stability of periodic motions is quite complicated. Instead, we use a single second-order differential equation obtained by one of us, by the maximal reduction of the order of equations of motion using their general integrals. This equation is satisfied by the trajectory of the trace of the vertical on the Poisson sphere fixed in the body. The orbital stability of a solution means that the perturbed trajectory remains near to the unperturbed, after perturbations preserving the values of general integrals. After classification of the two possible families of trajectories, equation in variation is obtained for each family. In the three-dimensional space of parameters affecting stability, we determine the surfaces carrying primitive periodic solutions, and thus separating stability and instability zones. Both equations in the variations were solved also numerically on certain sections of the parameter space. Numerical results accomplish the identification of zones lying between surfaces as stability or instability zones and do not show any traces of other zones, rather than those detected by analytical study.

Об устойчивости частных решений приближенных уравнений движения тяжелого твердого тела с вибрирующей точкой подвеса

Об устойчивости частных решений приближенных уравнений движения тяжелого твердого тела с вибрирующей точкой подвеса

Об устойчивости частных движений тяжелого твердого тела, обусловленных быстрыми вертикальными вибрациями одной из его точек

Об устойчивости частных движений тяжелого твердого тела, обусловленных быстрыми вертикальными вибрациями одной из его точек

Four-dimensional generalization of the Grioli precession

A particular solution of the four-dimensional Lagrange top on e(4) representing a four-dimensional regular precession is constructed. Using it, a four-dimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.

A kinematic interpretation of the motion of a heavy rigid body with a fixed point

A kinematic interpretation of the motion of a rigid body with a fixed point is proposed using the rolling of a mobile hodograph, which describes, on the ellipsoid of inertia, a vector collinear with the vector of the angular velocity of the body, with respect to a fixed vector. On the basis of this, an interpretation of the motion of the body in the Steklov, Grioli, Dokshevich and Bobylev – Steklov solutions is obtained. A new formula is derived indicating the connection between the angle of precession and the polar angle of the equations of the fixed hodograph, indicated by Kharlamov.

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.

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.

On the full list of finite‐valued solutions of the Euler‐Poisson equations having four first integrals

AbstractOnly 13 general and partial solutions of the problem of a heavy rigid body's movement have been found from the discovery of the first one by Euler in 1758 till now. All these solutions are finite‐valued in the classical Euler and Poisson's variables except the one in the Hess case. In this paper the problem of finding the full list of the finite‐valued solutions is considered.

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position.

A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid

We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid. We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity, or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier–Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder, but may be amplified for other geometries, such as in the case of a square cylinder.

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.

On the orbital stability of pendulum-like 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 the geometry of the mass of the body corresponds to the Bobylev-Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.

Cartesian approach for constrained mechanical systems with three degrees of freedom

In the history of mechanics, there have been two points of view for studying mechanical systems: The Newtonian and the Cartesian. According the Descartes point of view, the motion of mechanical systems is described by the first-order differential equations in the $N$ dimensional configuration space $\textsc{Q}.$ In this paper we develop the Cartesian approach for mechanical systems with three degrees of freedom and with constraint which are linear with respect to velocity. The obtained results we apply to discuss the integrability of the geodesic flows on the surface in the three dimensional Euclidian space and to analyze the integrability of a heavy rigid body in the Suslov and the Veselov cases .

A Poincaré Section for the General Heavy Rigid Body

A general procedure is developed for the study of rigid body dynamics using Poincaré sections. A section condition is chosen which captures every trajectory on a given energy surface. This section condition is given by extremal heights of the center of mass of the top. The possible topological types of the corresponding sections are orientable surfaces of genus up to 4 and unions thereof. The flow is not transverse to the section, and the set where tangency occurs is not invariant under the flow. In the standard construction of Poincaré sections the section is cut along the tangency set. In the present case this would lead to a complicated presentation of the section. Instead the section is cut into two pieces in such a way that each piece can be projected 1:1 onto a torus. This torus is obtained from the twice-punctured Poisson sphere by adding two circles. Examples of Poincaré maps illustrating the different topologies of the section are presented.

The necessary conditions for the existence of an additional integral in the problem of the motion of a heavy ellipsoid on a smooth horizontal plane

The equations of motion of an ellipsoid on a smooth horizontal plane are similar to the equations of motion of a heavy rigid body with a fixed point. In general, one integral is also lacking in order to integrate them. For a triaxial ellipsoid, the centre of mass of which coincides with the geometrical centre, it is proved that an additional integral is lacking (in the generic case).

On detachment conditions in the problem on the motion of a rigid body on a rough plane

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body from the plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk. We have showed the following. 1) In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle θ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments. 2) The same conclusion holds for a thin disk that rolls on the support without sliding. 3) For a disk of nonzero thickness in the absence of sliding, the detachment conditions hold on stationary motions in some domain in the space of parameters; in this case, the angle θ is not less than 49 degrees. For small values of θ, the contact between the body and the support does not break in a neighborhood of stationary motions.