On the dynamical symmetry points and the orientations of the principal axes of inertia of a rigid body

For an arbitrary rigid body, all dynamical symmetry points are found, and the directions of the axes of dynamical symmetry are determined for these points. We obtain conditions on the principal central moments of inertia under which the Lagrange and Kovalevskaya cases can be realized for the rigid body. We also analyze the set of orientations of the bases formed by the principal axes of inertia for various points of the rigid body.     

Global qualitative analysis of tippe top dynamics

The tippe top is a dynamically and geometrically symmetric body supported by a horizontal plane. If one twists the tippe top rapidly about the symmetry axis so that the symmetry axis is vertical and its center of mass takes the lowest position, then it turns upside down by 180° and stats to rotate about the same symmetry axis with the center of mass occupying the highest position. A local analysis of tippe top dynamics (in a neighborhood of its rotations about the vertical symmetry axis) is given in [1, 2]. The simplest model of the tippe top is a dynamically symmetric inhomogeneous ball whose center of mass lies on the dynamic symmetry axis but does not coincide with its geometric center. Such a model allows global qualitative analysis of the top dynamics.     

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.     

Four-Body Central Configurations¶with some Equal Masses

We prove firstly that any convex non-collinear central configuration of the planar 4-body problem with equal opposite masses β >α > 0, such that the diagonal corresponding to the mass α is not shorter than that corresponding to the mass β, must possess a symmetry and therefore must be a kite. Then by a recent result of Bernat, Llibre and Perez-Chavela, this kite is actually a rhombus. Secondly we prove that a convex non-collinear planar 4-body central configuration with three equal masses must be a kite too. We also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center, or an isosceles triangle with the fourth mass on the symmetry axis.     

Stability of the cylindrical precession of a satellite in an elliptic orbit

We study the linear problem on the stability of rotation of a dynamically symmetric satellite about the normal to the plane of the orbit of its center of mass. The orbit is assumed to be elliptic, and the orbit eccentricity is arbitrary. We assume that the Hamiltonian contains a small parameter characterizing the deviation of the satellite central ellipsoid of inertia from the sphere. This is a resonance problem, since if the small parameter is zero, then one of the frequencies of small oscillations of the symmetry axis in a neighborhood of the unperturbed rotation of the satellite about the center of mass is exactly equal to the frequency of the satellite revolution in the orbit. We indicate a countable set of values of the angular velocity of the unperturbed rotation for which the resonance is even double. The stability and instability domains are obtained in the first approximation with respect to the small parameter.     

On the choice of physically realizable parameters when studying the dynamics of spherical and ellipsoidal rigid bodies

The paper presents necessary and sufficient conditions whose must be satisfied by the main geometric and dynamic parameters of spherical, ellipsoidal, or parabolic rigid bodies for their physical realization. The main parameters are both the geometric characteristics of the body boundary (radius of the sphere, semiaxes of the ellipsoid, principal curvatures at the vertex, and the paraboloid center location on its symmetry axis) and the body mass and dynamic characteristics (body mass, displacement of the body center of mass from the center on the paraboloid symmetry axis or from the sphere or ellipsoid center of symmetry, the orientation of the principal central axes of inertia with respect to the principal geometric axes of the shell, and the values of the principal central moments of inertia). The physical realization is understood as the existence of an actual distribution of positive masses inside the sphere, ellipsoid, or paraboloid for which the above-listed characteristics of the body are equal to the chosen ones. Several examples from earlier-published papers dealing with the dynamics of spherical, ellipsoidal, or parabolic bodies with physically unrealizable parameters are given.     

New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.     

The global flow of the hyperbolic restricted three-body problem

Two mass points of equal masses m ₁ = m ₂ > 0 move under Newton's law of attraction in a non-collision hyperbolic orbit while their center of mass is at rest. We consider a third mass point, of mass m ₃ = 0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their center of mass. Since m ₃ = 0, the motion of m ₁ and m ₂ is not affected by the third, and from the symmetry of the motion it is clear that m ₃ remains on the line L. The hyperbolic restricted 3-body problem is to describe the moton of m ₃. Our main result is the characterization of the global flow of this problem.     

Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision

The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates. Partially supported by Dirección de Investigación UBB, 064608 3/RS.     

Kovalevskaya弹性杆

应用Kirchhoff比拟讨论Kovalevskaya情况弹性细杆的平衡稳定性问题．导出Kirchhoff方程的解析积分．对于杆截面的主轴与Frenet坐标轴重合的无扭转杆的特殊情形作定性分析，讨论其平衡状态的稳定性与分岔．证明了判断受拉扭作用的圆截面直杆平衡稳定性的Greenhill公式也适用于Kovalevskaya情形的非圆截面杆．     

Poincaré-Chetaev bifurcation diagrams in the problem of motion of an inhomogeneous dynamically and geometrically symmetric ellipsoid on a smooth plane

The existence, stability, and bifurcation of steady motions of an inhomogeneous dynamically and geometrically symmetric ellipsoid is considered. The mass center of the ellipsoid is shifted and located on its symmetry axis. The ellipsoid moves on a perfectly smooth horizontal plane.     

Sensitivity analysis for a type of statically stable sailcrafts

Two types of sensitivities are proposed for statically stable sailcrafts.One type is the sensitivities of solar-radiation-pressure force with respect to position of the center of mass,and the other type is the sensitivities of solar-radiation-pressure force with respect to attitude.The two types of sensitivities represent how the solar-radiationpressure force changes with the position of mass center and the attitude.Sailcrafts with larger sensitivities undergo larger error of the solar-radiation-pressure force,leading to larger orbit error,as demonstrated by simulation.Then as a case study,detailed formulas are derived to calculate the sensitivities for sailcrafts with four triangular sails.According to these formulas,in order to reduce both types of sensitivities,the angle between opposed sails should not be too large,and the center of mass should be as close to the axis of symmetry of the four sails as possible and as far away from the center of pressure of the sailcraft as possible.     

Effect of movability of the resonator center on the operation of a hemispherical resonator gyro

It was established in [2] that resonator deformation according to the second mode shape of a thin hemispherical shell results in a displacement of the center of mass if the resonator is unbalanced, i.e., if the distribution of mass over the surface of the hemisphere deviates from axial symmetry. In the same paper, it was shown that this displacement of the center of mass makes the instrument sensitive to linear vibrations. The present paper deals with linear vibration caused in the presence of unbalance by the working vibrations themselves and by the forces used to maintain the latter. The linear vibration is considered in the form of beam vibrations of the resonator stem. The study is aimed at determining the influence of the coupling between the working and beam vibrations on the instrument readings. We obtain a formula relating the hemispherical resonator gyro drift to the unbalance and the eccentricity, which, in particular, can be caused by the gravity component normal to the sensitivity axis. The drift considered here is essentially caused by the fact that deformation of the resonator supports also results in deformation of the electric control field in the gap between the electrodes. The resulting additional forces cause the effect studied in this paper. The drift magnitude depends on how the control of the phase state of the resonator is chosen. In what follows, to be definite, we consider the control in fast-time mode, i.e., at the natural vibration frequency. A similar effect takes place for any other type of control of waves in the resonator.     

Chaotic dynamics of a body–vortex pair

We study an idealized model of body–vortex interaction in two dimensions. The fluid is incompressible and inviscid and assumed to occupy the entire unbounded plane except for a simply connected region representing a rigid body. There may be a constant circulation around the body. The fluid also contains a finite number of point vortices of constant circulation but is otherwise irrotational. We assign a mass distribution to the body and let it move and rotate freely in response to the force and torque exerted by the fluid. Conversely, the fluid moves in response to the body motion. We study the occurrence of chaos in the system of ODEs emerging from these assumptions. It is well-known that the system consisting of a circular body with uniform mass distribution interacting with a single point vortex is integrable. Here we investigate how this integrability breaks down when the body center-of-mass is displaced from its geometrical center. We find two distinct regions of chaos and discuss how they relate to the topology of the trajectories of body and vortex.     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

Various friction models in two-sphere top dynamics

The dynamics of a two-sphere tippe top on a rough horizontal plane is considered. The top is bounded by a nonconvex surface consisting of two spherical segments of distinct radii and a cylinder; the cylinder axis coincides with the common symmetry axis of the segments. If the top is initially placed so that its center of mass is almost in the lowest position, the symmetry axis is almost vertical, and a high angular velocity of rotation about the vertical symmetry axis is imparted to the top, then it turns upside down from its base to the leg and starts to rotate on the leg. Then the top gradually returns to the stable equilibrium. The problem of the top motion is often used to demonstrate the efficiency of various proposed friction models [1–3]. The effects arising in the two-sphere top dynamics with various dry friction models are compared in the present paper. Both analytic methods based on the theory of stability and bifurcations and numerical calculations are used. The numerical study is performed under the assumption that the supporting plane is deformable, which permits one to describe transient processes accompaniedwith impacts by using a single system of equations.     

On limit motions of a system of control moment gyros with intrinsic dissipation in a homogeneous gravitational field

We study the limit motions of a free rigid body bearing n two-degree-of-freedom control moment gyros with dissipation in the gyro gimbal suspension axes. We show that, in the absence of dynamic symmetry, the limit motions of the system are only steady rotations at a constant angular velocity. In the case of dynamic symmetry, the gyros can be arranged so that, in addition to steady rotations, the system exhibits limit motions that are regular precessions.     

Mathematical Model for the Numerical Solution of Nonstationary Problems in Solid Mechanics by a Modified Godunov Method

A mathematical model of substance behavior under developed elastoplastic strains is worked out for solving onedimensional problems of solid mechanics. The model is based on the fundamental laws of conservation of mass, momentum, and total energy, Wilkins model, kinetic model of substance destruction, and modified Godunov method for the numerical solution of problems in mathematical physics. A hybrid difference scheme is constructed, which approximates acoustics equations with constant coefficients in smooth flows for the case of plane symmetry with the second order in time and space.     

Planar oscillations of a dumb-bell of a variable length in a central field of Newtonian attraction. Exact approach

Dynamics of a dumb-bell of a variable length in a central field of Newtonian attraction is considered. It is assumed that the body moves in a plane fixed in the absolute space and passing through an attracting center. The law of length׳s variation providing an existence of stationary configurations is pointed out. For these configurations the dumb-bell forms a constant angle with a local vertical passing through the center of mass of the dumb-bell, which moves in an elliptic orbit similar to the Keplerian. In particular, the mentioned constant angle may be equal to zero. In contrast to previous investigations (Burov and Kosenko, 2011 [8], [10], Burov, 2011 [9]) the problem is solved within the exact formulation, without supplementary simplifying assumptions concerning smallness of the dumb-bell in comparison to its distance from the attracting center.     

On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent $a\le -1$ a ≤ ? 1 .