Symmetric Rigid Body Research Articles

The stability of the steady motion of a gyrostat with a liquid in a cavity

A symmetrical rigid body with a spherical base, carrying a rotor and having a cavity in the shape of an ellipsoid of revolution, completely filled with an ideal incompressible liquid in uniform vortex motion, is moving along an absolutely rough plane. It is shown that this system admits of an energy integral, Jellett's integral, the integral of constant vorticity and a geometric integral. The construction of a Lyapunov function as a linear combination of first integrals [1] yields the sufficient conditions for the rotation of the gyrostat about the vertically positioned axis of symmetry to be stable. The conditions for the gyrostat's rotation to be unstable are found. It is shown that the rotor may prove to have either a stabilizing or destabilizing effect on the system and that the gyrostat admits of motions of the type of regular precession. The sufficient conditions for the stability of these motions are obtained.

Internal resonance in an autonomous Hamiltonian system close to a system with a cyclic coordinate

The motion of an autonomous Hamiltonian system with two degrees of freedom, close to a system with a cyclic coordinate, is considered. It is assumed that the generating system admits of a steady rotation, the corresponding equilibrium position of the reduced system being stable in the linear approximation. It is also assumed that there is an internal resonance in the system: the ratio of the natural frequency of small oscillations of the reduced system to the frequency of variation of the cyclic coordinate is close to an integer. Non-linear oscillations of the complete system in the neighbourhood of this steady rotation are investigated. Periodic motions are constructed and their bifurcation and stability are examined. Methods of KAM theory are used to study quasi-periodic motions of the system. As an example, the problem of the motion of a nearly dynamically symmetrical heavy rigid body along an absolutely smooth horizontal plane is investigated in the case of internal resonance.

Non-linear resonance evolutionary effects in the motion of a rigid body about a fixed point

Evolutionary effects in the motion of a heavy dynamically symmetrical rigid body about a fixed point in cases close to the Lagrange case are investigated in the non-linear formulation. Evolutionary effects are due to perturbations acting on the body, namely, a small displacement of the centre of mass with respect to the axis of dynamic symmetry, a perturbing moment, constant in a connected system of coordinates, and dissipative moments. It is shown that the presence of perturbations leads to the existence of “attracting” or “repulsing” resonances, which determine the evolution of the system in non-resonance parts of the motion. The method of integral manifolds is used for singularly perturbed systems and the method of averaging. A qualitative representation of possible motions of a statically stable top in the phase plane, taking a lower-order resonance into account, is given.

Regular and chaotic motions of the fast rotating rigid body: a numerical study

We numerically investigate the dynamics of a symmetric rigid body with a fixed point in a small analytic external potential (equivalently, a fast rotating body in a given external field) in the light of previous theoretical investigations based on Nekhoroshev theory. Special attention is posed on "resonant" motions, for which the tip of the unit vector $\mu$ in the direction of the angular momentum vector can wander, for no matter how small $\varepsilon$, on an extended, essentially two-dimensional, region of the unit sphere, a phenomenon called "slow chaos". We produce numerical evidence that slow chaos actually takes place in simple cases, in agreement with the theoretical prediction. Chaos however disappears for motions near proper rotations around the symmetry axis, thus indicating that the theory of these phenomena still needs to be improved. An heuristic explanation is proposed.

Generalized factorization for N×N Daniele–Khrapkov matrix functions

AbstractA generalization to N×N of the 2×2 Daniele–Khrapkov class of matrix‐valued functions is proposed. This class retains some of the features of the 2×2 Daniele–Khrapkov class, in particular, the presence of certain square‐root functions in its definition. Functions of this class appear in the study of finite‐dimensional integrable systems. The paper concentrates on giving the main properties of the class, using them to outline a method for the study of the Wiener–Hopf factorization of the symbols of this class. This is done through examples that are completely worked out. One of these examples corresponds to a particular case of the motion of a symmetric rigid body with a fixed point (Lagrange top). Copyright © 2001 John Wiley & Sons, Ltd.

On integrability of the motion of an axisymmetric rigid body

We consider the problem of the existence of additional analytical first integrals in some Hamiltonian systems, which are close to integrable, namely, the motion of a rigid body is close to a dynamically symmetric one. The problem of motion of a rigid body in an ideal liquid (the Kirchhoff problem) and the similar problem of rotation of a rigid body with a fixed point in an axisymmetric force field with a quadratic potential are investigated. The existence of hyperbolic periodic and asymptotic trajectories is shown. It is proved that perturbed trajectories are crossed but do not coincide. This is the reason for the absence of an additional analytical first integral in the perturbed problem. The problem of perturbed motion of a dynamically symmetric rigid body along an absolutely smooth horizontal plane is considered. Non-integrability of this problem is proved by the method of splitting asymptotic surfaces.

Attitude determination and stabilization of a spherically symmetric rigid body in a magnetic field

Observation and stabilization problems for a rigid body rotating about its mass centre in a time-periodic magnetic field is considered. The moment of the applied forces about the mass centre is a vector product of the field and the vector of control. The only measurement is of the field and its time derivatives in the body co-ordinates. The model describes the angular motion of a satellite in the geomagnetic field with the control carried out by a magnetic moment of current coils (magnetorquers) mounted on the satellite body. It is shown that generically the knowledge of the field and its derivatives is sufficient for the attitude determination. A problem of stabilization along one direction is considered. The stabilizer constructed in this work drives the system to a given motion from all initial points that do not belong to an unstable manifold. It is shown that all stabilizers have such a manifold of initial conditions.

Analysis of Low-Frequency Microaccelerations onboard the Foton-11Satellite

The method and the results of investigating the low-frequency component of microaccelerations onboard the Foton-11satellite are presented. The investigation was based on the processing of data of the angular velocity measurements made by the German system QSAM, as well as the data of measurements of microaccelerations performed by the QSAM system and by the French accelerometer BETA. The processing was carried out in the following manner. A low-frequency (frequencies less than 0.01 Hz) component was selected from the data of measurements of each component of the angular velocity vector or of the microacceleration, and an approximation was constructed of the obtained vector function by a similar function that was calculated along the solutions to the differential equations of motion of the satellite with respect to its center of mass. The construction was carried out by the least squares method. The initial conditions of the satellite motion, its aerodynamic parameters, and constant biases in the measurement data were used as fitting parameters. The time intervals on which the approximation was constructed were from one to five hours long. The processing of the measurements performed with three different instruments produced sufficiently close results. It turned out to be that the rotational motion of the satellite during nearly the entire flight was close to the regular Eulerian precession of the axially symmetric rigid body. The angular velocity of the satellite with respect to its longitudinal axis was about 1 deg/s, while the projection of the angular velocity onto the plane perpendicular to this axis had an absolute value of about 0.2 deg/s. The magnitude of the quasistatic component of microaccelerations in the locations of the accelerometers QSAM and BETA did not exceed 5 × 10–5–10–4m/s2for the considered motion of the satellite.

A Changing-Chart Symplectic Algorithm for Rigid Bodies and Other Hamiltonian Systems on Manifolds

We revive the elementary idea of constructing symplectic integrators for Hamiltonian flows on manifolds by covering the manifold with the charts of an atlas, implementing the algorithm in each chart (thus using coordinates) and switching among the charts whenever a coordinate singularity is approached. We show that this program can be implemented successfully by using a splitting algorithm if the Hamiltonian is the sum H1 +H2 of two (or more) integrable Hamiltonians. Profiting from integrability, we compute exactly the flows of H1 and H2 in each chart and thus compute the splitting algorithm on the manifold by means of its representative in any chart. This produces a symplectic algorithm on the manifold which possesses an interpolating Hamiltonian, and hence it has excellent properties of conservation of energy. We exemplify the method for a point constrained to the sphere and for a symmetric rigid body under the influence of positional potential forces.

Optimal stabilization of the rotational motion of a rigid body with the help of rotors

Optimal stabilization of the rotational motion of a symmetrical rigid body with the help of internal rotors is studied. In such a study the asymptotic stability of this motion is proved by using Barbachen and Krasovskii theorem. The optimal control moments which stabilize this motion are obtained using the conditions of ensuring the optimal asymptotic stability as non-linear functions of phase coordinates of the system. These moments stabilize asymptotically one type of the rotational motions of the rigid body. The other rotational motions are unstable in the Lyapunov sense. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia coincide with the inertial axes, is proved to be asymptotically stable. This study is characterized by that non-linear equations of motion which are used to prove the asymptotic stability and derivation of the control moments.

Unrestricted Planar Problem of a Symmetric Body and a Point Mass. Triangular Libration Points and Their Stability

We present an analysis of the model introduced by Kokoriev and Kirpichnikov (1988) for the study of unrestricted planar motion of a point mass and a symmetric rigid body whose gravity field is approximated by two point masses (a dumb-bell model). To show possible generalization of the model, we give a systematic derivation of equations of motion for a more general unrestricted problem of a point and a rigid body possessing a plane of dynamical symmetry. We give a simple description of bifurcation of triangular libration points, and we perform an analysis of their linear stability. We propose to extend the model of Kokoriev and Kirpichnikov (1988) to a case when the symmetric body is oblate. In the proposed model the gravity field of moving and rotating body is approximated by two complex masses at complex distance (a complex dumb-bell model). An analysis of bifurcation of the triangular libration points in this model is also presented.

Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. Part II: Gyroscopic rotations

We continue the analysis which began in part 1 of this paper of the long-time behaviour of the fast rotations of a rigid body in an external analytic force field. Specifically, we consider the motions of a symmetric rigid body whose angular velocity is nearly parallel to the symmetry axis of the ellipsoid of inertia, which were excluded from the previous analysis because of the singularity of the action-angle coordinates. By suitably implementing the techniques of Nekhoroshev's theorem, so as to overcome this difficulty, we provide a description of these motions on timescales which grow with the angular velocity as ; such a description confirms the general properties of fast motions established in part 1.

Stability Analysis of Force Feedback and Force Control of a Constrained Arm with a Symmetric Rigid Tip Body Considering Bending and Torsional Flexibility

Abstract In this paper, stability analysis and robust control of a force-controlled arm having a symmetric rigid tip body, of which the mass center lies on the central axis of the arm, are discussed. We consider link flexibility as uncertainty and derive dynamic equations of the force-controlled arm. As the obtained boundary condition of the vibraton equation of the arm is nonhomogeneous, we introduce a change of variables to derive homogeneous boundary conditions. On the basis of a finite-dimensional modal model of distributed-parameter systems, stability of the force feedback is analyzed by using the root locus technique, and an optimal controller with low-pass property is constructed. Simulations have been carried out.

Optimal Regulation and Passivity Results for Axisymmetric Rigid Bodies Using Two Controls

Wepresent a partial solution to theproblem of optimal feedback reorientation of the symmetry axisofan axially symmetric rigid body. The performance index is quadratic in the state and the control variable, and the optimal reorientation maneuver requires the use of only two control torques. Because of the passivity characteristics and the cascade structure of the system we e rst state two optimal regulation problems for the dynamics and the kinematics subsystems, separately. In this case one is able to e nd explicit solutions to the associated Hamilton ‐ Jacobi equations. For thecompletesystem, wepresent solutions fortwo partial cases. Thee rst case is when there is no penalty on the control input. In this case, one can asymptotically recover the cost for the kinematics by making the dynamics sufe ciently fast. The second case investigates restrictions imposed by optimality considerations on the aforementioned control law to avoid high gain.

Numerical analysis of relative equilibrium states and its stability of a string system with a rigid body

In this paper a mechanical system is studied in which a rotor rotates around a fixed axis with a string suspended symmetric rigid body. All relative equilibrium states and their stability are discussed. Considering the spinning angular velocity ω around the fixed vertical axis as a parameter, algebraic equations with this parameter are obtained. Every solution of the equations is relevant to a relative equilibrium state of the system. The existence of two important relative equilibrium states is discussed by numerical method developed in bifurcation theory in this paper. In addition, The Lagrange's Theorem is used to determine the stability of the relative equilibrium state relevant to the solution of the algebraic equations.

Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. Part I

In this paper we study the dynamics of a fast rotating symmetric rigid body in an arbitrary analytic potential, by means of the techniques of Hamiltonian perturbation theory, specifically Nekhoroshev's theory. For a rigid body with a fixed point, this approach allows us to get an accurate description of the motion (with rigorous estimates) on time scales which increase very fast with the angular velocity of the body, precisely as . As we show, on such time scales, the body performs an approximately free motion around the instantaneous direction of the angular momentum vector m, which in turn moves slowly in space, with speed of order . The properties of such a slow drift of m strongly depend on the resonance properties of the initial angular velocity. For nonresonant initial conditions, the unit vector in the direction of m closely follows, on the unit sphere, the level curves of an averaged external potential, so the motion is essentially regular (chaotic effects, if any, are confined to small-amplitude oscillations around such motion). On the contrary, in case of resonance, can perform large-scale chaotic motions, which spread over genuinely two-dimensional regions of the unit sphere; numerical evidence of this phenomenon in an oversimplified model is also provided. Similar results are proven for prolate rigid bodies with no fixed point; an important difference is that, because of the presence of additional `slow' degrees of freedom, a large-scale chaotic behaviour of m is now possible in nonresonant motions, too. Because of the presence of singularities of the action-angle coordinates, our description is not optimal for motions very close to gyroscopic rotations; the forthcoming part II will be devoted to these special motions.

Nutational stability and passive control of spinning rockets with internal mass motion

Necessary and sufficient conditions for asymptotic stability are developed for a symmetric spinning rocket with axial forcing and dissipative internal mass motion. Results show that spin about the axis of least inertia is asymptotically stable provided that three conditions are satisfied: 1) the axial thrust magnitude is sufficiently large, 2) the restoring forces for the internal mass motion are sufficiently large, and 3) the internal mass motion occurs aft of the system mass center. Under these same conditions, spin about the axis of largest inertia may be unstable. Numerical studies show that a passive control device with these characteristics can be used to attenuate the coning of prolate spinners such as those transfer orbit boost vehicles that have been plagued by unexpectedly pronounced coning instabilities. Nomenclature c = damping constant for particle motion F = magnitude of thrust (assumed constant) /i, /i, 73 = moments of inertia of (symmetric) rigid body for axes #1, #2, and 53, respectively k = spring constant for elastic restraint on particle M = mass of rigid body m = mass of particle li = m/(m -f M) Q = nominal spin rate of rigid body about #3 0)1,0)2 = components of rigid-body angular velocity for axes B and B2, respectively

The dynamics of the precessional motions of a system of two rigid bodies in a gravitational field

The conditions for the edstence of precessions in a system consisting of two symmetric rigid bodies in a gravitational field are derived, on the assumption that one of the bodies is rotating uniformly about the vertical and the other is precessing about the vertical. Regular precession of asymmetric bodies linked by a spherical hinge is considered.

Controllability of a System of Two Symmetric Rigid Bodies in Three Space

Consider a mechanical system consisting of two completely symmetric, three-dimensional rigid bodies, each with inertia tensor the identity matrix and mass center at $0 \in \mathbb{R}^3 $. Imposing the constraint that this system have zero total angular momentum, the angular velocities of these bodies are negatives of one another, and the transfer of this system from one position to another is nonholonomic. While Chow’s theorem establishes the fixed-endpoint controllability of this system, this result does not explicitly exhibit any motions between a given set of endpoints. This paper shows how to explicitly construct simple motions of this system on [0,1] with arbitrary endpoints in $SO(3)^2 $. In particular, using normalized quaternions to describe rotations in terms of elementary functions, a continuous motion with the given endpoints is constructed; it consists of at most three successive motions during each of which the bodies rotate on fixed axes.

Extremal geometry for an optimal control problem with a coupled system of two symmetric rigid bodies

We consider motions of a mechanical system consisting of two completely symmetric rigid bodies constrained to rotate with opposite angular velocities, and the following optimal control problem: find the motions of this system on [0, I] with given endpoints in SO(3) x SO(3) which minimize the integral of the system's kinetic energy. We show that the extremals for this problem can be obtained explicitly as products of exponentials, and give a geometric interpretation of these extremals