Asymmetric Rigid Body Research Articles

Quantum Persistent Tennis Racket Dynamics of Nanorotors.

Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate moment of inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect, quickly averages to zero in classical ensembles since the flipping period varies significantly upon approaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermal asymmetric nanorotors and show that classically forbidden tunneling gives rise to persistent tennis racket dynamics, in stark contrast to the classical expectation. We characterize this effect, demonstrating that quantum coherent flipping dynamics can persist even in the regime where millions of angular momentum states are occupied. This persistent flipping offers a promising route for observing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles.

New cases of regular precession of an asymmetric liquid-filled rigid body

To describe the rotation of a rigid body with an ellipsoidal cavity filled with an ideal vorticated liquid, the Poincare–Hough–Zhukovsky equations are used. It is obtained constraints (hereinafter referred to as configuration conditions) on the mass distribution and cavity dimensions of an asymmetric liquid-filled rigid body under which the rigid body can perform the regular precession. Two possible nontrivial cases are indicated when one or two components of the direct vector of the axis of proper rotation are equal to zero. It is shown that if the axis of proper rotation coincides with one of the principal axes of inertia of the system, it suffices to fulfill one configuration condition. The ratio between the periods of proper rotation and precession is found. The regular precession of a system in which the principal moments of inertia are close to each other and the cavity is close to spherical is considered. For the case when the difference between the two equatorial moments of inertia is an order of magnitude smaller than the difference between the equatorial and polar moments, the main part of the ratio between the periods coincides with the Euler period, and the correction is due to the inequality between the equatorial moments of inertia. In the case when the axis of proper rotation is orthogonal to a principal axis and is not coincident with any other principal axis, the number of configuration conditions is two. Expressions for the rates of precession and proper rotation are obtained, and the position in the body of the axis of proper rotation is indicated. Special cases are given that allow simplification of the configuration conditions.

Fractional analysis of induced resonance effects in the spherical motion of a rigid body with flywheels

The evolution of the motion of an asymmetric rigid body with flywheels can be explored by the methods of asymptotic analysis. It is shown, that the effect of small moments resulting from the displacement of the center of mass and of small control moments from the flywheels lead to the emergence of non-linear resonance evolutionary effects. The aim of this work is to investigate the resonance effects and analyze their influence on the change of the fast phase in the spherical motion of the asymmetrical rigid body with flywheels. To explore these resonance effects we applied the method of integral manifolds and the averaging method. The averaged equations showed that resonance effects can lead to the regular precession, to the formation of stationary points, or to the prolonged resonance. In addition, the characteristic features of the influence of secondary resonance effects on the change in the fast phase are revealed.

On the Regular Precessions of an Asymmetric Liquid-Filled Rigid Body

The conditions for the existence of a linear invariant system of Poincare–Zhukovsky equations have been found. In a linear invariant system, three configuration conditions have been obtained that are sufficient to allow a mechanical system without dynamic symmetry to undergo regular precession. The explicit expression of the moments of inertia of a system consisting of a rigid body with an ellipsoidal cavity filled an ideal vorticity fluid is given in terms of the cavity dimensions; the velocities of precession and self-rotation are found. The particular case of the permanent rotation of an asymmetric rigid shell around the angular momentum vector is considered; in this case, any axis rigidly bound to the shell can be used as the axis of permanent rotation.

Analogies of the classical Euler top with a rotor to spin squeezing and quantum phase transitions in a generalized Lipkin-Meshkov-Glick model

We show that the classical model of Euler top (freely rotating, generally asymmetric rigid body), possibly supplemented with a rotor, corresponds to a generalized Lipkin-Meshkov-Glick (LMG) model describing phenomena of various branches of quantum physics. Classical effects such as free precession of a symmetric top, Feynman’s wobbling plate, tennis-racket instability and the Dzhanibekov effect, attitude control of satellites by momentum wheels, or twisting somersault dynamics, have their counterparts in quantum effects that include spin squeezing by one-axis twisting and two-axis countertwisting, transitions between the Josephson and Rabi regimes of a Bose-Einstein condensate in a double-well potential, and other quantum critical phenomena. The parallels enable us to expand the range of explored quantum phase transitions in the generalized LMG model, as well as to present a classical analogy of the recently proposed LMG Floquet time crystal.

On the application of KBM method for the 3-D motion of asymmetric rigid body

This article aims to study the three-dimensional motion of a rigid body that rotates about one of its fixed points. The effects of both a Newtonian force field and a moment of a gyrostat are taken into account. In the present work, we have assumed that the body has a spindle initial speed about one of the principal axes of the ellipsoid of inertia. The approximate solutions of the nonlinear problem are obtained by utilizing the Krylov–Bogoliubov–Mitropolski (KBM) technique and its amendments. Such solutions are demonstrated and then illustrated graphically in order to provide an extensive description of the body motion at any time. It is justified that the attained results are in a well consistency with those of the previous works, which are considered as limiting cases. The numerical solutions for governing system of motion are achieved using the fourth-order Runge–Kutta method and represented graphically. The comparison between the analytical solutions and the numerical ones reveals high consistency between them.

Classification of the classical [formula omitted] gauge transformations in the rigid body

In this paper we revisit the classification of the gauge transformations in the Euler top system using the generalized classical Hamiltonian dynamics of Nambu. In this framework the Euler equations of motion are bi-Hamiltonian and SL(2,R) linear combinations of the two Hamiltonians leave the equations of motion invariant, although belonging to inequivalent Lie–Poisson structures. Here we give the explicit form of the Hamiltonian vector fields associated to the components of the angular momentum for every single Lie–Poisson structure including both the asymmetric rigid bodies and its symmetric limits. We also give a detailed classification of the different Lie–Poisson structures recovering all the ones reported previously in the literature.

Про стійкість обертання несиметричного твердого тіла зі збудженням в середовищі, що чинить опір.

Assuming that the center of mass of a solid is on the third principal axis of inertia of a rigid body, on the basis of the Lienard-Shipar criterion, written in an innocent form, the asymptotic stability condition for the uniform rotation of a dynamically asymmetric rigid body with a fixed point as a system of three inequalities. The solid is under the action of attraction forces, dissipative torque and constant moment in the inertial and non-reference frames. The carried out studies of the asymmetry of a solid body and the dissipative moment, as well as of two constant moments on the stability conditions for uniform rotation of a rigid body. The results for a symmetric rigid body in the case of dynamic asymmetry of a rigid body and dissipative asymmetry are known.

Regular precession of a rigid body (gyrostat) acted upon by an irreducible combination of three classical fields

Abstract We show that a heavy, magnetized and electrically charged asymmetric rigid body moving about a fixed point while carrying a rotor and acted upon by three uniform fields can perform a regular precession about a nonvertical axis, of the type described for the case of a single field by Grioli in 1947. This is the first, and the only known by now, non-equilibrium solution of the problem of motion of a body in the presence of three classical fields, which are irreducible to a less number of fields.

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 dynamics of the eye: geodesics on a configuration manifold, motions of the gaze direction and Helmholtz’s theorem

The simplest model with which to examine the dynamics of the human eye consists of a rigid body which is free to rotate about a fixed point. Two classical laws governing monocular vision, which are known as Listing’s law and Donders’ law, can be enforced in this model using a single holonomic constraint. While there has been considerable attention paid to the kinematics of the eye, the dynamics of the eye predicted by rigid body models has not received the same level of attention. In the present paper, the unforced dynamics of the resulting rigid body model are examined with particular emphasis placed on the geodesics of the configuration manifold. A comprehensive portrait of these motions is presented, and the insight gained is related to the dynamics of the gaze direction and saccadic motions of the eye. Among our results, we find that modeling the eye as an asymmetric rigid body produces a non-integrable system of governing equations and that the geodesics on the configuration manifold provide a wealth of potential motions for the gaze direction.

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.

Modeling and experimentation with asymmetric rigid bodies: a variation on disks and inclines

We study the ascending motion of a disk rolling on an incline when its centre of mass lies outside the disk axis. The problem is suitable as laboratory project for a first course in mechanics at the undergraduate level and goes beyond typical textbook problems about bi-dimensional rigid body motions. We develop a theoretical model for the disk motion based on mechanical energy conservation and compare its predictions with experimental data obtained by digital video recording. Using readily available resources, a very satisfactory agreement is obtained between the model and the experimental observations. These results complement previous ones that have been reported in the literature for similar systems.

Modelo simplificado para el estudio del balanceo asimétrico de cuerpos rígidos esbeltos

This paper deals with the numerical modelling of rigid blocks by means of a rigid body spring model (RBSM). The hypothesis of small rotations has been considered in order to simplify the mathematical formulation. The asymmetric rocking motion is described. Extensive numerical simulations have been carried out in order to validate the numerical model, as well as to study the dynamical behaviour of asymmetric rigid bodies. The maximum response depends on the type of asymmetry and the direction of the load, which should become new variables in the study of asymmetric rocking motion.

Flat turn as an admissible extremal in the optimal rotation control problem for an asymmetric body

The problem of optimal control of rotation of an asymmetric rigid body is studied. An integrally quadratic functional characterizing the total energy costs is taken as the criterion. It is shown that, under certain conditions, the problem has a nontrivial extremal corresponding to a 180-degree flat turn, i.e., rotation about an axis fixed in the inertial space. The obtained results are based on an analysis of the equations arising after the application of the Pontryagin maximum principle (PMP) formalism.

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.

Kinetic energy of an asymmetric rigid body moving around a fixed point: invariant representation in terms of quaternion matrices

The similarity transform and the properties of quaternion matrices are used to derive invariant expressions for the inertia matrix and kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in body-fixed and inertial frames of reference. The elements of the quaternion matrixes are Rodrigues–Hamilton parameters and quasi-velocities

Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body

A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler–Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler’s equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed

Axial Velocity Solution for Spinning-Up Rigid Bodies Subject to Constant Forces

An approximate analytical solution is presented for the axial velocity of a spinning-up spacecraft subject to constant body-fixed forces and moments. We assume that the rigid body spins about the maximum or the minimum principal moment of inertia. Also, it is assumed that the deviation of the spin axis from an inertially fixed direction is small. Our closed-form solution for axial velocity applies to axisymmetric, nearly axisymmetric, and in some cases asymmetric rigid bodies. Furthermore, asymptotic limits are determined for large spin rates and for some geometric limits such as a sphere, a thin rod, and a flat disk. When the axial body-fixed force is nonzero, the axial velocity is dominated by the expected secular term of axial acceleration multiplied by time. However, when the axial force is zero, then the much more complicated effects of the transverse forces come into play: there are no longer any secular terms, but rather there is an asymptotic limit to the axial velocity accrued from the projection of the transverse accelerations onto the inertial direction of the initial spin axis. The axial velocity solution consists of Fresnel integrals, integrals of Fresnel integrals, and many new families of related integrals. Numerical simulation demonstrates the accuracy of the analytical solution in a practical example.

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.