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1.
Marcello Romano 《Celestial Mechanics and Dynamical Astronomy》2008,101(4):375-390
New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of
inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for
the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular
to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial
angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed
with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case
of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix
in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of
a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of
an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body
with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic
solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion
time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an
exact solution of the rotational motion of a rigid body exists. 相似文献
2.
Although analytic solutions for the attitude motion of a rigid body are available for several special cases, a comprehensive theory does not exist in the literature for the more complicated problems found in spacecraft dynamics. In the present paper, analytic solutions in complex form are derived for the attitude motion of a near-symmetric rigid body under the influence of constant body-fixed torques. The solution is very compact, which enables efficient and rapid machine computation. Numerical simulations reveal that the solution is very accurate when applied to typical spinning spacecraft problems. 相似文献
3.
We present an algorithm to compute the incomplete elliptic integral of a general form. The algorithm efficiently evaluates some linear combinations of incomplete elliptic integrals of all kinds to a high precision. Some numerical examples are given as illustrations. This enables us to numerically calculate the values and the partial derivatives of incomplete elliptic integrals of all kinds, which are essential when dealing with many problems in celestial mechanics, including the analytic solution of the torque-free rotational motion of a rigid body around its barycenter. 相似文献
4.
Hiroshi Kinoshita 《Celestial Mechanics and Dynamical Astronomy》1992,53(4):365-375
Torque-free motion of a rigid body is integrable and its solution is expressed in terms of elliptic functions and elliptic integrals. The conventional analytical expression of the solution, however, is complicated and not suitable for hand-calculation. Recently the rotational motions of small celestial bodies in the solar system are frequently investigated by numerically integrating the equations of motion instead of using the analytical solution, since the numerical evaluation of the analytical and exact solution is a little bit difficult. As the observational accuracy of the rotational motions of the small bodies in the solar system is quite low, what we need for the reduction of these observations are rough estimates of the period of Eulerian motion ( or the free precession period) and the amplitudes of the main periodic terms. Here we give simple analytical expansions of torque-free motions for short- and long-axis modes, which are correct up to the second-order of a small parameter. These expressions include only trigonometric functions and are easily evaluated by hand calculation for estimates of the essential quantities from which we can determine a global rotational motion of the torque-free motion. They can also be used as the zero-th order solution in a perturbation method, when the motion is perturbed by external torques. 相似文献
5.
We use the alternative MEGNO (Mean Exponential Growth of Nearby Orbits) technique developed by Cincotta and Simó to study the stability of orbital—rotational motions for plane oscillations and three-dimensional rotations. We present a detailed numerical—analytical study of a rigid body in the case where the proper rotation of the body is synchronized with its orbital motion as 3: 2(Mercurian—type synchronism). For plane rotations, the loss of stability of the periodic solution that corresponds to a 3: 2 resonance is shown to be soft, which should be taken into account to estimate the upper limit for the ellipticity of Mercury. In studying stable and chaotic translational—rotational motions, we point out that the MEGNO criterion can be effectively used. This criterion gives a clear picture of the resonant structures and allows the calculations to be conveniently presented in the form of the corresponding MEGNO stability maps for multidimensional systems. We developed an appropriate software package. 相似文献
6.
M. K. Das 《Astrophysics and Space Science》1985,114(2):295-302
The nonlinear pseudo-radial mode of oscillation of a rotating magnetic star is studied. It is shown that for a general rotational field, the coupling between magnetic field and rotation tends to reduce the average rotational energy parameterT. This result in a lowering of the maximum pulsation amplitudeq
max, which depends on strength of rotation and magnetic field. The configuration tends, therefore, to a new equilibrium state at lower value ofq
max. The analytic solution of the pulsation equation for the case ofy=5/3 in the presence of rotation and magnetic field has also been derived in the Appendix. 相似文献
7.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》1990,50(2):165-187
Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches. 相似文献
8.
John L. Junkins Ira D. Jacobson Jeffrey N. Blanton 《Celestial Mechanics and Dynamical Astronomy》1973,7(4):398-407
A rigorously valid nonlinear oscillator analog of the torque-free rotational dynamics of a general rigid body is presented. The analog consists of threeuncoupled nonlinear oscillators, the motion of each being governed by a second order nonlinear ordinary differential equation of the Duffing type. The nonlinear oscillator analog and three associated phase planes, as established herein, provide a new basis for analysis and visualization of rigid body dynamics. The phase planes are particularly useful in providing complete visibility of the motion's limiting cases and stability properties. 相似文献
9.
Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions. 相似文献
10.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1974,9(1):3-20
It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem.In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles. 相似文献
11.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):221-234
A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter. 相似文献
12.
F. M. F. El-Sabaa 《Astrophysics and Space Science》1992,193(2):309-315
The equation of motion of a rigid body in the Kovalevskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré the existence of a periodic solution is established. 相似文献
13.
F. M. F. El-Sabaa 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):71-79
The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré, the existence of a periodic solution is established. 相似文献
14.
Gwenaël Boué Nicolas Rambaux Andy Richard 《Celestial Mechanics and Dynamical Astronomy》2017,129(4):449-485
We revisit the rotation dynamics of a rigid satellite with either a liquid core or a global subsurface ocean. In both problems, the flow of the fluid component is assumed inviscid. The study of a hollow satellite with a liquid core is based on the Poincaré–Hough model which provides exact equations of motion. We introduce an approximation when the ellipticity of the cavity is low. This simplification allows to model both types of satellite in the same manner. The analysis of their rotation is done in a non-canonical Hamiltonian formalism closely related to Poincaré’s “forme nouvelle des équations de la mécanique”. In the case of a satellite with a global ocean, we obtain a seven-degree-of-freedom system. Six of them account for the motion of the two rigid components, and the last one is associated with the fluid layer. We apply our model to Titan for which the origin of the obliquity is still a debated question. We show that the observed value is compatible with Titan slightly departing from the hydrostatic equilibrium and being in a Cassini equilibrium state. 相似文献
15.
Keiji Ohtsuki 《Icarus》2006,183(2):373-383
We derive an equation for the evolution of rotational energy of Keplerian particles in a dilute disk due to mutual collisions. Three-dimensional Keplerian motion of particles is taken into account precisely, on the basis of Hill's approximation. The Rayleigh distribution of particles' orbital eccentricities and inclinations, and the Gaussian distribution of their rotation rates are also taken into account. Performing appropriate variable transformation, we show that the equation can be expressed with two terms. The first term, which we call collisional stirring term, represents energy exchange between rotation and random motion via collisions. The second term, which we call rotational friction term, tends to equalize the mean rotational energy of particles with different sizes. The equation can describe the evolution of rotational energy of Keplerian particles with an arbitrary size distribution. We analytically evaluate the rates of stirring and friction for the random kinetic energy and rotational energy due to inelastic collisions, for non-gravitating particles in a dilute disk. Using these results, we discuss equilibrium states in a disk of spinning, non-gravitating Keplerian particles. 相似文献
16.
We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
Rotational perturbations of charged cosmological viscous-fluid universes coupled with a scalar field
Manihar Singh Koijam 《Astrophysics and Space Science》1989,162(1):129-150
The dynamics of a slowly rotating charged viscous-fluid Universe coupled with a zero-mass scalar field is investigated; and the rotational perturbations of such models are studied in order to substantiate the possibility that the Universe is endowed with slow rotation, in the course of presentation of several new analytic solutions. The effects of charged field and scalar field on the rotational motion are discussed. Except for perfect dragging, the scalar field as well as the charged field is found to have a damping effect on the rotation of matter. Rotating models which are expanding as well are obtained, in which cases the rotational velocities are found to decay with the time, and these models may be taken as good examples of real astrophysical situations. The periods of physical validity of different models are also obtained. 相似文献
18.
We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc. 相似文献
19.
Joseph J. F. Liu Philip M. Fitzpatrick 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):463-487
Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w
0)2, wheren is the satellite's orbital mean motion andw
0 its initial rotational angular speed. 相似文献
20.
Paolo Lanzano 《Astrophysics and Space Science》1969,5(3):300-322
We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123. 相似文献