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1.
For use in numerical studies of rotational motion, a set of elements is introduced for the torque-free rotational motion of a rigid body around its barycenter. The elements are defined as the initial values of a modification of the Andoyer canonical variables. A computational procedure is obtained for determining these elements from the combination of the spin angular momentum vector and a triad defining the orientation of the rigid body. A numerical experiment shows that the errors of transformation between the elements and variables are sufficiently small. The errors increase linearly with time for some elements and quadratically for some others. 相似文献
2.
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
3.
Marcello Romano 《Celestial Mechanics and Dynamical Astronomy》2008,100(3):181-189
The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of
inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary
initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular,
the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points
on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type,
which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent
hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic
map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element 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
4.
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
5.
6.
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
7.
Andrzej J. Maciejewski 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):47-57
The Euler-Poinsot equations of rigid body motion are considered. The Euler parameters attitude parametrization is assumed. The Hamiltonian form of these equations is obtained. Two different solutions are shown. A short discussion of free motion in this new treatment is also given. 相似文献
8.
F. M. El-Sabaa 《Celestial Mechanics and Dynamical Astronomy》1983,29(3):249-253
Equations of motion of a heavy rigid body about a fixed point in the Kowalevskaya's case are reduced to those of the plane motion of a particle under the action of potential force. In the new form of equations of motion, elliptic coordinates λ, μ are used, and motion has taken the Hamilton-Jacobi form. 相似文献
9.
In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies. 相似文献
10.
The restricted 2+2 body problem is considered. The infinitesimal masses are replaced by triaxial rigid bodies and the equations of motion are derived in Lagrange form. Subsequently, the equilibrium solutions for the rotational and translational motion of the bodies are detected. These solutions are conveniently classified in groups according to the several combinations which are possible between the translational equilibria and the constant orientations of the bodies. 相似文献
11.
Taeyoung Lee Melvin Leok N. Harris McClamroch 《Celestial Mechanics and Dynamical Astronomy》2007,98(2):121-144
Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under
their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s
principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion,
referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method
for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good
energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the
gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without
the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell
bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational
integrator compared with integrators that are not symplectic or do not preserve the Lie group structure. 相似文献
12.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》1979,19(3):215-241
Effects of an interaction between the mantle and the core of the Earth on its rotational motion are investigated. Assuming that the Earth consists of a rigid mantle and a rigid core with a frictional coupling and a kind of inertial coupling between them, the equations of motion are derived, and they are solved in a close approximation. The solution gives the expressions for the precession, the nutation, the secular changes in the obliquity and the rotational speed, the polar motion and so on as functions of the magnitudes of these forces. A numerical estimation shows that the effect of the friction on the amplitude and phase of the nutation is small for a reasonable intensity of the friction while inertial coupling force has a decisive influence on the amplitude, and an appropriately chosen value of the latter force gives a nutation which closely agrees with observations. It is also indicated that this torque remarkably lessens the rates of the secular changes in the obliquity and the rotational speed. The possibility of a periodical change in the amplitude of the polar motion is suggested as a result of the interaction between the two consituents. 相似文献
13.
Jason W. Mitchell David L. Richardson 《Celestial Mechanics and Dynamical Astronomy》2001,81(1-2):13-25
Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques. 相似文献
14.
The field equations and the equation of motion for stationary, axisymmetric, rotating ideal magnetohydrodynamic flow are presented. Exact solutions are obtained for rigid rotation, for vanishing and nonvanishing Lorentz force. 相似文献
15.
F. M. El-Sabaa 《Astrophysics and Space Science》1989,162(2):235-242
The equations of motion of a rigid body about a fixed point in a central Newtonian field is reduced to the equation of plane motion under the action of potential and gyroscopic forces, using the isothermal coordinates on the inertia ellipsoid.The construction of periodic solutions nearby equilibrium points, by using the Liapunov theorem of holomorphic integral are obtained and the necessary and sufficient conditions for the stability of the system are given. 相似文献
16.
Richard Moeckel 《Celestial Mechanics and Dynamical Astronomy》2017,128(1):3-18
Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis in \(\mathbb {R}^3\). Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of \(n\ge 3\) disjoint rigid bodies is never an energy minimizer. This generalizes a known result about point masses to the case of rigid bodies. 相似文献
17.
Yoshimasa Hosokawa 《Astrophysics and Space Science》1985,115(2):403-407
Dynamics of the apsidal motion in close binary systems are discussed. A comparison between the solution for the perfect fluid system and the solution for the rigid system reveals that some overall viscosity in the interior of distorted star has a right tendency to reconcile the observation of apsidal motion with the theory of internal structure. 相似文献
18.
19.
The motion of a gyrostat in a circular orbit in a Newtonian field of force is considered. The gyrostat has four homogeneous viscoelastic bars attached to it. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Bending deformations of the bars, accompanied by dissipation of energy, are the cause of the evolution of the system's rotational motion. Approximate equations describing this evolution are derived, together with averaged equations in Andoyer variables. 相似文献
20.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system. 相似文献