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Friday, May 15, 2020 | History

2 edition of Equilibria of orbiting gyrostats with internal angular momenta along principal axes found in the catalog.

Equilibria of orbiting gyrostats with internal angular momenta along principal axes

Richard W. Longman

# Equilibria of orbiting gyrostats with internal angular momenta along principal axes

## by Richard W. Longman

Written in English

Subjects:
• Artificial satellites -- Attitude control systems.,
• Artificial satellites -- Orbits.,
• Gyrostabilizers.,
• Angular momentum.

• Edition Notes

The Physical Object ID Numbers Statement Richard W. Longman. Series Paper / Rand -- P-3916, P (Rand Corporation) -- P-3916. Contributions Rand Corporation. Pagination 40 p. : Number of Pages 40 Open Library OL18554618M

The following classification of the possible relative equilibria of a gyrostat satellite is assumed (see, for example, Ref. 2): a) class 1 are relative equilibria for which the unit vectors of the Ox k axes are directed to one or other side along the Oy k axes, that is, each unit vector x . I am following Landau. Here $\mathbf{L}$ is angular momentum and $\mathbf{\Omega}$ is the angular velocity. The qualitative treatment for symmetric top in absence of gravity starts by choosing principal axes of body such that $\mathbf{L\cdot e_2}=0$, where {$\mathbf{e_1,e_2,e_3}$} are the principal axes directions and $\mathbf{e_3}$ is axis of symmetry.

This is the angular velocity of the wheel (with respect to ground) resolved along the local xyz axes. Furthermore, This is the angular acceleration of the wheel (with respect to ground) resolved along the local xyz axes. Thus, the second and third of Euler's equations are equal to zero, therefore ΣM Gy = M y = 0, and ΣM Gz = M z = 0. As a. 3. Angular Velocity with Euler Angles [9 points] (a) [2 points] Show that the components of angular velocity along the body axes (x',y ',z ') are given in terms of Euler angles by ω x l = φ˙ sin θ sin ψ + θ ˙ cos ψ, ω ˙ y l = φ sin θ cos ψ − θ ˙ sin ψ, ω ˙ ˙ z. l = φ cos θ + ψ. This is done in the text!

bodies, in particular, gyrostats. The basic methods and principles of control of rotational motions of bodies and systems were studied, for example, in notes []. Modern domestic and foreign writers investigate tasks about stability of equilibrium positions and stationary motions of gyrostats in orbits [4, 5], about resonant and chaotic modes of. In physics, when you calculate an object’s moment of inertia, you need to consider not only the mass of the object but also how the mass is distributed. For example, if two disks have the same mass but one has all the mass around the rim and the other is solid, then the disks would [ ].

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### Equilibria of orbiting gyrostats with internal angular momenta along principal axes by Richard W. Longman Download PDF EPUB FB2

At small rotor speeds, an orbiting gyrostat with rotor axis along a principal body axis has 24 discrete orientations that are in equilibrium relative to gravitational torques. With increasing rotor speed, these disappear four at a time, leaving only those equilibria that have the rotor axis aligned with the perpendicular to the orbital plane.

Add tags for "Equilibria of orbiting gyrostats with internal angular momenta along principal axes". Be the first. These are functions of the moments of inertia of the gyrostat and the relative angular momentum of the rotor(s) with respect to the satellite.

The matrix equation is put into a form from which it is found that orbital stability can be achieved by proper choice of the internal angular momentum by: The Equilibria of Orbiting Gyrostats with Internal Angular Momenta Along Principal Axes.

General Solution for the Equilibria of Orbiting Gyrostats Subject to Gravitational Torques. Gravity Assist from Jupiter's Moons for Jupiter-Orbiting Space Missions. The 3×1 system angular momentum vector may be written in F b as h = Iω +AI sω s (1) where ω is the angular velocity of F b, and ω s is an N ×1 matrix containing the axial angular velocities of the rotors relative to F b.

The N × 1 matrix h a contains the absolute axial angular momenta of the wheels and may be written as h a = I sA Tω +I. Thus, in the present paper all equilibria of a gyrostat in a circular orbit are found in the case where internal angular momentum of a gyrostat is collinear to one of its principal axes of inertia.

Sufficient conditions for stability of these equilibria are also derived. Hamiltonian mechanics and relative equilibria of orbiting gyrostats Article in Journal of the Astronautical Sciences 55(1) March with 16 Reads How we measure 'reads'.

Those configurations for which the internal angular momentum vector (or the rotor axis) is aligned with a principal axis have been treated in a separate work, where it is shown that at one, and only one, rotor speed there exists a continuum of equilibrium orientations.

A relative equilibria of a gyrostat satellite with internal angular momentum along a principal axis was determined. The rotors angular velocities relative to the satellite body were found to be. The problem of the motion of rigid bodies with equal central moments of inertia was investigated in Refs 1, 2, 3 for a homogeneous cone and a homogeneous cylinder with specially chosen dimensions, and also for solids, the mass distribution of which allows of a regular polyhedron symmetry group, all the steady motions were obtained and their stability was investigated.

The equilibria of orbiting gyrostats with internal angular momenta along principal axes. In: Proc Symp on Gravity Gradient Attitude Stabilization. El Segundo, CA, Los Angeles, The dynamics of a satellite-gyrostat moving in the central Newtonian force field along a circular orbit is studied.

In the particular case when the vector of gyrostatic moment is parallel to one of the satellite’s principal central axes of inertia, all the equilibrium states are determined.

Longman, R. W.:‘The Equilibria of Orbiting Gyrostats with Internal Angular Momenta along Principal Axes’,Proceedings of the Symposium on Gravity Gradient Attitude Stabilization, Air Force Report No. SAMSO-TR (Also appears as RAND. Cremonese Roma.

Longman, R. W.:The Equilibria of Orbiting Gyrostats with Internal Angular Momenta Along Principal Axes, Proc. of the Symposium on Gravity Gradient Stabilization Aerospace Corporation, El Segundo, Calif.

The internal angular momentum vector due to these rotors is parallel to one of the principal axes of the entire satellite; this axis is aligned with (or close to) the normal to the orbit plane. The Equilibria of Orbiting Gyrostats with Internal Angular Momenta along Principal Axes R W Longman Equilibrium orientations of a gyrostate satellite - on a circular orbit with inner motions fixed.

Longman, R. W.:‘The Equilibria of Orbiting Gyrostats with Internal Angular Momenta along Principal Axes’,Proceedings of the Symposium on Gravity Gradient Attitude Stabilization.

 Longman, R. W.:‘Stable Tumbling Motions of a Dual-Spin Satellite Subject to. A gyrostat is a rigid body containing an internal source of angular momentum which does not alter the geometry of the system. Because of their relative simplicity, gyrostats are frequently used to. The problem of determining all equilibria of a satellite in a circular orbit is solved in the case where the satellite is subjected to gravitational and aerodynamic torques.

The number of isolated equilibria is shown to be no less than eight and no more than The existence proof of one-parameter families of stationary solutions is given.

Using Lyapunov's method sufficient conditions for. Longman, “The Equilibria of Orbiting Gyrostats with Internal Angular Momenta along Principal Axes,” Proceedings of the Symposium on Gravity Gradient Attitude Stabilization, DecemberAerospace Corporation, El Segundo, California, published as Air Force SAMSO-TR (Aerospace Corporation Report No.

TR()-1), pp. In spectroscopy: Total orbital angular momentum and total spin angular momentum quantum numbers giving the total orbital angular momentum and total spin angular momentum of a given state.

The total orbital angular momentum is the sum of the orbital angular momenta from each of the electrons; it has magnitude Square root of √ L(L + 1) (ℏ), in which L is an integer.The attitude motion is studied of asymmetric dual-spin gyrostats which may be modeled as free systems of two rigid bodies, one asymmetric and one axisymmetric.

Exact analytical solutions of the attitude motion are presented for all possible ratios of inertia moments of these bodies. The dynamics of free gyrostats with zero internal torque is considered.1.

Introduction. The general study of the dynamics of rigid bodies and gyrostats has been presented extensively in the classic literature. Eulerian, Lagrangian and Hamiltonian formulations of such dynamics have been the main tools used in the formulation of these problems (see for instance, or).

It is known that a gyrostat is a mechanical system S made of a rigid body S 1 to which other.