Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Sufficient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also sufficient, for some configurations of the gyrostat

Stability Conditions for Permanent Rotations of a Heavy Gyrostat with Two Constant Rotors

In this paper, we consider the motion of an asymmetric heavy gyrostat, when its center of mass lies along one of the principal axes of inertia. We determine the possible permanent rotations and, by means of the Energy-Casimir method, we give sufficient stability conditions. We prove that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space, by the action of two spinning rotors, one of them aligned along the principal axis, where the center of mass lies. We also derive necessary stability conditions that, in some cases, are the same as the sufficient ones

Assessment of corneal biomechanical properties and intraocular pressure in myopic Spanish healthy population

Purpose. To examine biomechanical parameters of the cornea in myopic eyes and their relationship with the degree of myopia in a western healthy population. Methods. Corneal hysteresis (CH), corneal resistance factor (CRF), Goldmann correlated intraocular pressure (IOP), and corneal compensated IOP (IOPcc) were measured using the ocular response analyzer (ORA) in 312 eyes of 177 Spanish subjects aged between 20 and 56 years. Refraction was expressed as spherical equivalent (SE), which ranged from 0 to -16.50 diopters (D) (mean: -3.88 ± 2.90 D). Subjects were divided into four groups according to their refractive status: group 1 or control group: emmetropia (-0.50 = SE 0.05); nevertheless, IOPcc was significantly higher in the moderatelymyopic (15.47±2.47mmHg) and highlymyopic (16.14± 2.59mmHg) groups than in the emmetropia (15.15 ± 2.06mmHg) and low myopia groups (14.53 ± 2.37mmHg). No correlation between age and the measured parameters was found. CH and IOPcc were weakly but significantly correlated with SE (¿ = 0.171, ¿ = 0.002 and ¿ = -0.131, ¿ = 0.021, resp.). Conclusions. Present study showed only a very weak, but significant, correlation between CHand refractive error, with CH being lower in both moderately and highlymyopic eyes than that in the emmetropic and low myopic eyes.These changes in biomechanical properties of the cornea may have an impact on IOP measurement, increasing the risk of glaucom

Exact solution of a triaxial gyrostat with one rotor

The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves. © 2008 Springer Science+Business Media B.V

Chaos in the reorientation process of a dual spin spacecraft with time dependent moments of inertia.

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances

On the stability of equilibria in two degrees of freedom Hamiltonian systems under resonances.

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under resonances. Determining the stability or instability is based on a geometrical criterion based on how two surfaces, related with the normal form, intersect one another. The equivalence of this criterion with a result of Cabral and Meyer is proved. With this geometrical procedure, the hypothesis may be extended to more general cases. © 2005 Springer Science+Business Media, Inc

Non linear stability in resonant cases: A geometrical approach.

In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables