On the stability of motion of gyrostats

Rumiantsev, V.V.

doi:10.1016/0021-8928(61)90094-6

Cited by 33 publications

(9 citation statements)

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“…The 11 is cosine of the angle between the X and the x b . We can transform the X-axis to the x b by (1) rotation of the XYZ about X through α, (2) rotation about y * 1,2 through β, (3) rotation about x * 2,3 through γ , (4) rotation about y * 3 through λ, thus:…”

Section: Resultsmentioning

confidence: 99%

“…The attitude dynamics of gyrostats has been considered in a pioneering work by Volterra [9]. The practical importance and complexity of the dynamics of gyrostats motivates researchers of the related fields to work more precise on the gyrostats [10][11][12][13]. Although most of these efforts are centered on analyzing the equilibrium states, some authors recently have investigated bifurcations and chaos in the gyrostat satellites [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation and Chaos in an Apparent-Type Gyrostat Satellite

Shirazi

Ghaffari-Saadat

2005

Nonlinear Dyn

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Attitude dynamics of an asymmetrical apparent gyrostat satellite has been considered. Hamiltonian approach and Routhian are used to prove that the dynamics of the system consists of two separate parts, an integrable and a non-integrable. The integrable part shows torque free motion of gyrostat, while the non-integrable part shows the effect of rotation about the earth and asphericity of the satellite's inertia ellipsoid. Using these results, theoretically when the non-integrable part is eliminated, we are able to design a satellite with exactly regular motion. But from the engineering point of view the remaining errors of manufacturing process of the mechanical parts cause that the non-integrable part can not be eliminated, completely. So this case can not be achieved practically. Using Serret-Andoyer canonical variable the Hamiltonian transformed to a more appropriate form. In this new form the effect of the gravity, asphericity, rotational motion and spin of the rotor are explicitly distinguished. The results lead us to another way of control of chaos. To suppress the chaotic zones in the phase space, higher rotational kinetic energy can be used. Increasing the parameter related to the spin of the rotor causes the system's phase space to pass through a heteroclinic bifurcation process and for the sufficiently large magnitude of the parameter the heteroclinic structure can be eliminated. Local bifurcation of the phase space of the integrable part and global heteroclinic bifurcation of whole system's phase space are presented. The results are examined by the second order Poincaré surface of section method as a qualitative, and the Lyapunov characteristic exponents as a quantitative criterion.Nomenclature a 1 , a 2 , a 3 = Directional cosines of the spin axis in the body fixed coordinate. α 1 , α 2 , α 3 , α 4 = Dimensionless parameters of the system. b i j , r i j , j i j = Components of the body, the rotor and the whole satellite moment of inertia tensors respectively, in the body fixed coordinate. β 1 , β 2 , β 3 = Euler angles. φ = Euler angle (rotation angle about xb-axis). G, g b , g r = The satellite, the body and the rotor center of masses, respectively. H = Hamiltonian of the system. h , h 0 , h 1 , h 2 = Normalized Hamiltonian. i 1 , i 2 , i 3 = Unit vectors of the body fixed coordinate. J b , J r , j s = The body, the rotor and the whole satellite moment of inertia tensors respectively, in the body fixed coordinate. J 1 , J 2 , J 3 = Principal moments of inertia of the satellite. K , K 0 K 1 , K 2 = Transformed Hamiltonian. L = Lagrangian of the system. λ i , i = 1, . . . , 6 = Characteristic Lyapunov Exponents. p σ = Generalized momentum associated with σ . θ = Euler angle (rotation about y-axis). R 3 = Moment of inertia of the rotor about the body fixed axis of spin. R = Radius of the satellite's orbit. R = Routhian of the system. 260 K. H. Shirazi and M. H. Ghaffari-Saadat σ,σ = Relative angle and relative angular velocity of the body and the rotor, respectively. T = Kinetic energy.V = Potential ener...

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bifurcation and Chaos in an Apparent-Type Gyrostat Satellite

Shirazi

Ghaffari-Saadat

2005

Nonlinear Dyn

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“…This problem is an integrable case, and its solution, in the case of one rotor, is given in terms of elliptic functions (see e.g. [Ruminatsev, 1961;Cochran et al, 1982]). …”

Section: Introductionmentioning

confidence: 99%

Spin Rotor Stabilization of a Dual-Spin Spacecraft with Time Dependent Moments of Inertia

Lanchares

Iñarrea

Salas

1998

Int. J. Bifurcation Chaos

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“…Here, these two unit vectors are located in the principal central plane of inertia of the gyrostat, containing the axis of the undeformed rod. The deformations of the rod, which naturally depend on the orientation and the gyrostatic moment which ensures equilibrium of the chosen orientation (non-trivial equilibrium since, in this case, generally speaking, the rod is deformed), and its stability in the Lyapunov sense are determined for the two single parameter families of uniaxial orientations of the system to an attracting centre which have been separated out in this way.The problem of the steady motions of a gyrostat [1,2], which is treated here as a rigid body with a rotating statically counterbalanced flywheel and a dynamically rotating flywheel positioned in it is customarily separated into a direct and an inverse problem. In the direct problem (the problem of analysis) it is necessary to find the steady motions (the equilibria, in particular) for a given gyrostatic moment of the system.…”

mentioning

confidence: 99%

“…The problem of the steady motions of a gyrostat [1,2], which is treated here as a rigid body with a rotating statically counterbalanced flywheel and a dynamically rotating flywheel positioned in it is customarily separated into a direct and an inverse problem. In the direct problem (the problem of analysis) it is necessary to find the steady motions (the equilibria, in particular) for a given gyrostatic moment of the system.…”

mentioning

confidence: 99%

The stability of a family of non-trivial equilibrium orientations to the attracting centre of a gyrostat with an elastic rod

Chaikin¹

2004

Journal of Applied Mathematics and Mechanics

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On the stability of motion of gyrostats

Cited by 33 publications

References 1 publication

Bifurcation and Chaos in an Apparent-Type Gyrostat Satellite

Bifurcation and Chaos in an Apparent-Type Gyrostat Satellite

Spin Rotor Stabilization of a Dual-Spin Spacecraft with Time Dependent Moments of Inertia

The stability of a family of non-trivial equilibrium orientations to the attracting centre of a gyrostat with an elastic rod

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