Chaotic rotations of an asymmetric body with time-dependent moments of inertia and viscous drag. (English) Zbl 1129.70304
Summary: We study the dynamics of a rotating asymmetric body under the influence of an aerodynamic drag. We assume that the drag torque is proportional to the angular velocity of the body. Also we suppose that one of the moments of inertia of the body is a periodic function of time and that the center of mass of the body is not modified. Under these assumptions, we show that the system exhibits a transient chaotic behavior by means of a higher dimensional generalization of the Melnikov’s method. This method give us an analytical criterion for heteroclinic chaos in terms of the system parameters. These analytical results are confirmed by computer numerical simulations of the system rotations.
MSC:
| 70E17 | Motion of a rigid body with a fixed point |
| 37N05 | Dynamical systems in classical and celestial mechanics |
| 70E20 | Perturbation methods for rigid body dynamics |
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
References:
| [1] | Bryson A. E., Control of Spacecraft and Aircraft (1988) |
| [2] | Deimel R. F., Mechanics of the Gyroscope. The Dynamics of Rotation (1952) · Zbl 0049.40603 |
| [3] | Deprit A., J. Astron. Sci. 41 pp 603– |
| [4] | Gradshteyn I. S., Table of Integrals, Series and Products (1980) · Zbl 0521.33001 |
| [5] | Gray A., A Treatise on Gyrostatics and Rotational Motion (1959) |
| [6] | DOI: 10.1115/1.2791660 · doi:10.1115/1.2791660 |
| [7] | DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 |
| [8] | DOI: 10.1512/iumj.1983.32.32023 · Zbl 0488.70006 · doi:10.1512/iumj.1983.32.32023 |
| [9] | Hughes P. C., Spacecraft Attitude Dynamics (1986) |
| [10] | Iñarrea M., Int. J. Bifurcation and Chaos 10 pp 997– |
| [11] | DOI: 10.1063/1.526276 · Zbl 0547.70006 · doi:10.1063/1.526276 |
| [12] | Lambert J. D., Computational Methods in Ordinary Differential Equations (1976) |
| [13] | DOI: 10.1142/S0218127498000401 · Zbl 0963.70524 · doi:10.1142/S0218127498000401 |
| [14] | DOI: 10.1007/978-3-642-88412-2 · doi:10.1007/978-3-642-88412-2 |
| [15] | DOI: 10.1007/978-1-4757-2184-3 · doi:10.1007/978-1-4757-2184-3 |
| [16] | Melnikov V. K., Trans. Moscow Math. Soc. 12 pp 1– |
| [17] | Peano G., Atti. R. Accad. Sci. Torino 30 pp 515– |
| [18] | Peano G., Atti. R. Accad. Sci. Torino 30 pp 845– |
| [19] | DOI: 10.1016/S0020-7462(99)00028-1 · Zbl 1068.70505 · doi:10.1016/S0020-7462(99)00028-1 |
| [20] | DOI: 10.1137/0147015 · Zbl 0629.34043 · doi:10.1137/0147015 |
| [21] | Serret J. A., Mémoires de l’académie des sciences de Paris 35 pp 585– |
| [22] | DOI: 10.1017/CBO9780511815652 · doi:10.1017/CBO9780511815652 |
| [23] | DOI: 10.1016/0020-7462(94)00049-G · Zbl 0837.70004 · doi:10.1016/0020-7462(94)00049-G |
| [24] | DOI: 10.1007/BF02417877 · JFM 29.0650.01 · doi:10.1007/BF02417877 |
| [25] | DOI: 10.1080/14786440408564240 · doi:10.1080/14786440408564240 |
| [26] | Wiesel W. E., Spaceflight Dynamics (1997) |
| [27] | DOI: 10.1007/978-1-4612-1042-9 · doi:10.1007/978-1-4612-1042-9 |
| [28] | DOI: 10.1016/0019-1035(84)90032-0 · doi:10.1016/0019-1035(84)90032-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
