Analysis of the behavior of cycles limit under the increase of the orbit eccentricity and under the change of aerodynamic parameters

Аuthors

Vorontsova V. L.

Institute of Management, Economics and Finance Kazan Federal University, KFU, 4, Butlerova str., Kazan, 420012, Russia

e-mail: VLVorontsova@yandex.ru

Abstract

Orbit is considered as «a bunch of bodies» two material points connected by flexible massless inextensible thread as a system with the released bond. The equation of associated motion with the forces of gravity gradient, aerodynamic pressure gradient and aerodynamic drag of friction are used [1, 2].
In this article were taken into consideration the behavior of the limit cycle for the equation of the relative motion of the gravitational effects of aerodynamic pressure, aerogradient and dissipative factors [2], depending on the growth of the eccentricity e and aerodynamic parameter a. The effect of the ellipticity of the orbit of a possible chaotization movement is considered.
The known methods of nonlinear mechanics are used for research: the method of Lagrange equations of the first kind, the method of phase plane, numerical implementation of the method of point mappings, theory of stability of motion.
A comparative analysis of the phase portraits of the orbit with a very small ellipticity e 0.001= (case 1) and with the increase of eccentricity to e = 0.01 (case 2) are carried out. If the eccentricity of the orbit is increased by an order for the same values of aerodynamic parameters a and values and the density gradient along the dumbbells k and aerodynamic dissipation b (k = 1; b = 0.01), the location of the limit cycle will be much higher. As a result of comparative analysis, as well as in [4], we can conclude that values of the angular velocity α' in turn we can consider as a result of the growth of the values of parameter. When approaching the maximum value a achieved, the values α' are differ more than one unit.

Keywords:

orbital «bunch of bodies», limit cycles, aerodynamic parameter, eccentricity of the orbit

References

1. Beletskii V.V., Vorontsova V.L., Kof L.M., Pankova D.V. Preprint Instituta prikladnoi matematiki im. M.V. Keldysha RAN, 1996, Part 1, no. 38, 32 p.
2. Beletskii V.V., Pivovarov M.L. Prikladnaya matematika i mekhanika, 2000, vol. 64, no. 4, pp. 721-731.
3. Goncharova E.N., Vorontsova V.L. Vestnik SGU, 2008, no. 57(4), pp. 14-17.
4. Vorontsova V.L. Vestnik SGU, 2011, no. 77 (1), pp. 304-306.