On the destabilization of some equilibriums of nonconservative system with three degrees of freedom

Аuthors

Krasilnikov P. S.^*, Amelin R. N.^*

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: kaf803@mai.ru

Abstract

This paper focuses on Zieglers effect. This effect is the destabilization of stable equilibrium of system ’$under the action of small friction forces. This effect exists if and only if the system is subject to the potential forces, the following forces and the linear forces of a viscous friction. The following force is a jet force of jet engine. The following force is a positional force that do not have a potential energy. Therefore the effect of destabilization can prove oneself when the jet engine of launch vehicles works.
The destabilization of stable equilibrium of some system of rods, which model a complex engineering structure is considered. It is suppose that the rod system is subject to the potential forces, the following forces and the linear forces of a viscous friction. First approximation equations are obtained. It is shown that these equations fall into two subsystems with one and two degree of freedoms accordingly. The conditions of asymptotic stability and instability as a first approximation, the conditions of the existence of Zieglers effect are obtained. It is shown that Zieglers zone is a narrow band close to a boundary of stability area. By means of theorems of partial asymptotic stability and instability, necessary and sufficient conditions of asymptotic stability in complete equations are obtained.

Keywords:

destabilization, small friction forces, Zieglers effect, asymptotical stability

References

1. Rabinovich B.I. Vvedenie vdinamiku raket-nositelei kosmicheskikh apparatov (Introduction indynamics oflaunch vehicles ofspace vehicles), Moscow, Mashinostroenie, 1975, 416p.
2. Baikov A.E., Krasilnikov P.S. Prikladnaya matematika imekhanika, 2010, vol.74, no.1, pp. 74-88.
   
3. Oziraner A.S. Prikladnaya matematika imekhanika, 1973, vol.37, no.4, pp. 659-665.
   
4. Prokopev V.P. Prikladnaya matematika imekhanika, 1973, vol.39, no.3, pp. 422-426.