The Ziegler effect in a non-conservative mechanical system. (English. Russian original) Zbl 1272.70081

J. Appl. Math. Mech. 74, No. 1, 51-60 (2010); translation from Prikl. Mat. Mekh. 74, No. 1, 74-88 (2010).
Summary: The destabilization of the stable equilibrium of a non-conservative system under the action of an infinitesimal linear viscous friction force is considered. In the case of low friction, the necessary and sufficient conditions for stability of a system with several degrees of freedom and, as a consequence, the conditions for the existence of the destabilization effect (Ziegler’s effect) are obtained. Criteria for the stability of the equilibrium of a system with two degrees of freedom, in which the friction forces take arbitrary values, are constructed. The results of the investigation are applied to the problem of the stability of a two-link mechanism on a plane, and the stability regions and Ziegler’s areas are constructed in the parameoter space of the problem.


70J25 Stability for problems in linear vibration theory
70F20 Holonomic systems related to the dynamics of a system of particles
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