On stability of gyroscopic systems

A. Yu. Aleksandrov ; A. A. Kosov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2013), pp. 3-13

Résumé

Mechanical systems described by the Lagrange equations of the second kind influenced by potential, dissipative, gyroscopic and nonconservative forces are considered. It is assumed that gyroscopic forces are dominant. This domination is accounted by in the presence of a large parameter as a coefficient at the vector of these forces. By the use of the Lyapunov function method the lower bounds for the large parameter values are obtained which permit to receive justified conclusions of asymptotic stability of the full system on the base of its decomposition on two subsystems. To provide this decomposition, two approaches are proposed. In the first approach, vector Lyapunov functions are applied, while the second one is based on using scalar Lyapunov functions. It is worth mentioning that compared with the known results the suggested approaches give opportunity to receive stability conditions in the case of nonstationary matrices of dissipative and positional forces. Moreover, systems with nonstationary unboundedly increasing with time parameter at gyroscopic forces are studied. The restrictions on the velocity of parameter increasing are found for which the evolution of the parameter does not destroy the asymptotic stability of the equilibrium position. The cases of linear dissipative forces and essentially nonlinear dissipative forces given by homogeneous Rayleigh functions are considered. Bibliogr. 18.

Détail

Keywords: mechanical systems, gyroscopic forces, stability, Lyapunov functions.
Mots-clés : decomposition