On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints

Abstract

The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V.V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example. © 2009 Elsevier Ltd. All rights reserved.