On the instability of equilibrium of a mechanical system with nonconservative forces

摘要

In this paper the stability of equilibrium of nonholonomic systems, on which dissipative and nonconservative positional forces act, is considered. We have proved the theorems on the instability of equilibrium under the assumptions that: the kinetic energy, the Rayleigh’s dissipation function and the positional forces are infinitely differentiable functions; the projection of the positional force component which represents the first nontrivial form of Maclaurin’s series of that positional force to the plane, which is normal to the vectors of nonholonomic constraints in the equilibrium position, is central and repulsive (with its centre of action in the equilibrium position). The suggested theorems are generalization of the results from [V.V. Kozlov, Prikl. Math. Mekh. (PMM), T58, V5, (1994), 31-36] and [M.M. Veskovic, Theoretical and Applied Mechanics, 24, (1998), 139-154]. The result obtained is analogous to the result from [D.R. Merkin, Introduction to theory of the stability of motion, Nauka, Moscow (1987)], which refers to the impossibility of equilibrium stabilization in a holonomic conservative system by dissipative and nonconservative positional forces in case when the potential energy in the equilibrium position has the maximum. The proving technique will be similar to that used in the paper [V.V. Kozlov, Prikl. Math. Mekh. (PMM), T58, V5, (1994), 31-36]. .