Gyroscopic stabilization in the presence of nonconservative forces

Abstract

This paper analyzes the stability of a linear autonomous nonconservative system with an even number of degrees of freedom in the presence of potential, gyroscopic, dissipative, and nonconservative positional forces. It is well known that, when applied separately, dissipative and nonconservative positional forces destroy gyroscopic stabilization [1, 3]. However, their combination can make a system asymptotically stable. It is found that the complexity of the choice of such a combination is associated with a Whitney umbrella singularity existing on the boundary of the gyroscopic stabilization domain of the nonconservative system. In this paper, an approximation to the boundary of the asymptotic stability domain near the singularity is explicitly found and an analytical estimate of the critical gyroscopic parameter is obtained. As an example, we analyze the stability of Hauger’s gyropendulum under the action of a follower torque. 1. Consider an autonomous nonconservative system of the form