Neutralization of the Destabilization Effect Caused by Small Damping Force in Non-conservative System

Abstract

For the first time, the destabilizing effect of small dissipative forces was marked by Thomson and Tait in 1879. They showed that a statically unstable conservative system stabilized by gyroscopic forces could be destabilized again by introducing small damping forces. Later on, in 1924, Kimball noticed the paradoxical effect of damping on dynamic stability for rotor systems that have stable steady motions for a certain range of speed but become unstable when the speed is changed to a value outside the range. Further progress in engineering and technology has witnessed that such phenomenon arises in numerous mechanical applications and may cause equipment malfunction. In the present article, this problem is studied by comparing the stability conditions for undamped and weakly damped systems. The aim is to find the interrelation between the matrix elements characterizing the damping force (assuming that the stiffness matrix is known), which allows to avoid or minimize the negative effect of small damping in the presence of non-conservative positional forces. For the case study of two Degrees-of-Freedom system, a simple analytical technique is suggested for this purpose. This technique is demonstrated on two mechanical systems: a system with friction-induced vibrations and a double pendulum under the action of the follower force.