Stabilizing and destabilizing effects of damping in non-conservative systems: Some new results

Previous work has amply demonstrated that non-conservative systems can be made unstable by the application of damping. Systems with two neutrally-stable damping levels, whereby the system initially gains stability but later loses stability as the level of damping is increased, have also been observed. The phenomenon of three damping-induced stability transitions has not been reported in the literature. Here we show that the addition of damping can cause non-conservative systems to become stable, then unstable, then stable again at the same value of the non-conservative forcing variable. This combination of stability transitions is found to exist for several example systems, including linkages with follower end forces and fluid-conveying pipes. Another interesting observation is that a given system can exhibit different forms of stability transitions in different regions of its parameter space. In a particular example, the neutral stability curves corresponding to two different modes are observed to intersect, such that the boundary separating the stable and unstable regions is piecewise continuous. This observation requires that the accepted definitions of “stabilizing” and “destabilizing” roles of damping be revised. All of these stability transition behaviors were found by applying the Routh-Hurwitz procedure, whereby the traditional procedure is first applied to the characteristic polynomial of the system, and then again to guarantee the existence of a second-order auxiliary polynomial in the Routh array. This procedure is developed in the context of examples, each of which concerns a classical apparatus who dynamics are more interesting than previously believed.