Optimal stability envelope of the damped hauger system

Abstract

By using the Frobenius' method, the authors obtain the virtually exact instability boundary of the externally damped viscoelastic Hauger system. The interaction between the two types of damping, yields results paradoxical to those obtained with each type of damping considered separately. An optimal stability envelope, which for a prescribed value of internal damping and any external damping gives the highest attainable critical load, is obtained. The addition of external damping for a prescribed internal damping, raises the critical load at best to the above optimal value; further addition of external damping diminishes the critical load. The Ziegler's jump or pseudo-destabilization due to infinitesimal internal damping, is shown to vanish with the introduction of external damping. For large external damping, the Ziegler's jump approaches a negative asymptotic value. These paradoxical results provide a deeper insight into the complex interaction between the internal and external damping in polygenic systems.