Langthjem M., Sugiyama Y.

Engineers are accustomed to think of damping mechanisms which dissipate energy as devices which stabilize vibrations. The fact that a small amount of internal (material) damping in a structure subjected to non-conservative (circulatory) forces (such as from jet engines, rocket motors, water jets, etc.) significantly reduces the critical load level, where a dynamic instability (flutter) sets in, has accordingly been considered as a paradox. The resolution of a paradox is an attractive research topic, and as such the destabilization paradox has received much attention since its discovery by Hans Ziegler in 1952. A variety of structural elements, boundary conditions, non-conservative loads, and damping mechanisms have been studied and many new destabilization phenomena have been reported. Much less has been done in explaining the actual mechanisms behind the destabilizing effects. Mathematical analyzes have been carried out, aiming at giving concise analytical expressions for the critical parameters in the vicinity of the stability bound. But a physical explanation of the destabilizing effect of damping in a circulatory system, i.e. how small damping affects the vibration modes and phase angles, has hitherto only been given for Ziegler's follower force-loaded double pendulum (Ziegler 1952, Semler et al. 1998); it has not yet been given for a continuous structure. It is the aim of the present work to provide such an explanation for the damped Beck's column, i.e. a cantilevered column subjected to a tangential �efollower�f force at the free end. We consider Kelvin-Voigt internal damping and viscous external damping, and study how vibration modes and phase angles change at the stability bound while the amount of damping is gradually reduced towards zero. We also give a detailed energy balance analysis. In this way we are able to add considerably, we think, to the understanding of the �edestabilization paradox�f in the theory of dynamic stability.