12 - Dynamic stability

Abstract

The term dynamic stability encompasses many classes of problems and many different physical phenomena; in some instances, the term is used for two distinctly different responses for the same configuration subject to the same dynamic loads. The class of problems falling in the category of parametric excitation, or parametric resonance, includes the best defined, conceived, and understood problems of dynamic stability. Examples of parametric resonance include a thin flat plate parametrically loaded by inplane forces, which may cause transverse plate vibrations; parametrically loaded shallow arches (symmetric loading), which under certain conditions vibrate asymmetrically with increasing amplitude; and long cylindrical, thin shells (or thin rings) under uniform but periodically applied pressure that can excite vibrations in an asymmetric mode. Thus, in parametric excitation, the loading is parametric with respect to certain deformation forms. This makes parametric resonance different from the usual forced vibration resonance. Based on these examples, it is realized that systems that exhibit bifurcational buckling under static conditions (regardless of whether the bifurcating static equilibrium branch is stable or unstable) are subject to parametric excitation. Moreover, there exists a large class of problems for which the load is applied statically but the system is nonconservative.