Stability Problems

Abstract

The stability of the equilibrium configuration of floating bodies has already been considered by simple static means, and proper conditions were found in the form of inequalities of the type (2.115). In this section, the stability of conservative systems at rest is observed by analyzing the motion that follows any perturbation of the equilibrium state (ie by the dynamic method of small perturbations) and, equivalents, by the static Dirichlet stability criterion. Simple applications are given, including the balance problem of rigid heavy bodies in contact, the structural problem of the buckling of slender columns and thin plates (a bifurcation problem), as well as the snapping of shallow arches when the lateral load becomes critical. The extension of the dynamic method of small perturbations to include the stability of a given (main) motion is illustrated through the analyses of a mechanical control device, the centrifugal governor, and the gyroscope without moment. In a third subsection, the stability of elastic-plastic structures is considered statically by limit load analysis and, quasistatically, for alternating loadings by the shake-down theorems of Melan and Koiter. Furthermore, the hydrodynamic stability of incompressible flow in open channels is discussed and the loss of energy in the hydraulic jump where the rapid flow changes “abruptly” to tranquil streaming is given. Finally, flutter instability is described well by the self-excited oscillations of a simplified model of an airfoil that occur above a critical speed of flight.