The Notion of Stability

Consider a structure undergoing external loads applied quasistatically. In such a case, static equilibrium is maintained as the elastic structure deforms. If now at any level of the external force a “small” external disturbance is applied, and the structure reacts by simply performing oscillations about the deformed equilibrium state, the equilibrium state is said to be stable. This disturbance can be in the form of deformation or velocity; by “small,” we mean “as small as desired.” As a result of this latter definition, it is more appropriate to say that the equilibrium is stable for a small disturbance. In addition, we stipulate that when the disturbance is introduced, the level of the external forces is kept constant. Conversely, if the elastic structure either (a) tends to and remains in the disturbed position, or (b) diverges from the equilibrium state, the equilibrium is said to be unstable. Some authors prefer to distinguish these two conditions and call the equilibrium “neutrally stable” for case (a) and “unstable” for case (b). When either of these two cases occurs, the level of the external forces is referred to as “critical.”

This is illustrated using the system shown in Fig. 2.8. This system consists of a ball of weight W resting at different points on a surface with zero curvature normal to the plane of the figure. Points of zero slope on the surface denote positions of static equilibrium (i. e., points A, B, and C). Furthermore, the character of equilibrium at these points is substantially different. At A, if the system is disturbed through infinitesimal disturbances (i. e., small displacements or small velocities), it simply oscillates, about the static-equilibrium position A. Such an equilibrium position is called stable for small disturbances. At point B, if the system is disturbed, it tends to move away from the static-equilibrium position B. Such an equilibrium position is called unstable for small disturbances. Finally, at point C, if the system is disturbed, it tends to remain in the disturbed position. Such an equilibrium position is called neutrally stable or indifferent for small disturbances. The expression “for small disturbances” is used because the definition depends on the small size of the perturbations and is the foundational reason we may use linearized equations to conduct the analysis. If the disturbances are allowed to be of finite magnitude, then it is possible for a system to be unstable for small disturbances but stable for large disturbances (i. e., point B, Fig. 2.9a) or stable for small disturbances but unstable for large disturbances (i. e., point A, Fig. 2.9b).[2]

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