Overview

In this chapter some mechanical effects associated with bifurcations of eigenvalues and singularities of the stability boundaries are studied. First, we analyze stability and catastrophes in one-parameter circulatory systems (with non-conservative positional forces). It is proven that flutter and divergence instabilities and transition of divergence to flutter are typical catastrophes for one-parameter circulatory systems.

Then two other interesting mechanical phenomena are considered. The first one is the phenomenon of transference of instability between eigenvalue branches. It turns out that a stable eigenvalue branch of a system subjected to non-conservative loading suddenly becomes unstable and vice versa with a change of problem parameters. The second phenomenon is the destabilization of a circulatory system by infinitely small damping. It turns out that the critical load parameter of the system with small damping is typically smaller than the critical load of the system with no damping. In this chapter, these two mechanical phenomena are explained from the point of view of behavior of eigenvalue branches in the vicinity of a double eigenvalue with a single eigenvector.

Finally, we discuss an exciting effect of disappearance of flutter instability in the problem of aeroelastic stability of an unswept wing braced by struts of two types. This problem was first considered by [Keldysh (1938)]. In this chapter it is shown that for one type of the strut the flutter instability is replaced by the static form (divergence) and the critical speed has a discontinuity: it jumps to a higher value. And for the second type...