The effect of small internal and external damping on the stability of distributed non-conservative systems

The effect of small internal and external damping on the stability of distributed non-conservative systems is investigated. A theory is constructed for the qualitative and quantitative description of the “destabilization paradox” in these systems, one manifestation of which is an abrupt drop in the critical load and frequency when small dissipative forces are taken ino account. The theory is base on an analysis of the bifurcations of multiple eigenvalues of non-self-adjoint differential operators that depend on parameters. Explicit formulae are obtained for the collapse of multiple eigenvalues with Keldysh chains of arbitrary length, for linear differential operators that depend analytically on a complex spectral parameter and are smooth functions of a vector of real parameters. It is shown that the “destabilization paradox” is related to the perturbation by small damping of a double eigenvalue of a circulatory system with a Keldysh chain of length 2, which is responsible for the formation of a singularity on the boundary of the stability domain. Formulae describing the behaviour of the eigenvalues of a non-conservative system when the load and disspitation parameters are varied are described. Explicit expressions are obtained for the jumps in the critical loads and frequency of the loss of stability. Approximations are obtained in analytical form of the asymptotic stability domain in the parameter space of the system. The stabilization effect, in which a distributed circularity system is stabilized by small dissipative forces and which consists of an increase in the critical load, is explained, and stabilization conditions are derived. As a mechanical example, the stability of a visco-elastic rod with small internal and external damping is investigated; unlike earlier publications, it is shown that the boundary of the stability domain has a “Whitney umbrella” singularity. The dependence of the critical load on the internal and external friction parameters is obtained in analytical form, yielding an explicit expression for the jump in critical load. On the basis of the analytical relations, the domains of stabilization and destabilization in the parameter space of the system are constructed. It is shown that the analytical formulae are in good agreement with the numerical results of earlier research.