Stability of Multidegree of Freedom Potential Systems to General Infinitesimal Positional Perturbation Forces

Abstract

The stability of general linear multi-degree of freedom stable potential systems that are perturbed by general arbitrary positional forces, which may be neither conservative nor purely circulatory/conservative, is considered. It has been recently recognized that such perturbed potential systems with multiple frequencies of vibration are susceptible to instability, and this paper is centrally concerned with the situation when potential systems have such multiple natural frequencies. An approach based on perturbation theory that includes nonlinear terms in the expansions of the perturbed eigenvalues is developed. Explicit conditions under which the system either remains stable or becomes unstable due to flutter are provided. These results show that the stability/instability picture that emerges is far subtler and more complex than what might be intuitively inferred. The manner in which prior results related to narrower classes of perturbation matrices, like circulatory matrices, get included in the more general results obtained here is pointed out. Several numerical examples illustrate the applicability of the analytical results. An engineering application is provided demonstrating the power of the analytical results.