Abstract
The stability of general linear multidegree 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.
Abstract
This paper explores the decomposition of linear, multi-degree-of-freedom, conservative gyroscopic dynamical systems into uncoupled subsystems through the use of real congruences. Two conditions, both of which are necessary and sufficient, are provided for the existence of a real linear coordinate transformation that uncouples the dynamical system into independent canonical subsystems, each subsystem having no more than two-degrees-of-freedom. New insights and conceptual simplifications...
Abstract
This paper deals with the uncoupling of linear damped multi-degree-of-freedom gyroscopic potential systems in which the damping is taken to have a specifically chosen form. Necessary and sufficient conditions are obtained that guarantee the uncoupling of such damped systems into independent subsystems with at most two degrees-of-freedom. Along with several other results, it is shown that when the potential (stiffness) matrix of the damped system has distinct eigenvalues—a...
Abstract
The stability of general linear multidegree 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....
The stability of linear multi-degree-of-freedom stable potential systems with multiple natural frequencies under the action of infinitesimal circulatory forces is considered. Contrary to the received view that such systems are inherently unstable, a careful study shows that such systems have a much more complex behaviour than previously recognized and could exhibit an alternation of stability and instability that depends on the structure of the potential system and its interaction with the circulatory...
Abstract
This paper provides a generalization of the celebrated Merkin theorem. It provides new results on the destabilizing effect of circulatory forces on stable potential systems. Previous results are described and discussed, and the paper uncovers a deeper understanding of the fundamental reason for the destabilization. Instability results in terms of rank conditions that deal with the potential and circulatory matrices that describe the system are obtained, thereby generalizing...
