On the Generalization of Merkin’s Theorem for Circulatory Systems: A Rank Criterion

ASME International

Ranislav M. Bulatovic , Firdaus E. Udwadia

2020

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 this remarkable theorem. These new results are compared with those obtained earlier.

ASME International

Uncoupling of General Linear Multi-degree-of-freedom Structural and Mechanical Systems Through Quasi-diagonalization

Abstract This paper develops a fundamental result in linear algebra by providing the necessary and sufficient conditions for the simultaneous quasi-diagonalization of two symmetric matrices and two skew-symmetric matrices by a real orthogonal congruence. This result is used to study the uncoupling of general linear multi-degree-of-freedom (MDOF) structural and mechanical systems described by arbitrary damping and stiffness matrices through quasi-diagonalization, and real orthogonal...

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ASME International

Uncoupling of Damped Linear Potential Multi-Degrees-of-Freedom Structural and Mechanical Systems

Abstract This paper provides the necessary and sufficient conditions for a multi-degrees-of-freedom linear potential system with an arbitrary damping matrix to be uncoupled into independent subsystems of at most two degrees-of-freedom using a real orthogonal transformation. The incorporation of additional information about the matrices, which many structural and mechanical systems commonly possess, shows a reduction in the number of these conditions to three. Several new results are...

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ASME International

Decomposition and Uncoupling of Multi-Degree-of-Freedom Gyroscopic Conservative Systems

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...

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ASME International

Uncoupling of Linear Multi-Degree-of-Freedom Damped Gyroscopic Potential Systems

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...

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ASME International

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

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....

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