A theory of the destabilization paradox in non-conservative systems
In the present paper, a theory is developed qualitatively and quantitatively describing the paradoxical behavior of general non-conservative systems under the action of small dissipative and gyroscopic forces. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. The movement of eigenvalues of the system in the complex plane is analytically described and interpreted. Approximations of the asymptotic stability domain in the space of the system parameters are obtained. An explicit asymptotic expression for the critical load as a function of dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. The classical Ziegler–Herrmann–Jong pendulum considered as a mechanical application demonstrates the efficiency of the theory.
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The author is grateful to Prof. A. Seyranian for valuable comments. The support of this work by the Russian Foundation for Basic Research and National Natural Science Foundation of China (grants RFBR-NNSFC 02-01-39004, RFBR 03-01-00161), and by the United States Civilian Research and Development Foundation for the Independent States of the Former Soviet Union and Russian Ministry of Education (grant CRDF-BRHE Y1-MP-06-19) is gratefully acknowledged.