ON THE BIFURCATION AND STABILITY OF THE STEADY-STATE MOTIONS OF COMPLEX MECHANICAL SYSTEMSстатья

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[1] On the bifurcation and stability of the steady-state motions of complex mechanical systems / V. M. MOROZOV, V. N. RUBANOVSKII, V. V. RUMIANTSEV, V. A. SAMSONOV // Journal of Applied Mathematics and Mechanics (English translation of Prikladnaya Matematika i Mekhanika). — 1973. — Vol. 37, no. 3. — P. 387–399. Many objects of modern technology (rockets, spaceships, airplanes, gyroscopic devices, centrifuges, etc. ) can be modelled in a number of cases by mechanical systems comprised of absolutely rigid bodies and material points and of deform- able (liquid and elastic) bodies connected with them. Mechanical systems con- taining among its parts both subsystems with a finite number of degrees of free- dom as well as units with distributed parameters, i. e. continuous media, are called complex systems for brevity. We consider the steady-state motions of complex systems. Stationary values of the potential energy V or of the altered potential energy FV of the system correspond to the steady-state motions. The stability problem for the steady-state motions leads to the investigation of the nature of the extremum of the potential energy V or W. The minimum of the potential energy corresponds to a stable motion. In a number of important cases the stability (instability) conditions can be obtained as conditions for the positive definiteness (for sign-alteration together with certain additional conditions) of the second variation baV or FW of the potential energy. These general results are applied to solving a number of concrete problems on the stability of the steady-state motions of complex systems. Stability conditions for the motion of a rigid body with liquid and elastic parts in various force fields are discussed. [ DOI ]

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