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Stability of circulatory systems under viscous friction forces. (English. Russian original) Zbl 1465.70064
Mech. Solids 55, No. 8, 1135-1141 (2020); translation from Prikl. Mat. Mekh. 84, No. 6, 677-686 (2020).
Summary: The problem on the spectrum structure of motion equations in a mechanical system linearized near equilibrium in a nonpotential force field is considered. Particular attention is paid to the case when the force field is circulatory and there are also viscous friction forces acting on the system. The stability problem solution is based on the search of invariant subspaces that are uniquely projected onto the configuration space of the system.

70J25 Stability for problems in linear vibration theory
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
70E18 Motion of a rigid body in contact with a solid surface
Full Text: DOI
[1] Bolotin, V. V., Nonconservative Problems of the Theory of Elastic Stability (1961), Moscow: Fizmatgiz, Moscow
[2] A. A. Mailybaev and A. P. Seiranyan, Multiparameter Stability Theory with Mechanical Applications (World Scientific, River Edge; NJ, 2003).
[3] Kirillov, O. N., Nonconservative Stability Problems of Modern Physics (2013), Berlin, Boston: Walter de Gruyter, Berlin, Boston · Zbl 1285.70001
[4] Merkin, D. R., Gyroscopic Systems (1974), Moscow: Nauka, Moscow · Zbl 0308.70008
[5] Baikov, A. E.; Krasil’nikov, P. S., The Ziegler effect in a non-conservative mechanical system, J. Appl. Math. Mech., 74, 51-60 (2010) · Zbl 1272.70081
[6] Bulatovic, R., A stability criterion for circulatory systems, Acta Mech., 228, 2713-2718 (2017) · Zbl 1401.70029
[7] Awrejcewicz, J.; Losyeva, N.; Puzyrov, V., Stability and boundedness of the solutions of multi-parameter dynamical systems with circulatory forces, Symmetry, 12, 1210 (2020)
[8] Ostrowski, A.; Schneider, H., Some theorems on the inertia of general matrices, J. Math. Anal. Appl., 4, 72-84 (1967) · Zbl 0112.01401
[9] Gantmakher, F. R., The Theory of Matrixes (1967), Moscow: Nauka, Moscow
[10] Kozlov, V. V., “Invariant planes, indices of inertia, and degrees of stability of linear dynamic equations,” Tr. Mat. Inst, V.A. Steklova, 258, 154-161 (2007)
[11] Seiranyan, A. P., “On the theorems of Metelitsin,” Izv. Ross. Akad. Nauk: Mekh. Tverd, Tela, No., 3, 39-43 (1994)
[12] Kliem, V.; Seiranyan, A. P., Metelitsyn’s inequality and stability criteria for mechanical systems, J. Appl. Math. Mech., 48, 199-205 (2004) · Zbl 1078.70021
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