Lyapunov-Kozlov method for singular cases
Abstract
Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.
Keywords:
singular case / potential / instability / dissipative force / asymptotic motionSource:
Applied Mathematics and Mechanics-English Edition, 2011, 32, 9, 1207-1220Publisher:
- Shanghai Univ, Shanghai
Funding / projects:
- Micromechanical criteria of damage and fracture (RS-MESTD-Basic Research (BR or ON)-174004)
- Mechanics of nonlinear and dissipative systems - contemporary models, analysis and applications (RS-MESTD-Basic Research (BR or ON)-174016)
DOI: 10.1007/s10483-011-1494-6
ISSN: 0253-4827
WoS: 000295303900012
Scopus: 2-s2.0-80052599047
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Cović, V. AU - Djurić, D. AU - Vesković, M. AU - Obradović, Aleksandar PY - 2011 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1219 AB - Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example. PB - Shanghai Univ, Shanghai T2 - Applied Mathematics and Mechanics-English Edition T1 - Lyapunov-Kozlov method for singular cases EP - 1220 IS - 9 SP - 1207 VL - 32 DO - 10.1007/s10483-011-1494-6 ER -
@article{ author = "Cović, V. and Djurić, D. and Vesković, M. and Obradović, Aleksandar", year = "2011", abstract = "Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.", publisher = "Shanghai Univ, Shanghai", journal = "Applied Mathematics and Mechanics-English Edition", title = "Lyapunov-Kozlov method for singular cases", pages = "1220-1207", number = "9", volume = "32", doi = "10.1007/s10483-011-1494-6" }
Cović, V., Djurić, D., Vesković, M.,& Obradović, A.. (2011). Lyapunov-Kozlov method for singular cases. in Applied Mathematics and Mechanics-English Edition Shanghai Univ, Shanghai., 32(9), 1207-1220. https://doi.org/10.1007/s10483-011-1494-6
Cović V, Djurić D, Vesković M, Obradović A. Lyapunov-Kozlov method for singular cases. in Applied Mathematics and Mechanics-English Edition. 2011;32(9):1207-1220. doi:10.1007/s10483-011-1494-6 .
Cović, V., Djurić, D., Vesković, M., Obradović, Aleksandar, "Lyapunov-Kozlov method for singular cases" in Applied Mathematics and Mechanics-English Edition, 32, no. 9 (2011):1207-1220, https://doi.org/10.1007/s10483-011-1494-6 . .