On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints
Abstract
The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V. V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of suc...h solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example.
Keywords:
Nonholonomic system / Lyapunov first method / Instability of equilibriumSource:
Mathematical and Computer Modelling, 2010, 51, 9-10, 1097-1106Publisher:
- Pergamon-Elsevier Science Ltd, Oxford
Funding / projects:
- Contemporary problems of mechanics of deformable bodies (RS-MESTD-MPN2006-2010-144019)
- Razvoj mašina visokih performansi i metoda za identifikaciju njihovog odziva na unutrašnje i spoljašnje poremećaje (RS-MESTD-MPN2006-2010-14052)
DOI: 10.1016/j.mcm.2009.12.017
ISSN: 0895-7177
WoS: 000275746300009
Scopus: 2-s2.0-77649233537
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Cović, V. AU - Vesković, M. AU - Obradović, Aleksandar PY - 2010 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1071 AB - The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V. V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example. PB - Pergamon-Elsevier Science Ltd, Oxford T2 - Mathematical and Computer Modelling T1 - On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints EP - 1106 IS - 9-10 SP - 1097 VL - 51 DO - 10.1016/j.mcm.2009.12.017 ER -
@article{ author = "Cović, V. and Vesković, M. and Obradović, Aleksandar", year = "2010", abstract = "The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V. V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example.", publisher = "Pergamon-Elsevier Science Ltd, Oxford", journal = "Mathematical and Computer Modelling", title = "On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints", pages = "1106-1097", number = "9-10", volume = "51", doi = "10.1016/j.mcm.2009.12.017" }
Cović, V., Vesković, M.,& Obradović, A.. (2010). On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints. in Mathematical and Computer Modelling Pergamon-Elsevier Science Ltd, Oxford., 51(9-10), 1097-1106. https://doi.org/10.1016/j.mcm.2009.12.017
Cović V, Vesković M, Obradović A. On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints. in Mathematical and Computer Modelling. 2010;51(9-10):1097-1106. doi:10.1016/j.mcm.2009.12.017 .
Cović, V., Vesković, M., Obradović, Aleksandar, "On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints" in Mathematical and Computer Modelling, 51, no. 9-10 (2010):1097-1106, https://doi.org/10.1016/j.mcm.2009.12.017 . .