TY  - JOUR
AU  - Cović, V.
AU  - Mitrović, Zoran
AU  - Rusov, Srđan
AU  - Obradović, Aleksandar
PY  - 2013
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1704
AB  - The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem.
PB  - Wiley-V C H Verlag Gmbh, Weinheim
T2  - Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik
T1  - On the instability of equilibrium position of a mechanical system with singular constraints
EP  - 936
IS  - 12
SP  - 928
VL  - 93
DO  - 10.1002/zamm.201200080
ER  - 

@article{
author = "Cović, V. and Mitrović, Zoran and Rusov, Srđan and Obradović, Aleksandar",
year = "2013",
abstract = "The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem.",
publisher = "Wiley-V C H Verlag Gmbh, Weinheim",
journal = "Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik",
title = "On the instability of equilibrium position of a mechanical system with singular constraints",
pages = "936-928",
number = "12",
volume = "93",
doi = "10.1002/zamm.201200080"
}

Cović, V., Mitrović, Z., Rusov, S.,& Obradović, A.. (2013). On the instability of equilibrium position of a mechanical system with singular constraints. in Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik
Wiley-V C H Verlag Gmbh, Weinheim., 93(12), 928-936.
https://doi.org/10.1002/zamm.201200080

Cović V, Mitrović Z, Rusov S, Obradović A. On the instability of equilibrium position of a mechanical system with singular constraints. in Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik. 2013;93(12):928-936.
doi:10.1002/zamm.201200080 .

Cović, V., Mitrović, Zoran, Rusov, Srđan, Obradović, Aleksandar, "On the instability of equilibrium position of a mechanical system with singular constraints" in Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik, 93, no. 12 (2013):928-936,
https://doi.org/10.1002/zamm.201200080 . .

TY  - JOUR
AU  - Vesković, M.
AU  - Cović, V.
AU  - Obradović, Aleksandar
PY  - 2011
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1154
AB  - The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative, conservative, and circulatory forces. The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity. It is assumed that the kinetic energy, the Rayleigh dissipation function, and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions. The results obtained here are partially generalized the results obtained by Kozlov et al. (Kozlov, V. V. The asymptotic motions of systems with dissipation. Journal of Applied Mathematics and Mechanics, 58(5), 787-792 (1994). Merkin, D. R. Introduction to the Theory of the Stability of Motion (in Russian), Nauka, Moscow (1987). Thomson, W. and Tait, P. Treatise on Natural Philosophy, Part I, Cambridge University Press, Cambridge (1879)). The results are illustrated by an example.
PB  - SHANGHAI UNIV, SHANGHAI
T2  - Applied Mathematics and Mechanics-English Edition
T1  - Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces
EP  - 222
IS  - 2
SP  - 211
VL  - 32
DO  - 10.1007/s10483-011-1407-9
ER  - 

@article{
author = "Vesković, M. and Cović, V. and Obradović, Aleksandar",
year = "2011",
abstract = "The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative, conservative, and circulatory forces. The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity. It is assumed that the kinetic energy, the Rayleigh dissipation function, and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions. The results obtained here are partially generalized the results obtained by Kozlov et al. (Kozlov, V. V. The asymptotic motions of systems with dissipation. Journal of Applied Mathematics and Mechanics, 58(5), 787-792 (1994). Merkin, D. R. Introduction to the Theory of the Stability of Motion (in Russian), Nauka, Moscow (1987). Thomson, W. and Tait, P. Treatise on Natural Philosophy, Part I, Cambridge University Press, Cambridge (1879)). The results are illustrated by an example.",
publisher = "SHANGHAI UNIV, SHANGHAI",
journal = "Applied Mathematics and Mechanics-English Edition",
title = "Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces",
pages = "222-211",
number = "2",
volume = "32",
doi = "10.1007/s10483-011-1407-9"
}

Vesković, M., Cović, V.,& Obradović, A.. (2011). Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces. in Applied Mathematics and Mechanics-English Edition
SHANGHAI UNIV, SHANGHAI., 32(2), 211-222.
https://doi.org/10.1007/s10483-011-1407-9

Vesković M, Cović V, Obradović A. Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces. in Applied Mathematics and Mechanics-English Edition. 2011;32(2):211-222.
doi:10.1007/s10483-011-1407-9 .

Vesković, M., Cović, V., Obradović, Aleksandar, "Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces" in Applied Mathematics and Mechanics-English Edition, 32, no. 2 (2011):211-222,
https://doi.org/10.1007/s10483-011-1407-9 . .

TY  - 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 . .

TY  - JOUR
AU  - Cović, V.
AU  - Vesković, M.
AU  - Obradović, Aleksandar
PY  - 2011
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1210
AB  - This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov's generalization of the first Lyapunov's method, as well as Rout's method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin's series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938-944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov's generalizations of the first Lyapunov's method are formulated in a form including one variable less than was the case in existing literature.
PB  - Springer, Dordrecht
T2  - Meccanica
T1  - On the instability of steady motion
EP  - 863
IS  - 4
SP  - 855
VL  - 46
DO  - 10.1007/s11012-010-9348-2
ER  - 

@article{
author = "Cović, V. and Vesković, M. and Obradović, Aleksandar",
year = "2011",
abstract = "This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov's generalization of the first Lyapunov's method, as well as Rout's method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin's series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938-944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov's generalizations of the first Lyapunov's method are formulated in a form including one variable less than was the case in existing literature.",
publisher = "Springer, Dordrecht",
journal = "Meccanica",
title = "On the instability of steady motion",
pages = "863-855",
number = "4",
volume = "46",
doi = "10.1007/s11012-010-9348-2"
}

Cović, V., Vesković, M.,& Obradović, A.. (2011). On the instability of steady motion. in Meccanica
Springer, Dordrecht., 46(4), 855-863.
https://doi.org/10.1007/s11012-010-9348-2

Cović V, Vesković M, Obradović A. On the instability of steady motion. in Meccanica. 2011;46(4):855-863.
doi:10.1007/s11012-010-9348-2 .

Cović, V., Vesković, M., Obradović, Aleksandar, "On the instability of steady motion" in Meccanica, 46, no. 4 (2011):855-863,
https://doi.org/10.1007/s11012-010-9348-2 . .

TY  - JOUR
AU  - Obradović, Aleksandar
AU  - Cović, V.
AU  - Vesković, M.
AU  - Dražić, Milan
PY  - 2010
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1110
AB  - The brachistochrone problem of the rheonomic mechanical system whose motion is subject to nonholonomic constraints is solved with nonlinear differential equations of motion. Apart from control forces, the system is influenced by the action of other known potential and nonpotential forces as well. The problem of optimal control is solved by applying Pontryagin's Maximum Principle and the singular optimal control theory. This procedure results in the two-point boundary value problem for the system of ordinary nonlinear differential equations of the first order, with a corresponding number of initial and end conditions. This paper determines the control forces that are realized by imposing on the system a corresponding number of independent ideal holonomic constraints, without the action of active control forces. These constraints must be in accordance with the previously determined brachistochronic motion. The method is illustrated with a single complex example that represents the first known concrete demonstration of brachistochronic motion of the nonholonomic rheonomic mechanical system.
PB  - Springer Wien, Wien
T2  - Acta Mechanica
T1  - Brachistochronic motion of a nonholonomic rheonomic mechanical system
EP  - 304
IS  - 3-4
SP  - 291
VL  - 214
DO  - 10.1007/s00707-010-0295-8
ER  - 

@article{
author = "Obradović, Aleksandar and Cović, V. and Vesković, M. and Dražić, Milan",
year = "2010",
abstract = "The brachistochrone problem of the rheonomic mechanical system whose motion is subject to nonholonomic constraints is solved with nonlinear differential equations of motion. Apart from control forces, the system is influenced by the action of other known potential and nonpotential forces as well. The problem of optimal control is solved by applying Pontryagin's Maximum Principle and the singular optimal control theory. This procedure results in the two-point boundary value problem for the system of ordinary nonlinear differential equations of the first order, with a corresponding number of initial and end conditions. This paper determines the control forces that are realized by imposing on the system a corresponding number of independent ideal holonomic constraints, without the action of active control forces. These constraints must be in accordance with the previously determined brachistochronic motion. The method is illustrated with a single complex example that represents the first known concrete demonstration of brachistochronic motion of the nonholonomic rheonomic mechanical system.",
publisher = "Springer Wien, Wien",
journal = "Acta Mechanica",
title = "Brachistochronic motion of a nonholonomic rheonomic mechanical system",
pages = "304-291",
number = "3-4",
volume = "214",
doi = "10.1007/s00707-010-0295-8"
}

Obradović, A., Cović, V., Vesković, M.,& Dražić, M.. (2010). Brachistochronic motion of a nonholonomic rheonomic mechanical system. in Acta Mechanica
Springer Wien, Wien., 214(3-4), 291-304.
https://doi.org/10.1007/s00707-010-0295-8

Obradović A, Cović V, Vesković M, Dražić M. Brachistochronic motion of a nonholonomic rheonomic mechanical system. in Acta Mechanica. 2010;214(3-4):291-304.
doi:10.1007/s00707-010-0295-8 .

Obradović, Aleksandar, Cović, V., Vesković, M., Dražić, Milan, "Brachistochronic motion of a nonholonomic rheonomic mechanical system" in Acta Mechanica, 214, no. 3-4 (2010):291-304,
https://doi.org/10.1007/s00707-010-0295-8 . .

TY  - JOUR
AU  - Cović, V.
AU  - Vesković, M.
AU  - Djurić, D.
AU  - Obradović, Aleksandar
PY  - 2010
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1076
AB  - Lyapunov's first method, extended by V. V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The motion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin's series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three cases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints (V. V. Kozlov (1986)) is generalized to the case of nonlinear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov (1994) to nonholonomic systems with nonlinear constraints.
PB  - SHANGHAI UNIV, SHANGHAI
T2  - Applied Mathematics and Mechanics-English Edition
T1  - On the stability of equilibria of nonholonomic systems with nonlinear constraints
EP  - 760
IS  - 6
SP  - 751
VL  - 31
DO  - 10.1007/s10483-010-1309-7
ER  - 

@article{
author = "Cović, V. and Vesković, M. and Djurić, D. and Obradović, Aleksandar",
year = "2010",
abstract = "Lyapunov's first method, extended by V. V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The motion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin's series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three cases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints (V. V. Kozlov (1986)) is generalized to the case of nonlinear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov (1994) to nonholonomic systems with nonlinear constraints.",
publisher = "SHANGHAI UNIV, SHANGHAI",
journal = "Applied Mathematics and Mechanics-English Edition",
title = "On the stability of equilibria of nonholonomic systems with nonlinear constraints",
pages = "760-751",
number = "6",
volume = "31",
doi = "10.1007/s10483-010-1309-7"
}

Cović, V., Vesković, M., Djurić, D.,& Obradović, A.. (2010). On the stability of equilibria of nonholonomic systems with nonlinear constraints. in Applied Mathematics and Mechanics-English Edition
SHANGHAI UNIV, SHANGHAI., 31(6), 751-760.
https://doi.org/10.1007/s10483-010-1309-7

Cović V, Vesković M, Djurić D, Obradović A. On the stability of equilibria of nonholonomic systems with nonlinear constraints. in Applied Mathematics and Mechanics-English Edition. 2010;31(6):751-760.
doi:10.1007/s10483-010-1309-7 .

Cović, V., Vesković, M., Djurić, D., Obradović, Aleksandar, "On the stability of equilibria of nonholonomic systems with nonlinear constraints" in Applied Mathematics and Mechanics-English Edition, 31, no. 6 (2010):751-760,
https://doi.org/10.1007/s10483-010-1309-7 . .

TY  - 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 . .