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 fo...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.展开更多
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...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 dissi- pation. 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.展开更多
The complete solutions of the upright and oblique permanent rotations of a symmetric heavy gyroscope with perfect dissipation are given.The asymptotic stability criteria and unstability criteria for these rotations in...The complete solutions of the upright and oblique permanent rotations of a symmetric heavy gyroscope with perfect dissipation are given.The asymptotic stability criteria and unstability criteria for these rotations in the sense of Liapunov and the sense of Movchan are also given on the basis of exact nonlinear motion equations respectively.The related oblique rotations are non-isolated.The main subdomains of the re- gions of asymptotic stability are obtained.The related bifurcation phenomena are discussed in detail.展开更多
基金supported by the Ministry of Science and Technological Development of the Republic of Serbia (Nos. ON174004,ON174016,and TR335006)
文摘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.
基金supported by the Ministry of Science and Technological Development of the Republic of Serbia (Nos.ON174016 and TR35006)
文摘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 dissi- pation. 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.
基金The project is supported by the National Natural Science Foundation of China
文摘The complete solutions of the upright and oblique permanent rotations of a symmetric heavy gyroscope with perfect dissipation are given.The asymptotic stability criteria and unstability criteria for these rotations in the sense of Liapunov and the sense of Movchan are also given on the basis of exact nonlinear motion equations respectively.The related oblique rotations are non-isolated.The main subdomains of the re- gions of asymptotic stability are obtained.The related bifurcation phenomena are discussed in detail.
