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Applied Mathematics and Mechanics (English Edition) ›› 2011, Vol. 32 ›› Issue (9): 1207-1220.doi: https://doi.org/10.1007/s10483-011-1494-6

• Articles • 上一篇    

Lyapunov-Kozlov method for singular cases

 V. ˇCOVI´C1, D.DJURI´C1, M. VESKOVI´C2, A.OBRADOV1   

  1. 1. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade 11000, Serbia;
    2. Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, Kraljevo 36000, Serbia
  • 收稿日期:2011-10-11 修回日期:2011-06-15 出版日期:2011-09-02 发布日期:2011-09-02

Lyapunov-Kozlov method for singular cases

 V. ˇCOVI´C1, D.DJURI´C1, M. VESKOVI´C2, A.OBRADOV1   

  1. 1. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade 11000, Serbia;
    2. Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, Kraljevo 36000, Serbia
  • Received:2011-10-11 Revised:2011-06-15 Online:2011-09-02 Published:2011-09-02

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2

摘要： 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.

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.

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