Lyapunov-Kozlov method for singular cases

• Published in: Applied mathematics and mechanics Vol. 32; no. 9; pp. 1207 - 1220
• Main Authors: Covic, V, Djuric, D, Veskovic, M, Obradovic, A
• Format: Journal Article
• Language: English
• Published: Heidelberg Shanghai University Press 01-09-2011; Faculty of Mechanical Engineering, University of Belgrade,Kraljice Marije 16, Belgrade 11000, Serbia%Faculty of Mechanical Engineering, University of Kragujevac,Dositejeva 19, Kraljevo 36000, Serbia
• Subjects: Applications of Mathematics; Classical Mechanics; Differential equations; Dissipation; Fluid- and Aerodynamics; Inertia; Instability; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Mechanical systems; Partial Differential Equations; Stability; Uniqueness; 奇异; 定性分析方法; 平衡位置; 应用程序; 李亚普诺夫; 李雅普诺夫; 运动微分方程; 非线性机械系统; dissipative force; 34D20; instability; asymptotic motion; potential; 70F25; O317; singular case
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-011-1494-6

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.