Stability analysis of dynamic systems with couplings and integrals of motions: PMM vol. 38, n≗4, 1974, pp. 607–615

Abstract

We give a method for obtaining the stability conditions for nonlinear systems, based on an analysis of the linearized coupling equations and of the linearized or quadratic expressions for the integrals of motion. Liapunov's method is usually employed in the investigation of the stability of dynamic systems. The investigation of the Hamiltonian function is a convenient tool for systems with internal energy dissipation. In fact, in the development of the Thompson (Lord Kelvin)-Tait-Chetaev theorem [1–4] it was shown that the positive definiteness of the Hamiltonian function provides the necessary and sufficient stability conditions in the case of complete dissipation. We have obtained just sufficient conditions for system with partial dissipation; moreover, the method does not yield the possibility of obtaining far-reaching inferences on stability on the basis of the analysis of the linearized equations. It should be noted also that in several cases it is convenient to introduce a number of variables, exceeding the number of degrees of freedom, and to examine the couplings. Then the equations can be simplified or represented in a form convenient for stability analysis.