On stability of equilibrium

of non-conservative mechanical systems

R.M.Bulatović

University of Montenegro

Cetinski put b.b., 81 000 Podgorica, Serbia and Montenegro

 

The stability of equilibrium of mechanical systems subjected to dissipative, gyroscopic, potential and circulatory forces is investigated. From a practical point of view it is of interest to find stability conditions which are in a simple way related to the properties of the system matrices. These may yield useful design constraints. Several conditions in this direction are derived which supplement and improve previously obtained results of the same type. In addition, simple examples are given to illustrate the stability conditions.

The dynamical behavior of a mechanical system with general types of forces can be described in the vicinity of the equilibrium by a vector differential equation in Lagrange form (for generalized coordinates) with assumption about dissipative, gyroscopic, potential and circulatory (positional non-conservative) forces. The equilibrium state of system, according to Lyapunov theorems on the stability in the first approximation, is asymptotically stable if all eigenvalues of the linearized system, i. e., all roots of the characteristic equation have negative real parts; but in case of roots with a positive real part the equilibrium state is unstable.

The Routh-Hurwitz criterion can be applied to determine whether or not all the roots of characteristic equation have negative real parts. However, since the Routh-Hurwitz criterion requires the knowledge of the coefficients in the characteristic equation and the evaluation of certain determinants this criterion is not suitable when the dimension of the system is large and the system's parameters are not fully specified. Therefore, stability conditions which are stated directly in terms of the matrices D, G, P and C are of practical interest and importance. Such criteria may yield design constraints in terms of the physical parameters of the system. An old result in this direction is the famous Thomson-Tait-Chetayev theorem: In the absence of circulatory forces (C=0), the equilibrium state is asymptotically stable (unstable) if the matrix P is positive definite (P has at least one negative eigenvalue). The presence of circulatory forces (C ¹ 0) considerably complicates the situation and this theorem cannot be directly used. On the other hand, such forces are very common in nature and in contemporary engineering, and many attempts have been made to explain the effect of these forces on the stability of equilibrium state.

In this paper several stability results are obtained both by analyzing the characteristic equation and by constructing the Lyapunov function.