Stability of matrix second-order systems: new conditions and perspectives

Modeling of many dynamic systems results in matrix second-order differential equations. In the paper, the stability issues of matrix second-order dynamical systems are discussed. In the literature, only sufficient conditions of stability and/or instability for a system in matrix second-order form are available. In this paper, necessary and sufficient conditions of asymptotic stability for time-invariant systems in matrix second-order form under different types of dynamic loadings (conservative/nonconservative) are derived and a physical interpretation is carried out. The stability conditions in the sense of Lyapunov (the jw-axis behavior of eigenvalues) are also analyzed. As the conditions are gained directly in terms of physical parameters of the system, the effect of different loadings on the system stability is made transparent by dealing with the stability issues directly in matrix second-order form.