Numerical analysis of the parametrically excited stability of multi-degree-of-freedom systems

A direct numerical method for the stability of multi-degree-of-freedom systems with period parameters was proposed. The perturbation equation of a parametrically excited system is first rewritten in the form of state equation. The perturbation solution is expressed as the product of exponential characteristic component and periodic component according to the Floquet theory. The periodic component and periodic system parameters are further expanded into the Fourier series. Then a series of algebraic equations are derived and the matrix eigenvalue problem is established. The stability of the parametrically excited system can be determined directly by using the eigenvalues solved numerically. The proposed method is applicable to damped systems with general period-parameter excitation and the final eigenvalue matrix has not any inverse sub-matrix. Also it is applied to the parametrically excited instability analysis of an inclined stay cable under periodic support motion excitation. Numerical results illustrate the effectiveness of the proposed direct numerical method for parametrically excited stability.