Eigenvalue analysis approach to random periodic parameter-excited stability: Application to a stay cable

A direct eigenvalue analysis approach to solving the stability problem of randomly and periodically parametrically excited linear systems is developed based on the response moment stability, Floquet theorem, Fourier series, and matrix eigenvalue analysis. A differential equation for the perturbation second moment of parametrically excited systems is obtained. Its solution is expressed as the product of exponential and periodic components based on Floquet theorem. By expanding the periodic component and periodic parameters into Fourier series, an eigenvalue equation is obtained. Then the stochastically and periodically parametrically excited vibration stability is converted into periodically parameter-varying response moment stability and determined directly by matrix eigenvalues. The developed approach to parametrically excited systems is applied to the stability analysis of an inclined stay cable under random and periodic combined support motion excitations. The unstable regions of stochastically and periodically parametrically excited cable vibration are obtained and compared to illustrate the stochastic instability and the effect of random excitation components on the instability. The developed approach is applicable to more general periodically and randomly parametrically excited systems.