Stability of Dynamic Systems under Parametric Excitations with Multiple Frequencies - Deng, Jian (Journal of Applied Nonlinear Dynamics, 2026)
The stability of dynamic systems under parametrically periodic excitations with a single frequency has been extensively investigated. However, parametrically arbitrary excitations with multiple frequencies are common in various sciences and engineering, including earthquakes and blasting. This paper proposes a new numerical method to study the stability of dynamic systems under parametric excitations with multiple frequencies. The critical step of the numerical method involves approximating the system with multiple frequencies by a system with a single principal frequency (or period) as closely as possible. Subsequently, a numerical algorithm is proposed to calculate both the state transition matrix on one principal period, which determines the dynamic stability, and the responses at any specific time. The efficiency and accuracy of the proposed numerical method are demonstrated. As an application example, the dynamic stability of a column under parametric loads with multiple frequencies is obtained through parametric studies involving the magnitude of incommensurate frequency, the number of frequencies, damping, and semi-rigid connections.