Stability analysis for parametric resonances of frame structures using dynamic axis-force transfer coefficient

For general frame structures subjected to periodic loads, the motion equation of structures is generally expressed as a nonhomogeneous Mathieu-Hill equation in the form of matrices. The parametric resonance stabilities of these structures are generally determined by calculating the Lyapunov exponents (or the energy-growth expo-nents) of structural time-history responses. The process of stability analysis is complicated and time-consuming. This paper proposes a new (simplified) method to solve dynamic stability problems. The concept of the dynamic axial-force transfer coefficient is first presented. A new simplified formula is derived to calculate the parametric resonance stability boundaries of the frame structures by using the dynamic axial-force transfer coefficient. In the special cases in which the first static (elastic) buckling mode of the frame structure is approximately equal to the first vibration mode, the stability boundaries of the parametric resonances can be more easily obtained with the aid of commercial software. The stability boundaries calculated by the present method are in good agreement with the results of the existing numerical method and experiment. The present method is simpler and more effective than the existing methods for stability analyses of the parametric resonance and autoparametric resonance of general frame structures.