Stability Analysis for Spatial Autoparametric Resonances of Framed Structures

The spatial dynamic instabilities of framed structures due to autoparametric resonances have been seldom investigated in the published literature. Based on the finite element method (FEM), the spatial parametric vibration equations are established for general framed structures. The Newmark's method and the energy-growth exponent (EGE) are used to determine the stability of the spatial autoparametric resonances of framed structures. A portal frame model is used to conduct a spatial (out-of-plane) autoparametric resonance experiment. The numerical results of the autoparametric resonances are found to agree with those of the test, which proves the validity of the present theoretical formulation. A numerical example for autoparametric resonance stability analysis of a spatial frame is presented to firstly predict the three instability modes of autoparametric resonances, i.e. global unidirectional translational instability, bidirectional (diagonal) translational instability and torsional instability. When the excitation frequency is approximately twice the modal frequency of spatial vibration of a framed structure, spatial dynamic instability will occur due to autoparametric resonance. A small excitation force can cause a strong autoparametric resonance of the framed structure. The potential risk of spatial dynamic instability is revealed for the framed structures under periodic loads.