Assessment of lateral dynamic instability of columns under an arbitrary periodic axial load owing to parametric resonance

The paper presents a method for assessing the dynamic stability of simply supported columns with damping under arbitrary periodic axial loading owing to parametric resonance. The equation for determining critical excitation frequencies causing lateral parametric instability of columns is derived using the Bolotin′s method. A method of matrix transformation is developed to solve the ill-condition problem in the even regions of dynamic instability of the column caused by inclusion of damping. The accuracy of the proposed method is carefully verified by comparing the results with those in the literature. New findings of this investigation are: (1) the critical excitation parameter for inducing dynamic instability under arbitrary periodic loading increases with increasing damping ratio; (2) in difference from columns under a simple harmonic load, the width of instability regions of columns under an arbitrary periodic load does not necessarily decrease as the number of region increases, so the instability regions can cover most of the parametric plane, and thus the dynamic instability is more likely to occur; (3) under an arbitrary periodic axial load, the first region of instability of columns may not be the critical one, and the number of harmonic waves adopted in the computation is important for the location of the most critical region of instability and the distribution of instability regions in the parametric plane; and (4) the dimension of determinant in the equation of boundary frequencies rather than the number of harmonic waves is responsible for computation efficiency, and the sufficient large dimension of determinant should be retained to achieve the computational accuracy.