This study investigates the dynamic instability behavior of a column carrying a concentrated mass with oscillating motion along the column axis. The dynamic equation of the column was derived based on the assumed-modes method. The derived dynamic equation, which contains parametrically excited terms associated with modal accelerations, modal velocities, and modal displacements, is a general form of Mathieu's equation. A new analytical method used to determine the instability regions of the column was directly applied to the transition state. This method is di!erent from the traditional perturbation method in which a criterion, involving the determination of the characteristic exponents, is used to yield the transition curves. The principal vibration frequencies, the ratio of principal amplitudes, and the phase di!erence between the parametrically excited force and the principal frequency response on the transition state were obtained systematically. The parametric instability behavior of a column carrying a periodically moving concentrated mass is di!erent from that of a column subjected to a periodic tangential inertia force. The present case contains the simple resonances and combination resonances of sum type only, while the case with tangential inertia force may contain the combination resonances of the di!erence type additionally. Four examples are given to demonstrate the instability behavior of various columns carrying concentrated oscillating mass along the column axis at varying positions.