Parametric instability of a column with an axially oscillating mass

This study investigates the dynamic instability behavior of a column carryi ng a concentrated mass with oscillating motion along the column axis. The d ynamic 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 displacem ents, 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 different from the traditional pertur bation method in which a criterion, involving the determination of the char acteristic exponents, is used to yield the transition curves. The principal vibration frequencies, the ratio of principal amplitudes, and the phase di fference between the parametrically excited force and the principal frequen cy response on the transition state were obtained systematically. The param etric instability behavior of a column carrying a periodically moving conce ntrated mass is different from that of a column subjected to a periodic tan gential inertia force. The present case contains the simple resonances and combination resonances of sum type only, while the case with tangential ine rtia force may contain the combination resonances of the difference type ad ditionally. Four examples are given to demonstrate the instability behavior of various columns carrying concentrated oscillating mass along the column axis at varying positions. (C) 1999 Academic Press.