The instability behavior of a parametrically excited column subjected to a periodic load at any axial position in the direction of the tangency coefficient was investigated analytically. For an inextensional neutral axis, the lateral and axial deflections of the column can be expressed as functions of natural mode shapes and the corresponding mode deflections of the column. The components of modal excitation force induced by the periodic load were found as well. The dynamic equation of the system was obtained by incorporating the modal excitation forces and the modal equations of the free transverse vibration of the column into the virtual work equation. The instability bandwidth of simple and combination resonances of a general column can be described systematically by the natural frequencies of the column, the amplitude of the excitation force and a set of instability bandwidth parameters. A general formula was obtained to determine directly the instability regions of the column system, while bypassing the procedures for reducing and solving the dynamic equation of the system.
Physical explanations are given for the behavior of simple and combination resonances. Examples for columns with various boundary conditions were described to indicate their instability regions and were found to agree quite well with the results by previous researchers.