The natural frequency of a uniform column at the buckling load is theoretically zero when the axial load equals the critical buckling load. This occurs because, at this point, less energy is required to initiate vibration due to lower kinetic energy associated with reduced velocity. The concept implies that once the frequency reaches zero, no additional energy is needed to start the "vibration," albeit at an infinitely slow rate. However, real-world factors like geometrical tolerances and off-center loads introduce destabilizing forces that can initiate motion. The behavior of the column becomes nonlinear once buckling occurs, contradicting the initial assumption of infinitely slow motion.
