Extract from authors' abstract: It is well known that if a column exceeds a certain critical length it will, when placed upright, buckle under its own weight. Recent experiments demonstrate that a column that is longer than its critical length can be stabilized by subjecting its bottom support point to a vertical vibration of appropriate amplitude and frequency. This paper proposes a theory for this phenomenon. Geometrically nonlinear dynamical equations are derived for a stiff rod (with linearly elastic constitutive laws) held vertically upwards via a clamped base point that is harmonically excited. Taking the torsion-free problem, the equations are linearized about the trivial response to produce a linear non-autonomous inhomogeneous differential equation (PDE). Solutions of this PDE are examined using two-timing asymptotics and numerical Floquet theory in an infinite-dimensional analogue of the analysis of Mathieu equation.