The 'Indian rope trick' for a parametrically excited flexible rod: linearized analysis

It is well known that if a column exceeds a certain critical length it will, when placed upright, buckle under its own weight. In a recent experiment Mullin has demonstrated 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 partial differential equation (PDE). Solutions to this PDE are examined using two-timing asymptotics and numerical Floquet theory in an infinite-dimensional analogue of the analysis of the Mathieu equation. Good agreement is found between asymptotics and numerics for the conditions on amplitude and frequency of vibration for stabilizing an upside-down column that is longer than the critical length. A simple condition is derived for the lower bound on frequency for stability in terms of amplitude and the column's length. An upper bound is more subtle, due to the presence of infinitely many resonance tongues inside the stability region of parameter space.