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 leng th can be stabilized by subjecting its bottom support point to a vertical v ibration of appropriate amplitude and frequency. This paper proposes a theo ry for this phenomenon. Geometrically nonlinear dynamical equations are derived for a,stiff rod (wi th linearly elastic constitutive laws) held vertically upwards via a clampe d base point that is harmonically excited. Taking the torsion-free problem, the equations are linearized about the trivial response to produce a linea r non-autonomous inhomogeneous partial differential equation (PDE). Solutio ns to this PDE are examined using two-timing asymptotics and numerical Floq uet theory in an infinite-dimensional analogue of the analysis of the Mathi eu equation. Good agreement is found between asymptotics and numerics for t he conditions on amplitude and frequency of vibration for stabilizing an up side-down column that is longer than the critical length. A simple conditio n is derived for the lower bound on frequency for stability in terms of amp litude and the column's length. An upper bound is more subtle, due to the p resence of infinitely many resonance tongues inside the stability region of parameter space.