Arnold tongues and mode–jumping in the supercritical post–buckling of an archetypal elastic structure

Abstract

The buckling and post–buckling of a classical simply supported elastic strut on a stiffening elastic foundation is treated as a nonlinear dynamical system. For supercritical loads this fourth–order system has a pair of imaginary eigenvalues, indicating the possibility of either a double or a quasi–periodic response. Finite lengths impose constraints that restrict such responses to the rational positions on the real line of the ratio of competing wavelengths: for irrational positions the response locks onto a single periodic wave. The extent of the mode–locking is described in parameter space by the geometry of Arnold tongues. Critical positions on the tongues are identified which mark conditions for mode–jumping from one wavelength to another.