Vibrational stabilization of the upright statically unstable position of a double pendulum

The phenomenon of stabilization by parametric excitation of an unstable, elastically restrained double inverted pendulum under its own weight is addressed. The solution is pursued by the Multiple Scale Method, as a perturbation of a critical Hamiltonian system, possessing a zero- and a real frequency. Several asymptotic expansions are carried out, which are able to capture the long-term behavior of the system, for generic (non-resonant) values of the excitation frequency, and some special (resonant) values of excitation-to-natural frequency ratio. It is shown that a proper ordering of the control parameters must be performed, and proper use of integer or fractional power expansions must be made, according to the resonance under study. In particular, a non-standard application of the Multiple Scale Method is illustrated for the 1:1 resonant case, requiring fractional powers and accounting for the ‘arbitrary constants’, generally omitted in regular cases. A comprehensive scenario of the stabilization regions is given in which lower-bound as well as upper-bound curves are evaluated, thus integrating results that recently appeared in the literature.