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

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