Vibrational stabilization of the upright statically unstable position of a double pendulumстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация: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.