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Transient analysis of slow motions in systems with inertially excited vibrations

Alexander Fradkov, Ilya I. BLEKHMAN, Dmitry Indeitsev
A nonlinear system with two degrees of freedom consisting of a rigid platform and mechanical vibroactuator is considered. The platform, connected to an immovable base by means of elastic and damping elements can move along a fixed direction. The mechanical vibroactuator is an unbalanced rotor, mounted on the platform and driven with an electric drive. The main contribution of this paper is analysis of existence and dynamics of internal pendulum. The problem of passage through resonance zone is solved by an iterative method combined with direct method of separation of motions. Though such an approach looks more primitive than the previous ones, it allows to obtain two autonomous second order equations for slow motions (for rotation frequency) and for semi-slow motions (for oscillations of rotation frequency) which can be solved separately. Both equations are valid both in below resonance and in above resonance area. Expression for the frequency of semi-slow oscillations (internal pendulum) in below resonance area can be derived from the obtained equations and provides an important contribution of the paper. This frequency depends essentially on rotation frequency ω and decreases down to zero when ω approaches the resonance frequency p.
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