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Slow motions in systems with inertially excited vibrations

Ilya I. BLEKHMAN, Dmitry Indeitsev
A nonlinear system with two degrees of freedom consisting of a rigid platform and mechanical vibroactuator is analyzed. 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. Such a system is a model of many vibrational machines and technological units.
During the speed up of the actuator to the working frequency ω* exceeding the free oscillation frequency p of the platform, a remarkable phenomenon can be observed: capture of the current frequency ω near resonance frequency
p. Further increase of the supply power of the drive leads to a jump transition from ω≈p to an above resonance frequency ω1>p. Such a phenomenon was first described by an eminent German physicist A.Sommerfeld. In 1953 one of the authors of this paper gave physical explanation and mathematical description of this phenomenon and coined the term “Sommerfeld effect”.
Later a comprehensive study of Sommerfeld effect was provided by numerous researchers. Particularly, it was discovered that “semi-slow” oscillations of rotor frequency may appear during start-up mode. Such an effect can be interpreted as appearance of “internal pendulum” in the system. Using internal pendulum and semi-slow oscillations of the rotor is important for a number of methods for control of vibrational units with inertia excitation of vibrations allowing to significantly reduced the motor power required for passage through resonance zone.
In the talk a brief survey of of such control methods is given. The main contribution of the talk is analysis of existence and dynamics of an internal pendulum. The problem of passage through resonance zone is solved by iteration 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 and semi-slow oscillations parameters which can be solved sequentially.
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