Slow motions in systems with inertially excited vibrations

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. 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 work gave physical explanation and mathematical description of this phenomenon and coined the term “Sommerfeld effect” [1]. Later a comprehensive study of Sommerfeld effect was carried on in numerous publications including a number of books [2 – 4]. In [5, 6, 7] (see also book [4]) it was discovered by means of classical methods of nonlinear mechanics and their modifications that “semi-slow” oscillations of rotor frequency may appear in the area of Sommerfeld effect. Such an effect can be interpreted as appearance of “internal pendulum” in the system. Natural frequency of internal pendulum is less than resonance frequency of the system. In the above cited papers representation of system solutions by expansions in powers of square root of a small parameter allowing to study multi-scale motions are used. Using internal pendulum and semislow oscillations of the rotor is important for a number of methods for control of vibration units with inertia excitation of vibrations allowing to significantly reduced the motor power required for passage through resonance zone [8, 9]. 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. A remarkable overturning property of an internal pendulum is discovered: in the below resonance area its equilibrium near the lower position is stable, while in the above resonance area its lower equilibrium becomes unstable and its equilibrium near upper position becomes stable. A comparison of the obtained analytical results with numerical results obtained by simulation of initial system equations is given demonstrating a good concordance of the results. The results of the paper can be used for improvement of control methods for vibration units in the start-up mode.