Energy and frequency ripple in devices with inertial excitation of oscillations

We consider vibration devices that consist of softly vibration-isolated rigid bodies subjected to vibrations transmitted by means of inertial vibration exciters (unbalanced rotors) driven into rotation by electric motors. Typically, when designing such devices, it is assumed that the rotors rotate uniformly with a certain circular frequency and the body performs small harmonic oscillations with the same frequency. The present work, using a second-order approximation of their nonlinear coupled differential equations, shows that the rotor and the oscillating body keep exchanging energy. At the same time, the angular velocity of the rotor oscillates with the working frequency as well as with its multiple frequencies during each revolution. As a result, the acceleration of the oscillating body also acquires harmonics with multiple frequencies. This may cause both unwanted and beneficial resonance phenomena. We obtain formulae describing the magnitudes of these ripples. We show that the magnitude of oscillations of the angular frequency can also be estimated using energy considerations. Such estimates are provided for the three most common schemes of dynamic devices. Available experimental data confirm the main conclusions of the theory. We discuss both the harmful effects of these phenomena as well as their possible applications. The latter include design of bi-harmonic vibration exciters and exciters based on vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.