Vibrating Tool with Relay Automatic Control
The purpose of the present paper is to study the dynamical modes of a vibrating tool with the unbalanced exciter that are controlled by a DC motor. Vibration tools of this type have found a broad range of applications in the industry, for instance, as vibrating technology machine and robots for various purposes. The considered tool is controlled by a relay system with hysteresis. This type of the switching control provides a smooth start of the DC motor and allows us to exclude the discontinuous conduction sate. The dynamics of such a system involves two oscillatory modes: a low frequency oscillations of the vibrating tool and a high frequency, relay switching operating regime. This interaction may give birth to a variety of unusual nonlinear dynamic phenomena. The behavior of such a system is described by a set of five coupled autonomous ordinary differential equations with discontinuous right-hand sides. We describe a simple approach for modeling that is based on integration of the motion equations by a fourth-order Runge-Kutta algorithm. The switching instants of the relay element and Poincare map are calculated using the time-step halving iterative scheme. Using this approach we perform a detailed numerical analysis of self-oscillations of a vibrating tool with hysteretic relay control. We give examples of transient and steady state behaviors, and also illustrate the variation of the oscillation frequency for a vibration tool with the reference signal.