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Coupled mode parametric resonance in a vibrating screen model

We consider a simple dynamic model of the vibrating screen operating in the parametric resonance (PR) mode. This model was used in the course of designing and setting of such a screen in LPMC. The PR-based screen compares favorably with conventional types of such machines, where the transverse oscillations are excited directly. It is characterized by larger values of the amplitude and by insensitivity to damping in a rather wide range. The model represents an initially strained system of two equal masses connected by a linearly elastic string. Self-equilibrated, longitudinal, harmonic forces act on the masses. Under certain conditions this results in transverse, finite-amplitude oscillations of the string. The problem is reduced to a system of two ordinary differential equations coupled by the geometric nonlinearity. Damping in both the transverse and longitudinal oscillations is taken into account. Free and forced oscillations of this mass-string system are examined analytically and numerically. The energy exchange between the longitudinal and transverse modes of free oscillations is demonstrated. An exact analytical solution is found for the forced oscillations, where the coupling plays the role of a stabilizer. In a more general case, the harmonic analysis is used with neglect of the higher harmonics. Explicit expressions for all parameters of the steady nonlinear oscillations are determined. The domains are found where the analytically obtained steady oscillation regimes are stable. Over the frequency ranges, where the steady oscillations exist, a perfect correspondence is found between the amplitudes obtained analytically and numerically. Illustrations based on the analytical and numerical simulations are presented.