  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.
