Transverse dynamical behaviors of axially moving nanoplates which could be
used to model the graphene nanosheets or other plate-like nanostructures with axial
motion are examined based on the nonlocal elasticity theory. The Hamilton's principle
is employed to derive the multivariable coupling partial differential equations
governing the transverse motion of the axially moving nanoplates. Subsequently, the
equations are transformed into a set of ordinary differential equations by the method
of separation of variables. The effects of dimensionless small-scale parameter, axial
speed and boundary conditions on the natural frequencies in sub-critical region are
discussed by the method of complex mode. Then the Galerkin method is employed to
analyze the effects of small-scale parameter on divergent instability and
coupled-mode flutter in super-critical region. It is shown that the existence of
small-scale parameter contributes to strengthen the stability in the super-critical
region, but the stability of the sub-critical region is weakened. The regions of
divergent instability and coupled-mode flutter decrease even disappear with an
increase in the small-scale parameter. The natural frequencies in sub-critical region
show different tendencies with different boundary effects, while the natural
frequencies in super-critical region keep constants with the increase of axial speed.
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