  We study the interaction among dispersion, nonlinearity, and disorder effects
in the context of wave transmission through a discrete periodic structure,
subjected to continuous harmonic excitation in its stop band. We consider a
damped nonlinear periodic structure of finite length with disorder. Disorder is
introduced throughout the structure by small changes in the stiffness
parameters drawn from a uniform statistical distribution. Dispersion effects
forbid wave transmission within the stop band of the linear periodic structure.
However, nonlinearity leads to supratransmission phenomenon, by which enhanced
wave transmission occurs within the stop band of the periodic structure when
forced at an amplitude exceeding a certain threshold. The frequency components
of the transmitted waves lie within the pass band of the linear structure,
where disorder is known to cause Anderson localization. There is therefore a
competition between dispersion, nonlinearity, and disorder in the context of
supratransmission. We show that supratransmission persists in the presence of
disorder. The influence of disorder decreases in general as the forcing
frequency moves away from the pass band edge, reminiscent of dispersion effects
subsuming disorder effects in linear periodic structures. We compute the
dependence of the supratransmission force threshold on nonlinearity and
strength of coupling between units. We observe that nonlinear forces are
confined to the driven unit for weakly coupled systems. This observation,
together with the truncation of higher-order nonlinear terms, permits us to
develop closed-form expressions for the supratransmission force threshold. In
sum, in the frequency range studied here, disorder does not influence the
supratransmission force threshold in the ensemble-average sense, but it does
reduce the average transmitted wave energy.
