Abstract
The frequency bands of perfect bi-periodic mass–spring systems and the localized modes in the same systems with one disordered subsystem are exactly analyzed using the U-transformation method. The linear bi-periodic system with an infinite number of subsystems may be considered as an equivalent cyclic bi-periodic system having infinite subsystems. The governing equation for such an equivalent system with cyclic bi-periodicity can be uncoupled by applying the U-transformation twice to form a set of single-degree-of-freedom equations. These equations can be used to analyze the pass bands and localized modes corresponding to the considered system with and without disorder, respectively. Some specific systems are taken as examples to demonstrate how to apply the formulas obtained in this paper and to find the localized modes and frequencies.
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