Exact analysis of localized modes in two-dimensional bi-periodic mass–spring systems with a single disorder

The frequency equation and localized modes in two-dimensional bi-periodic mass–spring systems with one disordered subsystem are exactly analyzed by means of double U-transformation. At first a plane distributed mass–spring system with 2 n 1 × 2 n 2 subsystems and cyclic periodicity in x - and y -directions is considered. Then by adopting a limiting process with n 1 , n 2 approaching to infinity, the limiting solution is applicable for the plane distributed bi-periodic mass–spring systems with boundary at infinity. The explicit frequency equation and localized modes are derived. Some specific systems are taken as examples to demonstrate how to apply the formulas and equations obtained in the present paper in order to find the localized modes and corresponding frequencies.