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With the purpose of obtaining complete stop bands which allow one isolate both longitudinal and transverse vibrations in functionally graded beams, this paper adopts the inertial amplification mechanisms via attaching them on the base beam whose material properties are described by the power-law function to study the nature of propagation and attenuation characteristics. The dynamic model of the inertial amplification mechanism is developed to capture inertial forces when the coupled axial-bending deformation is present in the base structure. By employing the principle of the virtual work and the Bloch theory, a discrete formulation is presented to exploit the complex band diagram. Numerical analyses show that multiple stop bands whose attenuation performances are comparable to that of resonance gaps can be obtained, demonstrating the efficiency on suppressing wave propagation over a wide frequency band, which is helpful to expand the range of engineering applications with functionally graded materials for vibration and sound reduction. Furthermore, the effects of the geometrical characteristics of attached substructure, power-law index and material damping are investigated. [ABSTRACT FROM AUTHOR]