Elimination of vibration localization: a mathematical justification

In this note, mistuned periodic structures are considered. Due to mistunings, some components of such structures may vibrate with small amplitudes, while some other components may vibrate with significantly large amplitudes. Such a behavior is known as vibration localization and is undesirable. In an earlier note, it was shown by the author that an effective passive technique to eliminate vibration localization in mistuned periodic structures is to add small components between the structure components. In this note, it is rigorously proved that the added components indeed eliminate vibration localization. The proof is established by using a dichotomy in the dynamics of the structure to which small components are added, and by applying the singular perturbation method.