Existing research on finite elastic metamaterial (EM) structures has primarily focused on the vibration characteristics of individual EM components under classical boundary conditions, revealing various extraordinary modal properties. In practical applications, however, EM components are typically coupled with other structural elements. The dynamic behavior of such coupled systems remains insufficiently explored. This study systematically investigates the dynamic behaviors of an EM rod link with attached end masses, with particular focus on modal properties within the bandgap. Through detailed analysis, we reveal the formation mechanisms and existence of unique localized modes emerging within the bandgap, examining their dependence on key parameters including attached masses and rod length. Furthermore, we characterize the vibration transmission properties of the EM link under the influence of these localized modes, highlighting their significant implications for engineering applications where vibration control is critical.