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Wave dispersion and topology of phononic crystals in classical mechanical systems are well understood and extensively studied subjects. However, the topological properties of acoustic metamaterials with more complex unit cells having several masses, internal resonators, and inerter elements is an insufficiently investigated topic. In this work, we study a class of locally resonant acoustic systems having diatomic- and triatomic-like mass-in-mass unit cells with inerter elements and different springs connecting outer masses. Winding numbers and signs of band gaps are investigated to assess the topological characteristics of a lattice band structure that support edge/interface modes and whether that property is affected when inerter elements are embedded into the system. The dynamics of finite undamped and damped chains constituted of two connected sub-lattices are investigated to demonstrate the existence of interface modes and their localization in space. We reveal that the presented diatomic-like and triatomic-like mass-in-mass chains can generate several interface modes that reside within both lower and higher frequency band gaps. The concept is illustrated through the investigation of the eigenvalue spectrum for varying and fixed stiffness of outer springs and frequency response function. The effect of arbitrary viscous damping is explored based on steady-state responses of the lattice interface mass points. Numerical analysis reveals that the introduction of inerters in combination with local resonators can significantly shift the band gaps and corresponding interface modes to lower frequency values while keeping the main topological properties of the initial configuration without inerters. The effect of damping is shown to be significant and capable to attenuate both lower and higher frequency interface mode amplitudes. We anticipate that this study will pave the wave for future works on the topic that include more reliable models of inerters in the topological mechanical metamaterial design.
inerters; interface modes; band inversion; locally resonant metamaterials; viscous damping