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Inerter-based devices have shown exceptional performance in high-acceleration scenarios but require grounding, limiting their use in superstructures. Inerters affect the modal response, especially with closely spaced modes. This study introduces a novel approach to enhance damping and peak response attenuation using mechanical network synthesis, combining an inerter, dashpot, and spring. The devices, modeled for a Single Degree of Freedom system under harmonic excitation, show significant peak response reduction with optimal parameters. The research extends to a shear building model under harmonic and seismic excitation, revealing substantial improvements, potentially surpassing Tuned Mass Dampers.