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Abstract Fish scale-like features on substrates arranged periodically produce peculiar mechanical behavior. These include nonlinear stiffness, anisotropy in deformation, and finally jamming behavior. These smart structures can be fabricated by partially embedding stiffer plate-like segments on softer substrates to create a bi-material system, with overlapping scales. The dynamic response shows remarkable geometrical-material interplay and anisotropies in damping. Especially interesting is the damping behavior that is distinct from typical damping found in mechanical structures which are often approximated as Rayleigh-Damping. Here we discuss some of these dynamic behaviors that include material-geometry distinction in damping, multiple damping scenarios and interplay of dissipation possibilities. We performed experimental analysis and compared the results with simple mathematical laws that govern architecture-dissipation relationships that can help understand the vibrating response of pillar/scale-covered membranes/thin plates. We conclude by noting the applicability these metastructure in structural damping with other forms currently in use in practice.