  In civil, mechanical, and aerospace engineering, structural dynamics is
commonly understood to be a discipline concerned with the analysis and
characterization of the vibratory response of structures. Key elements of the
response are the amplitude, phase, and damping ratio, which are quantities that
vary with the excitation frequency. In this paper, we extend the discipline of
structural dynamics to the realm of materials engineering by intrinsically
building localized substructures within, or attached to, the material domain
itself$-$which is viewed as an extended medium without defined external
boundaries. Our system is essentially a locally resonant elastic metamaterial,
except here it is viewed from the perspective of unique dissipation
characteristics rather than subwavelength effective properties or band gaps, as
widely done in the literature. We provide a theory, validated by experiments,
for substructurally synthesizing the dissipation under the conditions of
free-wave motion, i.e., waves not constrained to a prescribed driving
frequency. We use an extended elastic beam with attached pillars as an example
of a metamaterial. When compared to an identical infinite beam with no attached
substructures, we show that within certain frequency ranges the metamaterial
exhibits either enhanced or reduced dissipation$-$which we refer to as positive
and negative metadamping, respectively. These regimes are rigorously identified
and characterized using the metamaterial's band structure and
wavenumber-dependent dissipation diagram. This theory impacts applications that
require a combination of high stiffness and high damping or, conversely,
applications that benefit from a reduction in loss without the need to change
the backbone constituent material.
