  Estimating the number of degrees of freedom of a mechanical system or an
engineering structure from the time-series of a small set of sensors is a basic
problem in diagnostics, which, however, is often overlooked when monitoring
their health and integrity. In this work, we demonstrate the applicability of
the network-theoretic concept of detection matrix as a tool to solve this
problem. From this estimation, we illustrate the possibility to identify
damage. The detection matrix, recently introduced by Haehne et al. in the
context of network theory, is assembled from the transient response of a few
nodes as a result of non-zero initial conditions: its rank offers an estimate
of the number of nodes in the network itself. The use of the detection matrix
is completely model-agnostic, whereby it does not require any knowledge of the
system dynamics. Here, we show that, with a few modifications, this same
principle applies to discrete systems, such as spring-mass lattices and
trusses. Moreover, we discuss how damage in one or more members causes the
appearance of distinct jumps in the singular values of this matrix, thereby
opening the door to structural health monitoring applications, without the need
for a complete model reconstruction.
