We consider the inverse problem of identifying a single open crack
in a longitudinally vibrating rod having non-uniform smooth
profile. Without any a-priori assumption on the smallness of the
damage and assuming that the rod profile is symmetric with respect
to the mid-point of the rod axis, we present a constructive
diagnostic algorithm from minimal frequency data. We show that
the crack can be uniquely identified, up to a symmetric position,
from the first two positive natural frequencies of the rod under
free-free end conditions. We also show that the non-uniqueness of
the damage location can be removed by using as data the first
positive resonant frequency of the free-free rod and the first
antiresonant frequency of the driving-point frequency response
evaluated at one end of the rod. The results of numerical
simulations and of applications of the method to experimental data
agree well with the theory.
