  A `flutter machine' is introduced for the investigation of a singular
interface between the classical and reversible Hopf bifurcations that is
theoretically predicted to be generic in nonconservative reversible systems
with vanishing dissipation. In particular, such a singular interface exists for
the Pfl\"uger viscoelastic column moving in a resistive medium, which is proven
by means of the perturbation theory of multiple eigenvalues with the Jordan
block. The laboratory setup, consisting of a cantilevered viscoelastic rod
loaded by a positional force with non-zero curl produced by dry friction,
demonstrates high sensitivity of the classical Hopf bifurcation onset {to the
ratio between} the weak air drag and Kelvin-Voigt damping in the Pfl\"uger
column. Thus, the Whitney umbrella singularity is experimentally confirmed,
responsible for discontinuities accompanying dissipation-induced instabilities
in a broad range of physical contexts.
