  This paper provides necessary and sufficient conditions for the existence of
a pair of complex conjugate roots, each of multiplicity two, in the spectrum of
a linear time-invariant single-delay equation of retarded type. This pair of
roots is also shown to be always strictly dominant, determining thus the
asymptotic behavior of the system. The proof of this result is based on the
corresponding result for real roots of multiplicity four, continuous dependence
of roots with respect to parameters, and the study of crossing imaginary roots.
We also present how this design can be applied to vibration suppression and
flexible mode compensation.
