  The mechanical analog of optical frequency combs, phononic frequency combs,
has recently been demonstrated in mechanical resonators and has been attributed
to coupling between multiple phonon modes. This paper investigates the
influence of mode structure on comb generation using a model of two nonlinearly
coupled phonon modes. The model predicts that there is only one region within
the amplitude-frequency space where combs exist, and this region is a subset of
the Arnold tongue that describes a 2:1 autoparametric resonance between the two
modes. In addition, the location and shape of the comb region are analytically
defined by the resonance frequencies, quality factors, mode coupling strength,
and detuning of the driving force frequency from the mechanical resonances,
providing clear conditions for comb generation. These results enable comb
structure engineering for applications in areas as broad as sensing,
communications, and quantum information science.
