Phonation threshold pressure and the elastic shear modulus: comparison of two-mass model calculations with experiments.

Ishizaka and Flanagan's classic two-mass model of vocal fold motion is applied to small oscillations where the equations become linear and the aerodynamic driving force is described by an effective stiffness. The solution of these equations includes an analytic formula for the two eigenfrequencies; this shows that conjugate imaginary parts of the frequencies emerge beyond eigenvalue synchronization and that one of the imaginary parts becomes zero at a pressure signaling the instability associated with the onset of threshold. Using recent measurements by Fulcher et al. of intraglottal pressure distributions [J. Acoust. Soc. Am. 129, 1548-1553 (2011).] to inform the behavior of the entrance loss coefficients, an analytic formula for threshold pressure is derived. It fits most of the measurements Chan and Titze reported for their 2006 physical model of the vocal fold mucosa. Two sectors of the mass-stiffness parameter space are used to produce these fits. One is based on a rescaling of the typical glottal parameters of the original Ishizaka and Flanagan work. The second requires setting two of the spring constants equal and should be closer to the experimental conditions. In both cases, values of the elastic shear modulus are calculated from the spring constants.