A translating and rotating mass model of the vocal folds

The motion of a vocal fold is modelled with two mechanical resonators. One is for the translational movement of the body of the fold and is driven by the spatial average pressure in the glottal constriction. The other is rotational, it models the covering mucosa and is driven by the axial pressure gradient. Unlike the classical Ishizaka-Flanagan two-mass model, these resonators are mechanically uncoupled. Since the excitations of the two resonators are interrelated they still oscillate at the same frequency, but for oscillation, the rotational resonance must be higher than the translational. Also a pair of moderately detuned folds generate a well defined pitch rather than a composite diplophonic signal. Apart from fixed anatomical configuration parameters and lung pressure, the model is controlled with two parameters only, for cord tension and adduction. Regimes of oscillation are shown for relevant pairs of parameters. The aerodynamic drive power is predominantly supplied by transglottal pressure in the opening phase while the Bernoulli force at closure gives a rather small contribution. INTRODUCTION The first self-oscillating model of the vocal folds was reported by Flanagan & Landgraf (1968), and by Flanagan (1969). In this, a fold is represented by a mechanical resonator built from a mass suspended on a non-linear stiffness. One surface of the mass element bounds the glottal passage, and the element moves laterally to operate as a shutter valve controlling the area open to glottal airflow. The resonator is driven by the pressure exerted on this surface as consequence of the air stream fed into the passage by the lung pressure. To overcome deficiencies in oscillation at certain vocal tract loads and to improve anatomical likeness, this one-mass model was later developed into the now classical two-mass model of Ishizaka & Flanagan (1972) (the IF model). In this, a fold is represented by two resonators in tandem downstream the glottal passage, sharing an elastic mutual coupling element between them. This layout suggests to regard the folds as transmission lines where waves propagate. Such waves do exist in the mucosal cover of the human folds, they are known to be important to the oscillation mechanism and have inspired much work in their own right. However, the waves carry only a minor part of the oscillatory energy in the folds and a corroborating observation is that the two-mass model works best when the second mass is much smaller than the first. The present model was developed with the particular aim to control boundary movements in a simulation of glottal aerodynamics (Liljencrants, 1989; 1991), but it also purports to be anatomically slightly more realistic than the IF model of which it is a variation. The proposition is that the two resonators are organized as a translational system and a superimposed rotational system. The primary model element is thus a mechanical resonator comprising a compliance C=llk, a mass M, and a damping resistance R. An essential articulatory pitch control comes via the compliance, predominantly defined by the vocal fold tension. The translation may be lateral like in the primitive one-mass model. We can also include one more degree of freedom to allow for longitudinal movement. The folds would then follow essentially elliptic paths, lateral and axial relative to the airflow. Such extension does not require more control parameters in the model, nor does it complicate its implementation significantly.