Ishizaki.K, 1972  “Two mass theory”

1,What concept is Two mass theory?

   The two-mass approximation is often presented as a simple but efficient model for the purpose of numerical synthesis of voiced sounds. However, the quality and the naturalness of the generated signal is still to be improved. A simple modification of the two-mass model is proposed in which the mechanical dynamics of that of two point masses. The movement of these point masses drives a two-parameter curve describing the smooth surface of the vocal folds. This representation allows description of unsteady flow separation within the glottis. A semiempirical boundary layer theory is used to calculate the position of flow separation. This theory is experimentally tested using stationary flow measurements and unsteady flow visualization. Furthermore, computed volume flux and Bernoulli force are presented and compared with classical models. The description also includes an inelastic collision which yields a prediction of an additional volume flux due to the deformation of the vocal folds after the through flux has been stopped by the closure of the glottis. This additional deformation appears to have a significant influence on the voiced sound production.

2, What is Oscillation?
Oscillation is a repeated back-and-forth movement. Many common objects will oscillate for a short period of time if set into vibration by being struck; bells, crystal glasses, and so forth. What makes vocal fold oscillation more interesting than a simple oscillator - such as a clock's pendulum - is the question of how the back-and-forth movement can be sustained over time. This phenomenon is called flow-induced oscillation. A steady stream of air passing by a wall or surface can cause that surface to vibrate; this can be seen in the way airplane wings vibrate in flight, and in the rattling of air ducts in heating systems.

As is the case for most scientific phenomena, vocal fold vibration was initially explained with a somewhat simple - and as we will see, incomplete - model. As more and more has become known about human phonation, subsequent models have evolved in complexity. One of the first simple explanations of vocal fold vibration relied on basic physical laws, particularly the Bernoulli effect, the same effect that describes the 'lift' on an airplane wing. Over the past few decades, scientists have increasingly had the benefit of computer systems to create complex models to mimic vocal fold oscillation.

3, Myoelastic-aerodynamic Theory
Early voice researchers in the 1950's and 1960's explained vocal fold oscillation with the myoelastic-aerodynamic theory. According to these theories, Bernoulli forces (negative pressure) cause the vocal folds to be sucked together, creating a closed airspace below the glottis. Continued air pressure from the lungs builds up underneath the closed folds. Once this pressure becomes high enough, the folds are blown outward, thus opening the glottis and releasing a single 'puff' of air.

The lateral movement of the vocal folds continues until the natural elasticity of the tissue takes over, and the vocal folds move back to their original, closed position. Then, the cycle begins again. Each cycle produces a single small puff of air; the sound of the human voice is nothing more than tens or hundreds of these small puffs of air being released every second and filtered by the vocal tract.

Let's further examine the myoelastic-aerodynamic theory. Myo- means muscle; the vocal folds, after all, are mostly comprised of muscle tissue. The -elastic suffix serves to remind us that the vocal fold is elastic and that we have active control over its elastic properties. Aerodynamic means that the theory deals with the motion of air and other gaseous fluids, and with the forces active on bodies in motion (such as the vocal folds) in relation to such fluids.

A depiction of this simple system is shown below:

A simple rectangular block represents one vocal fold. A spring is useful for portraying the tissue stiffness or restoring force in the vocal fold. Finally, we've added a damping constant to represent the viscosity (energy absorption) of the tissue. The damping constant is similar to the shock absorber on a car or a tubular damper on a screen door.

So, how well does this simple model explain how the vocal folds sustain oscillation? Not well at all, researchers have found. Bernoulli forces alone cannot account for continual energy conversion from airstream to tissue. Soon, oscillation would damp out.

4, The One-Mass Model
A crucial component must be added to our simple model. An acoustic tube (to represent the vocal tract) is now attached to our model. Why is the acoustic tube needed? For the vocal folds to sustain oscillation, we know there must be a negative pressure within the glottis. But, pressure from the lungs cannot be negative; it is always positive. So, how does the air pressure at the level of the glottis become negative? Make a mental image of the the air from the lungs moving uni-directionally upward. When the glottis is closing, the airflow begins to decrease, but the air that is above the glottis does not "know" this, so it continues to move with its same speed (because of inertia). This creates a region just above the vocal folds where the air pressure decreases, because air is not coming from the bottom through the glottis as fast as it is leaving above. When the vocal folds are opening, fluid pressure against the walls is greater than when the vocal folds are close together. Thus, it is the asymmetry of driving force (air) that sustains oscillation.

Although our one-mass model is a closer representation of actual vocal fold oscillation than the myoelastic-aerodynamic model, some refinements will make the model even more like human phonation.

An increased use of videostroboscopy in clinics and research labs has allowed vocologists to observe many sets of vocal folds in slow motion. These observations have shown that vocal folds rarely move in a uniform block as depicted in our one-mass model. Rather, the vocal folds move in a wave-like motion from bottom to top, with the bottom edge leading the way. A more sophisticated version of the model can mimic the motion.

5, The Three-Mass Model
So, let's continue building our model. In order to model the shape of the vocal folds more accurately, we add two small masses (depicted as m₁ and m₂), one on top of the other, to represent the cover of the vocal fold. A large mass (depicted as m) represents the thyroarytenoid muscle. Although independent of one another in movement, all three masses are connected by springs and damping constants. Here is how the model looks and moves now:

Note that at some points in the cycle, the bottom of the vocal folds are farther apart than the upper part of the folds. We call this a convergent shape because the airflow is converging. On the other hand, the airflow diverges when the lowermost parts of the vocal folds are closer together; this is a divergent glottal shape. Average air pressures within the glottis tend to be larger in the convergent glottal configuration than in the divergent shape, resulting in the asymmetry of air pressures needed to sustain oscillation.

As technical capabilities in computer software and hardware increase and as improved imaging techniques allow researchers to study vocal folds in motion, models are increasingly becoming more realistic. Dr. Ingo Titze and his colleagues at The University of Iowa routinely use 16-mass models in their studies.