The mechanical properties of the vocal folds are represented in the model by stiffness and viscous damping. The properties of the approximated tissue depend primarily on the muscular tension and elongation of the folds. The tissue of the folds is depicted as a ribbon fixed at both ends (at the arythenoid and thyroid cartilages), but in the middle it can bend, contort and flex freely (Titze, 1994:94), although it is attached to the body of the larynx. The displacement between both masses is then equivalently described by the flexion of a ribbon in its cross-section.
The coupling stiffness kc expresses the vocal fold stiffness while bending in the direction perpendicular to the direction of the vibrations (the flextural stiffness in the lateral direction of the vocal folds). The kc parameter is highly nonlinear and changes in accordance with the tension and width of the folds because of the action of the cricothyroid and vocalis muscles (Gubrynowicz, 1997:17). The contraction force Fc is then represented as:
which under the assumption of linear elastic deformation of the folds can be approximated by:
Under normal physiological conditions the coupling stiffness
kc can vary substantially. The average
value1, used in the literature
(e.g. Ishizaka & Flanagan, 1972:1250), is about
25 kdyn/cm (0.25 N/m). As the coupling stiffness grows, the
two-mass model approaches the one-mass model (
). The
effect of kc on the model-generated waveform will be described in
section 27.3.
The spring elements s1 and s2 represent the tensions of the vocal folds. Their values depend on the deflection of the mass (in a non-linear manner) as well as on the elastic deformation of the fold during collision with the second fold.
The non-linear characteristic of the springs s1,2 is represented in equation (26) which is related to the force required to produce the deflection x from the initial position (Ishizaka & Flanagan, 1972: 1237):
(26)
where the force fki is needed to produce a deflection xi, ki is the linear stiffness and Etaki is the coefficient approximating the nonlinearity of the spring si.
An additional force acting on the vibrating masses is caused by a collision with the opposite fold and corresponds to the restoring force of the elastic deformation of the tissue structures. In terms of the model, this means that the total restoring force at the collision instant is equal to the sum of the forces caused by the deflection and the forces caused by deformation restoration:
(27)
where fki is defined as before and fhi denotes the force representing the non-linear spring shi
As follows from equation (26) - (28) the resulting force acting on mi during the closure of the glottis consists of the sum of both components, whereas during the motion without collision with the opposite fold, the restoring force comprises only the deflection component fki (note the direction of the X-axis: as the folds are closing the values of x become negative). The term Ag0i/2lg describes the pre-phonatory position of the fold and is usually called x0i.
Typical values of the tension parameters are (Ishizaka & Flanagan, 1972:1250):
These values were estimated for the human larynx and are thus in accordance with physiological measurements. The influence of the changing spring constants on the generated glottal waveform will be discussed in section 27.3.
.
