# Influence of the Nasometric Instrument Structure on Nasalance Score
## 2 Conventional Nasometric Instruments
### Lumped-Element Model for Closed Mask-Type Nasometric Instrument  
Recently, a closed mask-type nasometric instrument was implemented to minimize the influence of the mentioned acoustic feedback. However, its utilization is low. Speech therapists in the field still rely on subjective judgment using simple instruments. In some cases, a separator-type nasometric instrument is used intermittently. In practice, there are many opinions about the theoretical considerations and effects of the acoustic feedback brought about by the structure of the nasometric instrument. In this study, the frequency characteristics of the mask-type nasometric instrument were analyzed through the electromechanical-equivalent model. A mask containing an acoustic hole was attached to the face surface. In this case, the attached mask could be attached to a Helmholtz resonator having a closed volume and a short tube. Generally, the Helmholtz resonator demonstrates low-pass-filter characteristics when the sizes of the closed volume and the short entrance tube are smaller than the applied acoustic wavelength [16]. Figure 5 shows the physical structure, mechanical model, and equivalent circuit corresponding to the Helmholtz resonator. This can be applied to the mask-type nasometric instrument. The Helmholtz resonator structure in Figure 5a can be represented by a single-degree-of-freedom resonance system, as shown in Figure 5b. When a driving force of \(F\)cos\(\omega\)t is applied to the center of the resonator, the equation of motion is expressed by Equation (2). \[m_{M}\frac{d^{2}x}{dt^{2}}+r_{M}\frac{dx}{dt}+s_{M}\mathrm{x}=F\cos\omega \mathrm{t}. \tag{2}\]  
Here, if the variable x is changed to the speed \(\mathrm{v}=\mathrm{d}\mathrm{x}/\mathrm{d}\mathrm{t}\), Equation (3) is obtained. \[m_{M}\frac{dv}{dt}+r_{M}v+s_{M}\int vdt=F\cos\omega\mathrm{t}. \tag{3}\]  
Figure 5: (**a**) Schematic of Helmholtz resonator, (**b**) mechanical model of Helmholtz resonator, and (**c**) equivalent circuit corresponding to the Helmholtz resonator. The single-degree-of-freedom resonance system in Figure 5b is expressed as an electrical equivalent circuit, as shown in Figure 5c. The equation for the current \(I\) in the series resonant circuit is shown in Equation (4). \[L\frac{di}{dt}+Ri\frac{1}{C}\int idt=E\cos\omega\textbf{t}. \tag{4}\]  
By comparing Equations (3) and (4), we have the force \(F\), velocity \(\upsilon\) corresponding to the equivalent circuit voltage E, current \(I\), which corresponds to the sound pressure P, and volume velocity \(\mu\) of the acoustic system. Furthermore, the mass \(m_{M}\), resistance \(r_{M}\), and stiffness \(s_{M}\) correspond to \(L\), R, and 1/C of the electrical circuit, respectively [17; 18; 19; 20]. The corresponding physical quantities of electrical and mechanical systems are shown in Table 1.  
Here, the velocity \(\upsilon\) of the mechanical system is expressed by Equation (5). \[v=\frac{P}{r_{M}+j\left(\omega m_{M}-\frac{s_{M}}{\omega}\right)}=\frac{P}{z_ {M}}[m/sec]. \tag{5}\]  
\(Z_{M}\) represents the mechanical impedance of the vibration system. The frequency \(f_{0}\) at which the reactance part of the mechanical impedance becomes zero is expressed by the following (Equation (6)). \[f_{0}=\frac{1}{2\pi}\sqrt{\frac{s_{M}}{m_{M}}}=\frac{c}{2\pi}\sqrt{\frac{A}{Vi}}. \tag{6}\]  
It is the same as the series resonance characteristics of the equivalent circuit in which the resonance occurs at \(f_{0}\) and the speed and amplitude become maximum. Therefore, the sound pressure \(P_{h}\) in the volume is expressed in Equation (7). \[P_{h}=\frac{s_{M}u}{j\omega}=\frac{P}{\left(1-\omega^{2}\frac{m_{M}}{a_{M}} \right)+j\omega\frac{r_{M}}{s_{M}}}. \tag{7}\]  
\(P_{h}\) corresponds to the voltage of the capacitance C corresponding to E = I/\(\omega\)C of the equivalent circuit. In addition, the parameters of the mechanical model can be expressed as a spring constant, mass, and the resistance of air, as shown in Equation (8).