## 2 Materials and Methods
### Effective Parameters of Vocal Fold Vibration  
Muscle activation affects the mass and stiffness by changing the above parameters, which have an impact on the vibration [19]. In our low-dimensional model, the mass of the vocal folds can be calculated by  
\[m=pldD \tag{9}\]  
Changes in muscle activation do not affect density of the vocal fold. Therefore, we only consider the effective length, depth, and width of the vibration, such that the effective mass of the vocal fold now become the following:  
\[m_{ia}=(\omega_{D}\omega_{d}\omega_{l})\cdot m_{ia}m_{3r}=(\omega_{D}\omega_{ d}\omega_{l})\cdot m_{3r} \tag{10}\]  
where \(m_{ia}\) and \(m_{3r}\) on the right side of the equation are the default values of vocal fold mass.  
For vocal fold stiffness, the following relationship is given by the definition of fundamental frequencies:  
\[k=\pi^{2}\sigma dD/l \tag{11}\]  
Likewise, for our model, we can similarly write,  
\[k_{ia}=(\omega_{D}\omega_{d}\omega_{\sigma}/\omega_{l})\cdot k_{ia}k_{3r}=( \omega_{D}\omega_{d}\omega_{\sigma}/\omega_{l})\cdot k_{3r} \tag{12}\]  
where \(k_{ia}\) and \(k_{3r}\) on the right side of the equation are the vocal fold stiffness when no vibration occurs. In addition, the coupling stiffness \(k_{ca}\) is different from others. According to the mass distribution rule of a previous study [19], the effective coupling stiffness becomes  
\[k_{ca}=\left[\left(\frac{18 lD_{c}}{7d}-\frac{2 ld}{D}\right)/\left(\frac{18l_{0 }D_{0}}{7d_{0}}-\frac{2l_{0}d_{0}}{D_{0}}\right)\right]\cdot k_{ca} \tag{13}\]  
The default parameters in the deterministic vocal fold models with steady airflow used in Table 1 are considered as common physiological conditions [19; 25]. The units of parameters are cm, g, ms, and their corresponding combinations.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Lower mass & \(m_{1a}\) & 0.125 \\ Upper mass & \(m_{2a}\) & 0.05 \\ Lower stiffness constant & \(k_{1a}\) & 0.08 \\ Upper stiffness constant & \(k_{2a}\) & 0.008 \\ Coupling stiffness constant & \(k_{ca}\) & 0.025 \\ Lower mass thickness & \(d_{1}\) & 0.25 \\ Upper mass thickness & \(d_{2}\) & 0.05 \\ Thickness of the vocal fold & \(d_{0}\) & 0.3 \\ Length of the vocal fold & \(l_{0}\) & 1.4 \\ Depth of the vocal fold & \(D_{0}\) & 0.3 \\ Lower mass spring constant during vocal fold collisions & \(c_{1a}\) & 0.24 \\ Upper mass spring constant during vocal fold collisions & \(c_{2a}\) & 0.024 \\ Air density & \(\rho\) & 0.00113 \\ Subglottal pressure & \(P_{s}\) & 0.006 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and values of the model.  
The damping constant of the vocal folds can be calculated as  
\[\begin{split} r_{1a}&=2\zeta_{1}\sqrt{m_{1}K_{1}}\, \zeta_{1}=0.1\\ r_{2a}&=2\zeta_{2}\sqrt{m_{2}k_{2}}\,\zeta_{2}=0.6 \end{split} \tag{14}\]  
where \(\zeta_{1}\), \(\zeta_{2}\) refer to damping ratios for the viscous resistances \(r_{1a}\) and \(r_{2a}\). Based on previous studies [11], we selected the following parameters for the vocal fold polyp: \(m_{3r}=m_{1r}3D_{3}/lD_{0}d_{1}\), \(k_{3r}=0.014\), \(c_{3r}=3k_{3r}\), \(r_{3r}=r_{1r}(m_{3r}k_{3r}/m_{1r}k_{1r})^{0.5}\), \(l_{3}=l_{0}/3\), \(D_{3}=0.1D_{0}\), \(d_{3}=(d_{1}+d_{2})/3\).