## 2 Materials and Methods
### Equations of Motion for the Three-Mass Model  
The differential equations for the pathological model are  
\[m_{ia}\ddot{x}_{ia}+r_{ia}\dot{x}_{ia}+k_{ia}x_{ia}+\Theta(-a_{i})c_{ia}\frac{ a_{i}}{2l}+k_{ca}(x_{ia}-x_{ja})+T_{ia}=F_{ia}\, \tag{15}\]  
and for the equation of the right vocal fold polyp,  
\[m_{3r}x_{3r}-T_{1r}+T_{1l}=F_{3} \tag{16}\]  
The function \(\Theta(x)\) describing the collision is approximated by  
\[\Theta(x)=\left\{\begin{array}{ll}\tanh(\frac{50x}{x_{0}})&\text{for }x>0\\ 0&\text{for }x\leq 0\end{array}\right. \tag{17}\]  
The polyp \(m_{3r}\) and the right lower mass \(m_{1r}\) are connected by nonlinear spring and the damping, which can be described as  
\[T_{1r}=r_{3r}\big{(}\dot{x}_{1r}-\dot{x}_{3r}\big{)}+k_{3r}[x_{1r}-x_{3r}+\eta _{l3}(x_{1r}-x_{3r})^{3}]\, \tag{18}\]  
while the effective nonlinear spring forces of the left lower mass \(m_{1l}\) can be written as  
\[T_{1l}=\Theta(-a_{3})c_{3r}[\frac{a_{3}}{2\omega_{l}l_{3}}+\eta_{l3}(\frac{a_ {3}}{2\omega_{l}l_{3}})^{3}]\, \tag{19}\]  
for which the parameters were set as \(\eta_{l3}=100\) and \(\eta_{l3}=500\). There is no coupling or collision effect between the right vocal fold polyp and the upper mass, so \(T_{2l}=0\) and \(T_{2r}=0\).  
To obtain the glottal airflow, it is necessary to calculate the minimal glottal area, using  
\[a_{\text{min}}=\min(a_{1l},a_{2l})+\min(a_{1r},a_{2r},a^{\prime}_{3r}), \tag{20}\]  
and the glottal areas of the lower mass pair m\({}_{1a}\) and the upper mass pair m\({}_{2a}\), using  
\[a_{i}=a_{0i}+l_{0}(x_{il}+x_{ir})=a_{il}+a_{ir} \tag{21}\]  
The area from the midline to m\({}_{ia}\) is \(a_{ia}\), and \(a_{01}=a_{02}=0.05\). Similarly, the equation of glottal area for the polyp can be written as:  
\[a^{\prime}{}_{3r}=a_{01}/2-l_{3}D_{3}+(l_{0}-l_{3})x_{1r}+l_{3}x_{3r} \tag{22}\]  
\[a_{3}=a_{03}+l_{3}(x_{1l}+x_{3r})\] \[a_{03}=l_{3}(a_{01}/l_{0}-D_{3})\] \[a^{\prime}_{3}=a_{01}/2+l_{0}x_{1l}+a^{\prime}_{3r}.\]\(P_{s}\) is the subglottal pressure, and \(P_{1}\) acts on \(m_{1a}\), \(P_{2}\) acts on \(m_{2a}\), and \(P_{3}\) acts on \(m_{3r}\). The volume flow velocity \(U\) can be described by the Bernoulli equation as  
\[\begin{split} P_{1}=P_{\rm s}[1-\Theta(a_{\rm min})(\frac{a_{\rm min }}{a_{1}})^{2}]\Theta(a_{1})\\ P_{2}=0\\ P_{3}=P_{\rm s}[1-\Theta(a_{\rm min})(\frac{a_{\rm min}}{a^{\prime }_{3}})^{2}]\Theta(a^{\prime}_{3})\\ U=\sqrt{\frac{2P_{\rm s}}{\rho}}a_{\rm min}\Theta(a_{\rm min}) \end{split} \tag{23}\]  
The effective driving forces \(F_{1a}\) acting on \(m_{1a}\), \(F_{2a}\) acting on \(m_{2a}\), and \(F_{3r}\) acting on \(m_{3r}\) can thus be deduced as  
\[\begin{split} F_{1l}=(P_{1}l_{0}(d_{1}-d_{3})+P_{3}l_{0}d_{3}) \cdot(\omega_{d}\omega_{l}),F_{2l}=0\\ F_{1r}=(P_{1}l_{0}(d_{1}-d_{3})+P_{3}(l_{0}-l_{3})d_{3})\cdot( \omega_{d}\omega_{l}),F_{2r}=0\\ F_{3r}=P_{3}l_{3}d_{3}\Theta(a_{3})\cdot(\omega_{d}\omega_{l}). \end{split} \tag{24}\]  
The equations in this section give the vibratory equations of the vocal folds coupled with a right vocal polyp, while the vocal fold model with a unilateral polyp is reducible to the normal two-mass model when \(D_{3}=0\), \(l_{3}=0\), \(d_{3}=0\), and \(T_{in}=0\).