# Theoretical Modeling of Piezoelectric Cantilever MEMS Loudspeakers
## Appendix A  
Generally, a plane \(z=t_{0}\) with zero stress is selected when no voltage is applied, and the displacement of this plane is regarded as the total displacement of the MPCA:  
\[\sum\
olimits_{n=1}^{i}\int_{h_{n-1}}^{h_{n}}\frac{1}{s_{11}^{n}}\frac{ \partial^{2}u}{\partial x^{2}}(z-t_{0})dz=0, \tag{1}\]  
\[t_{0}=\frac{\sum\
olimits_{n=1}^{i}\frac{h_{n}^{2}-h_{n-1}^{2}}{2s_{11}^{n}}} {\sum\
olimits_{n=1}^{i}\frac{h_{n}-h_{n-1}}{s_{11}^{n}}}. \tag{2}\]  
Upon applying a voltage U in the z-direction on an MPCA, the bending moment \(M_{x}\) of the cross-section at \(x\) relative to the \(z=t_{0}\) plane is  
\[M_{x}=-\int\sigma_{x}wzdz. \tag{3}\]  
According to Equations (1) and (3),  
\[\begin{split} M_{x}=&\left\{\sum\
olimits_{n=1}^{i} \frac{w\left[\left(h_{n}-t_{0}\right)^{3}-\left(h_{n-1}-t_{0}\right)^{3} \right]}{3s_{11}^{n}}\right\}\frac{\partial^{2}u}{\partial x^{2}}\\ &+\left\{\sum\
olimits_{n=1}^{i}\frac{d_{31n}w}{2s_{11}^{n}} \left[\left(h_{n}-t_{0}\right)^{2}-\left(h_{n-1}-t_{0}\right)^{2}\right] \right\}E_{n},\end{split} \tag{4}\]where \(E_{n}=U/(h_{n}-h_{n-1})\). The shear force \(F_{z}\) and bending moment \(M_{x}\) satisfy  
\[F_{z}dx+dM_{x}=0.\] (A5)  
Substituting Equation (A4) into (A5) yields  
\[F_{z}=-\frac{dM_{x}}{dx}=-\left\{\sum_{n=1}^{i}\frac{w\Big{[}(h_{n}-t_{0})^{3}- (h_{n-1}-t_{0})^{3}\Big{]}}{3s_{11}^{n}}\right\}\frac{\partial^{3}u}{\partial x ^{3}}.\] (A6)  
According to [PERSON]'s second law, the equation of motion is  
\[\frac{dF_{z}}{dx}dx=\sum_{n=1}^{i}\rho_{n}w(h_{n}-h_{n-1})\frac{\partial^{2}u }{\partial t^{2}}dx,\] (A7)  
where \(\rho_{n}\) denotes the density of the nth layer. Owing to the harmonic vibration of the MPCA, it follows that  
\[u=u(x)e^{i\omega t}.\] (A8)  
Substituting Equations (A6) and (A8) into Equation (A7) yields  
\[P\frac{\partial^{4}u(x)}{\partial x^{4}}-Q\omega^{2}u(x)=0,\] (A9)  
with  
\[P=\sum_{n=1}^{i}\frac{w[(h_{n}-t_{0})^{3}-(h_{n-1}-t_{0})^{3}]}{3s_{11}^{n}},\] (A10)  
\[Q=\sum_{n=1}^{i}\rho_{n}w(h_{n}-h_{n-1}),\] (A11)  
\[R=\sum_{n=1}^{i}\frac{d_{31n}}{2s_{11}^{n}}(h_{n}+h_{n-1}-2t_{0}).\] (A12)  
A general solution for Equation (A9) is  
\[u(x)=A\cosh kx+B\sinh kx+C\cos kx+D\sin kx,\] (A13)  
where \(k=\sqrt{\frac{\omega}{\sqrt{P/Q}}}\). In piezoelectric cantilever MEMS loudspeakers, the MPCA is clamped at \(x=0\) and connected to a middle coupling element at \(x=l\). Therefore, the boundary conditions can be expressed as  
\[u(0)=0,\left.\frac{\partial u(x)}{\partial x}\right|_{x=0}=0,\] (A14)  
\[F_{z}\big{|}_{x=l}=F,\left.M_{x}\right|_{x=l}=0,\] (A15)  
Then, the coefficients are calculated as follows:  
\[A=\frac{-\frac{F}{k^{3}p}(\sinh kl+\sin kl)+\frac{RUlw}{k^{2}P}(\cosh kl+\cos kl )}{-2-2\cosh kl\cos kl},\] (A16)  
\[B=\frac{\frac{F}{k^{3}p}(\cosh kl+\cos kl)-\frac{RUlw}{k^{2}P}(\sinh kl+\sin kl )}{-2-2\cosh kl\cos kl},\] (A17)  
\[C=-A,D=-B.\] (A18)  
Then, the steady-state displacement at \(x=l\) can be deduced.  
According to Equations (1)-(3), the electric displacement through a piezoelectric layer is  
\[D_{3}=-d_{31}\frac{\frac{\partial^{2}u}{\partial x^{2}}(z-t_{0})}{s_{11}^{p}}+( \epsilon_{33}^{t}-\frac{d_{31}^{2}}{s_{11}})E_{3}. \tag{11}\]  
The mean electric displacement of an MPCA is  
\[\begin{split}\overline{D_{3}}&=\sum_{n=1}^{i}\int _{h_{n-1}}^{h_{n}}D_{3}\frac{dz}{h_{n}-h_{n-1}}\\ &=\sum_{n=1}^{i}[-\frac{d_{31n}}{2s_{11}^{n}}(h_{n}+h_{n-1}-2t_{0} )\frac{\partial^{2}u}{\partial x^{2}}+(\epsilon_{33n}^{t}-\frac{d_{31n}^{2}}{s _{11}^{n}})E_{3}].\end{split} \tag{12}\]  
The charge passing through the beam \(Q_{3}\) is [21; 22]  
\[Q_{3}=\int_{0}^{w}dy\int_{0}^{l}\overline{D_{3}}dx. \tag{13}\]  
Substituting Equation (12) into (13) yields Equation (5).