## 2 Structure of the Dynamic Mesh Atomizer and the Theoretical Model of the Atomizing Sheet
### Theoretical Model of the Atomizing Sheet
#### 2.2.2 Vibration Analysis of the Atomizing Sheet
By substituting \(ka=\lambda_{mms}\) into \(k^{4}=\omega^{2}\rho h/D\), the natural frequency of the circular thin plate under the simply supported state can be obtained as follows:  
\[\omega_{mms}=\frac{\lambda_{mms}^{2}}{a^{2}}\sqrt{\frac{D}{\rho h}} \tag{8}\]  
where \(m=0,1,2,\cdots\) represents the number of the nodal circles in the mode shape, s represents the boundary conditions of the peripheral simply supported, and \(\lambda_{mms}\) is the constant of the natural frequency. By substituting \(\lambda_{mms}\) back to the above formula, the mode shape of the circular thin plate corresponding to natural frequency \(\omega_{mms}\) can be obtained as:  
\[\phi_{mms}(r,\theta)=A_{n}[J_{n}(kr)+(J_{n+1}(\lambda_{mms})/I_{n+1}(\lambda_{ mms}))I_{n}(kr)](S\sin n\theta+C\cos n\theta) \tag{9}\]  
\(\bullet\) Forced vibration displacement response of the circular thin plate  
The problem of the forced vibration displacement response of the uniform plate with a small deflection under PZT excitation is analyzed. The displacement response of small deflection forced vibration \(w_{mms}(r,\theta,t)\) is expressed by the normalized displacement modes of the thin plate, as shown as follows:  
\[w_{mms}(r,\theta,t)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\phi_ {mms}(r,\theta)q_{mms}(t) \tag{10}\]  
where \(q_{mms}(t)\) is the generalized coordinates of \(\phi_{mms}(r,\theta)\). \(\phi_{mms}(r,\theta)\) satisfies the orthogonal condition of the thin plate mode, shown as follows:  
\[\iint\limits_{G}\rho h\phi_{m;n;n_{\text{S}}}(r,\theta)\phi_{m;n;n_{\text{S}} }(r,\theta)drd\theta=\left\{ \begin{array}{c}M_{m;n_{\text{S}}},i=j\\ 0,i\
eq j\end{array} i,j=1,2,\cdots,\infty\right. \tag{11}\]  
where \(M_{m;n_{\text{S}}}\) is the modal mass of the plate. The integral area \(G\) covers the entire board. When the thin plate is forced to vibrate, the effective damping force on a unit area perpendicular to the surface of the thin plate is proportional to the vibration velocity. A distributed exciting force \(f(r,\theta,t)=F(r,\theta)e^{i\omega t}\) with frequency \(\omega_{mms}\) is applied to the uniform plate. Then the differential equation that the generalized coordinate \(q_{mms}(t)\) should satisfy is:  
\[M_{mms}\ddot{q}_{mms}+C_{mms}\dot{q}_{mms}+K_{mms}q_{mms}=F_{mms}e^{i\omega t} \tag{12}\]  
where \(M_{mms}\), \(C_{mms}\), \(K_{mms}\), and \(F_{mms}\) are the modal mass, modal damping, modal stiffness and modal force, respectively. According to Equation (12), the steady-state forced response of the uniform thin plate under PZT element excitation is:  
\[w_{mms}(r,\theta,t)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}F_{ mms}\phi_{mms}(r,\theta)e^{i\left(\omega t-\varphi_{mms}\right)}/\left[K_{mms} \sqrt{\left(1-\omega_{mms}^{2}\right)^{2}+\left(2\zeta_{mms}\omega_{mms} \right)^{2}}\right] \tag{13}\]where \(\varpi_{mms}=\omega/\omega_{mms}\) is the frequency ratio, which is the ratio of the excitation frequency \(\omega\) to the system natural frequency \(\omega_{mms}\), \(\zeta_{mms}=C/2M_{mms}\omega_{mms}\) is the relative damping coefficient of the system, and \(\varphi_{mms}=\arctan(2\zeta_{mms}\varpi_{mms})/(1-\varpi_{mms}^{2})\) is the steady-state displacement response hysteretic phase angle of the excitation force. According to Reference [29], the functions \(\omega_{mnt}\) and \(w_{mnt}(r,\theta,t)\) under the condition of peripheral fixed support are consistent with those under the peripheral simply supported, as shown in Equation (9) and Equation (13), respectively, except for the mode function \(\phi_{mnt}(r,\theta)\), which is:  
\[\phi_{mnt}(r,\theta)=A_{n}[J_{n}(kr)-(J_{n}(\lambda_{mnt})/I_{n}(\lambda_{mnt} ))I_{n}(kr)](S\sin n\theta+C\cos n\theta) \tag{14}\]  
where the subscript \(f\) represents the boundary conditions of the peripheral fixed support.