## 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  
The structure of the tapered aperture metal sheet is a type of thin plate structure. Thus, the vibration of the atomizing sheet is used to study the forced vibration displacement response of the uniform thin plate with a small deflection under the PZT excitation. * Natural vibration of the circular thin plate  
To better analyze the forced response of the thin plate, the natural vibration of the circular thin plate is initially analyzed. The coordinate system on the neutral surface of the plate is _Oxyz_, and the polar coordinate system \((r,\theta)\) is established, as shown in Figure 4, where the displacement along the \(z\) axis is \(\omega\). Figure 3: Change process of the micro-tapered aperture in the atomizer. Suppose the thickness of the thin plate is \(h\), the material density is \(\rho\), the elastic modulus is \(E\), and the Poisson's ratio is \(\mu\). According to the small deflection bending theory of the elastic thin plate and based on [PERSON]'s hypothesis, the free vibration differential equation of the thin plate is as follows [29]:  
\[\frac{\partial^{2}w}{\partial t^{2}}+\frac{D}{\rho\hbar}\
abla^{4}w=0 \tag{1}\]  
where \(\
abla^{4}=\
abla^{2}\
abla^{2}=\left(\frac{\partial^{2}}{\partial t^{2}}+ \frac{1}{r}\frac{\partial}{\partial t}+\frac{1}{r^{2}}\frac{\partial^{2}}{ \partial\partial^{2}}\right)^{2}\) is the Laplace operator and \(D=\frac{Et^{3}}{12(1-\mu^{2})}\) is the bending stiffness of the plate. Given the symmetry of the circular thin plate, Equation (1) can be written as:  
\[w(r,\theta,t)=R(r)(S\sin n\theta+C\cos n\theta)e^{i\omega t} \tag{2}\]  
where \(n=0,1,2,\cdots\) represents the number of nodal diameters in the mode shape and \(S\), \(C\) are determined by the initial conditions. Substituting Equation (2) into Equation (1) yields  
\[\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{n^{2}}{r^{2}}\right) R(r)-k^{4}R(r)=0 \tag{3}\]  
where \(k^{4}=\omega^{2}\rho h/D\). The general solution of Equation (3) can be obtained as  
\[R(r)=A_{n}J_{n}(kr)+B_{n}Y_{n}(kr)+C_{n}I_{n}(kr)+D_{n}K_{n}(kr) \tag{4}\]  
where \(J_{n}(kr)\) is the first-type \(n\)th-order Bessel function; \(Y_{n}(kr)\) is the second-type \(n\)th-order Bessel function; \(I_{n}(kr)\) is the first-type \(n\)th-order corrected Bessel function; \(K_{n}(kr)\) is the second-type \(n\)th-order corrected Bessel function; and \(A_{n}\), \(B_{n}\), \(C_{n}\), and \(D_{n}\) are constants to be determined. For a circular and solid plate, \(B_{n}\) and \(D_{n}\) must be 0 because \(w\) and \(\partial w/\partial r\) at the center \((r=0)\) must have a limited value. For a circular thin plate simply supported by its periphery, two boundary conditions exist as follows:  
\[\left\{ \begin{array}{l}R(a)=0\\ M_{r}|_{r=a}=\left[\frac{\mathrm{d}^{2}R}{\mathrm{d}r^{2}}+\mu(\frac{1}{r} \frac{dR}{dr}-\frac{n^{2}}{r^{2}}R)\right]|_{r=a}=0\end{array} \right. \tag{5}\]  
where \(M_{r}\) is the bending moment acting on the circular thin plate. By substituting Equation (5) and the condition that \(B_{n}\) and \(D_{n}\) are equal to zero into Equation (4), we can obtain:  
\[\left\{ \begin{array}{l}A_{n}J_{n}(ka)+C_{n}I_{n}(ka)=0\\ A_{n}L_{n}+C_{n}Q_{n}=0\end{array} \right. \tag{6}\]  
Figure 4: Atomizing sheet model and coordinate system. where  
\[L_{n}(ka)=\left[ \begin{array}{c}\frac{(1-\mu)^{2}n^{2}+(\mu-1)n-k^{2}n^{2}}{a^{2 }}J_{n}(ka)-\frac{\mu k}{a}J_{n+1}(ka)\\ Q_{n}(ka)=\left[\frac{(1-\mu)^{2}n^{2}+(\mu-1)n-k^{2}n^{2}}{a^{2}}I_{n}(ka)+ \frac{\mu k}{a}I_{n+1}(ka)\right]\end{array} \right.\]  
Equation (6) is a linear system of equations concerning \(A_{n}\) and \(C_{n}\). If the equation has a non-zero solution, then the determinant of its coefficients must be 0. Thus  
\[ \begin{vmatrix}J_{n}(ka)&I_{n}(ka)\\ L_{n}(ka)&Q_{n}(ka)\end{vmatrix} =0 \tag{7}\]  
For each \(n\), this equation has an infinite number of roots of \(ka=\lambda_{mms}\).