# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 3 Finite Element Analysis  
The dynamic performance of the piezoelectric atomizer and its vibration optimization were studied using a FE analysis. The device was considered as a multi-degree-of-freedom continuum system, which can be represented in a matrix notation as [6]:  
\[[M]\bar{\mathbf{u}}+[C]\mathbf{\dot{u}}+[K]\mathbf{u}=\mathbf{F} \tag{6}\]  
where \(\mathbf{u}\), \(\mathbf{\dot{u}}\), and \(\bar{\mathbf{u}}\) represent the vector of displacement, velocity, and acceleration, respectively, for every node; [M] is the mass matrix; [C] is the damping matrix; [K] is the stiffness matrix; and \(\mathbf{F}\) is the external excitation force. Two types of analyses were performed--namely, free and forced vibrations. The modal (free) analysis is an eigenvalue problem assuming no external forces, \(F=0\). A forced harmonic analysis can also be carried out by applying a frequency sweep for different applied input voltages to the piezoelectric ring. Figure 2: Frequencies calculated as a function of (**a**) the plate thickness and (**b**) [PERSON]’s modulus for the first three axisymmetric modes. For the first example, a numerical modal analysis on a stainless-steel circular thin plate was carried out. The mechanical properties of the plate are given in Table 1. In these calculations, 16,746 elements (SOLID186 with 20 nodes that exhibit quadratic displacement in ANSYS) were used. The boundary conditions of the thin plate were clamped at the edge to the piezoelectric ring, as is shown in Figure 3. Thus, the displacements in the circumferential area were assumed to be zero (\(u_{x}=u_{y}=u_{z}=0\) at \(r=8\) mm). The simulation was carried out using an FE model implemented in ANSYS; the pseudocode for the analysis is given in Appendix A. Figure 4 shows the results for the lowest axisymmetric modes when the largest displacement is at the center of the plate. To check the validity of the calculations, Figure 5 shows the first nine resonant frequencies using the modal analysis of a single, clamped, circular, and homogeneous thin plate. These results are compared with the theoretical approximation given by Equation (5). The results shows that the FE analysis only deviated by \(\sim\)0.12% on average from the theoretical predictions. The mechanical behavior of the thin plate is expected to be affected by the loading from the liquid that will be atomized. However, the theoretical and experimental results show that the shape of the vibration modes is not modified; that is, the presence of liquid loading only shifts the values of the resonant frequencies with the invariant shapes of the vibration modes [21]. In a preliminary numerical harmonic analysis (see the description of the pseudocode in Appendix A), a stainless-steel circular thin plate coupled to a piezoelectric ring was modeled. The mechanical, piezoelectric, and physical properties of the materials are given in Tables 1 and 2. The thin plate was modeled as a deformable plate using fully integrated elements with 20 nodes. Due to the symmetry of the model, only a one-quarter model was needed, thus reducing the number of grids and the overall computational demand (see Figure 6) [22]. Next, micro-apertures with three different geometrical shapes: cylindrical, pyramidal, and conical were added to the model of the circular thin plate to form a meshed thin plate (thin-plated with micro-apertures), which was then coupled to the piezoelectric ring. The meshed plate contained 551 micro-apertures with the dimensions shown in Figure 7, where \(\text{d}=10\)\(\upmu\)m and \(D=80\)\(\upmu\)m. These were distributed over the thin plate following a rectangular array. Figure 4: (**a**) Geometry used for the FEM modal analysis. (**b**–**d**) Contour plots of the first three axisymmetric vibrational modes estimated for the homogeneous thin plate. Figure 5: Comparison between the analytical and FEM results of the resonant frequency of the vibration modes from a clamped circular plate.