## 3 Numerical Investigation of the Actuator  
Numerical investigations were performed to find the optimal dimensions of the actuator, to verify the operation principle, and to investigate the mechanical and electrical characteristics of the actuator. Comsol Multiphysics 5.4 software (v5.4, Comsol AB, Stockholm, Sweden) was used for the modeling. The material properties used for the modeling are presented in Table 2.  
Firstly, the dimensions of the actuator were optimized in order to minimize the difference between the natural frequencies of the first longitudinal and second bending vibration modes. The optimal dimensions of the actuator allow us to achieve superposition of the vibration modes. The height of the waveguide H and the upper diameter of the conical cavity \(D_{in}\) were selected as design variables, while other parameters were set as shown in Table 1. The optimization problem can be written as follows:  
\[\min_{\Delta f}(\Delta f=\big{|}w_{L_{1}}(H,D_{in})-w_{B_{2}}(H,D_{in})\big{|}); \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Property** & **PZT8** & **6061 Aluminum Alloy** \\ \hline Density [kg/m\({}^{3}\)] & 7800 & 2700 \\ Young’s modulus [N/m\({}^{2}\)] & \(7.6\times 10^{10}\) & \(7\times 10^{10}\) \\ Poisson’s ratio & - & 0.33 \\ Isotropic structural loss factor & - & 0.001 \\ Relative permittivity & \(\varepsilon_{33}\)T/\(\varepsilon_{0}=1200\) & - \\ Piezoelectric constant d\({}_{33}\) [\(10^{-12}\) m/V] & 225 & - \\ Piezoelectric constant d\({}_{31}\) [\(10^{-12}\) m/V] & \(-97\) & - \\ Piezoelectric constant d\({}_{15}\) [\(10^{-12}\) m/V] & 330 & - \\ Elastic stiffness coefficient C\({}_{30}\) [N/m\({}^{2}\)] & \(1.6\times 10^{10}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the materials. Figure 1: Schematics and excitation scheme of the traveling wave actuator: (**a**) Schematics of the traveling wave actuator: 1, the upper part of the waveguide; 2, the piezoceramic rings placed at the node; 3, the lower part of the waveguide; 4, the piezoceramic ring placed at the bottom of the waveguide; 5, the ball-bearing bed; 6, M3 fixing bolt; 7, the supporting cylinder; 8, the fixing ring; 9, the ball bearing. (**b**) Excitation scheme of the actuator. subject to  
\[h_{\min}\leq H\leq h_{\max}; \tag{2}\]  
\[d_{\min}\leq D_{in}\leq d_{\max}; \tag{3}\]  
\[H_{c}(H)=H-h_{offset} \tag{4}\]  
where \(H\) is the height of the waveguide; \(D_{in}\) is the diameter of the conical hole; \(\Delta f\) is the difference in the natural frequencies of the first longitudinal and second bending vibration modes; \(w_{LI}\) is the natural frequency of the first longitudinal vibration mode; \(w_{B2}\) is the natural frequency of the second bending vibration mode; \(h_{min}\) and \(h_{max}\) are the minimum and maximum values of waveguide height; \(d_{min}\) and \(d_{max}\) are the minimum and maximum values of the base diameter of the conical hole; \(H_{c}\) is the height of the conical cavity; and \(h_{offset}\) is 8 mm, and it is the offset of the conical cavity from the bottom surface of the waveguide. The following boundary values of the design variables were used: \(h_{min}\) = 35 mm, \(h_{max}\) = 57 mm, \(d_{min}\) = 12 mm, \(d_{max}\) = 18 mm. Modal frequency analysis of the actuator was performed during optimization. The boundary conditions were as follows: A fixed constraint was added in order to simulate the clamping system of the actuator. The actuator was loaded by gravity force. The linear search algorithm was applied for the calculations, while a step size of 0.5 mm was used for both design variables. The plot shown in Figure 2 presents the results of calculations. The difference between the natural frequencies varied from 43 Hz to 3850 Hz, and it depended on the combination of values of the design variables (Figure 2). The four best sets of design variable values are listed in Table 3. The difference in the natural frequencies for these sets varied from 43 Hz to 66 Hz.