## 4 Optimization of Effective Electromechanical Coupling Coefficients
With the outer radius of 60 mm, inner radius of 5 mm, and a total thickness of 7.5 mm of the trilaminar bending ring with unequal diameters unchanged, the effective electromechanical coupling coefficient (\(k_{\mathit{eff}}\)) of the first-order bending vibration versus thickness ratio (\(h_{m}/h\)) and radius ratio (\(r_{1}/r_{2}\)) under different boundary conditions are calculated to get the optimal thickness ratio and outer radius ratio, and the results are shown in Figure 8.  
Figure 9 shows that the changing trend of \(k_{\mathit{eff}}\) with the outer radius ratio (\(r_{1}/r_{2}\)) is different, and the maximum value appears in different positions under different boundary conditions. As the outer radius ratio (\(r_{1}/r_{2}\)) increases, the \(k_{\mathit{eff}}\) under different boundary conditions first increases to the maximum value and then decreases at different speeds. The fixed inner and free outer boundary first appear as the maximum, the free inner and free outer boundary is second, the free inner and supportive outer boundary is next, and the free inner and fixed outer boundary finally appear as the maximum. Further analysis shows that under the free inner and fixed outer boundary, when the thickness ratio (\(h_{m}/h\)) = 0.3 and the outer radius ratio (\(r_{1}/r_{2}\)) = 0.88, \(k_{\mathit{eff}}\) reaches a maximum value of 0.45. Under the fixed inner and free outer boundary, when the thickness ratio (\(h_{m}/h\)) = 0.5 and the outer radius ratio (\(r_{1}/r_{2}\)) = 0.65, \(k_{\mathit{eff}}\) reaches a maximum value of 0.45. Under the free inner and free outer boundary, when the thickness ratio (\(h_{m}/h\)) = 0.5 and the outer radius ratio (\(r_{1}/r_{2}\)) = 0.74, \(k_{\mathit{eff}}\) reaches a maximum value of 0.47. Under the free inner and supportive outer boundary, when the thickness ratio (\(h_{m}/h\)) = 0.5 and the outer radius ratio (\(r_{1}/r_{2}\)) = 0.89, \(k_{\mathit{eff}}\) reaches a maximum value of 0.50. Figure 9: The \(k_{\mathit{eff}}\) of first-order bending resonance versus thickness ratio (\(h_{m}/h\)) and radius ratio (\(r_{1}/r_{2}\)) under different boundary conditions. (**a**) the free inner and fixed outer; (**b**) the fixed inner and free outer; (**c**) the free inner and free outer; (**d**) the free inner and supportive outer. In particular, when used as a driving source for the vibration of the liquid cavity in the piezoelectric circular tube, the outer boundary of the trilaminar bending ring transducer with unequal diameters is clamped on one end of the piezoelectric circular tube and its boundary condition is approximated as free inner and fixed outer. As shown in Figure 9a, the \(k_{\mathit{eff}}\) decreases monotonically with the increase in \(r_{0}/r_{1}\), which is possibly due to the reduction in the piezoelectric ceramic. Therefore, to maximize \(k_{\mathit{eff}}\), a smaller inner radius for the trilaminar bending ring transducer with unequal diameters should be considered. As shown in Figure 9a, with the increase in \(r_{1}/r_{2}\), the \(k_{\mathit{eff}}\) initially increases to the maximum value and then rapidly decreases. This is because the first-order bending vibration has a nodal circle. When the outer radius of the piezoelectric ceramic does not cross the nodal circle, both the piezoelectric ceramic vibration and electric field excitation are in phase, resulting in an increase in \(k_{\mathit{eff}}\) as the outer radius of the piezoelectric ceramic increases. However, when the outer radius of the piezoelectric ceramic crosses the nodal circle, the piezoelectric ceramic vibration on both sides of the nodal circle is out of phase, while the electric field excitation is in phase. Consequently, the excitation of the piezoelectric ceramic is partially suppressed, and \(k_{\mathit{eff}}\) subsequently decreases. Hence, it is preferable to avoid the outer radius of the piezoelectric ceramic from crossing the vibration nodal circle as much as possible to maximize \(k_{\mathit{eff}}\).