## 5 Conclusions  
In this study, a rapid calculation method based on [PERSON]'s hypotheses and classical electroelasticity theory was proposed. This method allows for the swift computation of the resonant frequency, antiresonant frequency, and effective electromechanical coupling coefficient of the trilaminar bending ring with unequal diameters under different boundary conditions. Remarkably, this approach can save significant time compared to using the finite element method.  
Moreover, the study optimizes the effective electromechanical coupling coefficients (\(k_{\mathit{eff}}\)) of the first-order bending vibration of the trilaminar bending ring transducer with unequal diameters under different boundary conditions by adjusting the thickness ratio and radius ratio between the metal substrate and the piezoelectric ceramic. Specifically, under the free inner and fixed outer boundary condition, \(k_{\mathit{eff}}\) initially increases to a maximum value and then rapidly decreases as the radius ratio (\(r_{1}/r_{2}\)) increases. This behavior can be attributed to the presence of a nodal circle in the first-order bending vibration of the trilaminar bending ring with unequal diameters. The vibration of the piezoelectric ceramic on both sides of the nodal circle is out of phase, whereas the electric field excitation is in phase, leading to partial suppression of the piezoelectric ceramic's vibration. This outcome distinguishes it from the trilaminar bending ring with equal diameters [13]. This finding is crucial for unveiling the bending vibration mechanism of the trilaminar bending ring with unequal diameters.  
Additionally, the study found that the maximum effective electromechanical coupling coefficient of the trilaminar bending ring with unequal diameters is obtained around a thickness ratio (\(h_{m}/h\)) of 0.3 and an outer radius ratio (\(r_{1}/r_{2}\)) of 0.9 under the free inner and fixed outer boundary condition. This result is crucial for the design of the crosswell seismic acoustic source and could contribute to improving their acoustic radiation performance.  
In summary, this study establishes a solid theoretical foundation for the design and optimization of trilaminar bending ring transducers with unequal diameters. By providing a comprehensive understanding of the resonant frequency, antiresonant frequency, and effective electromechanical coupling coefficient, our research offers valuable insights into improving the performance and efficiency of the transducers for actuating the crosswell seismic source. The findings have practical implications for enhancing the accuracy and resolution of crosswell seismic surveys, thereby facilitating more precise subsurface imaging and exploration. Ultimately, this study paves the way for advancements in the field of crosswell seismic acoustic source technologies.  
**Author Contributions:** Conceptualization, C.Z.; methodology, C.Z.; software, C.Z.; validation, C.Z.; and Y.Z.; formal analysis, C.Z.; investigation, C.Z. and Y.Z.; resources, Y.D. and Y.Z.; data curation, C.Z.; writing--original draft preparation, C.Z.; writing--review and editing, C.Z., Y.Z. and X.Z.; visualization, J.Z.; supervision, X.Z.; project administration, Y.D. and Y.Z.; funding acquisition, Y.D. and Y.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (Grant Nos. 52227901, 12274432, and 42127807).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.