**6. Conclusions**

In this study, we used a multi-parameter perturbation method to solve the bending deformation problem of piezoelectric cantilever beams under combined loads. And we compared the solution presented here with the existing solution from Yang and Liu [17] to validate the rationality of the presented solution. In addition, we carried out the related experiments of the piezoelectric cantilever beam, and compared the experimental results with the theoretical solution presented here, and also investigated the influence of the piezoelectric properties on the deformation of piezoelectric cantilever beams. The following main conclusions can be drawn.

(i) The theoretical results are in good agreemen<sup>t</sup> with the experimental results, which means that the analytical solution given in this paper is correct and the multi-parameter perturbation method is e ffective.

(ii) From the perturbation expansion, it is easy to find that the zero-order perturbation solution is a pure mechanical solution, in which the piezoelectric e ffect has not been incorporated. From the first-order, second-order, and higher order perturbation solutions, the piezoelectric e ffect is gradually reflected. This structural form of the multi-parameter perturbation solution presented here is beneficial to the analysis and understanding of the solved problem.

(iii) The deformation magnitude of a piezoelectric cantilever beam is smaller than that of a cantilever beam without piezoelectricity, due to the well-known piezoelectric sti ffening e ffect.

Unfortunately, the numerical simulation for the physical system studied here has not been carried out in this study. In our previous study [34], we used ABAQUS software to simulate the problem of functionally graded piezoelectric cantilever beams with di fferent properties in tension and compression. Similarly, the problem studied here may also be simulated by ABAQUS, which is our follow-up research. In summary, the multi-parameter perturbation method presented in this paper provides a new way to solve complex non-linear structural problems. The analytical solution of the bending problems of piezoelectric cantilever beams under combined loads can provide a theoretical basis and reference for the analysis and design of sensors or actuators made of piezoelectric polymers.

**Author Contributions:** Conceptualization, X.-T.H. and J.-Y.S.; funding acquisition, X.-T.H. and J.-Y.S.; methodology, X.-T.H. and Z.-X.Y.; data curation, Z.-X.Y. and H.-X.J.; writing—original draft preparation, X.-T.H. and Z.-X.Y.; writing—review and editing, H.-X.J. and J.-Y.S.

**Funding:** This project is supported by National Natural Science Foundation of China (Grant No. 11572061 and 11772072).

**Conflicts of Interest:** The authors declare no conflicts of interest.
