In this work, an experimental model and a simplified analytical model of U-shaped atomic force microscopy (AFM) probe are described. The effects of surface elasticity, surface piezoelectricity and thermal loading on U-shaped probe are considered for the first time. The obtained experimental results are for commercial U-shaped probe AN2-300 when it is mounted on the afm+ system. The results indicate that modeled beams on the basis of the couple stress theory have higher stiffness than the modeled ones based on the classical continuum theory. It is shown that both the surface elasticity and surface residual stress influence the natural frequencies. Also, it can be observed that the piezoelectric field shows both decreasing and increasing effects depending on the sign of the voltage. In order to check the accuracy of the presented theory for the U-shaped AFM probe, the analytical results achieved by the modified couple stress theory are compared with experimental results, and it is found that they are in a very good agreement. This work will improve understanding the vibrational behavior of U-shaped probes and may be of special interest in design and application of nanoscale structures.

在这项工作中，描述了U型原子力显微镜（AFM）探针的实验模型和简化的分析模型。首次考虑了表面弹性，表面压电性和热负荷对U形探头的影响。当安装在afm +系统上时，获得的实验结果适用于商用U型探头AN2-300。结果表明，基于偶应力理论的模型梁的刚度高于基于经典连续理论的模型。结果表明，表面弹性和表面残余应力都会影响固有频率。另外，可以观察到，压电场根据电压的符号同时显示出减小和增大的效果。为了验证所提出的U型AFM探针理论的准确性，将改进的耦合应力理论获得的分析结果与实验结果进行了比较，发现它们之间有很好的一致性。这项工作将增进对U形探针振动特性的理解，并且在纳米级结构的设计和应用中可能会引起特殊关注。