However, it also suffers from many problems such as uneven machined surface, large cutting force and easy clogging of the tool with chips. Therefore, trepanning is also difficult to meet the high-quality machining requirements of the vertical surface in honeycomb core components.  
In recent years, ultrasonic cutting technology has attracted a lot of attention for its advantages of good cutting quality and high cutting efficiency. Compared with traditional machining methods, ultrasonic vibration significantly reduces the cutting force [11; 12], and the machined surface is basically free of machining defects [12; 13], such as burr and tear. The current studies on the ultrasonic cutting of Nomex honeycomb core mainly focus on cutting force modeling, cutting quality, finite element simulation and new processing methods. [PERSON] et al. [14] established the dynamic force model for ultrasonic-assisted cutting with straight blade and analyzed the reasons for the reduction in cutting force in ultrasonic-assisted cutting. [PERSON] et al. [15] established a finite element simulation model of the ultrasonic vibration-assisted cutting force of the disc-cutter on the honeycomb core material. [PERSON] et al. [16] summarized the characteristics of the surface microscopic of Nomex honeycomb by the location of the feature and analyzed the influence of feed rate and ultrasonic vibration amplitude to the damage form. In addition, [PERSON] et al. [17] developed a new ultrasonic longitudinal-torsional vibration-assisted cutting method, in which the machined surface with better quality can be obtained when compared to ultrasonic longitudinal vibration-assisted cutting. Although studies on the ultrasonic cutting technology of Nomex honeycomb core have been performed in various ways, there are only two kinds of special tools for ultrasonic cutting, include straight blade and disc-cutter [2]. However, due to the structural limitations of the straight blade, the arc vertical surface is difficult to achieve with ultrasonic machining. Moreover, due to the length limitations of the straight blade, the deeper straight vertical surface is difficult to process at one time.  
In addition, due to the special structure and mechanical properties of the honeycomb core, the studies on its machining quality mainly focus on burrs and tear [2; 12], which are significantly different from the metal material [18; 19]. Moreover, the selection of the processing track is of great significance to the final machining precision and quality [20; 21], but there is no relevant research on the trepanning trajectory at present.  
In summary, the existing processing methods do not meet the high-quality machining requirements of the vertical surface. To improve the machining quality, a novel ultrasonic trepanning method for the vertical surface is proposed, and a special ultrasonic trepanning tool was developed for the first time. Moreover, in order to further clarify the key factors affecting the trepanning quality, the theoretical model of the trepanning dimensional error was established and verified, and the quantitative analysis of the trepanning incision quality was carried out innovatively.  
Figure 1: Typical vertical surface: (**a**) internal arc vertical surface; (**b**) external arc vertical surface; and (**c**) straight vertical surface.