## 3 Theoretical Analysis of Ultrasonic Trepenning Quality
### Dimensional Error in Trepenning  
#### 3.1.1 Theoretical Analysis of Dimensional Error in Trepenning  
In the actual trepenning processing, compared with the design vertical surface, the actual vertical surface between the two adjacent trepenning positions has a regular \"tower tip\" shape remaining unremoved part. The trepenning processing of a straight and curved vertical surface is shown in Figure 10, the shaded area is the remaining unremoved part in the trepenning processing. The area of the remaining unremoved part affects the dimensional accuracy of the honeycomb core component. If the area of the remaining unremoved part is too large, the vertical surface will deviate from the design size. If it exceeds the allowable range, the dimension of the vertical surface will be out of tolerance. In order to ensure the trepenning quality of the vertical surface, the size of the remaining unremoved part in trepenning should be limited to a reasonable range.  
The area of the residual unremoved part is the most direct way to characterize its size, but it cannot intuitively reflect the influence of the size of the unremoved part on the actual dimensional error. In the calculation of the dimensional error of the vertical surface, the \"peak\" of the residual unremoved part has the greatest influence on the actual dimensional error. The height of the \"peak\" position of the residual unremoved part along the direction perpendicular to the vertical surface directly affects the actual dimensional error. Therefore, it is selected as the evaluation parameter of the trepenning dimensional error, referred to as the residual height \(h\).  
Figure 9: Schematic of trepenning quality.  
The residual height of the arc vertical surface in trepenning is demonstrated in Figure 10a,b, respectively. The radius of the arc vertical surface to be machined is \(R\). The trepenning tool performs the cutting along the circular arc trepenning path, which is concentric with the designed arc vertical surface. The step length between two adjacent trepenning positions is the rotation angle \(\theta\) between two adjacent trepenning positions passing through the circle center \(O\). The residual height \(h_{I}\) of the internal arc vertical surface in trepenning and the residual height \(h_{E}\) of the external arc vertical surface in trepenning can be expressed as follows:  
\[h_{I}=R-(R-r)\cos\frac{\theta_{I}}{2}-\sqrt{r^{2}-(R-r)^{2}\sin^{2}\frac{ \theta_{I}}{2}} \tag{5}\]  
\[h_{E}=(R+r)\cos\frac{\theta_{E}}{2}-R-\sqrt{r^{2}-(R+r)^{2}\sin^{2}\frac{ \theta_{E}}{2}} \tag{6}\]  
The residual height of the straight vertical surface in trepenning is demonstrated in Figure 10c. The trepenning tool performs the cutting along the trepenning path parallel to the straight vertical surface. The distance between the two adjacent trepenning positions is the step length \(Ls\). The residual height \(h_{S}\) of the straight vertical surface in trepenning can be expressed as  
\[h_{S}=r-\sqrt{r^{2}-\frac{L_{S}{}^{2}}{4}} \tag{7}\]  
Figure 10: Dimensional error in trepenning: (**a**) internal arc vertical surface; (**b**) external arc vertical surface; (**c**) straight vertical surface.