## 3 Results and Discussions
### Analysis of Force Change and Amplitude Effect during Reaming
Compared with CR, the total reaming time is the same, but the proportion of the \"empty cutting\" process in ultrasonic vibration reaming reaches about 50%, and the average reaming thrust force of B-C and D-E stages is small. The thrust force of UVR is 40% smaller than CR in A-F stages. The average reaming axial forces--24 N of D-E stages in the stabilization stage of CR and 12 N in the D-E stages of UVR--are lower than the former by 50%. This is mainly due to the fact that during UVR, the chips are all debris-like and the process of removing the outside of the hole along the chip guide groove is relatively smooth. In addition, in ultrasonic vibration reaming, the reamer has been in the state of \"empty cutting\" for a long time, while the cutting edge of the reamer has been in the state of cutting in CR. This is consistent with the trend of force variation in ultrasonic assisted drilling. However, due to the large cutting thickness in drilling, the extrusion of the cutting edge on the hole wall limits the exertion of ultrasonic effect. In ultrasonic reaming, the effect of ultrasound is obvious because of the small cutting thickness; thus, the effect of reducing force is more obvious. Taking the selectable points on the tool as the research object, the motion equation of ordinary hinge cutting is as follows:  
\[\left\{ \begin{array}{l}x=\frac{D}{2}\cos(2\pi nt/60)\\ y=\frac{D}{2}\sin(2\pi nt/60)\\ z=v_{a}t\end{array} \right. \tag{1}\]  
In Formula (1), \(v_{a}\) is the feed speed of the spindle. During ultrasonic reaming, ultrasonic vibration is applied axially and the equation of trajectory motion changes to  
\[\left\{ \begin{array}{l}x=\frac{D}{2}\cos(2\pi nt/60)\\ y=\frac{D}{2}\sin(2\pi nt/60)\\ z=v_{a}t+A\sin(2\pi ft)\end{array} \right. \tag{2}\]  
In Equation (2), \(A\) is the ultrasonic amplitude and \(f\) is the ultrasonic vibration frequency. Figure 7a shows the trajectories of the selectable points on the main cutting edge of the reamer of CR and Figure 7b shows that of UVR. Figure 7: Cutting trajectory of the selectable points on the cutting edge of the reamer: (**a**) CR; (**b**) UVR. In UVR for this test, the equation of motion tracking for the four-edged reamer blade on the hole wall during reaming is:  
\[\left\{ \begin{array}{l}Z_{\text{A}}(\theta)=\frac{f_{\text{z}}}{2\pi}\theta+A \sin(W_{\text{f}}\theta)\\ Z_{\text{B}}(\theta)=\frac{f_{\text{z}}}{2\pi}(\theta+\pi)+A\sin[(W_{\text{f}}( \theta+\frac{\pi}{2})]\\ Z_{\text{C}}(\theta)=\frac{f_{\text{z}}}{2\pi}(\theta+\pi)+A\sin[(W_{\text{f}}( \theta+\pi)]\\ Z_{\text{D}}(\theta)=\frac{f_{\text{z}}}{2\pi}(\theta+\pi)+A\sin[(W_{\text{f}}( \theta+\frac{3\pi}{2})]\\ W_{\text{f}}=\frac{60f_{\text{z}}}{n}\end{array} \right. \tag{3}\]  
In Formula (3), \(Z_{\text{A}}\), \(Z_{\text{B}}\), \(Z_{\text{C}}\), and \(Z_{\text{D}}\) are the displacements of each cutting edge in the feed direction with the angle of rotation of the reamer. The feed quantity per tooth of the tool is \(f_{\text{z}}\). \(W_{\text{f}}\) represents the number of vibrations per rotation of the reamer. It can be obtained from Formula (3). The axial thickness between the two cutting edges is expressed as follows:  
\[ \begin{array}{l}f_{1}=\frac{f}{4}+2A\sin(\frac{W_{\text{f}}}{2}\pi)\cos[W_{ \text{f}}(\theta+\frac{(2k-1)\pi}{4})]\\ (k=1,2,3,4)\end{array}  \tag{4}\]  
The phase difference between the two blades when vibration occurs is:  
\[\Phi=\frac{\pi}{2}W_{\text{f}} \tag{5}\]  
Minimum cutting thickness in the axial direction of both blades is:  
\[f_{\text{f}}=\frac{f}{4}-2A\left|\sin(\frac{W_{\text{f}}}{4}\pi)\right| \tag{6}\]  
It can be seen from Equation (6) that the cutting thickness is related to the amplitude and feed. The greater the amplitude \(A\), the more obvious the chip breaking effect is. Chips press the front face intermittently, reducing sliding friction. Therefore, the larger the amplitude, the smaller the thrust force.