## 3 Experimental Setup
### Principle of Ultrasonic Vibration Extrusion  
The operating frequency is determined by the ultrasonic vibration device; it is the series resonant frequency or resonant frequency of the ultrasonic vibration device. The series resonant frequency and resonant frequency are the inherent properties of the ultrasonic vibration device, which can be detected by the ultrasonic power supply and displayed on the ultrasonic controller. The amplitude can reach 5 \(\upmu\)m at 100% power according to the product manual, and 100% power was used in the experiment.  
Ultrasonic vibration extrusion superimposes the longitudinal vibration at the ultrasonic frequency on the traditional extrusion, and the vibration magnitude is generally on  
Figure 5: Experimental setup. (**a**) Numerical control machine; (**b**) workpiece and extrusion tool; (**c**) Control and support devices; (**d**) execution device.  
the micrometer scale [27]. Assuming that the section which contains the current contact point of the tool and workpiece is perpendicular to the workpiece axis, the intersection of the axis and the section is the coordinate origin \(O\), and the workpiece axis is the \(X\) axis. The positive \(Y\) direction is the feed direction of the tool, and the section is the \(YZ\) plane. The schematic diagram of the motion trajectory of the tooltip relative to the workpiece is shown in Figure 6.  
Assuming that the ultrasonic vibration is sinusoidal and the workpiece is stationary, the tool moves along the workpiece surface in the form of spiral feed, accompanied by the sinusoidal ultrasonic vibration with the direction of motion pointing to the axis of the workpiece (i.e., \(x\) axis in Figure 6). In this case, the motion equation of the tip of the tool is given by [28]  
\[\left\{\begin{matrix}x=vr=vnt\\ y=\left[r+A_{ut}\sin(2\pi ft)\right]\cos(2\pi nt)\\ z=\left[r+A_{ut}\sin(2\pi ft)\right]\sin(2\pi nt)\end{matrix}\right. \tag{4}\]  
where \(v\), \(n\), \(r\), \(A_{ut}\), \(f\) and \(t\) represent the feed, rotation speed, workpiece radius, ultrasonic amplitude, frequency, and time, respectively.  
The operating frequency 41.243 kHz and ultrasonic amplitude 5 \(\upmu\)m correspond to f and \(A_{ut}\) in Formula (4). A larger operating frequency will cause the tool head to produce more hammering times, and a larger ultrasonic vibration amplitude will increase the hammering depth of the tool head, making it easier to produce plastic deformation on the machined surface.  
With ultrasonic vibration of the tool, the high instantaneous strain rate and the high-frequency cyclic load of dynamic ultrasonic loading cause stress waves in the material. Take one period of the ultrasonic vibration loading cycles for analysis. When the tool vibrates forward, the material is in the loading stage, and the dislocations in the deformation region slide, multiply and plug up rapidly, leading the plastic deformation stress increases rapidly. At this time, the compressive stress wave is transmitted along the axial direction; when the punch vibrates backwards, the material is unloading, and the stress wave is emitted at the bottom of the material, transformed into a tensile stress wave, and then transmitted in the reverse direction. Thus, the transmission of the stress wave significantly changes the stress state in the deformation region, causing the internal structure of the material to slide in the reverse direction, and the dislocation also diffuses in multiple directions, significantly reducing the dislocation accumulation. At the same time, the dislocation moving in the opposite direction is easier to meet and annihilate, which can not only improve the plastic deformation limit of the material but also significantly reduce the deformation stress. After 41,243 loading cycles (i.e., the ultrasonic vibration frequency) in 1 s, the material undergoes significant plastic deformation under high-frequency dynamic loading and produces significant strengthening on the surface.  
Figure 6: Motion trajectory of the tooltip relative to the workpiece.