## 4 Experimental Results and Discussion
### Analysis of the Surface Roughness and Hardness
If the rotational speed, the feed and static extrusion amount remain unchanged, to achieve the same roughness of single ultrasonic extrusion, more ultrasonic energy injection is required to cause the machined surface to produce more plastic deformation. At the same time, the surface hardness of the composite process is higher than the ultrasonic vibration extrusion process, as shown in Figure 8b, since both the low-temperature chromium penetration and the ultrasonic vibration extrusion significantly increase the surface hardness, and its surface hardness is increased by 2.55 times compared with the ultrasonic extrusion process. Comparing the effects of process parameters on surface roughness and hardness after the two processes, the results of the range analysis are shown in Tables 3 and 4. Table 4 shows the roughness of the composite process, and the descending order of the factor influence is ACB, which stands for spindle speed, static extrusion amount, and feed. A3B4C1 is the best solution for surface roughness. The descending sequence of factor influence for hardness is CBA, i.e., static extrusion quantity, feed, spindle speed, and its ideal solution is C4B4A2. The optimal solution considering both the surface roughness and the hardness is analyzed. For factor A, it is the main factor of roughness and the last factor of hardness; then, A3 is selected. Factor B is the second factor of both roughness and hardness, and B4 is the optimal factor for both the surface roughness and the hardness; then, B4 is selected. Factor C is the main factor of the hardness and the last factor of the roughness; then, C4 is selected. Thus, the optimal solution considering both roughness and hardness is A3B4C4. The visual analysis of the two processes is shown in Figures 9 and 10. The composite process can obtain higher hardness than the ultrasonic extrusion, and its surface roughness remains at a low level, although it is relatively large. To study comprehensively the influences of the machining parameters, i.e., spindle speed, feed and static extrusion on the machined surface roughness and hardness, the second-order response surface method is used to establish the relationship between surface properties and machining parameters [31]. The equation of this method is given by  
\[y=\beta_{0}+\sum_{i}^{n}\beta_{i}x_{i}+\sum_{i}^{n}\beta_{ii}x_{i}^{2}+\sum_{i}^ {n}\beta_{ij}x_{i}x_{j}+\varepsilon, \tag{5}\]  
where, \(\beta_{0}\) is a constant, \(n\) is the number of factors, \(\beta_{i}\) is the linear coefficient, \(x_{i}\) and \(x_{j}\) are processing parameters, \(\beta_{ii}\) and \(\beta_{ij}\) is the coefficient of quadratic term, \(\varepsilon\) is the error of experimental data. Figure 10: Visual analysis of the composite process. (**a**) Surface roughness. (**b**) Surface hardness. The model coefficients are determined by multivariate data analysis of the orthogonal test data in Table 2. The model of the ultrasonic extrusion and the composite process is given by  
\[\left\{ \begin{matrix}R_{u}=12n-26326f+146P+232812f^{2}-P^{2}-48 nf-378 fP-590\\ H_{u}=-564n+2178486f+7988P-3n^{2}-7015625f^{2}\\ +57P^{2}+12757 nf+26 nP-171926 fP+1907587\end{matrix} \right., \tag{6}\]  
\[\left\{ \begin{matrix}R_{cu}=-21n+88913f+330P+79687f^{2}-3P^{2}-234 nf-2662 fP-518 \\ H_{cu}=n-8942f+12P+44914f^{2}+18 nf-3 fP+512\end{matrix} \right., \tag{7}\]  
where subscripts \(u\) and \(cu\) represent the ultrasonic extrusion and the composite process, respectively. \(R\) and \(H\) represent the surface roughness and hardness, respectively. Through these mathematical models, the roughness and hardness under composite process and ultrasonic process can be well predicted and analyzed, the influence of spindle speed, feed and static extrusion amount on workpiece surface roughness and surface hardness can be analyzed, and the process parameters can also be optimized by using this model.