## 3 Influencing Factors and Interaction Identification of the Machined Surface Error
### Interaction of Influencing Factors of the Machined Surface Error
Using this as a research idea, the height of the slope of the analogous curve within a specified interval reflects the specific bending degree of the surface; using this as an analogy for the interaction-specific quantitative values, the impact indicator \(C\) can be derived by bringing Equations (19) and (20) into Equation (21); the solution process is as follows:  
\[C=\left|\frac{\text{tr}(\text{adj}H_{1})-\text{det}(\text{Hess}\,F)}{2|\
abla F |^{3}}\right|\cdot\Delta x \tag{21}\]  
\[H_{1}=\left( \begin{array}{cc}\text{Hess}F&\
abla F^{\text{T}}\\ \
abla F&0\end{array} \right) \tag{22}\]  
\[F(x,y,z,1)= \begin{pmatrix}x&y&z&1\end{pmatrix}\begin{pmatrix}a_{11}&a_{12}&a_{ 13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ 1\end{pmatrix}  \tag{23}\]  
\[\text{Hess}F= \begin{pmatrix}2a_{11}&2a_{12}&2a_{13}\\ 2a_{21}&2a_{22}&2a_{23}\\ 2a_{31}&2a_{32}&2a_{33}\end{pmatrix}  \tag{24}\]  
\[\left|\
abla F\right|^{3}=8\sqrt{\left(F_{1}{}^{2}+F_{2}{}^{2}+F_{3}{}^{2} \right)^{3}} \tag{25}\]  
where Hess\(F\) denotes the Hessian matrix with \(F(x,y,z)=0\)--that is, the Hessian matrix of Equations (19) and (20)--\(\
abla F\) denotes the gradient vector, \(\text{adj}H_{1}\) represents the accompanying matrix of matrix \(H_{1}\), tr() represents the trace of the matrix, and det(Hess\(F\)) represents the determinant of the matrix's solution. Equations (19) and (20) can be brought into Equation (21) and solved using MATLAB; taking the rotational speed of the milling cutter range 1290-2150 rpm as an example, by 100 equal points, the region with the highest curvature--i.e., the highest degree of bending--was selected as the characterization object, and the results are shown in Figure 10.  
Figure 9: Response surface of design variable interactions with positional error: (**a**) The feed rate per tooth and rotational speed of the milling cutter. (**b**) The rotational speed of the milling cutter and milling vibration level. (**c**) The feed rate per tooth and milling vibration level. The maximum index of interaction between the rotational speed of the milling cutter and the milling vibration on the surface positional error of milling machining is 0.36. The maximum index of interaction between the feed rate per tooth and milling vibration on the machined surface positional error is 0.16. The maximum index of the interaction between the rotational speed of the milling cutter and the feed rate per tooth on the machined surface positional error is 0.092. The reason for these results is that the more dramatic the effect of tool vibration on the peaks, valleys, and surface slope of the machined surface topography in the high-speed cutting process, the greater the effects on the formation of the machined surface which, in turn, affects the machined surface positional error, while for other influencing factors on the machined surface positional error the influence is small. Among the factors that influence the angular error distribution characteristics of the machined surface, the interactions between the rotational speed of the milling cutter and cutter tooth error and milling vibration, between the feed rate per tooth and cutter tooth error and milling vibration, and between milling vibration and cutter tooth error, have an obvious effects on the distribution characteristics of the error in the angle between the tangent plane and the reference planes; we constructed its response surface as shown in Figure 11.  
From Figure 11, it can be seen that the angular error distribution characteristics of the machined surface are jointly influenced by five types of interactions of the four design variables. The response surface bending is more evident in the fourth set of combinations of the feed rate per tooth and milling vibration interaction effects, showing that the interaction of the feed rate per tooth and milling vibration has an obvious effect on the correlation of angular error on the machined surface of the feature point.