## 3 Modeling
### Model for Determining Cutting Depth Based on Structural Integrity
=\frac{d_{m}}{2}F_{c}\hat{i}+\frac{d_{m}}{2}F_ {a}\hat{k} \tag{24}\]  
where \(M_{res}\) takes the form given by Equation (29):  
\[M_{res}=\sqrt{\left(d_{m}F_{c}\right)^{2}+\left(l_{T}F_{p}\right)^{2}+\left(d_{m} F_{a}-l_{T}F_{p}\right)^{2}} \tag{29}\]  
Substituting Equation (1) into the magnitude of the resulting force \(F_{res}\) gives the expression given by Equation (30):  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]  
Simplifying Equation (30) gives the form given by Equation (31):  
\[F_{res}=\frac{\pi D_{c}a_{c}k_{s}F_{v}}{V_{c}}\sqrt{\left(\frac{1}{5000}\right) ^{2}+\left(\frac{1}{3000}\right)^{2}+\left(\frac{1}{1000}\right)^{2}} \tag{31}\]  
The reduced form of the magnitude of the resultant force takes the form of Equation (32):  
\[F_{res}\approx\frac{\pi D_{c}a_{c}k_{s}F_{v}}{V_{c}}K_{F} \tag{32}\]  
For the constant \(K_{F}\) defined by Equation (33):  
\[K_{F}\approx 0.001073 \tag{33}\]  
The magnitude of the bending moment \(M_{res}\), expressed in terms of its force components given by Equation (1), has the form given by Equation (34):  
\[M_{res}=\frac{\pi D_{c}a_{c}k_{s}F_{v}}{V_{c}}\sqrt{\left(\frac{d_{m}}{1000} \right)^{2}+\left(\frac{l_{T}}{3000}\right)^{2}+\left(\frac{d_{m}}{5000}-\frac {l_{T}}{3000}\right)^{2}} \tag{34}\]  
In a simplified way, we have the expression for the bending moment \(M_{res}\) given by Equation (35):  
\[M_{res}=\frac{\pi D_{c}a_{c}k_{s}F_{v}}{V_{c}}K_{M} \tag{35}\]  
For the term \(K_{M}\) defined by Equation (36):  
\[K_{M}=\sqrt{\left(\frac{d_{m}}{1000}\right)^{2}+\left(\frac{l_{T}}{3000} \right)^{2}+\left(\frac{d_{m}}{5000}-\frac{l_{T}}{3000}\right)^{2}} \tag{36}\]  
Substituting Equations (32) and (35) into the average stress given in Equation (28) gives Equation (37):  
\[\sigma_{prom}=\frac{4F_{res}}{\pi\left(d_{m}+2a_{c}\right)^{2}}+\frac{32M_{res }\left(d_{m}+2a_{c}\right)}{\pi\left(d_{m}+2a_{c}\right)^{4}} \tag{37}\]  
Simplifying, we have the expression given in Equation (38):  
\[\sigma_{prom}=\left[\frac{D_{c}a_{c}k_{s}F_{v}}{V_{c}\left(d_{m}+2a_{c}\right) ^{2}}\right]\left[4K_{F}+\frac{32K_{M}}{d_{m}+2a_{c}}\right] \tag{38}\]  
The stress manifested in the component to be machined will depend on the parameters: \(d_{m}\), \(D_{c}\), \(R_{c}\); arrangements defined by the design of the machining process, which leads to geometric inequality in the axial section, such that the stress concentrator described above appears; therefore, the maximum stress, applied in the stress concentrator and of interest to maintain the structural integrity, is defined by Equation (39) as:  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]0  
Substituting Equation (38) in (39) gives the form given by Equation (40):  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]1  
The structural integrity of the piece to be machined will be maintained as long as the maximum stresses (\(\sigma_{max}\)) present in the piece, due to the machining process, are less than the elastic resistance module (\(S_{y}\)), and related, in a simple way, by a safety indicator (\(N\geq 1\)) in the way shown in Equation (41):  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]2  
The critical value of elastic failure will occur when \(N=1\), such that the structural integrity expression is given by Equation (42):  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]3  
Substituting Equation (40) into Equation (42) gives the structure given by Equation (43):  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]4  
Thus, for there to be no failure in the material due to the machining process, the inequality shown in Equation (44) must be met:  
\[F_{res}=\sqrt{\left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{5000V_{c}}\right)^{2}+ \left(\frac{\pi D_{c}a_{c}k_{s}F_{v}}{3000V_{c}}\right)^{2}+\left(\frac{\pi D_{ c}a_{c}k_{s}F_{v}}{1000V_{c}}\right)^{2}} \tag{30}\]5  
Which will be the form finally adopted by the structural restriction for the piece to be machined, depending on the machining parameters and the cutting depth of the working tool.