## 4 Validation of Surface Roughness Model and Process Parameter Optimization
### Establishment of Surface Roughness Model Considering Milling Cutter Vibration
#### 4.1.1 Establishment of the Vibration Model of Milling Cutter  
In the modeling process of the vibration of the milling cutter, the effect of tool runout on the machining process is ignored. The milling cutter is assumed to be a uniform continuous system (uniform simply supported beam of equal cross-section), which means that the density, cross-sectional area, and stiffness of the milling cutter do not vary with position. During the milling process, the milling cutter is held by the knife clip at one end and is in contact with the workpiece at the other end. Therefore, the vibration model of the milling cutter can be simplified to a fixed-hinged beam system subjected to concentrated forces. The force model of the milling cutter is shown in Figure 22.  
Let the intrinsic frequency at a given boundary condition be \(w_{r}\) and the corresponding formation function be \(Y_{r}(x)\). For a homogeneous beam of equal cross-section, the intrinsic frequency \(w_{r}\) can be expressed as [37, 38]:  
\[w_{r}=\frac{{{\lambda_{r}}^{2}}}{L^{2}}\sqrt{\frac{EJ}{\rho A}} \tag{2}\]  
where \(E\) is the modulus of elasticity, \(J\) is the moment of inertia of the section, \(A\) is the cross-sectional area, and \(\rho\) is the mass per unit volume. \(L\) is the overhanging length of the milling cutter, and \(\lambda_{r}\) is the characteristic root of Equation (2). For the fixed-hinged beam system, the characteristic roots of Equation (2) can be expressed as [39]:  
\[\lambda_{r}=\left(r+\frac{1}{4}\right)\pi \tag{3}\]  
Figure 21: The microhardness distribution of the surface layer. where \(r=1,2,\ldots\), denotes the order. Based on the above assumptions about the vibration model, the formation function of the milling cutter can be expressed as:  
\[Y_{r}(x)=sh\beta_{r}x-\frac{sh\lambda_{r}}{\sin\lambda_{r}}\sin\beta_{r}x \tag{4}\]  
where \(sh\) denotes the hyperbolic function and \(\beta_{r}\) is a function related to the intrinsic frequency and material properties:  
\[{\beta_{r}}^{4}=\frac{{w_{r}}^{2}\rho A}{EJ} \tag{5}\]  
Since the stiffness of the milling cutter in the axial direction is much greater than that in the radial direction, the milling machining system can be assumed as a two-degree-of-freedom system (\(x\) and \(y\) directions). As the solution process of the vibration model in the \(x\) and \(y\) directions is the same, for the sake of simplicity, only the solution process in the \(x\)-direction is discussed. Assume that the force applied to the milling cutter in the \(x\)-direction is the concentrated force \(F_{x}(t)\), which can be expressed as:  
\[f(x,t)=F_{x}(t)\delta(x-L) \tag{6}\]  
where \(\delta(x-L)\) denote the \(\delta\) function. For a homogeneous beam of equal cross-section, the vibration differential equation can be expressed as [40]:  
\[\rho A\frac{\partial^{2}d(x,t)}{\partial t^{2}}+\frac{\partial^{2}}{\partial x ^{2}}\bigg{[}EJ\frac{\partial^{2}d(x,t)}{\partial x^{2}}\bigg{]}=f(x,t) \tag{7}\]  
Introducing the regular coordinates \(q_{r}(t)\), based on the formation superposition method, the solution \(d(x,t)\) for a given boundary condition can be expressed as [41]:  
\[d(x,t)=\sum_{r=1}^{\infty}Y_{r}(x)q_{r}(t) \tag{8}\]  
The vibration differential equation (Equation (7)) can be expressed as:  
\[\sum_{r=1}^{\infty}\rho AY_{r}(x)\tilde{q}_{r}(t)+\sum_{r=1}^{\infty}\frac{d^ {2}}{dx^{2}}\bigg{[}EJ\frac{d^{2}Y_{r}(x)}{dx^{2}}\bigg{]}q_{r}(t)=f(x,t) \tag{9}\]  
\[w_{r}=\frac{{{\lambda_{r}}^{2}}}{L^{2}}\sqrt{\frac{EJ}{\rho A}} \tag{2}\]0 Force model of the milling cutter. The boundary conditions of the fixed-hinged beam system can be expressed as [42]:  
\[w_{r}=\frac{{{\lambda_{r}}^{2}}}{L^{2}}\sqrt{\frac{EJ}{\rho A}} \tag{2}\]1  
\[w_{r}=\frac{{{\lambda_{r}}^{2}}}{L^{2}}\sqrt{\frac{EJ}{\rho A}} \tag{2}\]2  
Regularize both sides of Equation (9) while integrating over the entire interval \(0<x<L\).