## 3 Dynamic Responses for Ball End Milling
### Responses of System Using GPEm  
The dynamic cutting forces expressed in Equation (10) are the external forces of the modeled flexible rotor-bearing system for ball end milling. The GPEm will be applied to solve the vibration problem of the tool-tip for ball end milling combined with the  
Figure 3: Cutting forces, chip thickness, and area of cutting segment in ball end milling. rotor-bearing system. Applying the transformation relation in Equation (1), the tool-tip deflections \(x(t_{\text{j}})\) and \(y(t_{\text{j}})\) can be transformed into the generalized coordinates \(a_{\text{n}}(t)\) and \(b_{\text{m}}(t)\) shown below:  
\[ \begin{array}{l}x(t_{\text{j}})=u(\frac{t}{t},t)=\sum\limits_{\text{n}=1}^{N_ {\text{p}}}a_{\text{n}}(t)(\frac{t}{t})^{\text{n}-1}\\ y(t_{\text{j}})=v(\frac{t}{t},t)=\sum\limits_{\text{m}=1}^{N_{\text{p}}}b_{ \text{m}}(t)(\frac{t}{t})^{\text{m}-1}\end{array}  \tag{12}\]  
Substitute Equation (12) into Equation (10) to yield the force terms of ball end mill in generalized coordinates. \[ \begin{array}{l}\begin{array}{l}F_{\text{x}}\\ F_{\text{y}}\\ F_{\text{z}}\end{array}\end{array} = \begin{bmatrix}F_{\text{x1}}(\theta)&F_{ \text{x2}}(\theta)\\ F_{\text{y1}}(\theta)&F_{\text{y2}}(\theta)\\ F_{\text{z1}}(\theta)&F_{\text{z2}}(\theta)\end{bmatrix} \left\{ \begin{array} []{l}\sum\limits_{\text{n}=1}^{N_{\text{p}}}a_{\text{n}}(t)\\ \sum\limits_{\text{m}=1}^{N_{\text{p}}}b_{\text{m}}(t)\end{array} \right\}+ \begin{array}{l}F_{\text{x3}}(\theta)\\ F_{\text{y3}}(\theta)\\ F_{\text{z3}}(\theta)\end{array} \right\} \tag{13}\]  
Applying the principle of virtual work to the generalized coordinate system and neglecting the \(z\)-direction force \(F_{\text{z}}\), the external forces \(F_{\text{x}}\) and \(F_{\text{y}}\) in Equation (13) can be rewritten as the right side of Equation (2) with generalized coordinates. Therefore, Equation (2) can be modified into the following equation for the whole tool spindle system. \[\begin{array}{l}\left[ \begin{array}{cc}M&0\\ 0&M\end{array} \right]\left\{ \begin{array}{l}\overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}+\Omega\!\left[ \begin{array}{ cc}0&G\\ -G&0\end{array} \right]\left\{ \begin{array}{l}\overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}+\Omega\!\left[ \begin{array}{ cc}0&G\\ -G&0\end{array} \right]\left\{ \begin{array}{l}\overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}+\left[ \begin{array}{cc}C_{ \text{xx}}&C_{\text{xy}}\\ C_{\text{yx}}&C_{\text{yy}}\end{array} \right]\left\{ \begin{array}{l} \overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}+\left[ \begin{array}{cc}K_{ \text{s}}+K_{\text{xx}}&K_{\text{xy}}\\ K_{\text{yx}}&K_{\text{s}}+K_{\text{yy}}\end{array} \right]\left\{ \begin{array}{l} \overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}\\ =\left[ \begin{array}{cc}F_{\text{x1}}(\theta)\;\Phi&F_{\text{x2}}(\theta)\; \Phi\\ F_{\text{y1}}(\theta)\;\Phi&F_{\text{y2}}(\theta)\;\Phi\end{array} \right] \left\{ \begin{array}{l}\overset{\rightharpoonup}{a}\\ \overset{\rightharpoonup}{b}\end{array} \right\}+\left[ \begin{array}{cc}F_{ \text{x3}}(\theta)\overset{\rightharpoonup}{d}\\ F_{\text{y3}}(\theta)\overset{\rightharpoonup}{d}\end{array} \right\}\\ \text{where }\Phi=\left[ \begin{array}{cccc}1^{0}&1^{1}&.&1^{N_{\text{p}}-2}&1^{N_{ \text{p}}-1}\\ 1^{1}&1^{2}&.&1^{N_{\text{p}}-1}&1^{N_{\text{p}}}\\ \overset{\rightharpoonup}{1^{N_{\text{p}}-2}}&1^{N_{\text{p}}-1}&.&1^{2N_{ \text{p}}-3}&1^{2N_{\text{p}}-2}\end{array} \right]\text{and}\overset{ \rightharpoonup}{d}=\left\{ \begin{array}{ll}1^{0}&1^{1}&\ldots&1^{N_{\text{p}}-2} &1^{N_{\text{p}}-1}\end{array} \right\}^{T}\). It should be noted that the first part of the milling force, on the right side of Equation (14), can be combined with the stiffness matrix to change the system parameters of the rotor-bearing system.