## 2 Instantaneous Bias of the Main Section of the Milling Cutter under Vibration
\right] \tag{1}\]  
\[Q_{3}=\left[ \begin{array}{cccc}\cos\theta_{s1}(t)&-\sin\theta_{s1}(t)&0&0\\ \sin\theta_{s1}(t)&\cos\theta_{s1}(t)&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right],Q_{4}=\left[ \begin{array}{cccc}\cos\theta_{s2}(t )&0&\sin\theta_{s2}(t)&0\\ 0&1&0&0\\ -\sin\theta_{s2}(t)&0&\cos\theta_{s2}(t)&0\\ 0&0&0&1\end{array} \right] \tag{2}\]  
\[M_{1}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&-\Delta z_{d}^{i}\\ 0&0&0&1\end{array} \right],M_{2}=\left[ \begin{array}{cccc}1&0&0&A_{x}(t)\\ 0&1&0&A_{y}(t)\\ 0&0&1&A_{z}(t)\\ 0&0&0&1\end{array} \right],M_{3}=\left[ \begin{array}{cccc}1&0&0&x^{\varphi_{ o}}(t)\\ 0&1&0&y^{\varphi_{o}}(t)\\ 0&0&1&z^{\omega_{o}}(t)\\ 0&0&0&1\end{array} \right] \tag{3}\]  
where \(Q_{1},Q_{2},Q_{3}\), and \(Q_{4}\) are rotation matrices and \(M_{1},M_{2}\), and \(M_{3}\) are translation matrices. \(\varphi_{d}(t)\) is the instantaneous angle between \(y_{d}\) axis and \(y_{s}\) axis in \(x_{o}s_{y}\) plane, which could be expressed as  
\[\varphi_{d}(t)=\varphi_{d}(0)+2\pi n\cdot t-\left[ \begin{array}{c}\varphi_ {d}(0)+2\pi n\cdot t\\ 2\pi\end{array} \right]\cdot 2\pi \tag{4}\]  
where \(\varphi_{d}(0)\) is the initial cutting time of the milling cutter, that is, the angle between \(y_{d}\) axis and \(y_{s}\) axis in \(x_{o}o_{y}s_{\rm{}}\) plane when \(t\) is \(0\). \(x^{o}(t),y^{o}(t),z^{o}(t)\) are the instantaneous position coordinate of the coordinate origin \(o_{\rm{o}}\) of the milling cutter cutting coordinate system without vibration in the workpiece coordinate system \(o\)-xyz, which could be solved as  
\[\left\{ \begin{array}{l}x^{o_{\rm{o}}}(t)=v_{f}\cdot t-x(0)\\ y^{o_{\rm{o}}}(t)=y_{L}-a_{e}+r_{\rm{max}}\\ z^{o_{\rm{o}}}(t)=z_{H}-a_{p}\end{array} \right. \tag{5}\]  
The instantaneous bias state of the main section of the milling cutter was characterized by the bias angle of the cutting coordinate system caused by milling vibration. The milling cutter trajectory \(o_{\rm{s}}(x,y,z)\) in cutting coordinate system under vibration could be written as  
\[o_{\rm{s}}(x,y,z)=\left[ \begin{array}{cccc}x&y&z&1\end{array} \right]^{T}=M_ {3}\cdot\left[ \begin{array}{cccc}A_{x}(t)&A_{y}(t)&A_{z}(t)&1\end{array} \right] ^{T} \tag{6}\]  
The speed \(v(t)\) of the milling cutter in the direction of the cutting vector along the \(o_{\rm{s}}(x,\)\(y,z)\) motion trajectory could be expressed as  
\[v(t)=\frac{do_{\rm{s}}(x,y,z)}{dt} \tag{7}\]  
The instantaneous angle between \(x_{\rm{s}}\) axis and \(x_{\rm{o}}\) axis \(\theta_{\rm{s}}(t)\) could be derived as  
\[\theta_{\rm{s}}(t)=\arccos\left(\frac{\overrightarrow{x}_{s}\cdot\overrightarrow {x}_{o}}{\left|\overrightarrow{x}_{s}\right|\cdot\overrightarrow{x}_{o}\right| }\right)=\arccos\frac{v_{x}(t)}{\sqrt{v_{x}^{2}(t)+v_{y}^{2}(t)+v_{z}^{2}(t)}} \tag{8}\]  
where \(v_{x}(t),v_{y}(t),\) and \(v_{z}(t)\) are the components of \(v(t)\) along the \(x\) axis, \(y\) axis, and \(z\) axis, respectively. The angle between the projection of \(x_{\rm s}\) axis on \(x_{\rm o}o_{\rm o}y_{\rm o}\) plane and \(x_{\rm o}o_{\rm o}z_{\rm o}\) plane and \(x_{\rm o}\) axis could be calculated as  
\[\theta_{\rm s1}(t)=\arcsin\frac{v_{y}(t)}{\sqrt{v_{x}^{2}(t)+v_{y}^{2}(t)}},\ \theta_{ \rm s2}(t)=\arcsin\frac{v_{z}(t)}{\sqrt{v_{x}^{2}(t)+v_{z}^{2}(t)}} \tag{9}\]  
The bias angle of main section of the milling cutter \(\theta(t)\) could be written as  
\[\theta(t)=\arctan\frac{v_{z}(t)}{\sqrt{v_{x}^{2}(t)+v_{y}^{2}(t)}} \tag{10}\]  
In order to investigate the effect of milling vibration on cutting posture of the milling cutter, the bias angle and the trajectory of milling cutter with/without vibration were compared by using Equations (1)-(10), as shown in Figure 2.  
Figure 2: Trajectory and the bias angle of the milling cutter, (**a**) trajectory of the milling cutter, (**b**) bias angle of the milling cutter.