# Investigation of the Machining Stability of a Milling Machine with Hybrid Guideway Systems
## 4 Results and Discussions
### Machining Stability  
To get insight into the machining performance of the milling machine at design stage, the machining stability was assessed based on frequency response functions of the tool tip. The machining stability of the milling machine was predicted using the analytical model developed in study [28]. In their approach, the time-varying force coefficient of the dynamic milling process model was approximated by Fourier-series components. The stability relationship between the chatter-free axial cutting depths (\(Z_{\text{min}}\)) and the spindle speed (\(n\)) in end-mill operation as follows. The speed-dependent transfer function \(H\)_(jw)_ representing the ratio of the Fourier transform of the displacement at the tool tip over the dynamic cutting force can be expressed as \(H\)_(jw) = R\({}_{e}\)(w) + jl\({}_{m}\)(w), where \(R_{e}\) and \(I_{m}\) are, respectively, the real and imaginary parts of the transfer function of the spindle tool tip. The limit cutting depth \(Z_{\text{min}}\) for stable machining at spindle speed \(n\) is defined as  
\[Z_{\text{min}}=\frac{-2\pi R_{e}}{NK_{t}}(1+\frac{I_{m}}{R_{e}}^{2} \tag{8}\]  
\[ \begin{split} n&=\frac{60\omega_{c}}{N(2k\pi+\phi) },\\ \phi&=\pi-2\text{tan}^{-1}(I_{m}/R_{e})\;\;k=lobes(0,1,2...)\end{split}  \tag{9}\]  
In the above equation, \(K_{t}\) is the cutting resistance coefficients in the tangential direction to the cutter, \(N\) is the number of cutter teeth, and \(k\) is the lobe number. The machining stability was calculated based on the vibration modes that cause the tool to deform greatly. Therefore, according to the tool end FRFs, the apparent peak vibrations at lower and high frequency were selected for prediction of the machining stability. The analysis of machining stability was used to evaluate how the linear guide preload will affect the dynamic behavior and machining. Considering that the yawing and twisting modes cause the tool to deform greatly and the two modes occur aligning in \(X\) and \(Y\) directions, therefore, the machining stability in \(X\) and \(Y\) directions are calculated, respectively. In this way, the  
Figure 8: Comparisons of the predicted frequency response functions at tool end of milling machine modelin \(X\) and \(Y\) directions. (**a**) Amplitude of the FRFs, (**b**) Real part of the FRFs. coupled effect of the dominant vibration modes was ignored when calculating the stability lobe curves. A two-tooth carbide cutter was employed to machine the stock material of Al7075 alloys and titanium alloys Ti-6 Al-4V alloys at high and low speed machining. The cutting resistance coefficients were calibrated as \(K_{t}\) = 796 N/mm\({}^{2}\) of for Al7075 alloys and \(K_{t}\) = 2000 N/mm\({}^{2}\) for Ti6 Al4V alloys [28; 29]. Figure 9 presents the stability lobes in the \(X\) and \(Y\) directions predicted at the vibration mode of machine tool structure, respectively. The limited depth is 4.2 mm in the \(X\)-direction and 10.1 mm in the \(Y\)-direction, respectively. Comparison of the results of stability analysis indicates that the machine tool has a different limited depth for stable machining in \(X\) and \(Y\) directions when the cutter is operated under lower speed. As found from the predicted tool end FRFs of milling machine, the results of stability analysis also indicate that the dynamic characteristic of the milling machine is greater in the \(Y\) direction than in the \(X\) direction, which further bring the higher limited depth in \(Y\) direction compared with the \(X\) direction. This result gives a suggestion for the improvement of the structure of the milling machine. On the other hand, the predicted stability lobes diagrams based of the vibration mode of spindle structure are illustrated in Figure 10. As can be seen, the limited depth for high speed stable machining in \(X\) direction is about 6.02 mm, almost equal to that in \(Y\) direction. This is an expected phenomenon since this stability is calculated based on the vibration mode associated with the bending motion of the spindle shaft.