# Enhancing Chatter Stability for Milling Thin-Walled Blades by Designing Non-Uniform Allowance
## 2 Description of the Methodology
### Non-Uniform Allowance Distribution of Titanium Alloy Thin-Walled Blade Finishing  
Under the condition of obtaining the sensitivity and setting the eigenvalue level constraints, such as taking the mean sensitivity of the blade as the level reference value, the amount of change in the thickness t of the grid cells can be calculated based on the non-uniform allowance design steps (3-5) in the finishing stage. The value of the thickness t is obtained by calculation and measurement as follows.  
\[t_{ij}=\begin{bmatrix}0.664489&1.598100&2.709500&3.371400&3.645500&3.561500&3. 125800&2.323200&1.112600\\ 0.528346&1.549600&2.802300&3.566800&3.822000&3.612200&3.019100&2.130200&0.998928\\ 0.612617&1.563900&2.767200&3.513000&3.738900&3.480900&2.846700&1.957200&0.907683 \\ 0.675002&1.577300&2.637900&3.297800&3.488400&3.231700&2.630300&1.796400&0.831464 \\ 0.705875&1.537800&2.44500&3.09100&3.168500&2.936400&2.397900&1.643100&0.761881 \\ 0.686540&1.460400&2.254700&2.742800&2.875200&2.660800&2.173500&1.490500&0.691834 \\ 0.647066&1.373300&2.082300&2.512400&2.618100&2.409500&1.957200&1.336800&0.6233 13\\ 0.598048&1.267600&1.888000&2.258000&2.339700&2.144200&1.737100&1.187200&0.562848 \\ 0.541569&1.146700&1.677800&1.988500&2.050700&1.876500&1.523000&1.047900&0.5 10375\\ 0.484355&1.023600&1.473000&1.732200&1.780800&1.630400&1.329200&0.924291 &0.465330\\ 0.436785&0.913686&1.295500&1.516000&1.556900&1.427500&1.169100&0.821592&0.42 7293\\ 0.393285&0.830623&1.163100&1.356400&1.393400&1.280200&1.052200&0.745560&0.397 133\\ 0.370252&0.784887&1.091300&1.267100&1.301900&1.198200&0.986841&0.701145&0.374 390\end{bmatrix}\]  
Next, the model simplification in Figure (a)a is carried out, and then the 2D cross-section shape meshing is shown in Figure (b)b. Here, with the standard grid cell (5 mm \(\times\) 5 mm), the calculation can be obtained due to the simplification and the shape of the blade body surface itself, the formation of a series of irregular grid cells, along the longitudinal direction of the blade body, the length of 5 mm, and the width of the following order, 3.4365 mm, 3.6309 mm, 3.7157 mm, 3.7582 mm, 3.8133 mm, 3.9092 mm, 4.0327 mm, 4.1640 mm, 4.3027 mm, 4.4499 mm, 4.6040 mm, 4.7625 mm, and 4.9228 mm. Finally, according to the mesh node thickness subdivided, the coordinates of the interpolation point of the intermediate part of each cross-section can be obtained to construct the cross-section curves, as shown in Figure (c)c.  
Finally, only the amount of height change of the grid cells at the three key sections is extracted, and first the cross-section curve of the blade performs the smooth processing is constructed as shown in Figure (a)a. Then, the surface reconstruction through the mesh surface is performed, the surface of the allowances distribution in the finishing stage is obtained, and the whole design process is as shown in Figure (b)b. The blade body is cut based on the actual blade. Lastly, the non-uniform allowance bias surface in the finishing stage can be further used as the driving surface for finishing programming to form a helical wrap-around tool position trajectory as shown in Figure (c)c.  
Figure 5: Mesh cell subdivision of the blade.  
Figure 6: Finish design with non-uniform allowances distribution.