# Enhancing Chatter Stability for Milling Thin-Walled Blades by Designing Non-Uniform Allowance
## 2 Description of the Methodology
### Design of Thin-Walled Blade Allowance Distribution Based on First-Order Sensitivity  
Suppose eigenvalues \(\lambda\) and eigenvectors \(\psi_{r}\) are multivariate functions of structural parameters \(m_{ij}\), \(k_{ij}\), and \(c_{ij}\). \[(\lambda_{r},\psi_{r})=f(m_{ij},k_{ij},c_{ij}) \tag{30}\]where \(m_{ij}\), \(k_{ij}\) and \(c_{ij}\) are the \(i\)-th row and \(j\)-th column elements of the mass matrix \(M\), stiffness matrix \(K\) and damping matrix \(C\) equations, respectively. The above equation can be expanded into a Taylor series. \[ \begin{array}{l}\lambda_{r}=\overline{\lambda}_{r}+\Delta\lambda_{r}= \overline{\lambda}_{r}+\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}(\frac{ \partial\lambda_{r}}{\partial m_{ij}})\Delta m_{ij}+\sum\limits_{i-1}^{N}\sum \limits_{j=1}^{N}(\frac{\partial\lambda_{r}}{\partial k_{ij}})\Delta k_{ij} \sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial\lambda_{r}}{ \partial c_{ij}})\Delta c_{ij}\\ +\frac{1}{2}\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial^{2} \lambda_{r}}{\partial m^{2}_{ij}})\Delta m^{2}_{ij}+\frac{1}{2}\sum\limits_{i- 1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial^{2}\lambda_{r}}{\partial k^{2}_{ij }})\Delta k^{2}_{ij}+\frac{1}{2}\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}( \frac{\partial^{2}\lambda_{r}}{\partial c^{2}_{ij}})\Delta c^{2}_{ij}\\ \frac{1}{2}\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}\sum\limits_{p=1}^{N} (\frac{\partial^{2}\lambda_{r}}{\partial m_{ij}k_{ij}})\Delta m_{ij}\Delta k_{ pq}+\frac{1}{2}\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}\sum\limits_{p=1}^{N} (\frac{\partial^{2}\lambda_{r}}{\partial m_{ij}c_{ij}})\Delta m_{ij}\Delta c_{ pq}\frac{1}{2}\sum\limits_{i-1}^{N}\sum\limits_{j=1}^{N}\sum\limits_{p=1}^{N} (\frac{\partial^{2}\lambda_{r}}{\partial k_{ij}c_{ij}})\Delta k_{ij}\Delta c_ {pq}\end{array}  \tag{31}\]  
In the actual calculation, due to the finishing process of thin-walled parts, the correction amount of structural parameters is very small. Therefore, the second-order correction term can be ignored, and the eigenvalue correction amount is as follows:  
\[\Delta\lambda_{r}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial \lambda_{r}}{\partial m_{ij}})\Delta m_{ij}+\sum\limits_{i=1}^{N}\sum\limits_{ j=1}^{N}(\frac{\partial\lambda_{r}}{\partial k_{ij}})\Delta k_{ij}+\sum \limits_{i=1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial\lambda_{r}}{\partial c_{ ij}})\Delta c_{ij} \tag{32}\]  
The derivatives in the equation are calculated for the initial eigenvalue \(\overline{\lambda_{r}}\) and eigenvector \(\overline{\psi_{r}}\). Via the above analysis, the sensitivity can be obtained given the condition of neglected damping. The correction \(\Delta m_{ij}\) can be determined by evaluating the change in \(\Delta\lambda_{r}\) and \(\Delta\psi_{r}\), and the change in \(\Delta k_{ij}\) results from each individual mass change. \[\Delta m_{ij}=\frac{1}{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}(\frac{ \partial\lambda_{r}}{\partial m_{ij}})}\left(\Delta\lambda_{r}-\sum\limits_{i= 1}^{N}\sum\limits_{j=1}^{N}(\frac{\partial\lambda_{r}}{\partial k_{ij}})\Delta k _{ij}\right) \tag{33}\]  
Once the change in the grid cell mass is determined, the corresponding change \(\Delta h_{ij}\) in height along the \(Z\)-axis can be ascertained using the geometric relationship as hexahedral grid cells are utilized. \[\Delta h_{ij}=h_{ij}-\frac{\Delta m_{ij}}{\rho\cdot S} \tag{34}\]  
The calculation using Matlab-based Equation (34) allows for obtaining the height change amount of all grid cells at the same level of stiffness. To achieve smooth non-uniform bias surfaces, it is possible to consider interpolation between grid nodes, as in Equation (35) to obtain the final cross-section bias curve. \[S_{k}(x)=y_{k}\alpha_{0}^{(k)}(x)+y_{k+1}\alpha_{1}^{(k)}(x)+m_{k}\beta_{0}^{( k)}(x)+m_{k+1}\beta_{1}^{(k)}(x) \tag{35}\]  
Through analogy, one can determine the thickness of a unit node within a cross-section of a thin-walled blade.