# Enhancing Chatter Stability for Milling Thin-Walled Blades by Designing Non-Uniform Allowance
## 2 Description of the Methodology
### First-Order Sensitivity Calculation for Thin-Walled Blades
If the structural parameter \(p_{m}\) is a derivative of Equation (19), then one can have  
\[\frac{\partial\lambda_{r}}{\partial p_{m}}A\overline{\psi}_{r}+\lambda_{r} \frac{\partial A}{\partial p_{m}}\overline{\psi}_{r}+\lambda_{r}A\frac{ \partial\overline{\psi}_{r}}{\partial p_{m}}+\frac{\partial B}{\partial p_{m}} \overline{\psi}_{r}+B\frac{\partial\overline{\psi}_{r}}{\partial p_{m}}=0 \tag{22}\]  
Multiplication \(\overline{\psi}_{r}^{T}\) of the above equation by A on the left gives  
\[\frac{\partial\lambda_{r}}{\partial p_{m}}\overline{\psi}_{r}^{T}A\overline{ \psi}_{r}+\lambda_{r}\overline{\psi}_{r}^{T}\frac{\partial A}{\partial p_{m}} \overline{\psi}_{r}+\overline{\psi}_{r}^{T}\frac{\partial B}{\partial p_{m}} \overline{\psi}_{r}+\left(\lambda_{r}\overline{\psi}_{r}^{T}A+\overline{\psi }_{r}^{T}B\right)\frac{\partial\overline{\psi}_{r}}{\partial p_{m}}=0 \tag{23}\]  
It can be obtained from Equations (17) and (21). \[\frac{\partial\lambda_{r}}{\partial p_{m}}=-\lambda_{r}\overline{\psi}_{r}^{ T}\frac{\partial A}{\partial p_{m}}\overline{\psi}_{r}-\overline{\psi}_{r}^{T} \frac{\partial B}{\partial p_{m}}\overline{\psi}_{r} \tag{24}\]  
The above equation can be further extended as follows:  
\[\frac{\partial\lambda_{r}}{\partial p_{m}}=-(\lambda_{r}^{2}\psi_{r}^{T}\frac{ \partial M}{\partial p_{m}}\psi_{r}+\lambda_{r}\psi_{r}^{T}\frac{\partial C}{ \partial p_{m}}\psi_{r}+\psi_{r}^{T}\frac{\partial K}{\partial p_{m}}\psi_{r}) \tag{25}\]  
where \(\psi_{r}\) is the upper half vector of \(\overline{\psi}_{r}\) and \(M\), \(C\) and \(K\) the mass, damping and stiffness matrices of order \(N\times N\). This gives an expression for the first order eigenvalue sensitivity. (1) Sensitivity of eigenvalues to \(m_{ij}\)  
\[\frac{\partial\lambda_{r}}{\partial m_{ij}}=-\lambda_{r}^{2}\psi_{r}^{T}\frac {\partial M}{\partial m_{ij}}\psi_{r} \tag{26}\]  
Since \(M\) is a symmetric matrix, it can be deduced as follows:  
\[\frac{\partial\lambda_{r}}{\partial m_{ij}}=\left\{ \begin{matrix}-2\lambda_{r }^{2}\varphi_{ir}\varphi_{ir},(i\
eq j)\\ -\lambda_{r}^{2}\varphi_{ir}^{2},(i=j)\end{matrix} \right. \tag{27}\]  
(2) Sensitivity of eigenvalues to \(k_{ij}\)  
\[\frac{\partial\lambda_{r}}{\partial k_{ij}}=-\psi_{r}^{T}\frac{\partial K}{ \partial k_{ij}}\psi_{r} \tag{28}\]  
It can be further deduced as follows:  
\[\frac{\partial\lambda_{r}}{\partial k_{ij}}=\left\{ \begin{matrix}-2\varphi_{ir }\varphi_{ir},(i\
eq j)\\ -\varphi_{ir}^{2},(i=j)\end{matrix} \right. \tag{29}\]  
Upon obtaining the sensitivity of the process system eigenvalues with respect to stiffness, the sensitivity at the blade's central axis position serves as a critical reference for further analysis. By adjusting the thickness of the grid cells on either side in a rational manner, the sensitivity of the blade's grid cells can be effectively homogenized, ensuring that the process of milling the material results in consistent stiffness removal trends. The adjustment amount \(\Delta_{i}\) of the thickness of the mesh nodes indirectly reflects the geometric distribution of the spatial nodes of the non-uniform allowances at each cross-section position.