# Enhancing Chatter Stability for Milling Thin-Walled Blades by Designing Non-Uniform Allowance
## 2 Description of the Methodology  
### Analysis of the Effect of Allowances Distribution on Blade Dynamic Characteristics  
The blade's complex structure is designed according to its own dynamic characteristics. The leading and trailing allowances incorporate a rounded surface transition that is very thin while the blade body comprises transverse cross-sections that vary across its surface. Its body surface consists of a base surface and a tongue-and-groove root surface cropping. The axial stack of the blade accumulates with a certain degree of torsion. To effectively present the proposed optimal design method while ensuring its applicability and facilitating meshing and finite element analysis, a simplification technique has been employed. Through the utilization of hexahedral cells, the blade has been simplified with utmost precision, minimizing any distortion that may compromise its integrity, as illustrated in Figure 1.  
Based on the finite element method, the physical blade is simplified and discretized into a continuum structure as illustrated in Figure 1. The control equations for the resulting multi-degree-of-freedom dynamical system can thus be formulated as shown bellow. \[[M]\big{\{}\ddot{x}(t)\big{\}}+[C]\big{\{}\dot{x}(t)\big{\}}+[K]\big{\{}x(t) \big{\}}=[f(t)] \tag{1}\]  
where the matrices \([M]\), \([C]\), and \([K]\), respectively, denote the mesh mass, damping, and stiffness distributions of the discretized blade. The vector distributions \(\big{\{}\ddot{x}(t)\big{\}}\), \(\big{\{}\dot{x}(t)\big{\}}\), and \(\{x(t)\}\), respectively, denote the mesh node acceleration, velocity, and displacement. Proportional damping can be assumed in this case. \[[C]=\alpha[M]+\beta[K] \tag{2}\]  
where \(\alpha\) and \(\beta\) are constants of proportionality and Equation (1) is transformed modally. \[\{x(t)\}=\sum_{r=1}^{N}q_{r}\{\phi_{r}\} \tag{3}\]  
\[\{\phi_{r}\}^{T}[C]\{\phi_{r}\}=\alpha\{\phi_{s}\}^{T}\{M\}\{\phi_{r}\}+\beta \{\phi_{s}\}^{T}\{K\}\{\phi_{r}\}= \begin{cases}0&r\
eq s\\ \alpha m_{s}+\beta k_{s}=c_{s},r=s\end{cases}  \tag{4}\]  
Figure 1: Grid cell division. In Equation(4) \(\{\phi_{r}\}\) is the \(r\)th order mode shape determined by \([M]\{\tilde{x}(t)\}+[K]\{x(t)\}\) and \(q_{r}\) is the corresponding mode coordinate obtained by substituting Equation (3) into Equation (1). \[[M]\Bigg{(}\sum_{r=1}^{N}\tilde{q}_{r}\{\phi_{r}\}\Bigg{)}+[C]\Bigg{(}\sum_{r=1 }^{N}\dot{q}_{r}\{\phi_{r}\}\Bigg{)}+[K]\Bigg{(}\sum_{r=1}^{N}q_{r}\{\phi_{r} \}\Bigg{)}=\{f(t)\} \tag{5}\]  
The above equation is left multiplied by \(\{\phi_{r}\}^{T}\), according to the assumed damping. Then, the above equation can be expressed as follows:  
\[m_{s}\ddot{q}_{s}+\dot{c}_{s}q_{s}+k_{s}q_{s}=\{\phi_{s}\}^{T}\{f(t)\} \tag{6}\]  
If \(\{f(t)\}=\{F\}e^{i\omega t}\) is given, then by substituting \(q_{s}=Q_{s}e^{i\omega t}\) into Equation (6), one can obtain Equation (7) as follows:  
\[\Big{(}-\omega^{2}m_{s}+j\omega c_{s}+k_{s}\Big{)}Q_{s}e^{i\omega t}=\{\phi_{ s}\}^{T}\{F\}e^{j\omega t} \tag{7}\]  
From this one can derive Equation (8) as follows:  
\[Q_{s}=\frac{\{\phi_{s}\}\{F\}}{-\omega^{2}m_{s}+j\omega c_{s}+k_{s}}\propto y =\frac{f}{k} \tag{8}\]  
From the above equation, it is clear that \(Q_{s}\) corresponds to the displacement response resulting from a single-degree-of-freedom system, which includes masses, damping, and stiffnesses \(m_{s}\), \(c_{s}\) and \(k_{s}\), respectively, when under the action of the modal force \(P_{s}=\{\phi_{s}\}^{T}\{F\}\). Meanwhile, \(\big{(}-\omega^{2}m_{s}+j\omega c_{s}+k_{s}\big{)}\) represents the stiffness of the system and is referred to as the dynamic stiffness. \[\{x\}= \begin{cases}X_{1}\\ X_{2}\\ \vdots\\ X_{N}\end{cases}