# Enhancing Chatter Stability for Milling Thin-Walled Blades by Designing Non-Uniform Allowance
## 1 Introduction  
Blades, constituting approximately 30% of the manufacturing process of an aircraft engine, are an essential component with regard to its performance [1; 2; 3; 4; 5]. Consequently, CNC precision machining technology has become the primary method for manufacturing blades worldwide. However, during the milling process of a thin-walled blade, its overhang gradually increases as the wall thickness is reduced. The dynamic characteristics of the blade, with changes in cutting position and material removal, display a strong time-varying nature. This feature can easily lead to the chatter phenomenon during milling, resulting in a decline in machining surface quality and accuracy, which can significantly impact on the blade's performance [6; 7; 8; 9; 10]. Chattering during the milling process of thin-walled blades remains to be a significant technical challenge within the aero-engine manufacturing industry. In recent years, researchers from domestic and international institutions have undertaken comprehensive investigations into the milling vibration problem related to thin-walled blades. The focus of these studies has primarily revolved around analyzing the dynamics of the milling process system model and employing the Floquet theory to establish stability criteria. The thin-walled cutting modal parameter, which is susceptible to varying conditions over time, has been taken into account to predict process systemstability with ultimate depth of cut and speed. This has contributed to stability lobe formation. For instance, [PERSON] et al. combines parametric model downscaling of time-lagged systems with stability analysis to improve computational simulation efficiency, predicting the stability domain while considering the influence of material removal on processing system dynamics [11]. [PERSON] and other scholars developed an effective method for predicting stability lobes in multimodal milling systems across all main chirp modes in the time domain [12; 13; 14; 15]. Recently, some scholars employed random variables to set the structural parameters of the milling processing system and spindle speed while using chattering frequency as an intermediate variable. These measures help establish a reliability model for locating the probability of side milling stability. This model replaces the traditional stability leaflet prediction diagram with a more accurate method [16; 17; 18]. For instance, [PERSON] et al. presents a useful probabilistic forecast approach to estimate the dynamic stability during the milling process. This method takes into consideration the uncertain system parameters of the milling process [19]. In addition, [PERSON] employs neural networks to carry out an extensive analysis of the influence of random factors in milling processing. Subsequently, he proposes a probabilistic assessment technique to predict the reliability of regenerative chattering stability during milling [20]. [PERSON] et al. considers the influence of random variables on the stability of overall blisk side milling machining to accurately predict the value [21]. By applying the reliability analytic approach to the dynamic structural system, a chatter reliability model for the side milling machining system of an overall blisk is established. Those previous researchers have commonly started by developing a dynamic model of the process system, obtaining its structural parameters, and adjusting the process parameters to achieve cutting stability. The accuracy of extracting process parameters and the influence of external factors significantly affect the predictive capability of the system. In light of chatter active control, adding an active controller to the milling process system reduces the chatter problem of the system. For instance, [PERSON] et al. suggests a robust controller based on linear matrix inequalities to model the chatter active control system for milling, and the results evidences significant enhancement in the milling surface finish and the chatter-free boundary [22]. [PERSON] et al.