## 1 Introduction
[30] while considering the dynamic characteristics of the machine tool and the milling force constraint. [PERSON] et al. [31] optimized the multi-axis machining strategy on the basis of the thought of variable depth-of-cut machining for thin-walled workpieces. [PERSON] et al. [32] obtained the effect of material removal on the dynamic characteristics of thin-walled parts in the milling process by the hammering method and established the stability limit diagram. [PERSON] et al. [33] proposed a structural dynamic modification method with equal mass to predict the dynamic stability of the thin-walled workpiece milling process. Based on the Ad-ams-Bashforth scheme, a numerical difference method is proposed by [PERSON] et al. [34] with considering material removal and position dependence for thin-walled workpiece stability prediction. All the above studies are based on the equal thickness thin-walled workpiece, in which material removal is conducted on an equal-mass basis. Therefore, as an extension, [PERSON] et al. [35] studied the dynamic response of a thin-walled component with variable thickness in the milling process. Nonetheless, the existing research mostly focuses on the three-axis side milling of thin-walled parts; the tool structure is simple, the workpiece is relatively regular, most of its parts are cuboid thin-walled parts, few of which involve the multi-axis milling process of thin-walled parts with a complex curved surface and variable thickness. [PERSON] et al. [36] developed a cyclic symmetry analysis method to solve the dynamic characteristic issues of the thin-walled integral impeller. In the study, [PERSON] assumed that the dynamic characteristic of the integral impeller was approximately equal to a single blade. [PERSON] et al. [37] obtained the dynamic characteristics of a jet-engine blade by the modal experiment while using a time-domain method to evaluate a chatter-free milling mode. [PERSON] et al. [38] presented a novel method of reduced-order workpiece dynamic parameters updated model for the thin-walled parts in the frequency domain and time domain, based on the perturbation and reduced-order substructuring methods. The dynamics of the workpiece in processing was predicted by using the structural dynamic modification method [39]. However, there is no in-depth research regarding the dynamic characteristics of the impeller. In the actual production and machining process, the processed thin-walled parts are mostly complex curved surfaces, and variable thickness is more common. In addition, most of the traditional methods (such as FEM or modal test) in the literature for the instantaneous dynamics prediction need to re-build and re-mesh the FEM model at each cutting step, which proves time-consuming and inefficient. Therefore, it is necessary to study the dynamic characteristics of multi-axis milling of thin-walled parts with a complex surface and variable thickness. Aiming at tackling the limitation, this paper adopts a structural dynamic modification method to predict the stability of a multi-axis milling thin-walled workpiece with variable mass removal. First,a time-varying parameter system model for multi-axis milling of thin-walled parts with a complex curved surface and variable thickness is established. Second, the instantaneous dynamic characteristics of workpiece processing are obtained using the extended Sherman-Morrison-Woodbury method with the dynamic characteristic parameters of the workpiece before and after the machining is completed. Subsequently, the variation curve of the dynamic characteristics of the workpiece in the process of machining with the cutting path is obtained using the least squares method. Finally, the stability of multi-axis milling is predicted using the extended numerical integrated method, and a typical multi-axis milling molding part of the aero-engine blade is employed to verify the proposed method. The rest of the paper is organized as follows.