## 1 Introduction  
Thin-walled structures are commonly applied in the aerospace, marine, and automotive industries owing to their incomparable advantages, such as lightweight and high specific strength. Considering the high material removal rate and sleek finished surface, bull-nose end milling has become a popular operation for the manufacturing of thin-walled parts, such as aero-engine blades, impellers, and casings [1; 2; 3]. These parts have complex and thin-walled structures and often bring many machining difficulties and challenges. Chatter is essentially a self-excited vibration and usually appears in the milling process. It not only pulls down the material removal rate but also results in a coarse finished surface, premature tool wear, excessive cutting noise, and so on [4; 5]. Chatter has long been one of the main problems during the milling operation. Thus, to avoid chatter, the milling operation must be performed in a steady state [6]. Furthermore, the calculation of SLD has become and still is an important research topic in high-quality machining. Presently, two primary mathematical solutions are employed to construct the SLD, namely the analytical and numerical approaches. In terms of the analytical approaches, [PERSON] and [PERSON] [7] proposed a single-frequency solution (SFS) on the basis of the truncated Fourier series of the cyclical cutting forces. The SFS is also known as the zeroth-order approximation (ZOA) approach, which can plot the SLD efficiently. However, it fails to compute the extra stability boundaries and the doubling bifurcations. For solving this question, [PERSON] and [PERSON] [8; 9] subsequently reported a multi-frequency solution (MFS), where the higher-order parts of the Fourier series are adopted. [PERSON] and [PERSON] [10] expanded the application of the MFS to machining conditions with small radial depths of cut. The disadvantage of the MFS is that it consumes more computing time. In comparison with the analytical approaches, more investigations have concentrated on numerical approaches [11]. [PERSON] and [PERSON] [12] reported an important SDM to evaluate milling stability. In this approach, the zeroth-order piecewise constant function is employed to approach the delayed part. To achieve better solution accuracy and quicker convergence rates, many high-order SDMs were developed. [PERSON] et al. [13] took into account the first-order approximation of the delayed term (namely the first SDM). [PERSON] et al. [14] investigated the second-order approximation of the delayed term (namely the second SDM). [PERSON] et al. [15] suggested a third SDM and a fourth SDM. On the other hand, [PERSON] et al. [16] pioneered the influential full-discretization method (FDM), which has a higher computational efficiency. In the FDM, in addition to the delayed term, the state term is also approximated by employing linear interpolation. Subsequently, many other approaches inspired by the FDM have been proposed, just like the 2 nd FDM [17], the 3 rd FDM [18], the updated FDM (UFDM) [19], the 2 nd UFDM [20], and the 3 rd UFDM [21]. During the bull-nose end finishing process of thin-walled or thin-floor parts, the influence of the workpiece dynamics cannot be neglected [22; 23; 24]. [PERSON] et al. [25; 26] modeled the dynamics of the thin-floor structure as a 1-DOF system and predicted the SLD in bull-nose end finish milling. Based on this study, the influence of the material removal on the bull-nose end milling stability is analyzed, and the three-dimensional (3D) SLD along the toolpath is calculated [27]. Later, a half-active magnetorheological damper is designed to dampen regenerative chatter for the bull-nose end finishing of thin floors [28]. [PERSON] et al. [29] simplified the dynamics of the aero-engine casings into a 1-DOF model along the workpiece's radial direction. The model only takes into account the workpiece flexibility and is able to evaluate the SLD for different small lead angles.