## 1 Introduction
[21] presented an improved semi-discretization algorithm to predict the stability lobes for variable pitch cutters, which were verified and compared with previous works such as the Altintas analytical method (zero-order method) [19]. [PERSON] and [PERSON] [22] showed the optimal design of a tool for milling operations with variable geometry to widen the stability zones using the semi-discretization method, validating it experimentally. They also used a design methodology to determine the optimal pitch angle geometry for a given cutting condition, allowing increased stability. [PERSON] et al. [23] extended the multifrequency solution proposed by [PERSON] and [PERSON] [24] to include the helix effect, they pointed out that the variation of the helix angle plays an important role in stability graphs due to repetitive vibrations driven by impact (flip), they found that the flip lobes became closed curves that are separated by horizontal lines where the depth of cut is equal to a multiple of the helix pitch. A similar phenomenon was confirmed using the semi-discretization method (SDM) in [25], meanwhile, [PERSON] et al. [26] considered the influence of the helix angle of the tool to obtain an analytical force model, they found that isolated islands of instability can occur in the milling processes, which are induced by the helix angle of the tool and lead to separate regions of period-doubling and quasi-period behavior. [PERSON] et al. [27] by using an adapted and time-averaged version of the SDM analyzed both the influence on the variation of the helix angle and the pitch angle of the tool to improve the prediction of vibrations and estimate predictions of surface errors. They used the semi-discretization method, the time-averaged semi-discretization method and the temporal finite element method to predict vibration stability for variable helix and variable pitch milling tools. [PERSON] et al. [28] modeled and compared stability for variable pitch and helix angle cutters, demonstrating that variable helix angle tools can have higher stability and productivity. [PERSON] and [PERSON] in [29] combined SDM with differential evolution to optimize variable helix milling tools to minimize vibration, their analysis predicted total vibration mitigation using the optimized variable helix milling tool at low radial immersion. Furthermore, [PERSON] and [PERSON] [30] introduced a general mechanical model based on SDM to predict the linear stability of specialty cutters with optional continuous variation of the helix angle. Using an extended second-order SDM, [PERSON] et al. [15] predicted the stability lobe diagrams for tools with variable pitch angles. Meanwhile, [PERSON] et al. [31] conducted a stability analysis for milling operations with variable pitch mills at variable speed,while [PERSON] et al. [32] proposed an integrated process machine model based on the computer graphics method to simulate the milling process of a variable pitch cutter. On the other hand, [PERSON] and [PERSON] in [33] led to the development of the enhanced multistage homotopy perturbation method (EMHPM) to solve differential delay equations (DDEs) with constant and variable coefficients and then this EMHPM was applied to predict the stability of a multivariate milling tool in which they consider the helix angle and the pitch angle variation of the cutting edges [34]. Based on the Laplace formulation, [PERSON] [35] studied the stability of milling operations with a variable helix angle. Using the multi-frequency solution, [PERSON] et al. [36] derived a dynamic process model where the non-linear shear force and the runout effect are included for milling with non-uniform pitch and variable helix tools. [PERSON] et al. [37] found that runout can significantly increase the stability limits regardless of spindle speed ranges, while [PERSON] et al. [38] in a study for a thin-walled workpiece demonstrated that by considering the effects of the runout, the helix angle and characterization dependent on the cutting speed, more precise stability boundaries are achieved.