# Improving Stability Prediction in Peripheral
## 1 Introduction  
Chatter is a dynamic instability phenomenon that diminishes the quality of parts and tool performance. It is an old adversary for machinists, and one of the most common issues in manufacturing. However, nowadays, due to the rapid growth of global competition to reduce cost and the increased dimensional accuracy in monolithic and thin geometries demanded by aeronautical industries, research has been focused on more accurate predictive models and methods for optimizing metal removal rates without chatter. Those models are delay-differential equations (DDE) with infinite dimensional state space. Particularly in milling, the equation of motiving is a DDE with a time-periodic coefficient, transformed in a finite dimensional system. According to the Floquet theory, the stability properties are determined by the system's monodromic operator. The literature provides various approaches for predicting stability lobes in various machining operations as a function of spindle speed and chip load. For instance, [PERSON] and [PERSON] et al. [1] presented a method for generating the stability lobes in milling operations by using time-domain numerical simulations. [PERSON] and [PERSON] [2] developed the first analytical solution that led to the milling operation prediction of stability lobes in the frequency domain by averaging the time-dependent periodic directional coefficients. This method provides accurate stability predictions. except for where there are cutting operations at low radial immersion. [PERSON] et al. [3] proved that the traditional regenerative stability theory could not accurately predict stability lobes at lowradial immersions. However, [PERSON] and [PERSON] [4] proved that the multi-frequency approach [5] was able to accurately predict low radial immersion milling stability lobes if the harmonics of the tooth-passing frequencies were included in the eigenvalue solution. The double-period bifurcation at small radial immersion was investigated by [PERSON] et al. [6] by applying the time finite element analysis (TFEA) in the solution of the governing equations of motion. The approximate solution obtained from this method was cast in the form of a discrete map that related position and velocity at the beginning and end of each element. Then, the eigenvalues of the discrete map were used to determine the stability bounds. Another method that forms a finite dimensional transition matrix as an approximation of the infinite dimensional monodromic operator is the semi-discretization (SD) method. In this method, first introduced by [PERSON] and [PERSON] [7], the delayed terms are discretized, the undelayed terms unchanged, and the time-periodic coefficients approximated by piecewise constant functions. An updated version of the SD method for periodic systems with a single discrete time delay was proposed by [PERSON] and [PERSON] [8]. In this approach, the time step is chosen as an integer fraction of the time period, meaning that the Floquet transition matrix is determined over a single period. [PERSON] et al. showed that the second and higher-order approximations of the delayed term did not provide better convergence than the first-order one [9; 10]. On the other hand, [PERSON] et al. [11] developed a new approximation technique for studying the stability properties of milling operations. This approach was based on the properties of Chebyshev polynomials and used a collocation representation of the solution at their endpoints. The stability bounds of the corresponding equations were determined by the eigenvalues of the approximate monodromic matrix, mapping function values at the collocation points from one interval to the next. They concluded that the Chebyshev collocation (CH) approach provided results similar to other methods which used equispaced points. This approach was extended by the authors in complex turning applications [12; 13] and compared to other computation techniques [14]. [PERSON] et al.