# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 5 Analysis of Stability with the 0-1 Test  
Instability in cutting appears as intermittent, forced conditions with fluctuating forces and irregular chip formation, causing vibrations, workpiece deflection, and reduced surface quality. The stochastic nature of \(h_{min}\) complicates process control, requiring robust monitoring and adaptive strategies. Understanding \(h_{min}\) variability is crucial for optimizing parameters and process stability in titanium machining to develop predictive models incorporating stochastic \(h_{min}\), thermal effects, and real-time monitoring. To analyze local instabilities, the 0-1 test for chaos was applied to cutting force data. This method distinguishes between regular and chaotic dynamics by analyzing time series  
Figure 14: Coefficient of variation in \(h_{min}\) relative to the feed rate for Ti Grade 2. Figure 13: Coefficient of variation in \(h_{min}\) relative to the feed rate for Ti Grade 5.  
directly, complementing classical approaches like maximal Lyapunov exponent calculation [77]. The data flow diagram is shown in Figure 15.  
The algorithm consists of several steps. As previously described, the acquired time series of the cutting force components was calibrated, trimmed, and filtered. Subsequently, the resultant cutting force value was calculated to sum the influence of individual components, so that regions where instabilities amplify or cancel each other indicate the system's dynamic behavior. For the cutting force \(F(j)\), where \(j\,=\,1,\ldots,\,N\), translation variables were computed using 100 randomly selected values of \(c\) from the interval \(c\in(0,\pi)\). This allowed for generating a basis with which to calculate the mean square displacement \(M(n)\). If the system dynamics are regular, the mean square displacement remains bounded over time. Otherwise, it increases linearly with time. To assess the time-bounded nature of \(M(n)\), the asymptotic growth rate coefficient \(k\) was computed and plotted as a function of the randomly selected \(c\) values. A median value of \(k\) close to one indicates chaotic dynamics. However, when analyzing the cutting force time series, it was found that both the sampling period and the number of samples influence the values of the asymptotic growth rate coefficient. A proper approach to applying the 0-1 test involves using the Discrete Fourier Transform to identify the maximum frequency present in the time series. In addition to the sampling interval \(dt\), another important parameter, \(T\)--a positive integer--plays a crucial role in the 0-1 test. From all available data points, every \(T\)-th sample is selected, and this newly created sequence is then subjected to the 0-1 test for chaos. Figure 15: Algorithm for applying the 0–1 chaos test to cutting force data. The issue is related to the choice of \(dt\) and the parameter \(T\), which is carefully selected to prevent oversampling [78]. The combination of \(dt\) and \(T\) values determines whether the 0-1 test yields valid or misleading results. This is especially important when analyzing cutting force signals that contain multiple frequencies in their spectrum, which may lead to an incorrect classification as chaotic, with the test output parameter \(K\) approaching 1. Thus, in cases of oversampling, even chaotic systems can be mistakenly classified as periodic by the 0-1 test. When analyzing the \(K\) values obtained from the 0-1 chaos test (Figures 16 and 17), we observe that for larger time delays (\(T\cdot dt\)), the values approach 1. This analysis was performed on real-time experimental data collected directly from the cutting process, confirming the presence of complex, nonlinear behavior in the system. A \(K\) value near 1 suggests that the process dynamics are chaotic rather than purely periodic or random. Such chaotic behavior is characteristic of mass-dissipative-elastic systems, where interactions between elastic deformation, energy dissipation, and material removal lead to unpredictable yet deterministic variations in cutting forces over time.