# Investigation of Titanium Alloy Cutting Dynamics in Thin-Layer Machining
## 5 Analysis of Stability with the 0-1 Test
The detailed analysis made it possible to determine a \(T\cdot dt\) criterion for which the 0-1 test identifies that the cutting process becomes chaotic. For titanium alloy Ti Gr5, this threshold is \(T\cdot dt=50\) s. In contrast, for titanium alloy Ti Gr2, the criterion is ten times smaller, with \(T\cdot dt=6\) s.  
The detailed analysis enabled the determination of a threshold criterion, expressed as the product \(T\cdot dt\), beyond which the 0-1 test detects chaotic behavior in the cutting process. For the titanium alloy Ti Grade 5, this threshold is approximately \(T\cdot dt=50\) s. In contrast, for titanium alloy Ti Grade 2, the threshold is significantly lower--about ten times smaller--at \(T\cdot dt=6\) s. This indicates that the onset of chaos in the machining process occurs much earlier for Ti Grade 2 compared to Ti Grade 5 under the tested conditions. The analysis established a threshold value of \(T\cdot dt\) for identifying the onset of chaotic behavior in the cutting process using the 0-1 test. For Ti Grade 5, chaos is detected when \(T\cdot dt\geq 50\) s, whereas for Ti Grade 2, the threshold is significantly lower at \(T\cdot dt=6\) s. This indicates that Ti Gr2 exhibits a higher susceptibility to chaotic behavior under similar cutting conditions. Figure 16: Results of the 0–1 test and \(T\cdot dt\) analysis for Ti Grade 5. For \(T\)-\(dt~{}=~{}50\) s, Figure 17 presents a sample time series of the cutting force and the corresponding phase plot for feed rates of 3 \(\upmu\)m/rev (below \(h_{min}\)) and 50 \(\upmu\)m/rev (above \(h_{min}\)) for Ti Grade 5. The differences in the force signals are significant. In the phase plot, circular trajectories typical of periodic behavior are visible. For the tested samples at a feed rate of 3 \(\upmu\)m/rev, the test output parameter \(K\), represented as the median of the asymptotic growth rate \(k\), is close to 1, indicating chaotic behavior. For all tested samples at this feed rate, the \(K\) value remained close to 1. In contrast, for a feed rate of 50 \(\upmu\)m/rev, the \(K\) value was below 0.5, indicating a periodic signal with disturbances. For \(T\)-\(dt~{}=~{}50\) s, Figure 18 presents representative time-series data of the cutting force alongside corresponding phase plots for two distinct feed rates: 3 \(\upmu\)m/rev, which is below the minimum undeformed chip thickness, and 50 \(\upmu\)m/rev, which exceeds this threshold, both tested on Ti Grade 5. The force signals recorded under these two conditions show marked differences in their behavior and characteristics. Figure 17: Results of the 0–1 test and \(T\) - \(dt\) analysis for Ti Grade 2. Figure 18: Results of the 0–1 test for \(T\)-\(dt~{}=~{}50\) s and variable feed for Ti Grade 5: (a) cutting force \(F\), (b) phase plot of the cutting force, (c) asymptotic growth rate \(k\), (d) output parameter \(K\) of 0–1 test. At the higher feed rate of 50 \(\upmu\)m/rev, the phase plot reveals circular trajectories, which are indicative of a stable, periodic cutting process. This suggests that when the feed per tooth is sufficiently large, the cutting process operates in a regular and predictable manner, with cutting forces fluctuating in a consistent and periodic way. The corresponding \(K\) values, derived from the 0-1 test for chaos and expressed as the median asymptotic growth rate \(k\), are below 0.5. Such values confirm the dominance of periodic, non-chaotic behavior, with only minor irregularities likely caused by normal process disturbances. In contrast, at the lower feed rate of 3 \(\upmu\)m/rev, which falls below the minimum chip thickness, the material is primarily deformed elastically or plastically rather than being effectively sheared into chips. This results in complex and irregular cutting force signals. Here, the \(K\) parameter approaches a value close to 1, consistently observed across all samples at this feed rate, which strongly indicates chaotic behavior in the cutting dynamics.