# Prediction Methods and Experimental Techniques for Chatter Avoidance in Turning Systems: A Review
## 2 Vibration Prediction in Turning Processes
They considered cases for face-turning and grooving were the workpiece was modeled as a Euler-Bernoulli type beam. The authors investigated the relationship between spindle speed and critical chip width under a variety of experimental setups. It was found that the critical chip width is always greater for the flexible case. At higher workpiece flexibility, a higher depth of cut is allowed. Other researchers analyzed complicated kinematics and complex effects. [PERSON] and [PERSON] [50] presented a method for chatter prediction in uniform and stepped workpieces. They determined the varying natural frequencies and varying mode shapes of workpieces during machining and gave some clues for a better understanding of gyroscopic and cross-coupling effects. [PERSON] and [PERSON] [51] used the method of multiple scales (MMS) to analyze nonlinear responses and the stability of the tool and workpiece. They considered the internal resonance and identified the most critical cutting parameters for chatter onset using time and frequency response curves from a theoretical point of view. [PERSON] et al. [52] studied the stability of turning systems when chatter can be influenced by periodic chip formation. These authors combined a model for the regenerative effect and a model for chip formation. They showed the time scale due to chip segmentation is shorter than the time scale in more common turning systems. Due to nonlinear effects, the cutting coefficients are lower  
Figure 2: (**a**) Scheme of the dynamic model for stability analysis of turning process considering workpiece flexibility; (**b**) stability lobes for workpiece considered as a rigid body and as a flexible body [48]. [PERSON] et al. [53] analyzed and identified the stability of a modulated tool path (MTP) turning strategy. Under this technique, sinusoidal oscillations are superimposed to a common constant feed. For the validation tests, they performed longitudinal cuts in a tube of Aluminum 6061-T6 showing the capabilities of the process by means of broken chips. The results from their periodic sampling method were compared with the traditional frequency-domain method with good agreement. [PERSON] et al. [54] predicted the stability lobes for cases in turning where spindle speed (not cutting speed) is low, between 50 and 100 rpm such as horizontal lathes, using multi-mode analysis. They confirmed positively their numerical models (Figure 3b) but found difficulties for reasonable computation times. They proposed an alternative based on a version of dde23 from Matlab(r) to analyze chatter cases [55]. [PERSON] et al. [56] made an approach to generalize turning cases were verification of their models was made with published experimental data from the literature. In a counterintuitive way, they stated that an increase in the stiffness of the workpiece can destabilize the tool modes producing an undesirable effect in the cutting stiffness and damping. [PERSON] et al. [57] were able to predict the stability boundaries in the turning of flexible workpieces with inserts considering their radius. They extended the Chebyshev collocation method to account for a variable side cutting edge angle. For the experimental tests, they designed and prepared a mass-concentrated workpiece which was held between the chuck and the tailstock. These authors verified the capacity of the method for different dynamic conditions since the piece continuously loses mass. Later, [PERSON] et al. [58] investigated the initiation mechanisms of chatter according to the position of the tool during the turning of flexible parts. [PERSON] et al. [59] presented a stability prediction model for the interrupted turning of cylindrical pieces. Previously, polygons were milled to create different interruption levels. They observed the gain of stability as a direct effect, but not proportional to the cutting interruption degree. [PERSON] et al. [60] combined a simulated annealing technique (SA) with numerical methods to find the optimal cutting conditions in turning operations.