# Prediction Methods and Experimental Techniques for Chatter Avoidance in Turning Systems: A Review
## 2 Vibration Prediction in Turning Processes
They used a 3D model to couple the equations of motion with a nonlinear cutting force and even considered whirling effect in their analysis. Recently, [PERSON] et al. [PERSON] et al. (2018) re-used the oriented transfer function concept from [PERSON] (2018) to solve stability in multi-dimensional cutting processes (with modes in different directions). These authors calculated the eigenvalues of the product transfer function and directional factors in a generalized model applicable to turning, milling, boring, or drilling. In the case of turning, they presented theoretical lobe diagrams depending on the rotation sense of the workpiece (clockwise and counterclockwise). These authors did not directly introduce the effect of the cutting speed in their analyses. Although the results were not validated, they expected different critical depths of cut, and chatter frequencies when turning with different rotational directions of the spindle. In an attempt to improve the accuracy of frequency-based methods, [PERSON] (2018) extended the force and stability models to more general tool geometries (variable position angle) and process damping (Figure 1b). Two approaches were compared: the first uses the hypothesis from [PERSON] (1961) this is, the chip flow is normal to the line connecting the ends of the engaged cutting edge; the latter forming an angle \(\varphi\) with feed direction. Under this approach, the components \(F_{n}\) and \(F_{r}\) act parallel and normal to this direction; a second one based on [PERSON]'s trapezoidal discretization approach. Both approaches show similar prediction errors in force, with errors lower than 15%. Although these approaches were known since the end of the 60 s, a significant advance was recently made in the modeling of process damping. At low cutting speeds, the free chatter zone may increase under certain conditions. It is a complex phenomenon depending on a number of parameters: cutting speed; workpiece material; tool flank angle; dominant frequency and its relationship with cutting frequency, relative with cutting speed; time (feed and tool wear). Process damping is mathematically simple to model but leads to laborious experimental tests. In practice, the stability lobes lead to higher depths of cut in comparison with high cutting speeds [39; 40]. [PERSON] et al. [41] identified the cutting coefficients including the process damping effect through a servo mechanism to control the oscillation frequency during experimental cutting tests. The study was applied to face turning but was later extended to the milling of low machinability materials [42; 43]. [PERSON] et al. [44] developed a very complete stability model for multiple degrees of freedom turning. They considered the process damping force as a function of depth of cut, cutting speed, tool velocity, and an empirical component. They mounted a device to account for process damping effects and then validate the stability lobes in an exhaustive picture of the turning system. [PERSON] and [PERSON] [45] studied the effects of tool position, workpiece dimensions, and tool's compliance on the stability charts. For model validation, they used a piece held at both ends, with good results applicable for roughing turning. They applied their calculations to experimental studies from other authors in the literature. Additionally, [PERSON] et al. [46] verified a 2-DOF model. Tool and workpiece were modeled as separate entities using single DOF models. Using their stability model for turning, stability under different operating conditions could be predicted, with and without the tailstock. [PERSON] et al. [47] proposed a model of various degrees of freedom taking into account the flexibility of the workpiece and tool separately or simultaneously. Estimations and trends were raised and observed, but the authors did not validate the models against experimental data (Figure 2). [PERSON] and [PERSON] [48; 49] developed 2-DOF models on flexible workpieces with and without the use of tailstock.