# Prediction Methods and Experimental Techniques for Chatter Avoidance in Turning Systems: A Review
## 2 Vibration Prediction in Turning Processes  
Some of the most important models are listed in Appendix A. [PERSON] [16] developed a one-dimensional orthogonal cutting model and obtained an approximate solution by projecting cutting forces and structural dynamics in the chip thickness direction. Later, [PERSON] [17; 18] carried out an experimental study where they concluded friction forces on the contact flank introduced energy on the cutting system and are responsible for maintaining the chatter vibration. [PERSON] [19] proposed a 2D model for the prediction of chatter marks based on tests on a cantilevered piece. They were capable of relating the behavior of the rotating workpiece with a certain force, inversely proportional to the cutting speed and proportional to the velocity of the vibration, and studied the phase shift of the vibration. [PERSON] [20] integrated the approach of the oriented transfer function with a cutting geometry in three dimensions but carried out the experimental validation for an orthogonal cutting. Then, he applied the Fourier series expansion to the periodic terms determining the Fourier coefficients of the corresponding milling transfer functions [21]. [PERSON] and [PERSON] [22] used the multidirectional approach of [PERSON] [23] applied to turning, including for the calculation of the chip section a coupling term between perpendicular directions. In a similar manner, [PERSON] and [PERSON] [24] introduced wear and non-linearity of process damping. These studies complicate the modeling and resolution. At the same time, [PERSON] and [PERSON] [25; 26] presented an analytical model for calculating the chip section in tools with radii. Applying the above procedure, [PERSON] [27] obtained the stability maps for the turning of a piece of low stiffness. Later, [PERSON] et al. [28] proposed some analytical models in the time domain for the prediction of boring stability. [PERSON] and [PERSON] [29; 30; 31] proposed a multi-dimensional frequency model that considered the effect of tool radius (Figure 1a). They discretized the chip section into small trapezoidal elements that resulted in a rise of the matrices' dimension of the system to be solved. The lower the size of these elements, the more accurate the model is. However, computational costs are also increased. In that work, the authors considered interesting combinations of flexible tools and flexible workpieces in turning and boring operations considering tool radius. They found that inserts with bigger nose radii tend to reduce the stability limits in the turning/boring of flexible workpieces while the opposite is true when the tool is the most flexible element. When turning a flexible workpiece, the dynamics are only controlled by the transfer function of the tool in the feed direction, if the side cutting edge angle or the nose radius is zero. The workpiece dynamics can only affect the system if there exists a side cutting edge angle (or a nose radius). If the workpiece is the most flexible part, the stability limit is dramatically reduced. If the tool is the most flexible part, the workpiece can increase the stability limit. [PERSON] and [PERSON] (2018) developed a very complete Two-Degree of Freedom (2-DOF) model for the cutting tool. They considered multiple time delays, cut-off possibility and process damping using a variation of the Runge-Kutta (RK) method. The chip thickness regenerative effect was studied including motions in tangential and radial directions. [PERSON] et al. [PERSON] et al. (2018) calculated the stability lobes when chatter comes from a dominant mode in the tangential direction. They analyzed the coupling with feed direction in a cylindrical turning operation. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) studied turning dynamics when chatter comes from the tool. In their original approach, the workpiece was modeled as three rotors (unmachined, being machined, and machined) connected by a flexible shaft.