## 2 Dynamic Model of Turning a Slender Shaft  
When a slender shaft is turned, one end is fixed to the lathe with a three-jaw chuck, the other end is usually supported by a tailstock centre to increase the stability of the processing and ensure the surface quality of the turning shaft [26; 27] (see Figure 1). A dynamic model for the slender shaft with one end fixed by the three-jaw chuck and the other end pinned by the tailstock, shown in Figure 1, was developed in this study by setting the end pinned by the tailstock as a flexible boundary, shown in Figure 2b. The three-jaw chuck, which limits the deflection and angle of the shaft, is approximated as a fixed support. The tailstock typically has two pairs of thrust bearings to withstand the axial force from the workpiece, as well as a radial bearing to support transverse loads; the thrust bearing enables the apex to provide a preloaded axial force [28] to the slender shaft, whereas the radial bearing provides lateral support. Therefore, the force on the pinned end is approximated as an axial force \(P\) and the bearing support is assumed as a spring that provides the model with external stiffness support \(\mathbf{Kt}\) in the Y direction. Since this model addresses the turning process of slender shafts, axial vibration, shear stress, and rotational inertia are not considered in this simplified model, therefore, the slender shafts are assumed to be continuous Eulerian-Bernoulli beam models [23]. In turning a shaft with high flexibility, it is common to assume the tool as rigid, and the workpiece as a single degree of freedom (SDOF) model, shown in Figure 2b. For a flexible workpiece, the modal stiffness of the workpiece is significantly larger in the axial direction than that in the bending direction. Hence, the vibration of transverse bending is the most significant dynamic response [29]. Thus, the workpiece vibration of transverse bending in the X direction is established as follows:  
\[M(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}+\frac{\partial^{2}}{\partial x^ {2}}\bigg{[}EI(x)\frac{\partial^{2}w(x,t)}{\partial x^{2}}\bigg{]}=F_{c}(x,t) +P(l,t) \tag{1}\]  
where \(E\) is workpiece modulus of elasticity, \(M(x)\) is the equivalent mass matrix per unit length, and \(I(x)\) is the moment of inertia at point \(x\) of the SDOF model. The mass matrix  
Figure 2: Dynamic model of turning a slender shaft, \(P\) = axial force; \(Kt\) = spring stiffness. (**a**) Classical simply supported beam model; (**b**) fixed-pinned model with flexible boundary. \(M\) and the moment of inertia \(I(x)\) are considered as homogenous and constant. For convenience of formulation, the mass \(M(x)\) and moment of inertia \(I(x)\) at each node are directly expressed as \(M\) and \(I\). \(w(x,t)\) is the lateral displacement for each point \(x\) of workpiece. \(P(l,t)\) is the axial force acting at the pinned end at point \(l\), for convenience, it is subsequently expressed as \(P\). \(F_{c}(x,t)\) is the cutting force at time \(t\) and at point \(x\) in the turning system and can be expressed as [30]  
\[F_{c}(x,t)=k_{s}(\text{DOC}-\text{D}(t)+\text{D}(t-\text{T}))a_{p} \tag{2}\]  
where \(k_{s}\) is cutting factor, DOC is the depth of cut, \(\text{D}(t),\text{D}(t-\text{T})\) are the current cutting diameter and the workpiece diameter of the previous turning, respectively. \(a_{p}\) is the feed rate.