## 1 Introduction  
Turning is a common manufacturing process that is used to create revolved parts by eliminating unwanted materials. Vibration occurs during the cutting processes in rotational machinery, particularly in the turning of slender workpieces. Many studies have been conducted to measure its dynamic characteristics, reduce its magnitude, and improve the quality and stability of machining [1, 2], in particular, by chatter avoidance [3, 4]. [PERSON] et al. [5] analysed the stability of the system during the cutting of a flexible workpiece supported by a tailstock; they studied the effect of the static deformation of the workpiece rather than the dynamics of the tool alone. [PERSON] et al. [6] investigated the dynamic characteristics of a tool-holder by simplifying it as a cantilever beam and formulated the natural frequency based on the Euler-Bernoulli beam theory. [PERSON] et al. [7] conducted a nonlinear dynamic analysis of a flexible workpiece and tool, obtaining the fundamental frequencies of a shell model of a thin cylindrical workpiece via finite element analysis (FEA). [PERSON] et al. [8] developed a model to predict chatter during the turning of slender workpieces. They considered the changes in the cutting position while the workpiece was supported by a tailstock; dynamic behaviour during chatter could be inferred from their model by considering linear stability and stiffness distribution. [PERSON] et al. [9] optimised process parameters such as cutting forces, cut depth, feed, and tool life for two-tool parallel turning but did not provide a theoretical dynamic model. All these stability studies of turning focused on the interaction of the workpiece and tool; they usually also simplified the tailstock-supported workpiece by representing it as a cantilever beam or as a simple supported beam, which deteriorates the estimation accuracy of the workpiece frequency and modal stiffness of the turning system. However, a slender workpiece located in a turning dynamic system (see Figure 1) is generally regarded as a combination of a cantilever and a simply supported beam, shown in Figure 2a. The mathematical models developed in these studies usually assumed certain ideal boundary conditions [10]: supported beams were treated as pinned-pinned, fixed-free (cantilever), fixed-pinned, or fixed-fixed. However, when a workpiece is clamped into a machining system, the loss of stiffness lowers the natural frequency of the first bending mode of the shaft compared to what it would be with any of these ideal boundary conditions [11]. This dramatically affects the dynamic response, especially for the resonant vibration response during machining process. [PERSON] et al. [12] constructed a one degree-of-freedom mechanical model considering the varying modal stiffness of the workpiece and improved the stability properties of turning a slender workpiece byplicating a compressive force at the tailstock. Research has been conducted to evaluate the dynamic responses of the beam model with different boundary conditions under moving loads [13]. Some theoretical and experimental studies of parallel turning stability have represented the tools by springs and damping elements [14]. For example, [PERSON] and [PERSON] [15] studied the forced harmonic motion of a non-uniform beam with dynamic vibration analytically, using the functional perturbation method [16]. This method can be applied to a cutting tool holder by optimising the real part of the frequency response function to enhance its ability to resist regenerative chatter. The natural frequencies and modes in the turning-point frequency range under various boundary conditions have been studied in this way for shells of revolution [17]. [PERSON] et al. established a two-dimensional (2D) dynamic model of the tool and part considering flexible boundary constraints [18]. They found that the dynamics and stability of a slender workpiece were affected by the connection stiffness of boundary constraints (chuck, tool nose, and tailstock).