## 2 Dynamic Model of Belt-Driven Electromechanical Spindle System  
### System Modelling  
Lumped parameter models are used to evaluate the belt transmission performance. This method simplifies the mathematical model. Previous studies [12; 26; 27; 28] have established simplified dynamic models of turning processes. For the study to be accurate, both the belt deformation and motor electromagnetic characteristics must be considered. Hence, in this paper, we present a complete electromechanical system model, in which the equations of motion associated with the mechanical components (pulleys, belt transmission, and spindle) are coupled and solved along with the equations that describe the electromechanical responses. Figure 3 shows the transmission system of a CNC lathe, which includes an industrial motor (IM), a driver pulley, a V-belt, a driven pulley, and a spindle. The workpieces were mounted on the spindle with a three-jaw calliper and turned through the cutting tool. \(r_{1}\) is the driving pulley radius, \(\theta_{r}\) is the radius and angular displacement of the motor shaft, \(r_{2}\) is the driven pulley radius, and \(\theta_{r}\) represents the radius and angular displacement of the main shaft. \(J_{m}\) represents the driven pulley and spindle moments of inertia, and \(J_{s}\) represents the driven pulley and motor moments of inertia. The belt branch was represented by the Kelvin model of a linear spring in parallel with a viscous damper. The stiffness \(K_{b}\) of the belt branches can be determined by the well-known relationship \(k_{b}=EA/l\), where E is the Young's modulus of the belt material, A is the cross-sectional area of the belt branch, \(l\) is its length, and \(c_{b}\) is the belt-pulley damping coefficient. The first step is to establish a mathematical IM model. According to [PERSON]'s voltage law, IM's stator voltage equation for the IM is [29]  
\[\mathbf{V}=\mathbf{I}\mathbf{R}+\frac{\mathbf{d}}{\mathbf{dt}}(\mathbf{LI})=\mathbf{I}\mathbf{R}+\mathbf{I}\frac{ \mathbf{dL}}{\mathbf{dt}}+\mathbf{L}\frac{\mathbf{d}\mathbf{I}}{\mathbf{dt}}, \tag{1}\]  
where voltage \(\mathbf{V}\) and current \(\mathbf{I}\) are the vector matrices. \(\mathbf{L}\) is the inductance matrix, and \(\mathbf{R}\) is the motor resistance matrix. Following Equation (1), the healthy motor's current differential equation is as follows:  
\[\dot{\mathbf{I}}=\left[\mathbf{V}-\mathbf{I}\left(\mathbf{R}+\omega_{e}\frac{\mathbf{d}}{d\theta_{ r}}(\mathbf{L}(\theta_{r}))\right)\right]\mathbf{L}(\theta_{r})^{-1}, \tag{2}\]  
where \(\omega_{e}\) is the electrical angular speed of the motor, and \(\omega_{e}=p\dot{\theta_{r}}\). \(p\) represent pole pairs of induction machines. The dots above current \(\mathbf{I}\) represent time derivatives. The electromagnetic torque \(T_{e}\) can be calculated for the motor [30] as  
\[T_{e}=\frac{P}{2}[\mathbf{l}]^{T}\frac{\partial\mathbf{L}(\theta_{r})}{\partial\theta_{r }}[\mathbf{l}] \tag{3}\]  
For the driven pulley, spindle, driving pulley, and motor, the equations of the rotating motions that link these components can be easily developed, as follows:  
\[\left\{ \begin{array}{l}J_{r}\ddot{\theta_{r}}=T_{e}-T_{sk1}-T_{sc1}\\ J_{s}\ddot{\theta_{s}}=T_{sk2}+T_{sc2}-T_{L}\end{array} \right., \tag{4}\]  
where \(T_{sc1}\) and \(T_{sc2}\) are the sticky moments of the belt at the driver and drive and are given by Equations (5) and (6), respectively. The stiffness moment of the belt at driver \(T_{sk1}\) and the stiffness moment of the belt at driven \(T_{sk2}\) are given in Equations (7) and (8), respectively. \[T_{sc1}=2c_{b}(r_{1}\dot{\theta_{r}}-r_{2}\dot{\theta_{s}})r_{1} \tag{5}\]  
\[T_{sc2}=2c_{b}(r_{1}\dot{\theta_{r}}-r_{2}\dot{\theta_{s}})r_{2} \tag{6}\]  
\[T_{sk1}=2k_{b}(P_{1}-P_{2})r_{1} \tag{7}\]  
\[T_{sk2}=2k_{b}(P_{1}-P_{2})r_{2}, \tag{8}\]  
where \(P_{1}\) and \(P_{2}\) represent the axial tension above and below belt span \(L_{1}\), respectively.