Considering the moving tool position in machining processes， the chatter model of a flexible workpiece is formulated to investigate the time-varying stability during a straight operation. The finite element model of the spindle-chuck-workpiece-tailstock system is established to analyze the effect of the supporting conditions on dynamics of the machined workpiece. Accordingly， the stiffness distribution and the natural frequency variation of the flexible workpiece under different boundary conditions is obtained. Finally， turning experiments are performed to verify the theoretical analysis. It has been shown that the stiffness of workpiece exhibits time-varying characteristics at different cutting positions and the minimum stiffness locates approximately at the midpoint close to the tailstock side. And increasing the spindle bearing or tailstock support stiffness can effectively improves the eigenfrequency and the dynamic characteristics of the system to a certain extent. The cutting observations demonstrated that the state of the turning operation of the clamped-simply supported flexible workpiece generally underwent shift from stable to chattering then back to stable states， showing the time-varying nature of the process. Furthermore， the workpiece is with chaotic trajectory of the shaft axis when chatter happened. The proposed model and analytical method could be useful to guide the turning operations of the flexible components in practice.