## 1 Introduction  
Thin-walled, lightweight parts are adopted in the aerospace and automobile industries to further improve fuel efficiency. However, vibrations can occur during the cutting process owing to the low rigidity of the thin-walled and lightweight parts. Recently, extensive research has been conducted to improve the machining of the thin-walled parts [1; 2; 3; 4; 5]. Chatter occurs under specific cutting conditions, and causes intensive vibrations, serious noise problems, deterioration of surface finish accuracy, tool damage, and tool wear. Several studies have attempted to predict the occurrence of chatter and develop countermeasures for the same [6; 7]. To suppress chatter occurrence, machine tool operators use various trial-and-error measures such as applying fixtures to the workpiece to increase its rigidity [8; 9; 10; 11; 12] and reducing the cutting speed and depth of cut. However, the reduction in cutting speed and depth of cut decreases the machining efficiency. In addition, the application of the workpiece fixture requires time, money, and empirical knowledge because the workpiece fixtures must be elaborately manufactured according to the workpiece dimensions and shapes. There are two methods of chatter control: active and passive. Studies on chatter suppression using active vibration control include reduction of the tool vibration during chatter generation by applying a controlled input to the tool using a piezo actuator [13; 14], an electrodynamic shaker [15], and a magnetic actuator [16]. Moreover, the chatter stability limit has been improved by applying damping to the tool using magnetorheological dampers [17], and via piezoelectric shunt damping [18]. Notably, regenerative chatter can be controlled by varying the spindle speed during the machining process [19; 20]. Another chatter suppression technique during the turning process of a slender workpiece involves varying the lateral stiffness of the workpiece with a time periodic axial load at a tailstock [21]. Moreover, passive countermeasures against chatter have been studied, including the addition of damping materials to the tool [22] and workpiece [23] and the addition of tuned mass dampers (TMDs) to a cutting tool [24; 25; 26], workpiece [27], and machine tool structure [28; 29; 30]. In particular, the TMD is an effective passive damping method owing to its large damping effect with a small mass. [PERSON] et al. [28] studied the chatter suppression effect on attaching a single TMD with variable stiffness to a machine tool structure with a long overhang ram. [PERSON] et al. [24] proposed the countermeasure against chatter during the turning process by mounting a piezoelectric inertia actuator on a cutting tool, acting as a TMD. [PERSON] [31] proposed the analytical solutions for evaluating the optimal natural frequency and damping ratio of a TMD by using the fixed point of the real part of the transfer function of the 1-DOF undamped vibration system for a single TMD. [PERSON] et al. [29] designed a nonlinear TMD containing an additional element of elastic support/dry friction to suppress chatter in the turning process and presented an optimum design of the proposed TMD by effectively suppressing the magnitude of the real part of the frequency response function of the damped machining system. In addition, [PERSON] et al. [30] proposed an optimum design of multiple TMDs for chatter reduction in the turning process by effectively suppressing the magnitude of the real part of the frequency response function of the damped machining system. [PERSON] et al. [26] proposed a receptance coupling based approach to optimally tune and place TMDs on boring bars for chatter suppression. [PERSON] et al. [25] designed multiple TMDs attached to a rotating tool holder to suppress tool chatter in the end milling process and presented optimum parameters of multiple TMDs by effectively increasing the critical depth of cut for each spindle speed.