## 1 Introduction  
Metal spinning is an advanced manufacturing process that involves little or no cutting. The tool, such as a spinning roller, is pressed against a metal sheet or metal tube to produce continuous local plastic deformation of the workpiece and gradually form the required rotary part [1]. Spinning technology is generally used for forming hollow parts with circular cross-sections. Since the 1980s, researchers around the world have proposed a variety of non-circular spin forming methods to broaden potential metal spinning applications. Current spinning methods for noncircular parts can be classified into two types: synchronous spinning and force-controlled spinning. In synchronous spinning, the rotation angle of the workpiece is synchronized with the radial feed of the roller. [PERSON] and [PERSON] (1984) achieved elliptical cone spinning using equipment comprised of a cam and links, representing the first proof-of-concept that metal spinning technology can be used to form non-circular products [2]. Later, [PERSON] et al. (1999) realized elliptical cone spinning using a cross-slot mechanism to generate the required radial motion of the mandrel. Relative movement between the rollers and mandrel resulted in an elliptical trajectory [3; 4]. Building on the work of [PERSON] and [PERSON] (1986), [PERSON] et al. (1999) replaced the motion of the roller with a mandrel. However, both of these early methods were limited to spinning workpieces with an elliptical cross-section. More recently, [PERSON] et al. (2010) introduced the profiling driving method. A pair of gears is used to synchronize the radial movement of the roller and the rotation of the mandrel, and the device was shown to successfully form parts with polygonal cross sections [5,6]. [PERSON] (2010) developed a non-circular spinning device that used stepping motors to synchronize the radial feed of the roller and the rotation of the mandrel, while errors in the process were tracked by software [7]. [PERSON] (2012) developed a synchronous, multi-pass metal spinning method and successfully obtained a square cup with a non-circular bottom [8,9]. Similarly, [PERSON] (2018) used an intermediate circular shape to form a square cup with an oblique bottom and vertical sidewalls, and the wall thickness distribution of the workpiece was relatively uniform [10]. Force-controlled spinning controls the spinning force in the radial direction to ensure the workpiece fits over the non-circular mandrel until the non-circular parts are complete. [PERSON] (2005) produced non-circular parts with a force feedback spinning device, in which the forming roller was driven by servo motors and the motors were controlled by a computer system to maintain a constant pushing force on the workpiece [11]. [PERSON] and [PERSON] (2005) developed a non-circular spinning device that used spring tension to control the forming force [12]. [PERSON] (2006) also introduced a non-circular spinning device that relied on linear servo motors to drive the forming roller, which was found to reduce the forming time [13]. In all of the methods described above, the workpiece requires a mandrel. In force-controlled spinning, control of the forming force results in an unstable roller motion trajectory. In synchronous spinning, the radial feed of the roller must be synchronized with the rotation angle of the workpiece during synchronous spinning, and the rotation speed of the workpiece must be lower than 60 rpm. In the present study, a non-circular spinning method without a mandrel is adopted. Two rollers are arranged on the principal axis of the roller, which is then synchronized with the principal axis of the workpiece. Relative motion between the roller and the workpiece is used to spin the workpiece. The radial feed of the principal axis of the roller does not need to be considered on each pass during the spinning process. Furthermore, the rotation speed of the workpiece and the accuracy of the roller trajectory are both guaranteed.