## 2 Working Principle and Kinematic Modeling of Extension-Dominated Mixer
### Kinematic Simulation  
To achieve the desired motion based on the epitrochoid principle, a coaxial input-output self-rotation and revolution transmission system was designed. The specific principles are detailed in the referenced literature [38,39]. Figure 5 is the relative position of the stator and rotor and the transmission device. In Figure 5b, the positions corresponding to Figure 2 are marked, where in the cross-section of Figure 2, it is a point, but in the 3D diagram, it becomes an axis. In order to better represent the relationship between the points on the cross-section and the transmission device, analysis is conducted using the straight section as an example. Point A corresponds to the geometric axis of the rotor's straight section, point B corresponds to the output shaft of the transmission device, and point C corresponds to the center point of the stator and the input shaft of the transmission device. The output shaft (point B) of the transmission device is located between the input shaft (point C) of the transmission device and the geometric axis of the straight section (point A), with a distance of eccentricity \(e\) from both. Using SOLIDWORKS Motion, the kinematic simulation of the internally meshing single-screw system was conducted. The eccentric shaft was designated as the motor, rotating uniformly at a given speed. The trajectories of the output shaft and the single screw were tracked, and the relative positions of the single screw and the helical cavity stator were analyzed. To visualize the motion trajectory of the single screw more intuitively, a reference plane was established on the single screw at every pitch interval starting from the helical surface: I, II, III, IV, V, as shown in Figure 5a.The motion trajectories of the output shaft center and the single-screw center at various cross-sections of the transmission system are depicted in Figure 6. The output shaft trajectory forms a circle with the eccentricity  
Figure 4: The trajectory of each point of the rotor in motion: (**a**) Motion trajectory of special points of rotor (**b**) Motion trace assembly at each point on the rotor. Figure 5: Relative position of stator and rotor and transmission device: (**a**) Axonometric drawing; (**b**) Cut way drawing. I, II, III, IV, V: the helical surface; A, B, C, D, E correspond to the points in Figure 2b. \(e\) is the eccentric distance. distance as its radius. The internal meshing screw is eccentrically connected to the output shaft (with an eccentricity distance of \(e\)). In this configuration, the center of the single screw lies on the trajectory of the output shaft, as represented by point \(O_{2}\) in Figure 2. The output shaft undergoes simultaneous revolution and self-rotation in opposite directions at equal speeds, fulfilling the condition for the epitrochoid \(O_{1}\) to undergo pure rolling within the stator. Consequently, the trajectory of the single-screw center is a straight line, with a distance of \(4e\) from the center of rotation. The track length of one revolution of the center of the transmission system output shaft is:  
\[L_{1}=2e\pi \tag{1}\]  
At the same time, the single screw in the stator extended distance of \(4e\) straight reciprocating movement once:  
\[L_{2}=8e \tag{2}\]  
It can be obtained that the average speed of the center of the output shaft is \(2e\pi/8e=\pi/4\) of the average speed of the center of the single screw. The relative position change diagram of the screw and stator during the rotation of the shaft is shown in Table 1. It can be seen that the relative position changes between the screw and the stator with a spiral cavity in each cross-section, the material is transported forward in the cavity formed by the screw and the stator, and the volume changes alternately, forming an extension-dominated flow field.