## 2 Working Principle and Kinematic Modeling of Extension-Dominated Mixer
### Kinematic Mathematical Analysis
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Point** & **Parameter Value** & **Trajectory Equation** & **Velocity Equation** \\ \hline B & \(-\) & \(x_{B}=-e\sin\omega t\) & \(V_{XB}=-e\omega\cos\omega t\) \\ A & \(L_{AB}=e\) & \(x_{A}=0\) & \(V_{xA}=0\) \\  & \(\varphi_{BA}=\frac{\pi}{2}\) & \(y_{A}=2e\cos\omega t\) & \(V_{yA}=-2e\omega\sin\omega t\) \\ C & \(L_{BC}=e\) & \(x_{C}=-2e\sin\omega t\) & \(V_{xC}=-2e\omega\cos\omega t\) \\  & \(\varphi_{BC}=-\frac{\pi}{2}\) & \(y_{C}=0\) & \(V_{yc}=0\) \\ D & \(L_{BD}=R-e\) & \(x_{D}=-R\sin\omega t\) & \(V_{xD}=-R\omega\cos\omega t\) \\  & \(\varphi_{BD}=-\frac{\pi}{2}\) & \(y_{D}=(2e-R)\cos\omega t\) & \(V_{yD}=(R-2e)\omega\sin\omega t\) \\ E & \(L_{BE}=R+e\) & \(x_{E}=R\sin\omega t\) & \(V_{yz}=R\cos\omega t\) \\  & \(\varphi_{BE}=\frac{\pi}{2}\) & \(y_{E}=(R+2e)\cos\omega t\) & \(V_{yz}=-(R+2e)\omega\sin\omega t\) \\ F & \(L_{BF}=\sqrt{e^{2}+R^{2}}\) & \(x_{F}=-e\sin\omega t-L_{BF}\sin(\varphi_{BF}-\omega t)\) & \(V_{xF}=-e\omega\cos\omega t+\omega L_{BF}\cos(\varphi_{BF}-\omega t)\) \\  & \(\varphi_{BF}=\frac{\pi}{2}+\arctan\frac{R}{e}\) & \(y_{F}=e\cos\omega t+L_{BF}\cos(\varphi_{BF}-\omega t)\) & \(V_{yF}=-e\omega\sin\omega t+\omega L_{BF}\sin(\varphi_{BF}-\omega t)\) \\ G & \(L_{BG}=\sqrt{e^{2}+R^{2}}\) & \(x_{G}=-e\sin\omega t+L_{BG}\cos(\varphi_{BG}-\omega t)\) & \(V_{xG}=-e\omega\cos\omega t+\omega L_{BG}\cos(\varphi_{BG}-\omega t)\) \\  & \(\varphi_{BG}=\arctan\frac{\pi}{2}\) & \(y_{G}=e\cos\omega t+L_{BG}\sin(\varphi_{BG}-\omega t)\) & \(V_{yG}=-e\omega\sin\omega t-\omega L_{BG}\cos(\varphi_{BG}-\omega t)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Trajectory equation and velocity equation of each point on rotor. rotor's highest point (position I), point A accelerates from zero velocity until reaching its maximum velocity at the midpoint (position III), after which it decelerates until reaching zero velocity at the lowest point (position V). Subsequently, point A reverses direction and accelerates until reaching its maximum velocity at position VII before decreasing back to zero velocity at position I, completing one cycle. The velocity trend of point C is opposite to that of point A, starting from deceleration at position I, reaching zero velocity at position III, then reversing direction and accelerating until reaching maximum velocity at position V. Similar to point A, point C then decelerates and reverses direction until returning to its initial position. At the rotor's highest point (position I), the rotor starts, and at the midpoint (position III), it reaches maximum velocity before decelerating to zero velocity at the lowest point (position V). Subsequently, point A's velocity reverses direction, gradually increasing until reaching its maximum at position VII, before decreasing back to zero velocity at position I, completing one cycle. Point C, being internal and situated on the moving circle, also traces a linear trajectory (horizontal line) with a stroke of 4e, zero vertical velocity, and periodic horizontal velocity variations. Point A exhibits only numerical velocity, while point C solely has horizontal velocity, with their velocity trends being inversely related. Points A and C have relatively lower velocities, differing by \(\pi/2\omega\) in phase. Point A, as the rotor's geometric center, exhibits the smallest velocity, while points D, E, F, and G, located on the rotor's surface, exhibit varying velocities. Point D, closest to point B, has the smallest velocity, while point E, farthest from point B, has the highest velocity. Points F and G, equidistant from point B, exhibit equal velocities, differing by \(\pi/2\omega\) in phase. At the beginning of each cycle, all points have equal absolute velocities.