# Simulation Analysis of a Three-Degree-of-Freedom Low-Frequency Resonant Mixer
## 3 Dynamic Analysis of the Three-Degree-of-Freedom Low-Frequency Resonant Mixer
### Harmonic Response Analysis of the Three-Degree-of-Freedom Low-Frequency Resonant Mixer
#### 3.2.2 Acceleration Response of the Three-Degree-of-Freedom Low-Frequency Resonant Mixer  
On the basis of the above work, the acceleration in a single direction for each of the three assemblies is solved in the software, and the direction is set as Z axis. Through computer simulation analysis, the acceleration response curve of the three-degree-of-freedom low-frequency resonant mixer near the third-order natural frequency can be obtained, as shown in Figure 5.  
From Figure 5, it can be observed that when the equipment operates at a frequency between 57 Hz and 62 Hz, the acceleration of the masses experiences a sudden change. Figure 4: Phase–frequency response curve of each assembly of the mixer: (**a**) phase–frequency response curve for the load assembly; (**b**) phase–frequency response curve of the vibration-isolating assembly; (**c**) phase–frequency response curve of the excitation assembly. Figure 5: Acceleration response curve of each assembly of the mixer. The equipment is resonating near the third-order natural frequency. At this point, the acceleration of the payload assembly is approximately 355.59% of that of the excitation assembly, achieving acceleration amplification. The maximum acceleration of the payload assembly is 617.13 m/s\({}^{2}\), which can facilitate efficient mixing of the material at the payload assembly location. During resonance, the excitation assembly still maintains a smaller acceleration, approximately 173.55 m/s\({}^{2}\), which is beneficial for the stable operation of the drive assembly connected to the excitation assembly. From Figure 4, it is known that near the third-order natural frequency, the excitation assembly and the payload assembly move in the same direction, while the vibration-isolating assembly moves in the opposite direction. When the equipment operates at a frequency slightly higher than the third-order natural frequency, the direction of motion of the excitation assembly is the same as that of the sine excitation force; when operating at a frequency slightly lower than the third-order natural frequency, the direction of motion is opposite to that of the sine excitation force. Considering the three assemblies as a whole, the following equations can be established:  
\[F_{4}+\sum_{i=1}^{3}F_{\text{II}}+F+\sum_{i=1}^{3}G_{i}=0 \tag{5}\]  
\[F_{\text{II}}=-m_{i}\cdot a_{i} \tag{6}\]  
The forces exerted by the three assemblies on the framework are given by the following:  
\[F_{5}=-F_{4}=\sum_{i=1}^{3}F_{\text{II}}+F+\sum_{i=1}^{3}G_{i}=-\sum_{i=1}^{3} m_{i}\cdot a_{i}+F+\sum_{i=1}^{3}m_{i}\cdot g \tag{7}\]  
where  
\(F_{4}\)--denotes the force exerted by the framework on the three assemblies, in N;  
\(F_{\text{II}},F_{\text{I2}},F_{\text{I3}}\)--denote the inertial force experienced by the payload assembly, vibration-isolating assembly, and excitation assembly, in N;  
\(F\)--denotes the sine excitation force, in N;  
\(m_{1},m_{2},m_{3}\)--denote the mass of the payload assembly, vibration-isolating assembly, and excitation assembly, in kg;  
\(a_{1},a_{2},a_{3}\)--denote the acceleration of the payload assembly, vibration-isolating assembly, and excitation assembly, in m/s\({}^{2}\);  
\(g\)--denotes the acceleration of gravity, \(g=9.8\text{m/s}^{2}\);  
From Figure 5, it can be seen that when the equipment operates near the third-order natural frequency, the accelerations of the assemblies are \(a_{1}=173.55\) m/s\({}^{2}\), \(a_{2}=564.53\) m/s\({}^{2}\), and \(a_{3}=617.13\) m/s\({}^{2}\).