# Simulation Analysis of a Three-Degree-of-Freedom Low-Frequency Resonant Mixer
## 4 Conclusions  
Based on the design of the three-degree-of-freedom low-frequency resonant mixer, this study conducted simulation and analysis of the equipment's operational performance and vibration isolation mechanism. Through theoretical analysis and simulation, the third-order natural frequency of the three-degree-of-freedom low-frequency resonant mixer was determined. When the equipment operates near the third-order natural frequency, the excitation assembly has a smaller amplitude and acceleration, ensuring the stability of the drive assembly at this location; the phase difference between the payload assembly and the vibration-isolating assembly remains at 180\({}^{\circ}\), and through force cancellation, the force exerted on the framework is reduced, demonstrating good vibration isolation performance; the equipment achieves amplification of both amplitude and acceleration, resulting in a larger amplitude and acceleration for the payload assembly. Specifically, the amplitude of the payload assembly is 4.42 mm, approximately 356.45% of the excitation assembly's amplitude. The acceleration of the payload assembly is 617.13 m/s\({}^{2}\), approximately 355.59%  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Third Order Natural** & **The Amplitude of Vibration of** & **The Amplitude of the Acceleration** \\  & **Frequency (Hz)** & **the Payload Assembly (mm)** & **of the Payload Assembly (m/s\({}^{2}\))** \\ \hline \(\varepsilon\) & \(-0.00024\) & \(-0.01144\) & \(-0.01031\) \\
**GCI (\%)** & \(0.04\) & \(2.53\) & \(2.28\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Table of data related to the analysis results associated with GCI analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number of Nodes**} & **Third Order Natural** & **The Amplitude of Vibration of** & **The Amplitude of the** \\  & **Frequency (Hz)** & **the Payload Assembly (mm)** & **Acceleration of the Payload** \\  & & & **Assembly (m/s\({}^{2}\))** \\ \hline
79,015 & 58.792 & 3.4 & 474.56 \\
98,036 & 58.947 & 4.42 & 617.13 \\
193,800 & 58.933 & 3.37 & 610.83 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Table of input data related to GCI analysis.  
of the excitation assembly's acceleration, which allows for efficient mixing of the material at the payload assembly location. This research can provide a reference for the design of related equipment.  
Conceptualization, J.L., Z.L. and B.C.; methodology, J.L.; software, Z.L. and B.C.; validation, J.L., Z.L. and B.C.; formal analysis, J.L. and Z.L.; investigation, J.L., Z.L. and B.C.; resources, J.L. and B.C.; data curation, J.L., Z.L. and B.C.; writing--original draft preparation, J.L. and Z.L.; writing--review and editing, J.L. and Z.L.; visualization, J.L., Z.L. and B.C.; supervision, J.L., Z.L. and B.C.; project administration, J.L. and Z.L.; funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Shaanxi Provincial Science and Technology Department (grant no. 2024 ZC-YDP-74) and the Shaanxi University of Science and Technology (grant no. 2020 BJ-12).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.