# Simulation Analysis of a Three-Degree-of-Freedom Low-Frequency Resonant Mixer
## 2 A Three-Degree-of-Freedom Low-Frequency Resonant Mixer
### The Design of a Three-Degree-of-Freedom Low-Frequency Resonant Mixer  
In the initial phase of design, the acceleration and amplitude of the payload assembly connected to the mixing container were designated as the design objectives. Referencing relevant theoretical research, the goal was to reduce energy input, enhance the efficiency of the equipment, and achieve lightweight design, all while ensuring structural integrity. The structural design of the various assemblies, including the mixing container and the excitation assembly, was carried out with structural optimization. After ascertaining the structure and material of each assembly, the final determination of the mass of each assembly is made. In conjunction with the desired inherent frequency of the equipment and referencing the known masses of the various components, the parameters such as spring stiffness were ultimately determined. The calculation results show that the parameters corresponding to Figure 1 are shown in Table 1. The quality data of the components are rounded to the nearest integer. It is evident that the three-degree-of-freedom low-frequency resonant mixer, when operating near its third-order natural frequency, exhibits a more gradual change in amplitude compared to the first and second-order natural frequencies. Additionally, the motion phases of the payload assembly and the excitation assembly are in opposition. This characteristic endows the mixer with the simultaneous benefits of amplitude amplification and vibration isolation [14]. In conjunction with existing research, the third-order natural frequency is adopted as the operating frequency of the equipment for further analysis. Subsequently, a comparison of the first three natural frequency will also be conducted from the perspective of vibration isolation. By programming the three-degree-of-freedom system using Matlab, the natural frequencies of the equipment were calculated [16]. The parameters for spring stiffness and the masses of the various components were input into the program to obtain the first three natural frequencies of the three-degree-of-freedom low-frequency resonant mixer. By observing whether the natural frequencies meet the design requirements, the rationality of the chosen parameters such as spring stiffness and the masses of the components was assessed. The obtained frequencies were sorted to determine the first three resonance frequencies of the equipment. \[\omega_{1}=15.30\mathrm{Hz},\omega_{2}=38.10\mathrm{Hz},\omega_{3}=59.12 \mathrm{Hz}\]  
To achieve resonance mixing, it is necessary to apply an external sine excitation force to the equipment in the vertical direction. Relevant research indicates that this three-degree-of-freedom low-frequency resonant mixer can amplify the sine excitation force by about three times when operating at the third-order natural frequency. The required working acceleration at the payload assembly is 600 m/s\({}^{2}\), and taking the inertial force at the payload assembly as three times the sine excitation force, the required sine excitation force is calculated to be 5800 N. The drive assembly consists of four sets of vibration components \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spring Rate (N/m)** & **Data** & **Quality (kg)** & **Data** \\ \hline \(k_{1}\) & 1.792 \(\times\) 10\({}^{6}\) & \(m_{1}\) & 81 \\ \(k_{2}\) & 1.260 \(\times\) 10\({}^{6}\) & \(m_{2}\) & 45 \\ \(k_{3}\) & 1.475 \(\times\) 10\({}^{6}\) & \(m_{3}\) & 29 \\ \(k_{4}\) & 8.910 \(\times\) 10\({}^{5}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of input parameters for the model. based on eccentric mass blocks, which can counteract horizontal forces and are connected to the excitation assembly. In this experiment, a four-axis synchronous control system was used, with four servo motors driving the drive assembly to generate the sine excitation force [17].