# Alfafa Stalk Crushing Simulation Test and Parameter Optimization Method
## 3 Key Steps in Modal Analysis and Results  
Modal analysis is a useful technique for determining the natural frequency and mode shape of a structure. Through obtaining this information, it is possible to prevent the external excitation frequency from overlapping with the natural frequency of the structure, thus avoiding the risk of resonance. This not only extends the life of the machine but also ensures its performance. The theoretical model for modal analysis is represented by Equation (1). \[[M]\big{\{}\tilde{x}\big{\}}+[C]\big{\{}\dot{x}\big{\}}+[K]\big{\{}x\big{\}}=\{F (t)\} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Alfalfa Poisson’s ratio & 0.4 \\ Alfalfa density/kg\(\cdot\)m\({}^{-3}\) & 256 \\ Alfalfa shear modulus/MPa & 5.00 \\ Steel 45 Poisson’s ratio & 0.3 \\ Steel 45 density/kg\(\cdot\)m\({}^{-3}\) & 7800 \\ Steel 45 shear modulus/MPa & 70,000 \\ Stalk-stalk collision recovery coefficient & 0.11 \\ Stalk–stalk static friction coefficient & 0.45 \\ Stalk–stalk rolling friction coefficient & 0.08 \\ Stalk–steel 45 collision recovery coefficient & 0.16 \\ Stalk–steel 45 static friction coefficient & 0.54 \\ Stalk–steel 45 rolling friction coefficient & 0.24 \\ Q235 elastic modulus/MPa & \(210\times 10^{3}\) \\ Q235 density/ton\(\cdot\)mm\({}^{-3}\) & \(7850\times 10^{-12}\) \\ Q235 Poisson’s ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. In the formula, \([M]\) is the mass matrix; \([C]\) is the damping matrix; \([K]\) is the stiffness matrix; and \(\{F(t)\}\) is the external load on the structure. The natural frequency is mainly related to its own material, mass, and dimensions and does not depend on the external load, so \(\{F(t)\}\) is zero, and the damping effect is considered to be approximately zero; therefore, Equation (1) can be expressed as Equation (2). \[[M]\big{\{}\ddot{\dot{x}}\big{\}}+[K]\{x\}=\{0\} \tag{2}\]  
The frame, which is the main load-bearing component of the crusher, is primarily constructed using square hollow steel that is welded together. To reduce the calculation running time in the modal analysis, the frame structure was simplified appropriately. The frame, which was welded using multiple parts, was considered as a single part. The finite element model used is shown in Figure 3.  
The vibration of the frame is determined by the superposition of various modes. Each mode represents a decoupling of the originally coupled modes. In theory, there are infinite orders of modes. However, the influence of lower-order vibration shapes on the structure is greater than that of higher-order vibration shapes. Therefore, the dynamic characteristics of the structure are primarily determined by the lower-order vibration shapes. In this study, ABAQUS finite element software was used for calculations, and the Lanczos solver was employed to extract the first six natural frequencies and vibration shapes. The specific vibration shapes and frequency information can be found in Figure 4 and Table 2.  
The rotation speed range of the pulverizing device was set to 1800-5400 r/min, which corresponds to an excitation frequency range of 30-90 Hz. Based on the modal analysis, it was observed that the first two natural frequencies of the rack fell within this excitation frequency range, indicating a need for optimization. Consequently, the thickness of the square tube steel in the frame was increased from the original 6 mm to 8 mm, and beam and column structures were added, and another modal analysis was conducted. The results showed that the natural frequencies of the first two modes were 98.21 Hz and 100.24 Hz, respectively, and the maximum displacement when resonance