## 4 Evaluation and Calculation Methods  
By analyzing the two-degree-of-freedom vibration system model, it was determined that the sensor system can be expressed as an undamped free vibration system, assuming that the damping coefficient is negligible. The differential equation of motion for the free vibration system can be expressed as Equation (6).  
\[[m]\Big{\{}\overset{\cdot}{x}(t)\Big{\}}+[k]\{x(t)\}=0 \tag{6}\]  
There into  
\[\left\{\begin{array}{l}\left\{x(t)=\left\{\begin{array}{l}x_{1}(t)\\ x_{2}(t)\end{array}\right\},[m]=\left[\begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array}\right]\\ \left[k\right]=\left[\begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array}\right]=\left[\begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array}\right]\end{array}\right. \tag{7}\]  
So the differential equation of motion is expressed as Equation (8),  
\[\left\{\begin{array}{l}m_{1}\overset{\cdot}{x}_{1}(t)+k_{11}x_{1}(t)+k_{12}x _{2}(t)=0\\ m_{2}\overset{\cdot}{x}_{2}(t)+k_{21}x_{1}(t)+k_{22}x_{2}(t)=0\end{array}\right. \tag{8}\]  
Assuming that the system undergoes simple harmonic vibration with the same frequency and phase Angle, the solution of vibration displacement of the system can be written as Equation (9).  
\[\begin{array}{l}x(t)=\left\{\begin{array}{l}x_{1}(t)\\ x_{2}(t)\end{array}\right\}=\left\{\begin{array}{l}X_{1}\\ X_{2}\end{array}\right\}\cos(\omega_{\mathrm{n}}t-\varphi_{0})\\ =\left\{X\right\}\cos(\omega_{\mathrm{n}}t-\varphi_{0})\end{array} \tag{9}\]  
where \(x_{1}\) and \(x_{2}\) are the maximum values of \(x_{1}(t)\) and \(x_{2}(t)\), \(m_{1}\) and \(m_{2}\) are the mass of the moving object, \(\{X\}\) is the amplitude column vector, \(\omega_{\mathrm{n}}\) is the natural frequency of free vibration of the system and \(\varphi_{0}\) is the initial phase of free vibration. Substitute Equation (9) into Equation (8) to obtain the algebraic system of amplitude column vector \(\{X\}\), as shown in Equation (10),  
\[\left\{\begin{array}{l}\left(k_{11}-m_{11}\omega_{n}^{2}\right)X_{1}+k_{12} X_{2}=0\\ k_{21}X_{1}+\left(k_{22}-m_{22}\omega_{n}^{2}\right)X_{2}=0\end{array}\right. \tag{10}\]  
By solving Equation (10), the natural frequency of the vibration system is Equation (11) and the amplitude response is Equation (12).  
\[\left\{\begin{array}{l}\omega_{n1,2}^{2}=\frac{-bx\sqrt{b^{2}-4 ac}}{2a}\\ a=m_{11}m_{22},b=-(m_{11}k_{22}+m_{22}k_{11})\\ c=k_{11}k_{22}-k_{12}k_{21}\end{array}\right. \tag{11}\]  
\[\left\{\begin{array}{l}\left|X_{1}\right|\\ \left|X_{2}\right|\end{array}\right\}=\frac{\delta_{\mathrm{st}}}{\left(1- \lambda_{1}^{2}\right)\left(\alpha^{2}-\lambda_{1}^{2}\right)-\mu\alpha^{2} \lambda_{1}^{2}}\left\{\begin{array}{l}\alpha^{2}-\lambda_{1}^{2}\\ \lambda_{1}^{2}\end{array}\right\} \tag{12}\]  
As proposed by [PERSON] et al. [18], the typical vibration frequency of the harvester vibrating screen ranges from 4 to 6 Hz. Moreover, the damping material rubber exhibits a stiffness of 30 \(\mathrm{KN/m}\), while the sponge has a stiffness of 80 \(\mathrm{N/m}\). To effectively showcase the vibration-damping effect, a vibration source frequency of 5 Hz was selected during the comparative test process. By combining the charge voltage collected by the charge amplifier and the vibration displacement data of the impact plate, the amplitude response results of \(x_{1}\) and \(x_{2}\) were derived from Equation (6). Subsequent analysis and comparison of these results demonstrate the vibration-damping effect.