# Study on Nonlinear Vibration of Vertical Lifting Section of Bulk Grain Entrainment Ship Unloader
## 3 Solving Vibration Differential Equations Based on Perturbation Method
### Solution Based on Galerkin Discrete Analysis  
The dynamic Equation (12) of the structure contains the variables of time and space coordinates, and it is a differential equation that is not convenient for analysis and calculation. Therefore, the equation was discretized and decoupled using the Galerkin method [6], which obtained a general ordinary differential equation that was easy to solve and calculate. According to Equation (27), the solution \(\mathrm{v}(x,t)\) of the vibration equation is obtained by using the time function \(q_{r}(t)\) and spatial functions \(\phi_{r}(x)\) represents as follows:  
\[\mathrm{v}(x,t)=\sum_{\sigma=1}^{n}q_{r}(t)\phi_{r}(x) \tag{28}\]  
The spatial function is the main mode of vibration \(\sin(r\pi x)\) of the lateral motion, where \(r=1,2,3,\ldots n\) in Equation (28), \(\mathrm{v}(x,t)\) is expressed as follows:  
\[\mathrm{v}(x,t)=\sum_{\sigma=1}^{n}q_{r}(t)\mathrm{sin}(r\pi x) \tag{29}\]  
Since the vibration of the vertical material-conveying system mainly affects the structural stability with low-frequency vibration, the first two modes of the system are mainly used. Equation (29) is substituted into Equation (14), the Galerkin of the first two orders is truncated, and the orthogonality of the modal function is used to multiply \(\sin(j\pi x)\) on both sides of the equation. The partial differential Equation (14) is converted into a second-order nonlinear ordinary differential equation after integral operation on the interval \([0,1]\) as follows:  
\[\frac{1}{2}\ddot{q_{1}}-\frac{8}{3}\gamma\dot{q_{2}}-\frac{1}{2}\Big{(}\gamma ^{2}-1\Big{)}\pi^{2}q_{1}-\frac{3}{16}E\pi^{2}q_{1}^{2}\ddot{q_{1}}-\frac{3}{ 2}E\pi^{2}q_{2}^{2}\ddot{q_{1}}=0 \tag{30}\]  
\[\frac{1}{2}\ddot{q_{2}}+\frac{8}{3}\gamma\dot{q_{1}}-2\Big{(}\gamma^{2}-1 \Big{)}\pi^{2}q_{2}-\frac{3}{8}E\pi^{2}q_{1}^{2}\ddot{q_{2}}-\frac{3}{4}E\pi^ {2}q_{2}^{2}\ddot{q_{2}}=0 \tag{31}\]  
Equations (30) and (31) are the second-order nonlinear ordinary differential equation of the transverse vibration of the vertical material-conveying system. The first- and second-order vibration responses of the structure are obtained based on differential equations. Based on the experimental method presented in [23], the elastic modulus of common varieties of soybean, wheat, and corn was measured in this study, and the average elastic modulus values obtained are shown in Table 1.  
Figure 6a,b show the first-order and second-order responses of different filling degrees k, respectively, where the conveyed material was soybean particles. And the fluctuation period changed significantly when the k value increased from 0.1 to 0.2. The frequency of \(q_{1}\) and \(q_{2}\) was calculated, which showed the frequency of \(q_{1}\) increases from 0.2823 Hz to 0.3283 Hz and the frequency of \(q_{2}\) increases from 0.5501 to 0.6462 Hz, following the increase in k value from 0.1 to 0.2. In addition, the wave patterns of \(q_{1}\) and \(q_{2}\) were consistent with different filling degrees, and the fluctuation period of \(q_{1}\) was approximately twice that of \(q_{2}\). Figure 6: (**a**) The first—order response \(q_{1}\) of different filling degrees k (soybean as a sample). (**b**) The second—order response \(q_{2}\) of different filling degrees k (soybean as a sample). Figure 7a,b show the first-order and second-order responses of different filling degrees k, respectively, where the conveyed material was wheat particles. And the fluctuation period change significantly when the k value increased from 0.2 to 0.3. The frequency of \(q_{1}\) and \(q_{2}\) was calculated, which showed that the frequency of \(q_{1}\) increases from 0.2815 Hz to 0.3205 Hz and the frequency of \(q_{1}\) increases from 0.5567 to 0.6349 Hz, following the increase in the k value from 0.2 to 0.3. In addition, the wave patterns of \(q_{1}\) and \(q_{2}\) were consistent with different filling degrees, and the fluctuation period of \(q_{1}\) was approximately twice that of \(q_{2}\).