# Study on Nonlinear Vibration of Vertical Lifting Section of Bulk Grain Entrainment Ship Unloader
## 3 Solving Vibration Differential Equations Based on Perturbation Method  
Where a small dimensionless parameter \(\delta\) was introduced, Equation (15) is transformed using multi-scale perturbation theory [1] as follows:  
\[\frac{\partial^{2}\
u}{\partial t^{2}}+2\gamma\frac{\partial^{2}\
u}{\partial x \partial t}+\left(\gamma^{2}-1\right)\frac{\partial\
u^{2}}{\partial x^{2}}= \delta N(\
u) \tag{16}\]  
where mass, gyroscope, and linear stiffness operators \(\mathrm{M}=\mathrm{I},\ \mathrm{G}=2\gamma\frac{\partial}{\partial x},\ \mathrm{K}=\left( \gamma^{2}-1\right)\frac{\partial}{\partial x^{2}}\) are introduced. By substituting it into Equation (16), the dynamic equation of the composite structure is obtained as follows:  
\[\mathrm{M}\frac{\partial^{2}\
u}{\partial t^{2}}+\mathrm{G}\frac{\partial\
u} {\partial t}+\mathrm{K}\
u=\delta\mathrm{N}(\
u) \tag{17}\]  
Let the uniformly asymptotic solution of Equation (17) be  
\[\
u(x,t,\delta)=\
u(x,T_{0},T_{1})+\delta\
u_{1}(x,T_{0},T_{1}) \tag{18}\]  
where \(T_{0}=t,T_{1}=\delta t\), and the time derivative represented by \(T_{0}\) and \(T_{1}\) becomes  
\[\frac{\partial}{\partial t}=\frac{\partial}{\partial T_{0}}+\delta\frac{ \partial}{\partial T_{1}} \tag{19}\]  
\[\frac{\partial^{2}}{\partial t^{2}}=\frac{\partial^{2}}{\partial T_{0}^{2}}+2 \delta\frac{\partial^{2}}{\partial T_{0}\partial T_{1}} \tag{20}\]  
Substituting Equation (18) into Equation (17) yields  
\[\mathrm{M}\left(\frac{\partial^{2}\
u_{0}}{\partial t^{2}}+\delta\frac{ \partial^{2}\
u_{1}}{\partial t^{2}}\right)+\mathrm{G}\left(\frac{\partial \
u_{0}}{\partial t}+\delta\frac{\partial\
u_{1}}{\partial t}\right)+\mathrm{ K}(\
u_{0}+\delta\
u_{1})=\delta\mathrm{N}(\
u_{0}) \tag{21}\]  
Substituting Equations (19) and (20) into Equation (21) yields  
\[\mathrm{M}\left[\left(\frac{\partial^{2}\
u_{0}}{\partial T_{0}^{2}}+2\delta \frac{\partial^{2}\
u_{0}}{\partial T_{0}T_{1}}\right)+\delta\left(\frac{ \partial^{2}\
u_{1}}{\partial T_{0}^{2}}+2\delta\frac{\partial^{2}\
u}{ \partial T_{0}T_{1}}\right)\right]+\mathrm{G}\left[\left(\frac{\partial\
u_{ 0}}{\partial T_{0}}+\delta\frac{\partial\
u_{0}}{\partial T_{1}}\right)+ \delta\left(\frac{\partial\
u_{1}}{\partial T_{0}}+\delta\frac{\partial\
u_ {1}}{\partial T_{1}}\right)\right]+\mathrm{K}(\
u_{0}+\delta\
u_{1})=\delta \mathrm{N}(\
u_{0}) \tag{22}\]  
Make both sides of the equation \(\delta\), the coefficients are equal, and the order of \(\delta^{0}\) is obtained as follows:  
\[\mathrm{M}\frac{\partial^{2}\
u_{0}}{\partial T_{0}^{2}}+\mathrm{G}\frac{ \partial\
u_{0}}{\partial T_{0}}+\mathrm{K}\
u_{0}=0 \tag{23}\]  
\(\delta^{1}\) is obtained as follows:  
\[\mathrm{M}\frac{\partial^{2}\
u_{1}}{\partial T_{0}^{2}}+\mathrm{G}\frac{ \partial\
u_{1}}{\partial T_{0}}+\mathrm{K}\
u_where \(\mathrm{N}(\mathrm{v}_{0})=\frac{3}{2}E^{*}\Big{(}\frac{\partial\mathrm{v}_{0}}{ \partial x}\Big{)}^{2}\frac{\partial^{2}\mathrm{v}_{0}}{\partial x^{2}}\), the boundary conditions for the simplified model are obtained as follows:  
\[\mathrm{v}(x|_{x=0},t)=0,\mathrm{v}(x|_{x=1},t)=0 \tag{25}\]  
The general solution of Equation (23) is expressed in the complex form of separated variables as follows:  
\[\mathrm{v}_{0}(x,t)=\phi_{n}(x)A_{n}(T_{1})e^{i\omega_{n}T_{0}}+aa \tag{26}\]  
where \(\phi_{n}(x)\) is the nth order modal function of the system; \(A_{n}(T_{1})\) is the \(n\)th order vibration amplitude of the system; \(\omega_{n}\) is the nth natural frequency; \(aa\) is the complex conjugate of the previous term.  
The natural frequency of the \(n\)th order \(\omega_{n}\) and modal function \(\phi_{n}(x)\) are obtained as follows:  
\[\omega_{n}=n\pi\Big{(}1-\gamma^{2}\Big{)},\phi_{n}(x)=\sqrt{2}\sin(n\pi x)e^{ i\pi\gamma x} \tag{27}\]