# Analysis of Dynamic Behavior of Spray Boom under Step Excitation
## 2 Materials and Methods
### Vibration Model of Spray Boom
The boundary conditions in which one end of the spray boom is fixed and the other end is free can be expressed as the deflection and rotation angle at the fixed end A (\(x=0\)) of the spray boom being equal to zero:  
\[m_{1}\ddot{Y}_{1}+c_{1}\left(\dot{Y}_{1}-\dot{Y}_{0}\right)+k_{1}(Y_{1}-Y_{0})=0 \tag{1}\]5By introducing condition (21) into Equation (20):  
\[m_{1}\ddot{Y}_{1}+c_{1}\left(\dot{Y}_{1}-\dot{Y}_{0}\right)+k_{1}(Y_{1}-Y_{0})=0 \tag{1}\]6  
The bending moment and shear force at the extremity of the spray boom (\(x=L\)) are equal to zero:  
\[m_{1}\ddot{Y}_{1}+c_{1}\left(\dot{Y}_{1}-\dot{Y}_{0}\right)+k_{1}(Y_{1}-Y_{0})=0 \tag{1}\]7  
Bringing condition (22) into Equation (20) yields:  
\[\left\{ \begin{array}{l}(ch\beta L+cos\beta L)c_{1}+(sh\beta L+sin\beta L)c_{2 }=0\\ (sh\beta L-sin\beta L)c_{1}+(ch\beta L+cos\beta L)c_{2}=0\end{array} \right.. \tag{23}\]  
The condition that Equation (23) has a nonzero solution is  
\[\left| \begin{array}{ll}ch\beta L+cos\beta L&sh\beta L+sin\beta L\\ sh\beta L-sin\beta L&ch\beta L+cos\beta L\end{array} \right|=0. \tag{24}\]  
After the above formula is expanded and simplified:  
\[cos\beta L\ ch\beta L=-1. \tag{25}\]  
The above equation is divided into \(y=cos\beta L\), \(y=-\frac{1}{ch\beta L}\). The two parametric equations draw curves separately in the same coordinate system. The intersection of the two equations is the solution of Equation (25), and its first three orders of eigenvalues are as follows:  
\[\beta_{1}L=1.875\ \ \beta_{2}L=4.694\ \ \beta_{3}L=7.855.\]  
For each characteristic root \(i\geq 3\), it can be accurately taken as:  
\[\beta_{i}L=\left(i-\frac{1}{2}\right)\pi,i=3,4,\cdots. \tag{26}\]  
The natural frequency of section AB can be calculated by Equation (14), and the natural frequencies of section AB are:  
\[\omega_{3i}=\beta_{i}^{2}\sqrt{\frac{EI}{\rho A}},i=1,2,\cdots. \tag{27}\]  
The ratio of coefficients can be determined by substituting each characteristic root into Equation (24):  
\[\mu_{3i}=\left(\frac{c_{2}}{c_{1}}\right)_{3i}=-\frac{ch\beta_{i}L+cos\beta_{i }L}{sh\beta_{i}L+sin\beta_{i}L}=-\frac{sh\beta_{i}L-sin\beta_{i}L}{ch\beta_{i }L+cos\beta_{i}L}. \tag{28}\]  
Therefore, the vibration mode function of section AB is:  
\[X_{i}(x)=ch\beta_{i}x-cos\beta_{i}x+\mu_{3i}(sh\beta_{i}x-sin\beta_{i}x). \tag{29}\]  
The orthogonality verification of Equation (29) is not elaborated here. The vibration solution of section AB under the transverse excitation of foundation is obtained by a vibration mode superposition method. It is assumed that the deflection of section AB is:  
\[W(x,t)=\sum_{i=1}^{\infty}X_{i}(x)q_{i}(t) \tag{30}\]  
where \(q_{i}(t)\) is the generalized coordinate of the system. The velocity at each point on the spray boom can be expressed as:  
\[\frac{\partial W}{\partial t}(x,t)=\sum_{i}X_{i}(x)\dot{q}_{i}(t). \tag{31}\]  
The kinetic energy of the system is:  
\[ \begin{array}{l}E_{K}=\frac{1}{2}\int_{0}^{L}\rho A\left[\frac{\partial W}{ \partial t}(x,t)\right]^{2}dx\\ =\frac{1}{2}\int_{0}^{L}\rho A\left[\sum_{i}X_{i}(x)\dot{q}_{i}(t)\right]^{2} \\ =\frac{1}{2}\sum_{i}\dot{q}_{i}^{2}\int_{0}^{L}\rho AX_{i}^{2}(x)dx\\ =\frac{1}{2}\sum_{i}M_{3i}\dot{q}_{i}^{2}\end{array}  \tag{32}\]  
where \(M_{3i}\) is the generalized mass of the spray boom. The bending moment of each section on the spray boom of section AB can be expressed as  
\[ \begin{array}{l}M(x)=EI\frac{\partial^{2}W}{\partial x^{2}}(x,t)\\ =EI\sum_{i}X_{i}^{\prime\prime}(x)q_{i}(t).\end{array}  \tag{33}\]  
Since only the bending potential energy of the system is considered, the potential energy of the system can be expressed as:  
\[ \begin{array}{l}E_{P}=\frac{1}{2}\int_{0}^{L}EI\left[\frac{\partial^{2}W}{ \partial x^{2}}(x,t)\right]^{2}dx\\ =\frac{1}{2}\sum_{i}\int_{0}^{L}\dot{q}_{i}^{2}EI\left[X_{i}^{\prime\prime}(x )\right]^{2}dx\\ =\frac{1}{2}\sum_{i}X_{i}\dot{q}_{i}^{2}\end{array}  \tag{34}\]  
where \(K_{i}=\omega_{3i}^{2}M_{3i}\). The virtual displacement of the unilateral spray boom can be expressed as:  
\[