# Analysis of Dynamic Behavior of Spray Boom under Step Excitation
## 2 Materials and Methods  
### Vibration Model of Spray Boom  
The spray boom is connected by trapezoidal suspension to the boom-lifting mechanism, which is fixed on the body of the boom sprayer. The body is rigidly connected to an inflatable tire through the vertical shaft and the bevel gear. There is no suspension system. The schematic diagram is shown in Figure 1. When the boom sprayer encounters obstacles in the field, it is equivalent to a step excitation to the body of the boom sprayer, causing the body to vibrate vertically, and the spray boom attached to the body will produce vertical oscillation and elastic deformation. According to the connection characteristics of the spray boom, the vertical shaft mechanism of the boom sprayer tire, the connecting tire, and the body is equivalent to a spring damping system. The spray boom can be regarded as an elastic long cantilever beam fixed on a flexible foundation linked by two hinge points. As shown in Figure 2.  
The balance position of the boom sprayer at rest is taken as the zero point, and the vertical axis is taken as the positive axis. When the front tire is excited by the step displacement \(Y_{0}(t)\), the response of the load-bearing mass \(m_{1}\) of the front axle of the boom sprayer is \(Y_{1}(t)\). \[m_{1}\ddot{Y}_{1}+c_{1}\left(\dot{Y}_{1}-\dot{Y}_{0}\right)+k_{1}(Y_{1}-Y_{0})=0 \tag{1}\]  
Let the movement of the vehicle body relative to the ground be \(Z=Y_{1}-Y_{0}\). Then, Equation (1) can also be expressed as  
\[m_{1}\ddot{Z}+c_{1}\dot{Z}+k_{1}Z=-m_{1}\ddot{Y_{0}}. \tag{2}\]  
If the excitation function \(F\) and variable \(x\) are used to replace \(-m_{1}\ddot{Y_{0}}\) and \(Z\), respectively, then Equation (2) is similar to the following equation:  
\[m_{1}\ddot{x}+c_{1}\dot{x}+k_{1}x=F. \tag{3}\]  
Therefore, the solution of the system excited by external force can be directly applied to the system excited by foundation. In other words, \(F\) in Equation (3) is used to replace \(-m_{1}\ddot{Y_{0}}\), and the solution of Equation (2) can be determined according to its solution. For an underdamped system subjected to base excitation, the relative displacement is obtained by convolution integral [14]:  
\[Z(t)=\frac{1}{m_{1}w_{1d}}\int_{0}^{t}F(\tau)e^{-\zeta_{1}w_{1n}(t-\tau)}sinw_ {1d}(t-\tau)d\tau \tag{4}\]  
where \(w_{1d}\), \(w_{1n}\), and \(\zeta_{1}\) are the damping vibration frequency, natural frequency, and damping ratio of the boom sprayer, respectively, and \(F(\tau)\) is any external force. Figure 1: Sprayer body and boom assembly. Figure 2: The multi-body vibration model of the boom sprayer; \(k_{1}\): stiffness of the boom sprayer tire, N\(\cdot\)m\({}^{-1}\); \(c_{1}\): tire damping of the boom sprayer, N\(\cdot\)s\(\cdot\)m\({}^{-1}\); \(k_{2}\): equivalent stiffness of the boom trapezoidal suspension and lifting mechanism, N\(\cdot\)m\({}^{-1}\); \(c_{2}\): equivalent damping of trapezoidal suspension and lifting mechanism of spray boom, N\(\cdot\)s\(\cdot\)m\({}^{-1}\); \(m_{1}\): bearing mass of the front bridge of the boom sprayer, kg; \(m_{2}\): mass of the fixed end of the spray boom, kg. Equation (4) can be simplified as follows:  
\[Z(t)=\frac{F_{0}}{k_{1}}\Bigg{[}1-\frac{1}{\sqrt{1-\zeta_{1}^{2}}}e^{-\zeta_{1}w_ {1d}t}\cos(w_{1d}t-\varnothing)\Bigg{]} \tag{5}\]  
where \(F_{0}\) is the external force applied to the double front wheels of the boom sprayer, \(\varnothing=arctan\frac{\zeta_{1}}{\sqrt{1-\zeta_{1}^{2}}}\). Therefore, the response of the front axle of the boom sprayer is:  
\[Y_{1}(t)=Z(t)+Y_{0}=\frac{F_{0}}{k_{1}}\Bigg{[}1-\frac{1}{\sqrt{1-\zeta_{1}^{2 }}}e^{-\zeta_{1}w_{1d}t}\cos(w_{1d}t-\varnothing)\Bigg{]}+Y_{0}. \tag{6}\]  
It is assumed that the front wheels of the boom sprayer encounter a 0.1 \(\times\) 0.1 m square obstacle during driving. Let the elapsed time be \(t_{0}\).