# Analysis of Dynamic Behavior of Spray Boom under Step Excitation
## 2 Materials and Methods
### Determination of Boom Sprayer Parameters  
To verify the correctness of the proposed continuous vibration model of the spray boom based on multi-body vibration, the SWAN3 WP-500 self-propelled boom sprayer (Essen Agricultural Machinery Changzhou Co., Ltd., Changzhou, China) was selected as a test model. The mass of this boom sprayer is 1250 kg, and the load-bearing mass m\({}_{1}\) of the front axle is 480 kg. The spray width is 12.5 m, and the extension length of the unilateral spray boom from the boom sprayer body is 5 m.  
The stiffness of the boom sprayer tire is related to the rim diameter, cross-sectional width, inflation pressure, service years of the tire, and other parameters. So far, a large number of experimental studies provide some valuable experience data and empirical formulas [15; 16; 17; 18]. The tire stiffness empirical formula obtained by the test method of [PERSON] et al. [13] was used here:  
\[k_{1}=172-1.77D+5.6H+34 WDP/10,000 \tag{49}\]  
where \(k_{1}\) is the tire stiffness, kN\(\cdot\)m\({}^{-1}\); \(D\) and \(W\) are the rim diameter and tire section width, respectively; \(H\) is the number of used years of the tire; and \(P\) is the tire inflation pressure, kPa. \(D\) and \(W\) were directly obtained from the tire model. The service life is one year, and the inflation pressure is 220 kPa. Hence, the tire stiffness is about 585.65 kN\(\cdot\)m\({}^{-1}\). Tire damping is related to tire deformation, tire inflation pressure, vibration velocity, and excitation frequency [19]. Herein, a nonlinear model of tire damping established by [PERSON] et al. through a harmonic excitation method [20] is used. The tire damping is about 2682.6 N\(\cdot\)s\(\cdot\)m\({}^{-1}\). The equivalent stiffness \(k_{2}\) of the trapezoidal suspension and lifting mechanism of the spray boom is mainly the bending deformation of the connecting rod and the hydraulic oil stiffness of the electric hydraulic cylinder. The equivalent damping \(c_{2}\) of the mechanism is mainly the relative motion of the connecting point at the end of the connecting rod. Many experimental studies provide valuable empirical data and empirical formulas [21; 22]. Herein, the suspension and lifting mechanism is simulated by dynamically analyzing the boom suspension performance of the boom sprayer [22]. The \(k_{2}\) and \(c_{2}\) are taken to be 84 kN\(\cdot\)m\({}^{-1}\) and 472.4 N\(\cdot\)s\(\cdot\)m\({}^{-1}\), respectively. The related parameters of the boom sprayer are shown in Table 1.  
The material of the spray boom is aluminum alloy 6063-T5 that has two cross-section structures (Figure 4). When the spray boom is working, the long side of the cross-section is vertical, and the short side is horizontal. The type A cross-section structure of the spray boom is used in this experiment. Related parameters are shown in Table 2. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Value** \\ \hline Boom sprayer mass, \(M_{1}/(\text{kg})\) & 1250 \\ Load-bearing mass of front axle, \(m_{1}/(\text{kg})\) & 480 \\ Tire stiffness of boom sprayer, \(k_{1}/(\text{kN}\cdot\)m\({}^{-1})\) & 585.65 \\ Tire damping of boom sprayer, \(c_{1}/\left(\text{N}\cdot\)s\(\cdot\)m\({}^{-1}\right)\) & 2682.6 \\ Suspension stiffness of spray boom, \(k_{2}/\left(\text{kN}\cdot\)m\({}^{-1}\right)\) & 84 \\ Suspension damping of spray boom, \(c_{2}/\left(\text{N}\cdot\)s\(\cdot\)m\({}^{-1}\right)\) & 472.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the boom sprayer. Figure 4: Cross-sectional structures of the spray boom. The damping ratio of the spray boom is determined by a free vibration test [23]. In brief, an acceleration sensor is installed on the extremity of the spray boom, and then a vertical upward pulling force is applied to make the extremity pull up vertically and stay static. Then, the pulling force is removed, and the acceleration data of the extremity are collected.