# Analysis of Dynamic Behavior of Spray Boom under Step Excitation
## 1 Introduction  
The proportion of small- and medium-sized boom sprayers with spray ranges of 10-14.9 m in China accounts for more than 70% according to statistics in 2017 [1]. A boom sprayer working in the field is affected by uncertain factors, such as uneven soil, speed change, uneven tire load, and liquid sloshing. Such uncertainty will easily cause harmful movements of the spray boom (e.g., rolling and yawing) and thereby induce a random change of nozzle height from the ground, seriously affecting the spray distribution form and leading to chemical fertilizer/pesticleakage and heavy spray. Due to the irregular movement of the spray boom, the variation coefficient of overall fog distribution fluctuates from 0% to 800% (the ideal droplet deposition is 100%). The rolling and vertical elastic deformation of the spray boom have a great influence, especially at its extremity when its motion amplitude is the largest, the deposition amount maximizes to 760% and minimizes to 0% [2, 3, 4]. Since the boom sprayer is a complex multi-body system, establishing an accurate motion model of the spray boom is of great significance to explore its vibration control. Researchers established dynamic models to describe the dynamic behavior of the spray boom when the boom sprayer walked in the field. [PERSON] derived the motion of the spray boom through a strongly coupled nonlinear time-varying differential equation, butthe calculation target was cumbersome, and the results were inconsistent [5]. Subsequently, [PERSON] proposed the modeling technology of a linear time-invariant non-gyro rigid multi-body system with tree structure. The motion equation of the spray boom was analytically derived on basis of the Roberson-Wittenburg method, but its versatility was lowered by the requirement of complex computer analysis and derivation program [6; 7]. [PERSON] derived motion equations of the spray boom through the finite element method and discrete element method separately [8]. With the first method, when the stiffness matrix of the flexible body was derived, consideration was given only to the bending caused by bending moment, but not to the tension or torsion of the spray boom [8]. With the second method, the mass distribution of the spray boom was completed by the shape function, and the mass, damping, and stiffness matrices that depended on the shape function were acquired, which decreased the modeling accuracy [8]. Later, [PERSON] established a linear flexible multi-body model of the spray boom by a discrete element method, but this model considered the boom sprayer body and boom suspension as rigid bodies, which seriously reduced the modeling accuracy [9; 10]. [PERSON] studied the motion modeling of a spray boom in the height difference between the left and right tire as the system input. The regression sliding average model (ARMAX) was used to obtain the incentive response model of the front and rear wheels, and then through convolution, the excitation response to the front and rear wheels is synthesized into the response model of spray boom. However, the real input of the boom sprayer is the vertical displacement of the tire [11]. [PERSON] built a suspension-boom model with spring dampers, but the vibration impact of the sprayer body was not considered during modeling [12]. He established a finite element model of the spray boom based on the ANSYS platform and performed a numerical mode analysis. The suspension-spray boom model was established in ADAMS platform based on the established model and modal information, and the elastic deformation at the extremity of the spray boom were analyzed. However, the finite element model established on the platform of ANSYS and ADAMS does not consider the flexibility relationship between components [13]. In this study, based on the Euler-Bernoulli beam theory, the boom sprayer body and boom suspension are regarded as equivalent to the spring damping system, and the influence of the flexible boom sprayer body and boom suspension on the elastic vibration of the spray boom is fully considered.