# Study of the Movement Dynamics of a Beet Leaves Harvester
## 1 Introduction  
Recent research as well as technological and production tests show that sugar beet root tops, which were previously used primarily as an organic fertilizer, can be a raw material for biogas production [1, 2]. Therefore, its harvesting and transportation is an important task in the field of agricultural mechanization [3, 4]. In addition, high-quality cutting of the sugar beet root collar reduces the loss of sugar-bearing upper parts; this significantly increases the yield of beet raw material, and therefore the sugar yield, from each hectare of crops [5, 6]. In recent years, the most widespread method of harvesting the beet heads in many beet-growing countries has been the multistage. During this stage, the entire mass of the leaves is cut and then harvested in a transport vehicle. Next, the beet root collars are cleaned or trimmed (or both are done simultaneously, using different tools) from the leaves' remains [7, 8]. These operations are carried out sequentially. Since leaves harvesting occurs before beet roots are pulled up, the topping tools must be front-mounted on a power vehicle (a tractor or front part of the composition). Topping tools can be independent agricultural machines or topper modules of a beet harvester [9]. However, experimental studies have established that the front-mounted leaves harvester makes oscillatory movements throughout the operative process in an area that isdetermined by: (i) the relief of the field surface; (ii) the tractor forward speed; (iii) the location of the feeler wheels relative to the machine's suspension system; and so on. The suspension system of the beet leaves harvester causes oscillation of the machine in the vertical plane during the operative conditions. This affects the quality of the technological process itself, i.e., the uniform cutting and collection of the leaves from the collar of root crops along the entire working width of the machine without significant losses [10]. Despite the wide distribution and use of front-mounted beet leaves harvester modules, fundamental analytical studies of their oscillatory motion over irregular soil surfaces are practically absent. [PERSON] et al. [11] presented a theoretical study of a tractor-mounted beet harvester's oscillations using the original Lagrange equations of the 2 nd kind. However, this work does not consider all the forces acting on this dynamic system, so the obtained mathematical model of its motion does not adequately describe the real process. In addition, this problem considers a system that consists of two differential equations with respect to two generalized coordinates, even though the problem can be solved accurately enough using only one such coordinate. Applying the same methodology as in [12], it is possible to build a more accurate mathematical model of the machine's motion over an uneven ground surface. This makes it possible to study the influence of construction parameters on the considered motion along the rows of sugar beet roots. The authors have developed a new universal machine that performs the technological process using the principle of a mower-shredder, which is front-mounted on a wheeled aggregating tractor. This machine uses a rotary cutting unit. The articulated cutting knives mounted on the drive drum have an arc shape; they rotate in the longitudinal-vertical plane and are supported by feeler wheels, providing cutting of the leaves' main mass across the working width of the machine [13; 14; 15; 16]. The green mass collected from the entire working width is loaded into the body of the nearby moving vehicle. The harvesting efficiency of this type of leaves harvester is strongly affected by the motion stability of its rotor cutting unit in the longitudinal-vertical plane, the capability of its feeler to determine the unevenness of the soil surface in its working width, and the parameters of the tractor connection mechanism, among others [17].