# Study of the Movement Dynamics of a Beet Leaves Harvester
## 3 Results and Discussion  
The values of coefficients \(A_{2}\), \(A_{1}\), \(A_{\omega}\), \(B_{1}\) and \(B_{\omega}\) of transfer function (25) are determined by the following construction parameters \(c\), \(\mu\), \(L\) and \(I_{oy}\) (see Equation (19)). In theoretical research we considered the following values: \(c\) in the range from 115 kN m\({}^{-1}\) to 515 kN m\({}^{-1}\); \(\mu\) in the range from 0.35 kN s m\({}^{-1}\) to 1.35 kN s m\({}^{-1}\); \(L\) in the range from 3.8 m to 4.8 m; \(I_{oy}\) in the range from 3000 to 3500 kg m\({}^{2}\). These values have been conditionally taken based on the corresponding ones reported in Table 1, then increased or decreased by an acceptable value. Furthermore, for the simulation, we have considered the oscillations frequency range to be 0-40 s\({}^{-1}\) as 40 s\({}^{-1}\) was the highest value we located when examining long term (more than 10 years) measurements of fluctuations carried out in the longitudinal profile of fields with sugar beet roots. It follows from the analysis of the calculation results that, at frequencies of external perturbation from 0 to 20 s\({}^{-1}\) change in the parameters \(\mu\) and \(I_{oy}\) in the above ranges of their values do not practically affect the corresponding amplitude-frequency and phase-frequency characteristics of this dynamic system. In the frequency interval from 20 to 40 s\({}^{-1}\), there is some qualitative difference between the corresponding frequency characteristics. However, the quantitative insignificance of this difference allows us to formulate the following conclusion: the front harvester with the values of parameters \(\mu\) and \(I_{oy}\), which range from 0.35 to 1.35 kN s m\({}^{-1}\) and from 3000 to 3500 kg m\({}^{2}\), respectively, is invariant to the influence of external vertical vibrations of the agricultural background's longitudinal profile, changing in the frequency range from 0 to 40 s\({}^{-1}\) by the harmonic law (1). On the other hand, the change in values of the stiffness coefficient (\(c\)) of the tyres of the harvester's supporting and feeler wheels has a significant effect on the frequency characteristics of this dynamic system in the process of its response to the input disturbance. In the case where \(c\) = 115 kN m\({}^{-1}\), we have the worst amplitude-frequency characteristic (curve 1 in Figure 3). Not only it is farthest from zero, but it also has an unwanted resonant peak at frequency 36 s\({}^{-1}\). Figure 3: Amplitude–frequency characteristics of the dynamic system when examining the disturbance at different values of the tyres stiffness coefficient \(c\) of the tyres of the harvester’s feeler wheels: curve 1, \(c\) = 115 kN m\({}^{-1}\); curve 2, \(c\) = 315 kN m\({}^{-1}\);curve 3. \(c\) = 515 kN m\({}^{-1}\). Increasing the stiffness coefficient of tyres of the topper harvester's support feeler wheels to 315 kN m\({}^{-1}\) is characterized by a sharp enough desirable decrease in the amplitude-frequency characteristics of this dynamic system (curve 2 in Figure 3). At the frequency of 36 s\({}^{-1}\), this decrease is 41.2%. This result is quite logical since the increase in the stiffness of the wheel tyres is accompanied by a simultaneous decrease in their elastic properties. The smaller the wheel tyres elastic properties, the lower the machine's propensity to effects from its own vertical vibrations. At a further increase of the coefficient \(c\) value by 200 kN m\({}^{-1}\), the estimated amplitude-frequency response also decreases (curve 3 in Figure 3). At the same time, the intensity of this process significantly decreases. Taking this into account, it can be stated that there is no need to take the value of coefficient \(c\) of tyre stiffness of the leaves harvester feeler wheels higher than 315 kN m\({}^{-1}\). We consider how this can be used in practice. The tyre stiffness factor can be calculated by the following equation [28; 29]:  
\[c=2\pi\rho_{t}\cdot\sqrt{r_{c}\cdot r_{o}}, \tag{32}\]  
where:  
\(\rho_{t}\) [Pa] is the inflation pressure in the tyre.