## 3 Results and Discussion
\begin{array}{l}x=\left\{\lambda_{2}\left[\frac{2g\ell^{2}\omega+\frac{k \gamma}{m}-2g\omega}{4\omega^{3}+\omega\left(2f\omega+\frac{k}{m}\right)^{2}}- \frac{(\cos\gamma+f\sin\gamma)kV_{n}}{m\omega^{2}}\right]-\frac{4g\ell\omega+ \frac{k\gamma}{m}}{4\omega^{2}+\left(2f\omega+\frac{k}{m}\right)^{2}}\right\} \times\\ \times(\lambda_{1}-\lambda_{2})^{-1}\e^{\lambda_{1}t}-\\ -\left\{\lambda_{1}\left[\frac{2g\ell^{2}\omega+\frac{k\gamma}{m}-2g\omega}{4 \omega^{3}+\omega\left(2f\omega+\frac{k}{m}\right)^{2}}-\frac{(\cos\gamma+f \sin\gamma)kV_{n}}{m\omega^{2}}\right]-\frac{4g\ell\omega+\frac{k\gamma}{m}}{4 \omega^{2}+\left(2f\omega+\frac{k}{m}\right)^{2}}\right\}\times\\ \times(\lambda_{1}-\lambda_{2})^{-1}\e^{\lambda_{2}t}+\frac{4g\ell\omega+ \frac{k\gamma}{m}}{4\omega^{3}+\omega\left(2f\omega+\frac{k\gamma}{m}\right) ^{2}}\sin\omega t+\frac{2g\ell^{2}\omega+\frac{k\gamma}{m}-2g\omega}{4\omega^ {3}+\omega\left(2f\omega+\frac{k}{m}\right)^{2}}\cos\omega t-\\ -\frac{(\cos\gamma+f\sin\gamma)kV_{n}}{m\omega^{2}}.\end{array}  \tag{50}\]  
Differentiating Expression (50) with respect to time \(t\), we obtain an expression for the determination of the relative velocity of a haulm particle for the particular case under consideration:  
\[\begin{array}{l}\dot{x}=\left\{\lambda_{2}\left[\frac{2g\ell^{2}\omega+ \frac{k\gamma}{m}-2g\omega}{4\omega^{3}+\omega\left(2f\omega+\frac{k}{m}\right) ^{2}}-\frac{(\cos\gamma+f\sin\gamma)kV_{n}}{m\omega^{2}}\right]-\frac{4g\ell \omega+\frac{k\gamma}{m}}{4\omega^{2}+\left(2f\omega+\frac{k}{m}\right)^{2}} \right\}\times\\ \times(\lambda_{1}-\lambda_{2})^{-1}\lambda_{1}\e^{\lambda_{1}t}\\ -\left\{\lambda_{1}\left[\frac{2g\ell^{2}\omega+\frac{k\gamma}{m}-2g\omega}{4 \omega^{3}+\omega\left(2f\omega+\frac{k\gamma}{m}\right)^{2}}-\frac{(\cos \gamma+f\sin\gamma)kV_{n}}{m\omega^{2}}\right]-\frac{4g\ell\omega+\frac{k \gamma}{m}}{4\omega^{2}+\left(2f\omega+\frac{k\gamma}{m}\right)^{2}}\right\} \times\\ \times(\lambda_{1}-\lambda_{2})^{-1}\lambda_{2}\e^{\lambda_{1}t}+\frac{4g \ell\omega+\frac{k\gamma}{m}}{4 So, a mathematical model of the movement of a particle of cut sugar beet haulm along the blade of the thrower of the loading mechanism has been built, taking into account the influence of the airflow created by the rotation of the thrower upon the movement of the haulm. As a result, the law of motion of a particle along the blade and the law of change in the relative velocity of its motion as functions of time, as well as the design, kinematic, and dynamic parameters of the thrower blades. For numerical simulation of the obtained mathematical models on a PC, we have compiled a program for numerical calculations in the MathLAB 9.5 program. Based on the results of the numerical calculations performed on a PC, graphs of dependences of the absolute speed \(V_{a}\) of the descent of the haulm particle \(M\) from the end of the blade upon the length \(l\) of the blade and upon the angular velocity \(\omega\) of rotation of the thrower blades were built. There are also obtained graphical dependences of the relative speed \(x\) of the particle \(M\) movement along the blade on the speed of the airflow \(V_{n}\) at different angular speeds \(\omega\) of rotation of the thrower. Figure 3 shows the dependence of the absolute speed \(V_{a}\) of descent of the haulm particle \(M\) from the end of the blade, obtained as a result of numerical simulation of the developed mathematical model on a PC, upon the angular velocity \(\omega\) of rotation of the thrower and the length \(l\) of the blade. Analysis of dependencies, presented in Figure 3, allows one to conclude that an increase in the angular velocity \(\omega\) of rotation of the thrower and the length \(l\) of its blade leads to an increase in the absolute speed \(V_{a}\) of descent of the haulm particle \(M\) from the end of the blade.