# Study of the Movement Dynamics of a Beet Leaves Harvester
## 2 Materials and Methods
### Mathematical Model  
To obtain the law of change of the angle \(\varphi\) in analytical form, let us apply differential equations in the Lagrangian form of the second kind [21]:  
\[\frac{d}{dt}\left(\frac{\partial T}{\partial\dot{\varphi}}\right)-\frac{ \partial T}{\partial\varphi}=\mathcal{Q}_{\varphi}-\frac{\partial P}{\partial \varphi}-\frac{\partial R}{\partial\dot{\varphi}}, \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Unit** & **Value** \\ \hline Machine weight with haulm (\(G\)) & N & 9300 \\ Operating forward speed (\(V\)) & m\(\cdot\)s\({}^{-1}\) & 2.2–2.5 \\ Performance & ha\(\cdot\)h\({}^{-1}\) & 1.08–1.35 \\ Beet leaves harvester inertia moment (\(I_{oy}\)) & kg\(\cdot\)m\({}^{2}\) & 3250 \\ Tyres for support feeler wheels & - & 7.50R16 \\ Radius for support feeler wheels & m & 0.365 \\ Inflation pressure of the support feeler wheels tyres & MPa & 0.125 \\ Stiffness coefficient (\(c\)) of the support feeler wheels tyres & kN\(\cdot\)m\({}^{-1}\) & 315 \\ Damping coefficient (\(\mu\)) of the support feeler wheels tyres & kN\(\cdot\)s\(\cdot\)m\({}^{-1}\) & 0.85 \\ Rotor radius of the leaves cutter & m & 0.30 \\ Distance from the machine suspension axis to the axis of the & m & 4.3 \\ support feeler wheels (\(L\)) & m & 3.0 \\ Distance from the machine suspension axis to the centre of & m & 0.01–0.03 \\ gravity (\(L\)-\(l\)) & m & 0.70 \\ Soil profile fluctuation amplitude (\(h_{0}\)) & m & 0.01–0.03 \\ Soil profile unevenness step (\(l_{1}\)) & m & 0.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main technical parameters of the front-mounted beet leaves harvester. where:  
\(T\): kinetic energy of the considered dynamic system. \(Q_{\varphi}\): generalized force. \(P\): potential energy of the system. \(R\): dissipative function (Rayleigh function). \(\varphi\): generalized coordinate. \(\dot{\varphi}\): the generalized speed. The kinetic energy \(T\) of the dynamic system under consideration is determined as follows [21; 22; 23]:  
\[T=\frac{I_{oy}\cdot\dot{\varphi}^{2}}{2}, \tag{3}\]  
where \(I_{oy}\) is the moment of inertia of the leaves machine relative to the axis \(Oy\), perpendicular to the longitudinal-vertical plane \(zOx\) and passing through the point \(O\), kg\(\cdot\)m\({}^{2}\). Using expression (3), we find that:  
\[\frac{\partial T}{\partial\varphi}=0;\quad\frac{d}{dt}\left(\frac{\partial T }{\partial\dot{\varphi}}\right)=I_{oy}\cdot\ddot{\varphi}. \tag{4}\]  
The potential energy of this dynamic system is formed by the work of the elastic forces of the supporting (feeler) wheels of the leaves harvester. In essence, this work is a function of the deflections of the wheels' tyres. In analytical terms, the specified deflections can be represented by the difference between the amplitudes \(z\) of vertical oscillations of the wheels' centres (point \(B\) in Figure 2) and the longitudinal profile of the field (\(h\)): i.e., \((z-h)\). Considering the above, the expression for calculating the potential energy of vibrations of the leaves harvester is as follows [24]:  
\[P=\frac{2\cdot c\cdot(z-h)^{2}}{2}=c\cdot(z-h)^{2}. \tag{5}\]  
In the presence of the angle \(\alpha\) (see Figure 2), the value of which can be considered small with sufficient accuracy for practice, the ordinate \(z\) is determined as follows:  
\[z=(L+r\cdot\sin\alpha)\cdot\varphi=(L+r\cdot\alpha)\cdot\varphi=L\cdot\varphi+ r\cdot\alpha\cdot\varphi. \tag{6}\]  
We emphasized above that the parameters \(\alpha\) and \(\varphi\) are small quantities (\(\alpha\leq\approx\)12\({}^{\circ}\); \(\varphi\leq\approx\)10\({}^{\circ}\)). It is quite clear that their product will have an even smaller value, which, without any harm, can be neglected.