## 3 Results and Discussion  
Substituting Expression (21) into (18) and the obtained result into Equation (17), we obtain a differential equation for the relative motion of the haulm particle \(M\) along the thrower blade of the following form:  
\[\begin{array}{l}m\cdot\ddot{x}=m\cdot r\cdot\omega^{2}\cos\psi+m\cdot g \cdot\cos\beta+k\cdot\big{(}V_{n}\cdot\cos\gamma-\dot{x}\big{)}-\\ -f\cdot\big{(}2m\cdot\omega\cdot\dot{x}+m\cdot g\cdot\sin\beta-m\cdot r\cdot \omega^{2  
We regroup the terms on the right side of Equation (27) as follows:  
\[ \begin{array}{l}\ddot{x}+\left(2f\cdot\omega+\frac{k}{m}\right)\cdot\dot{x}- \omega^{2}\cdot x=g\cdot\cos\omega t\cdot(\cos\psi_{1}+f\cdot\sin\psi_{1})+\\ +g\cdot\sin\omega t\cdot(\sin\psi_{1}-f\cdot\cos\psi_{1})+r_{0}\cdot\omega^{2} \cdot(\cos\psi_{0}+f\cdot\sin\psi_{0})+\\ +\frac{k}{m}\cdot V_{n}\cdot(\cos\gamma+f\cdot\sin\gamma).\end{array}  \tag{28}\]  
For abbreviations and convenience of integrating Equation (28), we introduce the following notation:  
\[ \begin{array}{l}K=g\cdot(\cos\psi_{1}+f\cdot\sin\psi_{1}),\\ L=g\cdot(\sin\psi_{1}-f\cdot\cos\psi_{1}),\\ C=\cos\gamma+f\cdot\sin\gamma,\\ D=\cos\psi_{0}+f\cdot\sin\psi_{0}.\end{array}  \tag{29}\]  
Taking into account Notation (29), Equation (28) is reduced to the following form:  
\[\ddot{x}+\left(2f\cdot\omega+\frac{k}{m}\right)\dot{x}-\omega^{2}\cdot x=L \cdot\sin\omega t+K\cdot\cos\omega t+C\cdot\frac{k}{m}\cdot V_{n}+r_{0}\cdot \omega^{2}\cdot D \tag{30}\]  
Equation (30) is a second-order linear differential equation with constant coefficients with the right-hand side [6]. Let us first find the general solution of the homogeneous equation:  
\[\ddot{x}+\left(2f\cdot\omega+\frac{k}{m}\right)\cdot\dot{x}-\omega^{2}\cdot x =0 \tag{31}\]  
We compose the characteristic equation:  
\[\lambda^{2}+\left(2f\cdot\omega+\frac{k}{m}\right)\cdot\lambda-\omega^{2}=0 \tag{32}\]  
We find the roots of quadratic Equation (32):  
\[\lambda_{1}=-\left(f\cdot\omega+\frac{k}{2m}\right)+\sqrt{\left(f\cdot\omega +\frac{k}{2m}\right)^{2}+\omega^{2}} \tag{33}\]  
and  
\[\lambda_{2}=-\left(f\cdot\omega+\frac{k}{2m}\right)-\sqrt{\left(f\cdot\omega +\frac{k}{2m}\right)^{2}+\omega^{2}} \tag{34}\]  
Then the general solution of Equation (31) has the form:  
\[x_{1}=C_{1}\cdot e^{\lambda_{1}t}+C_{2}\cdot e^{\lambda_{2}t} \tag{35}\]  
where \(C_{1}\) and \(C_{2}\) are arbitrary constants. We find a partial solution of the differential Equation (30) in the form of its right side:  
\[x_{2}=S\cdot\sin\omega t+T\cdot\cos\omega t+Q \tag{36}\]  
where \(S\), \(T\), and \(Q\) are constant coefficients to be determined. Substituting Expression (36) and its first and second derivatives into Equation (30), and equating the coefficients of the corresponding functions in the left and right parts of the resulting equation, we obtain a system of linear algebraic equations for the unknowns \(S\), \(T\), and \(Q\):  
\[\left. \begin{array}{l}-S\cdot\omega^{2}-\left(2f\cdot\omega^{2}+\frac{k\cdot \omega}{m}\right)\cdot T-\omega^{2}\cdot S=L,\\ -T\cdot\omega^{2}+\left(2f\cdot\omega^{2}+\frac{k\cdot\omega}{m}\right)\cdot S -\omega^{2}\cdot T=K,\\ -\omega^{2}\cdot Q=C\cdot\frac{k}{m}\cdot V_{n}+r_{0}\cdot\omega^{2}\cdot D.\end{array} \right\} \tag{37}\]Solving the system of Equation (37), we will have:  
\[Q=-\frac{C\cdot k\cdot V_{n}}{m\cdot\omega^{2}}-r_{0}\cdot D \tag{38}\]  
\[S=\frac{2K\cdot f\cdot\omega+\frac{k\cdot K}{m}-2L\cdot\omega}{4\omega^{3}+ \omega\cdot\left(2f\cdot\omega+\frac{k}{m}\right)^{2}} \tag{39}\]  
\[T=\frac{-2L\cdot f\cdot\omega-\frac{k\cdot L}{m}-2K\cdot\omega}{4\omega^{3}+ \omega\cdot\left(2f\cdot\omega+\frac{k}{m}\right)^{2}} \tag{40}\]  
Thus, the general solution of Equation (30) has the following form:  
\[x=C_{1}\cdot e^{\lambda_{1}t}+C_{2}\cdot e^{\lambda_{2}t}+S\cdot\sin\omega t+T \cdot\cos\omega t+Q \tag{41}\]  
where \(S\), \(T\), and \(Q\) are determined by Formulas (39), (40), and (38), respectively. The arbitrary constants \(C_{1}\) and \(C_{2}\) are determined from the following initial conditions: at \(t=0:x=0\), \(x=0\).