# Mathematical Model of the Plane-Parallel Movement of the Self-Propelled Root-Harvesting Machine

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Research

#### 2.1.1. Equivalent Scheme of a Self-Propelled Root Harvesting Machine

^{−1}range; and a performance in the 1.5 to 2.5 ha h

^{−1}range (Figure 1).

_{L}and Ψ

_{P}. These coordinates can be taken as generalized coordinates of the dynamical system under consideration.

#### 2.1.2. Mathematical Model

_{x}takes the form:

_{y}can be represented:

_{rot.}Is the moment of forces turning the car through an angle $\psi $ and M

_{res.rot.}is the torque moment of resistance.

#### 2.2. Experimental Tests

## 3. Results and Discussion

#### 3.1. Numerical Results

#### 3.2. Experimental Results

^{−1}) for the given construction and power parameters of the root-harvesting machine. Thus, the theoretical study of the trajectory of the center of the outside digging tool and the experimental evaluation of its work (in terms of the quality of harvesting with deviations in its trajectory of motion) in the experimental part formally confirm the coincidence of all the studies—both theoretical and experimental. However, the steady movement in the horizontal plane is provided by the whole set of operational and technical conditions. So, the pressure in the tires of the front driven wheels should be in the set limits by the machine-maker and also the clearances in the steering must not exceed the set values. In connection with this, the root harvesting machine must have precise and strict adjustments (clearances in the mechanical and hydraulic servo-mechanisms should be absent), adjusted to respect the translational speed of the machine (close to 3.0 m s

^{−1}) and by taking into account the characteristics of the conditional center lines of rows of crops, the average size of the upper parts of the root crops (the size and shape of the heads of root crops after cutting to the top of the foliage), and so on.

## 4. Conclusions

- It was determined that the dynamic system of the self-propelled harvesting aggregate can be considered as conservative with practically absent dissipative processes.
- Since the self-propelled root harvester is a complex, multi-mass dynamic system, the method of compiling systems of differential equations in the Lagrange form of the second kind is most suitable for investigating the motion of such a system. The resulting system of second-degree differential equations is nonlinear, the solution of which is produced by numerical methods on the PC. The obtained system of differential equations contains constructive parameters of the root-harvesting machine. By changing these parameters, it is possible to achieve such values that will contribute to a more stable movement of the self-propelled root crop machine in the horizontal plane.
- The use of the obtained mathematical model makes it possible to optimize the parameters of the plane-parallel motion and to improve the quality parameters of the technological process.
- Further analytical studies on this model can be focused on other solutions of this mathematical model of the PC, for cases when disturbing effects on the elements of a self-propelled root harvester can be represented in the form of statistical functions. The results obtained in the general form create prerequisites for the theoretical study of other mobile machine aggregates.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

B | center of mass of the root harvesting machine |

C | pole (a fixed point on the machine body located on its longitudinal axis) |

A, D | midpoints of the interaxial distances between the front driven and rear drive wheels, respectively |

B_{11}, B_{12}, B_{21}, B_{22} | centers of mass points of the front driven and rear drive wheels |

N | intersection point of the suspension axis of the digging sections and the longitudinal axis of the machine |

A | hinge point of the frame of the feeler wheels |

AB = a | distance from the center of mass B to the front axle |

DB = b | distance from the center of mass B to the rear axle |

DC = l | distance from pole C to the rear axle |

BK = h | distance from pole C to the axis of the suspension of the digging sections |

p | distance from the front axle to the axis of the feeler wheels |

n | distance from the front axle to the axis of the feelers |

m | distance between the suspension points of the outside feelers |

2e | distance between the axis of turns of steerable wheels |

u | distance between the axis of symmetry of neighboring excavating working bodies |

f | length of the swivel pin of the steering gear |

2r_{1} and 2r_{2} | diameters of the steerable and driving wheels, respectively |

2d and 2s | width of the track of the driving and feeler wheels |

AD = a + b | longitudinal base of root harvesting machine |

J_{M} | moment of inertia of the root harvesting machine relative to the vertical axis Cξ |

M | mass of the root harvesting machine |

M_{res.rot.} | torque moment of resistance |

M_{rot.} | moment of forces turning the root harvesting machine through an angle $\psi $ |

q_{i} | generalized coordinates |

Q_{i} | generalized forces |

R_{y}_{1}, R_{y}_{2} | soil reactions acting on the steerable wheels |

R_{x}_{1}, R_{x}_{2} | soil reactions acting on the driving wheels |

R_{O}_{1}, R_{O}_{2} | soil reactions acting on the feeler wheels |

R_{A}_{1}, R_{A}_{2}, R_{A}_{3} | soil reactions acting on the feelers |

R_{B}_{1}, R_{B}_{2}, …, R_{B}_{6} | total resistance reactions during the movement of excavated working organs in the soil |

R_{S}_{1}, R_{S}_{2}, R_{S}_{3}, R_{S}_{4} | lateral reactions of the soil, acting on the steerable and driving wheels of the root-harvesting machine when moving along the axis Y |

R_{OS}_{1}, R_{OS}_{2} | lateral reactions of the soil, acting on the feeler wheels of the root-harvesting machine when moving along the axis Y |

R_{BS}_{1}, R_{BS}_{2}, …, R_{BS}_{6} | lateral reactions of the soil, acting on the digging working organs |

R_{AS}_{1}, R_{AS}_{2}, R_{AS}_{3} | lateral reactions of the soil, acting on the sensors of the feelers |

T | energy of the system |

T_{1} and T_{2} | kinetic energy, respectively, of the translational motion of the root harvesting machine and its rotation around the pole C |

V_{B} | velocity of the center of mass |

$\psi $ | heading angle |

${\psi}_{L}$ and ${\psi}_{P}$ | angles to determine the orientation of the axes of the left and right steered wheels relative to the body of the machine |

$\dot{\psi}$ | angular speed of rotation of the machine around the pole C |

δq_{i} | variations of the corresponding generalized coordinates |

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**Figure 1.**The self-propelled root harvesting machine. (

**a**) view in the transport position; (

**b**) working scheme: 1: frame, 2: driving machine in rows; 3: front steerable wheels; 4: digging section; 5: feeler wheels for digging sections; 6: root lifts; 7: excavated working bodies of rotary type; 8: lobed bitter-excavator of root crops; 9: receiving biter conveyor cleaner; 10: screw conveyor-cleaner; 11: longitudinal conveyor; 12: driving wheel; 13: power unit mounted on the frame; 14: discharge conveyor; 15: bunker-storage; 16: lower conveyor of bunker-storage.

**Figure 2.**The equivalent scheme of the plane-parallel motion of the self-propelled root harvesting machine in the horizontal plane.

**Figure 3.**The trajectory of the motion of the center of mass of the root harvesting machine and the center of the outside digging working body at different velocities (m·s

^{−1}): 1 = 0.75; 2 = 1.68; 3 = 3.00.

**Figure 4.**The dependence of the sugar beetroot crop damage from the magnitude of the deviation of the longitudinal axis of the outside digging unit from the conditional centerline of the row of crops: 1 = total damage; 2 = heavy damage.

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**MDPI and ACS Style**

Bulgakov, V.; Pascuzzi, S.; Santoro, F.; Anifantis, A.S.
Mathematical Model of the Plane-Parallel Movement of the Self-Propelled Root-Harvesting Machine. *Sustainability* **2018**, *10*, 3614.
https://doi.org/10.3390/su10103614

**AMA Style**

Bulgakov V, Pascuzzi S, Santoro F, Anifantis AS.
Mathematical Model of the Plane-Parallel Movement of the Self-Propelled Root-Harvesting Machine. *Sustainability*. 2018; 10(10):3614.
https://doi.org/10.3390/su10103614

**Chicago/Turabian Style**

Bulgakov, Volodymyr, Simone Pascuzzi, Francesco Santoro, and Alexandros Sotirios Anifantis.
2018. "Mathematical Model of the Plane-Parallel Movement of the Self-Propelled Root-Harvesting Machine" *Sustainability* 10, no. 10: 3614.
https://doi.org/10.3390/su10103614