# A Mathematical Model of Plane-Parallel Movement of the Tractor Aggregate Modular Type

^{1}

^{2}

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## Abstract

**:**

^{−1}does not lead to a deterioration in the stability of the movement of either the technological or, especially, the power modules. The delay in the reaction of the power module of the machine-and-tractor aggregate of the modular type is practically invariant with respect to the change in the mode of movement of this aggregate within the range 1–3 m∙s

^{−1}. It was also found that the values of the tire slip resistance coefficients of the wheels of the power module do not have a noticeable impact upon the development of fluctuations of the disturbing moment.

## 1. Introduction

- –
- Increasing the working width of the technological part of the aggregate;
- –
- Increasing the working speed of the aggregate.

_{e}, kW) is accompanied by a corresponding increase in its mass (M

_{t}, ton). As a result of this, the energy saturation of the tractor (E

_{t}, kW∙kg

^{−1}), determined by ratio ${E}_{t}=\frac{{N}_{b}}{{M}_{t}}=\mathrm{const},$ remains approximately the same for almost all types and designs of the wheeled tractors.

_{t}at which it can fully realise the engine power N

_{e}through the generated tractive effort is equal to 0.014–0.015 kW∙kg

^{−1}[1]. Consequently, according to this concept, almost the entire value of N

_{e}is actually realised in the traction version through the running system of the power unit (the tractor). However, in this case in our soil conditions, it is impossible to realise all the installed power of the tractor engine, because of the nonlinearity of its regulatory characteristics and the oscillatory nature of the external traction load [1].

^{−1}. So, for example, for the power unit Steyr 8300 (St. Valentin, Austria) this figure reaches 42 (www.konedata.net).

_{t}value, the more acute the problem of using the full power of the energy-saturated tractor. Of the many ways how to solve this problem one of the most efficient, in this study, is the modular construction (design) and operation of aggregating tractors. Such a tractor, as a rule, consists of two modules: the power module (the wheeled tractor itself) and the technological module (the additional, drive axle).

^{−1}. The technological module is an axle, additionally attached to the tractor, with drive wheels, its own three-point hitch, a semitrailer, a brake system and its own power take-off shaft. The wheels of the technological module are driven by the synchronous power take-off shaft of the power module (i.e., the tractor). More perspectives are hydraulic or electric drive options for the wheels of the technological module.

- If you have one tractor of a certain traction class (for example, 1.4 or 2) and a technological module for it, you can successfully do without a tractor of traction class 3. That is, instead of two conventional tractors, a manufacturer of agricultural products (for example, a farmer) only needs one tractor and one technological module for it. It is quite efficient from the economic point of view.
- The use of a technological module as part of a modular machine-and-tractor aggregate does not increase the soil compaction. This is due to the fact that the wheels of the technological module are already following the compacted track, left by the wheels of the power module. The relatively light weight power module compacts the soil insignificantly [3]. Besides, there is practically no more or less significant additional soil compaction by the wheels of the technological module. Thus, a modular-type machine-and-tractor aggregate, for example, weighing 7.5 tons will compact the soil less than the conventional tractor of the same mass (weight). The reason is that the mass of a modular-type machine-and-tractor aggregate is distributed over 3 axles while the mass of a conventional tractor of the same mass is distributed only over two. Moreover, a modular 3-axle machine-and-tractor aggregate has a more efficient “multi-pass” effect than a conventional 2-axle tractor. Due to this, such an aggregate differs from the conventional tractor by lower wheel slip and lesser specific fuel consumption.
- Owing to the technological module, the annual load of the power module (tractor) increases significantly. For the power module (tractor) of the traction class 3, according to our calculations, it can reach 1700 h. Part of the time of the year the technological module may not be used but the losses from its downtime is about 5–7 times less than similar losses from the downtime of an idle tractor.

^{−1}[2].

^{6}N∙m∙s∙rad

^{−1}[4].

## 2. Materials and Methods

_{o}(Figure 3). As the practice of using many agricultural aggregates shows, the assumption of the constancy of their working speed is quite correct.

_{t}; the middle of the frontal driven axle is point A; the middle of the rear driving wheel axle—point B; the point of pivotal connection of the technological module to the power module—point K; the point of connection of the aggregated agricultural machine (plough) to the technological module (the middle of the axle of the technological module)—point C.

_{K}of the power module by a certain angle $\beta $ as a result of the action of the unfolding moment from the side of the technological module and the aggregated agricultural machine.

- The control impact in the form of the turning angle α of the driven wheels of the power module of the machine-and-tractor aggregate of a modular type;
- The disturbing impact in the form of a summary unfolding moment ${M}_{o}={M}_{m}-{R}_{g}\cdot {b}_{m}.$

_{K}of point K— the reduced centre of mass of the modular machine-and-tractor aggregate; the bearing angle φ of the power module; the turning angle β of the technological module relative to the power module.

_{o}relative to the bearing angle φ of the power module has the following form:

_{o}) but relative to the turning angle β of the technological module is more complex and has the following form:

_{t}and calculated by using the transfer function (37); and the second, A

_{e}, was obtained by us as a result of the experimental field study of the machine-and-tractor aggregate of a modular type during the process of ploughing.

_{a}that this aggregate passed the testing section of 250 m was recorded using an FS-8200 electronic stopwatch (China) with a measurement accuracy of 0.1 s. Subsequently, the working speed V

_{o}of this aggregate was calculated by the formula: ${V}_{o}=250\cdot {\left({t}_{a}\right)}^{-1}$.

^{−1}; the transmission capacity 150 Hz.

_{a}, k

_{b}and k

_{c}are important. Their values depend on the vertical load Q upon each tire, its diameter D, width b and air pressure ρ in it.

_{a}, k

_{b}, and k

_{c}were calculated by us according to expression (41), taking into account the data contained in Table 1. The numerical values of these coefficients obtained in this case were used as one of the analysed parameters, adopted when carrying out theoretical calculations of the developed mathematical model (35). This approach is explained as follows. Knowing the tendency of the impact of these coefficients upon the movement stability of the modules of the machine-and-tractor aggregate of a modular type, from Equation (41) it is easy to determine the required value of the air pressure in the tires of all its running wheels. As a result, this can be efficiently implemented under practical operating conditions of this modular machine-and-tractor aggregate.

_{o}was varied from 1 to 3 m∙s

^{−1}(3.6–10.8 km∙h

^{−1}). A lesser value of movement is technologically ineffective, but a greater value is limited by the technical requirements for the modular machine-and-tractor aggregate.

## 3. Results and Discussion

^{−1}. It was this value of velocity V

_{o}of the movement that was used to calculate by the transfer function (37) the theoretical (A

_{t}) amplitude–frequency characteristic of processing of the disturbing impact (moment M

_{o}) by the machine-and-tractor aggregate studied.

_{o}, expressed in N∙m, on the deviation angle of the technological module with the agricultural machine β, expressed in radians (rad). In fact, this frequency response, like any other, is the distribution of the gain of the input signal over frequencies. In our case the input signal is the M

_{o}moment (N∙m), and the output signal is angle β (rad). With the same dimension of the input and the output signals, this coefficient (i.e., the frequency response) is dimensionless. In our case, the frequency response has the dimension—rad∙(N∙m)

^{−1}and it is represented only in the frequency domain. Comparison of this characteristic with the experimental (А

_{е}) one demonstrated their satisfactory agreement: the maximum difference between the calculated and the field data does not exceed 14% (Figure 4). Moreover, this unambiguously indicates that the mathematical model (35) of the movement of the modular-type machine-and-tractor aggregate with an agricultural machine (in this case, a plough) attached to it is adequate and, therefore, reliable, and fully suitable for further theoretical research.

^{−1}, the amplitude–frequency characteristic of the turning angle β of the technological module during the production of the disturbing impact (moment M

_{o}) has a resonance peak at a frequency of 5 s

^{−1}(Figure 5).

_{o}= 2 m∙s

^{−1}, and more. This fact can be explained by the inertness of the technological module with a plough attached to it, which, when the speed of the aggregate is increased, manifests itself more effectively.

^{−1}(Figure 6).

^{−1}. In the time calculation this means that the reaction of a modular-type machine-and-tractor aggregate to the disturbing impact will occur with a delay of 0.3 s. However, the speed of movement of this aggregate has little impact upon this process.

^{−1}has very little impact upon the process of fluctuations in the turning angle of the technological module of the modular traction device during the disturbing impart upon it in the form of an unfolding moment.

_{a}and rear k

_{b}wheels of the power module of this modular machine-and-tractor aggregate.

_{c}of the tire slip resistance of the wheels of the technological module, there is a different result. With its increase from 160 to 210 kN∙rad

^{−1}, the maximum value of the amplitude–frequency characteristic of the dynamic system increases (Figure 7, Curve 2). With a further increase in the value of coefficient k

_{с}, these characteristics decrease. In this case, the resonance peaks of the amplitude–frequency characteristic (Curves 3 and 4) are shifted towards higher frequencies.

^{−1}, this difference is minimal.

^{−1}and more helps to reduce the amplitude of its fluctuations in a horizontal plane.

_{o}upon the dynamics of variations in the bearing angle φ of the power module of the machine-and-tractor aggregate of a modular type. In this case the transfer function, defined by Equation (36), can be used to build the amplitude phase frequency characteristics.

^{−1}affects insignificantly the nature of processing of the disturbing impact by the power module in the form of fluctuations of the M

_{o}moment (Figure 8). Here, as in the case of the technological module, the resonance peaks of the amplitude–frequency characteristic fall at a frequency of 5 s

^{−1}.

^{−1}, the maximum value of the amplitude–frequency characteristic of fluctuations in the bearing angle of the power module (Figure 8, Curve 1) is about 2.4 times less than the same value of the amplitude–frequency characteristic of fluctuations in the turning angle of the technological module (Figure 5, Curve 1). At a speed of the movement of the machine-and-tractor aggregate of a modular type, equal to 3 m∙s

^{−1}, the amplitude of fluctuations of angle φ, in comparison with the fluctuations of angle β, decreases 1.6 times.

_{o}acts directly upon the technological module with the plough but not on the power module.

_{o}, is such that, when the disturbance fluctuates with a frequency of up to 8.5 s

^{−1}, a change in the speed of the aggregate from 1 to 3 m∙s

^{−1}has little impact upon the fluctuation dynamics of the bearing angle of the power module of the modular machine-and-tractor aggregate (Figure 9).

^{−1}it is preferable for the machine-and-tractor aggregate of a modular type to move at a lower speed since in this case the delay in the reaction of the power module to the disturbing impact is large. Especially at the disturbance fluctuation frequencies, close to 10 s

^{−1}(Figure 9, Curve 1).

^{−1}it is preferable for the investigated modular-type machine-and-tractor aggregate to move at a speed of up to 3 m∙s

^{−1}(Figure 9, Curve 2). In this case, especially at frequencies greater than 15 s

^{−1}, the desired delay in the response of the power module of the modular machine-and-tractor aggregate to the disturbing impact increases by more than 0.5 rad.

## 4. Conclusions

- –
- A change in the working speed of this aggregate during ploughing from 1.0 to 3.0 m∙s
^{−1}does not lead to the deterioration in the stability of the movement of either the technological or, all the more, the power module of the machine-and-tractor aggregate of a modular type. The amplitude–frequency characteristic of processing the disturbing unfolding moment by them when parameter V_{o}is increased, although insignificantly, improves. The phase–frequency processing of the technological module of the modular-type machine-and-tractor aggregate somewhat deteriorates but only at relatively high frequencies of its fluctuations—more than 10 s^{−1}. The delay in the reaction of the power module of the machine-and-tractor aggregate of a modular type is practically invariant with respect to the change in the mode of the movement of this aggregate in the range 1–3 m∙s^{−1}. - –
- The values of the coefficients of resistance to tire slip of the wheels of the power module do not have a noticeable impact upon the processing of fluctuations of the disturbing moment. At the same time, the value of the coefficient of resistance to tire slip of each wheel of the technological module of the machine-and-tractor aggregate of a modular type must be not less than 160 kN∙rad
^{−1}. - –
- As a result, equipment of the technological module with tires, with a withdrawal resistance coefficient of at least 160 kN∙rad
^{−1}, along with the installation in its hydraulic cylinder of a throttle, with a resistance coefficient of 1.03∙10^{6}N∙m∙s∙rad^{−1}allows to ensure stability of the working movement of the aggregate, based on the modular means, in the range of speeds from 1 to 3 m∙s^{−1}.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The machine-and-tractor aggregate of a modular type. 1—the power module (the wheeled tractor); 2—the technological module; 3—the aggregated agricultural machine (plough); 4—cardan transmission.

**Figure 2.**The connection of the power and the technological modules. 1—the power module; 2—the technological module; 3—the articulated joint; 4—the hydraulic cylinder, connecting the modules.

**Figure 3.**An equivalent scheme of the modular-type machine-and-tractor aggregate in its plane-parallel movement.

**Figure 4.**The amplitude–frequency characteristics of the investigated machine-and-tractor aggregate of a modular type. 1—theoretical A

_{t}; 2—experimental A

_{e}.

**Figure 5.**The amplitude–frequency characteristic of angle β during the processing of the disturbing impact by the technological module at different speeds of movement of the machine-and-tractor aggregate of a modular type (V

_{o}): 1—1 m∙s

^{−1}; 2—2 m∙s

^{−1}; 3—3 m∙s

^{−1}.

**Figure 6.**The phase–frequency characteristic of angle β during the processing of the disturbing impact by the technological module at different speeds of movement of the machine-and-tractor aggregate of a modular type (V

_{o}): 1—1 m∙s

^{−1}; 2—2 m∙s

^{−1}; 3—3 m∙s

^{−1}.

**Figure 7.**The amplitude–frequency characteristic of angle β during the processing of the disturbance impact by the technological module for different values of coefficient k

_{с}: 1—160 kN∙rad

^{−1}; 2—210 kN∙rad

^{−1}; 3—260 kN∙rad

^{−1}; 4—310 kN∙rad

^{−1}.

**Figure 8.**The amplitude–frequency characteristic of angle φ during the processing of the disturbing impact at different values of speed V

_{o}: 1—1 m∙s

^{−1}; 2—3 m∙s

^{−1}.

**Figure 9.**The phase–frequency characteristic of angle φ during the processing of the disturbing impact at various values of speed V

_{o}: 1—1 m∙s

^{−1}; 2—3 m∙s

^{−1}.

Parameter | Designation | Unit of Measurement | Value |
---|---|---|---|

Mass (weight) of the power module | M_{t} | kg | 3820 |

Longitudinal base of the tractor | L | m | 2.37 |

Mass (weight) of the technological module | M_{m} | kg | 2500 |

Mass (weight) of the plough PLN-5-35 | M_{p} | kg | 800 |

Moment of inertia of the power module | J_{t} | kN∙m∙s^{2} | 15.7 |

Moment of inertia of the technological module | J_{m} | kN∙m∙s^{2} | 15.9 |

Front wheel tires of the power module: | 9.00R20 | ||

– width | b_{a} | m | 0.24 |

– diameter | D_{a} | m | 0.95 |

– air pressure | ρ_{a} | MPa | 0.10 |

– vertical load on the axle | ${Q}_{a}$ | kN | 12.70 |

Rear wheel tires of the power module: | 15.5R38 | ||

– width | b_{b} | m | 0.40 |

– diameter | D_{b} | m | 1.57 |

– air pressure | ρ_{b} | MPa | 0.12 |

– vertical load on the axle | ${Q}_{b}$ | kN | 25.30 |

Wheel tires of the technological module: | 16.9R38 | ||

– width | b_{c} | m | 0.43 |

– diameter | D_{c} | m | 1.69 |

– air pressure | ρ_{c} | MPa | 0.13 |

– vertical load on the axle | ${Q}_{c}$ | kN | 32.70 |

Rolling resistance force of the frontal wheels of the power module | P_{fa} | kN | 1.27 |

Traction force of the rear axle of the power module | F_{b} | kN | 10.10 |

Rolling resistance force of the wheels of the technological module: | P_{fc} | kN | 3.27 |

Design parameters, shown in Figure 3 | a_{m} | m | 1.22 |

b_{m} | m | 1.22 | |

Resistance coefficient of the throttle of the hydraulic cylinder of the technological module | K_{m} | N∙m∙s∙rad^{−1} | 1.03 × 10^{6} |

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**MDPI and ACS Style**

Bulgakov, V.; Aboltins, A.; Ivanovs, S.; Holovach, I.; Nadykto, V.; Beloev, H.
A Mathematical Model of Plane-Parallel Movement of the Tractor Aggregate Modular Type. *Agriculture* **2020**, *10*, 454.
https://doi.org/10.3390/agriculture10100454

**AMA Style**

Bulgakov V, Aboltins A, Ivanovs S, Holovach I, Nadykto V, Beloev H.
A Mathematical Model of Plane-Parallel Movement of the Tractor Aggregate Modular Type. *Agriculture*. 2020; 10(10):454.
https://doi.org/10.3390/agriculture10100454

**Chicago/Turabian Style**

Bulgakov, Volodymyr, Aivars Aboltins, Semjons Ivanovs, Ivan Holovach, Volodymyr Nadykto, and Hristo Beloev.
2020. "A Mathematical Model of Plane-Parallel Movement of the Tractor Aggregate Modular Type" *Agriculture* 10, no. 10: 454.
https://doi.org/10.3390/agriculture10100454