A Mathematical Model of the Plane-Parallel Movement of an Asymmetric Machine-and-Tractor Aggregate
Abstract
:1. Introduction
2. Materials and Methods
- ○
- The surface of the field across which the particular harvesting aggregate moves is strictly horizontal, and therefore its inclination is neglected;
- ○
- The wheeled aggregating tractor is a solid body that has a longitudinal plane of symmetry passing through its center of mass;
- ○
- Fluctuations of the tractive resistance of the trailed windrower have no significant effect on the speed of the forward movement of this aggregate, which is then considered constant in first approximation;
- ○
- Interaction of the pneumatic tires of the aggregating tractor wheels and the supporting wheels of the trailed windrower with the surface of the soil (in the lateral directions) does not go beyond the limits of the well-known hypothesis of the “lateral drift” of tires;
- ○
- Because of the small values, the gyroscopic and stabilizing moments of the pneumatic tires of the tractor and windrower wheels are not considered;
- ○
- The slip angles of the pneumatic tires of the travelling wheels of the aggregating tractor, located on the same geometric axis, as well as the lateral forces acting on them, will be considered sufficiently small;
- ○
- The turning angles of the left-side and the right-side driven wheels of the aggregating tractor are considered small and equal, taking into account that the basic movement of this harvesting machine-and-tractor aggregate is close to rectilinear, when performing the technological process of cutting grain crops.
2.1. Mathematical Model of the Aggregating Wheeled Tractor
- (a)
- the driving force of the rear wheels summing up from the two driving wheels of the tractor is applied at point and forms a drift angle with the longitudinal axis of symmetry of the tractor;
- (b)
- the rolling resistance force of the front wheels of the tractor, which also sums up from the two wheels, is applied at the intersection point of their axis with the longitudinal axis (point ) and deviated from the direction of the movement of its propellers by the drift angle ;
- (c)
- lateral forces, respectively, from two wheels: and , applied at points and , respectively;
- (d)
- the traction resistance force of the windrower movement, applied at point and deviated from the longitudinal axis of the aggregating tractor, i.e., from axis , by angle .
2.2. Mathematical Model of the Trailed Windrower
- (a)
- the longitudinal and the transverse components of the resistance forces of the harvested stubble mass that are applied to the windrower at point ;
- (b)
- the rolling resistance force of the left-side wheel of the windrower applied at point and deviated from the plane of the wheel by the drift angle ;
- (c)
- the rolling resistance force of the two right-side wheels of the windrower (in the diagram they are represented as one equivalent wheel), which is applied at point and is deviated from the plane of the wheel by the drift angle ;
- (d)
- lateral forces and , applied respectively to the left-side and the right-side wheel of the windrower at points and .
3. Results and Discussion
3.1. Mathematical Model of the Entire Asymmetric Machine-and-Tractor Aggregate
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- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;;
- ;
- ;
3.2. Experimental Verification of the Adequacy of the Mathematical Model
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Bulgakov, V.; Pascuzzi, S.; Nadykto, V.; Ivanovs, S. A Mathematical Model of the Plane-Parallel Movement of an Asymmetric Machine-and-Tractor Aggregate. Agriculture 2018, 8, 151. https://doi.org/10.3390/agriculture8100151
Bulgakov V, Pascuzzi S, Nadykto V, Ivanovs S. A Mathematical Model of the Plane-Parallel Movement of an Asymmetric Machine-and-Tractor Aggregate. Agriculture. 2018; 8(10):151. https://doi.org/10.3390/agriculture8100151
Chicago/Turabian StyleBulgakov, Volodymyr, Simone Pascuzzi, Volodymyr Nadykto, and Semjons Ivanovs. 2018. "A Mathematical Model of the Plane-Parallel Movement of an Asymmetric Machine-and-Tractor Aggregate" Agriculture 8, no. 10: 151. https://doi.org/10.3390/agriculture8100151