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Technological peculiarities of cultivation and harvesting of some agricultural crops make it necessary to use asymmetric machine-and-tractor aggregates. However, for the time being there is no sufficiently complete, analytical study of the steady movement of such machine-and-tractor aggregates. This necessitates the development of a theory of stable movement of the aggregates which would allow choosing their optimal kinematic and design parameters. On the basis of the results of mathematical simulation, a system of linear differential equations of the second order is obtained describing transverse displacement of the center of masses of the aggregating wheeled tractor and turning of its longitudinal axis of symmetry by some angle around the indicated center of mass, as well as the deviation angle of the rear-trailed harvester from the longitudinal axis of the tractor at any arbitrary moment of time. This system of differential equations can be applied for numerical calculations on the PC, which will make it possible to evaluate the stability of the movement of the asymmetric machine-and-tractor aggregate when it performs the technological process.

Turning time occupies a significant part of the operations carried out by implement-and-tractor aggregates, especially in fields with short runs. Incorrectly executed turns increase the width of the turning strips, significantly increasing the idle path of the implement-and-tractor aggregate, with negative effect on its efficiency. The objective of this paper was to theoretically analyse the turning agility of an asymmetric implement-and-tractor aggregate, taking into account its forward speed and design parameters. Considering a trailed asymmetric swath reaper and tractor aggregate, the obtained equations allowed a numerical simulation in order to evaluate the headland turning agility of this implement-and-tractor aggregate. The minimal radii of the trailed asymmetric swath reaper and tractor aggregate are, respectively, 8.33 m for right-side turn and 4.90 m for left-side turn. Furthermore, the optimal angle between the longitudinal axis of the aggregating tractor and the hitch bar of the trailed asymmetric implement exists only in the case of left-side U-turns and its value is 1.12 rad (64°). It is not possible to cover right-side U-turns or both right- and left-side pear-shaped loop-turn in the optimal mode.

The paper presents a dynamic model of four-element machine-and-tractor designed on the basis of a combined soil-cultivating-sowing aggregate and a tractor with an articulated frame. The pseudo velocity of the first tractor half-frame was considered a generalised coordinate. The control impact upon the tractor was obtained by means of the bending angle of the tractor half-frames. The trajectories and movement speeds of the aggregate elements were calculated as a function of time for the driving mode along a straight-line path and the tractor movement according to the harmonic control law. By means of the measuring system, trajectories of the aggregate elements were obtained. It was established that the movement trajectories of the aggregate elements represent a complex assessment parameter of the mathematical model adequacy of the aggregate plane-parallel movement. There was found a regularity of the aggregate turning radius and the bending angle of the tractor half-frames.