Existing technical means for loading silos with grain material do not fully meet the needs of production. The issue related to grain injury remains urgent, which predetermines the need to design a gravitational loader of another principle of operation with the appropriate theoretical justification for the movement of grain material in it. This paper has presented and substantiated the model of the gravitational movement of grain in the peripheral open screw channel with two variable angles of inclination. The model is based on the system of forces in the cylindrical coordinate system, acting on the volume of grain flow in the peripheral screw channel. The grain speed at the end of the braking section of the channel should be as low as possible but not less than the initial flow rate at the beginning of the acceleration section. The model takes into consideration this condition and ensures the optimal passage of grain along any part of the channel. The reported model makes it possible to obtain the speed of grain movement at any time, takes into consideration the height of the bunker hole and the dependence between the angles of inclination of the spirals of acceleration and brake sections. A mathematical dependence is given for these angles that ensures the passage of grain without its discharge and, at the same time, prevents injury to the grain mass due to a controlled decrease in the resulting speed. A separate dependence is provided to find the time at which the grain increases its speed on the acceleration section, reaching the maximum value. Based on the model, a peripheral open screw channel with two angles of inclination of spirals α and β has been proposed. For this channel, the relationships between its key parameters have been established, in particular, values have been substantiated for the recommended angles of 41°…45° for the acceleration section and 39°…35° for the brake section, respectively, as well as the hо/r ratio not less than 0.6...0.7.

The paper focuses on the possibility of using a systematic approach to information-analytical system. Using mathematical modeling for optimization problems of environmentally hazardous situations was studied. The methodological aspects of monitoring of background concentrations of toxic contaminants were investigated for emissions of environmental facilities and technology systems. The principles of construction of information-analytical models on the analytical description of points set were proposed to create simulation systems for environmental systems management tasks. The most widely used computer-aided tools were analyzed. Examples of problems that can be solved with the help of various software (spreadsheet MS Excel, package SPSS for Windows, Gran 2D et al.) were presented. Thus it allows to carry out environmental monitoring and calculation of damage from natural and man-made factors in the environment with using low level of software operation complexity.

An exact analytical solution to the problem of plane transverse bending of a section of a narrow multilayer beam under the action of a normal load on the longitudinal faces, distributed according to the law of the trapezoid, is presented. The solution is constructed using the principle of superposition on the basis of the authors’ general solutions to the problems of bending multilayer consoles under the action of loads at the free end, uniformly and linearly distributed load on longitudinal faces. On its basis, separate interchanges for multilayer beams with different methods of fixing the ends were obtained: hinged, rigid and combined. The obtained relations make it possible to determine the stress-strain state of multilayer beams with an arbitrary number of homogeneous (orthotropic, isotropic) and functional-gradient layers, taking into account transverse shear and compression deformations.

This research investigates the effectiveness of visualisation techniques in teaching mathematical modelling fundamentals to agricultural university students. We examine the hypothesis that dynamic vector diagrams representing mechanical motion characteristics (velocity, acceleration, and force) enhance student learning outcomes. The study compares three instructional approaches: using Excel spreadsheets, utilising GeoGebra dynamic geometry software, and employing both tools simultaneously. Our methodology involved 167 engineering students divided into three homogeneous groups, each completing identical modelling tasks concerning projectile motion under various conditions. Results demonstrate that students who simultaneously employed Excel and GeoGebra with dynamic vector diagram visualisation achieved significantly higher academic performance (mean score 78.25) compared to those using either Excel (73.36) or GeoGebra (73.85) exclusively. Statistical analysis through ANOVA confirms these differences are significant (p=0.010613). We observed that Excel users demonstrated stronger quantitative analytical skills, while GeoGebra users excelled in qualitative assessment tasks. This research extends previous findings on visualisation in mathematical education and provides practical insights for enhancing STEM education in agricultural universities through appropriate technology integration.