Mathematical model of a stacker crane with a slewing boom

The widespread use of racking stacker cranes is due to the spread of automated warehouses. Reduction of energy costs when using stacker cranes is one of the main directions of their improvement. The development of mathematical simulation models of stacker cranes allows to determine the energy costs of cargo movement at the stage of design and research. For a new design of the racking crane-stacker, having a rotary boom with a cargo carriage moving along it in the radial direction, a mathematical model in the form of a system of Lagrange differential equations of the second order has been developed. The system of differential equations took into account the energy losses according to the model of viscous friction in the moving links of the boom and the load carriage of the crane. For this purpose, when deriving the differential equations, the Rayleigh function was used, as well as analytical expressions of the potential and kinetic energies of the moving links of the crane, which were differentiated. With the help of typical blocks of the Russian simulation modeling system SimInTech, the obtained system of differential equations was realized in the form of a block diagram. The use of blocks of proportional-integral-differentiating regulators in the simulation model allowed to realize the principle of working out the given trajectory of movement of the crane links, and the numerical integration of the current powers of the boom and carriage drives allowed to determine the accumulated total energy consumption of the drives at a given displacement. The current values of drive powers were determined as products of moments and forces of drives on the speeds of the corresponding links of the boom and carriage. As an example, when running the simulation model, the time dependences of the coordinates of the boom and load carriage of the crane, as well as the moments and forces of the drives, power and work during the movement from a point with zero coordinates to a point with specified coordinates and back. The developed simulation mathematical model is intended to determine the energy costs in the drives of a new design crane with a rotating boom at given displacements of links, as well as, in the future, to verify the control algorithms.