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The task of controlling the load spatial oscillations of a hoisting crane using a spherical pendulum model with two angular degrees of freedom is considered. In order to limit uncontrolled load oscillations, its suspension point movements are optimised. The system of differential nonlinear equations for oscillations of a spherical pendulum with suspension point acceleration along the Cartesian axes was first applied to the curvilinear load moving in the limiting oscillations. Sigmoidal dependences of two load deviation angles and rotation of the pendulum are proposed, providing for expressions of the first two derivatives of the indicated angles, as well as linear accelerations of the load suspension point along two Cartesian horizontal axes. Numerical methods are applied to obtain time dependences of the velocities and displacements of the load suspension point. The solution provides the load moving along a curvilinear trajectory at specified distances along the specified axes, subject to the maximum acceleration and crane speed limitations. Optimal time dependences of the rope deflection angles, suspension point displacements and their first two derivatives in limited oscillations are presented. The scope of the methodology is the modelling of crane working processes and the automatic movement control for the bridge and gantry cranes.

Spherical pendulum model with a moving suspension point in the problem of spatial load movement by a hoisting crane with oscillation limiting