The use of hermite polynomials for the analytical solution of the problem of cargo movement on flexible pendulum suspension on a given trajectory

Abstract

The method of complete analytical solution of the problem of cargo movement on a pendulum suspension of constant length on a set trajectory in a flat statement is developed. To set the required trajectory as a time dependence of the linear horizontal coordinate of the cargo in a limited time interval, two-point Hermite polynomials are used. When a certain amount of the first derivatives of the linear coordinate of the load is equal to zero in the horizontal direction at the ends of the segment, a vertical arrangement of the crane’s cargo rope is provided at the boundary points and there are no uncontrolled angular swings of the cargo rope during the entire movement. Analytical dependences of the horizontal coordinate of the cargo, its first and second derivatives, the angle of the deviation of the cargo rope from the vertical, its first and second derivatives, as well as the horizontal coordinate of the upper point of the load suspension, its first and second derivatives from the time of displacement are obtained. Scope of the method – automatic control systems for the movement of bridge and gantry cranes, as well as modelling of crane work processes. The application area of the procedure is limited to the small angles of deflection of the cargo rope of the bridge crane from the gravitational vertical. The method opens the possibility of rapid synthesis of the optimal trajectory of the suspension point, which accurately provides the required trajectory of cargo movement. There is no need to use a comparatively complex mathematical apparatus of the theory of optimal management, or resource-intensive algorithms of multidimensional and iterative optimization.