Movement of a Heavy Rigid Body Suspended on a Cable of Variable Length with Oscillation Elimination

A nonlinear problem of terminal movement of a heavy rigid body suspended on an inextensible inertialess cable of variable length with controlled horizontal movement of the suspension point is considered. It is required to move the body for a certain time from an initial resting position to a given final resting position with oscillation elimination in the end of the operation. The law of variation in the cable length is assumed to be given, and the controlled movement of its suspension point is unknown. An approximate solution to the problem of kinematic control of the oscillations of a system described by two nonlinear differential equations with variable coefficients for moderately large angles of rotation of the strained cable and the body is sought in the form of series with unknown coefficients by the Bubnov-Galerkin method using given basic functions of time that satisfy some initial and final conditions. The acceleration of the suspension point of the cable is sought in the form of a finite series of sines with unknown coefficients. A coupled system of nonlinear algebraic equations for all unknown coefficients is obtained, which includes the equations of the Bubnov-Galerkin method and initial and final data that are not fixed when choosing the basis functions. This system of equations is solved by the method of successive approximations using, in the first approximation, solutions of the linearized equations. In the examples of a system with a cable of constant and variable length, calculations are carried out with an analysis of the accuracy of the solutions by comparing them with the numerical solutions of the nonlinear differential equations of the direct problem by the Adams method with the control laws found.