Elimination of a rigid body oscillations suspended on a variable-length cable with a controlled horizontal suspension movement

The paper considers a passive force (dynamic) and kinematic control problem of a heavy cargo movement (an undeformed solid) suspended on an inextensible inertia-free variable length cable with a controlled horizontal displacement of the suspension point. Differential equations with variable coefficients for small translational-rotational vibrations of the body are obtained. The following problem is stated: to move the body from the initial rest position to a given final equilibrium rest position for a preset time with oscillations elimination at the stop. In this case, the law of changing the cable length is considered to be prescribed, and the law of displacement of its suspension point is unknown. The integral conditions are established for required unknown control actions (force or acceleration of the suspension point), which should be satisfied. An approximate solution of the kinematic control problem described by two differential equations with variable coefficients for the angles of rotation of the cable and body is sought in series with unknown coefficients by the Bubnov-Galerkin method with the use of the given approximating functions of time satisfying certain initial and final conditions. Acceleration of the suspension point of the cable is sought in the form of a series of sines with unknown coefficients. A coupled system of linear algebraic equations for all unknown coefficients is obtained, which includes equations of the Bubnov-Galerkin method, equations for the initial and final conditions that are not satisfied in the choice of given functions, and one equation representing the integral condition in the form of the dependence of the acceleration of the cable suspension point on its specified finite displacement. The proposed approach for solving the problem of the oscillations finite control for a system with variable parameters is new. By using the examples of a system with a cable of constant and variable length, we performed the calculations with an analysis of the convergence and accuracy of solutions for two different sets of given functions and for different numbers of them by comparing them with numerical solutions of differential equations of the direct problem by the Adams method with the control laws found.