Propagation of longitudinal deformation wave along a lifting rope of variable length

The problem of motion of the rope of variable length consists of solving the boundary value problem with a variable boundary for the one-dimensional wave equation. A change of the rope length is caused by the force acting at the upper cross section of the rope. Studying the wave propagation process along the rope is based on the known integro-differential relation. The problem is reduced to solving two ordinary differential equations with a retarded argument that describe the variable length of the rope and the position of its lower end. The value of argument for functions involved in the right-hand side of the equations lag behind the argument value in the left-hand side of the equations by a time interval it takes for a propagation of the deformation wave throughout the current rope length. The problem is solved by using a technique of the sequential continuation of solution in the cyclic mode for each of the equation alternately. A computer realization of this technique presents no problem. A pilot computer program has been developed for solving the problem. Results of the numerical solution are presented in the case that the active force varies with time according to a piecewise linear relation.

► Oscillations of the lifting rope of variable length are determined by solving the wave equation. ► A change of the rope length is caused by a force at the upper cross section of the rope. ► Problem is reduced to solving two ordinary differential equations. ► Method of the sequential continuation of solution is used for numerical solution.