Propagation of Longitudinal Deformation Wave Along a Hoisting Rope Carrying an Intermediate Concentrated Load

Abstract

Motion of the hoisting rope carrying an intermediate concentrated load is described by the one-dimensional wave equation in the region consisting of two sections separated by a moving boundary condition. The system is moved by the driving force acting at the upper cross section of the rope. Position of the intermediate load and consequently the lengths of the rope sections vary in the time depending on the magnitude of driving force. Solution of the wave equation is represented as a sum of integrals with variable limits of integration. The problem is reduced to solving the sequence of ordinary differential equations which describe a motion of the load in the fixed coordinate system and the paths of the rope ends in the moving coordinate system connected with the load. The argument of functions involved in the right-hand side of these equations lag behind the argument of the derivatives in the left-hand side of equations by a short time interval. A description of the unknown functions in a parametric form makes possible to eliminate retarded arguments from the equations. The problem is solved by using a technique of the sequential continuation of solution for time intervals corresponding to propagation of the deformation wave in the opposite directions. A computer program has been developed for solving the problem. Results of the numerical solution are presented in the case that the driving force is a piecewise linear function of time and is discontinuous at the peak point.