Analytical Determination of the Vibrations of a Load, Twice Elastically Suspended From an Elastic Beam via Another Load

Abstract

This paper analyzes the dynamics of mechanical systems with suspended loads, such as bridge cranes, monorail hoists, mining machinery, etc. The considered mechanical system is composed of a load, twice elastically suspended from an elastic beam via another load. Two dynamical models which respect the kinetic energy of the elastic ropes are built for the system and their corresponding differential equations of motion are obtained. The vibration of the mechanical system is described by a coupled system of two ordinary and partial differential equations. The nonlinear restoring forces are linearized via the method of equivalent linearization and an analytical solution is obtained for the differential equations of both dynamical models simultaneously, using general initial conditions. The constants of integration are determined analytically for a specific instance of the initial conditions, which reflects an important practical case. The mechanical system is simulated numerically with initial conditions corresponding to the typical regimes of operation of real systems with suspended loads.