An Approximate Method for the Dynamic Analysis of Elastic Linkages

A new numerical procedure based on an iterative technique is progressively developed in this paper for obtaining an approximate particular solution from the equations of motion of an elastic linkage with small damping and at subresonant speeds. The method is introduced by employing a simple vibrating system, a single degree-of-freedom mass-dashpot-spring model under both harmonic forcing and periodic forcing. A harmonically excited two degree-of-freedom model is also solved by the suggested approach. Error functions are developed for each case to give an estimation of the order of error between the exact analytical solution and the approximate technique. The suggested technique is then extended to solve an elastic linkage problem where the uncoupled equations of motion are treated as a series of single degree-of-freedom problems and solved. These are retransformed into the physical coordinate system to obtain the particular solution. The first and second derivatives of the forcing functions (involving rigid-body inertia) are approximated utilizing the finite difference method.