Dynamic analysis of planar flexible mechanisms

Abstract

The traditional approach to dynamic analysis of mechanisms and machines is based on the assumption that the systems are composed of rigid bodies. The assumption however, does not always hold. Members are elastic, so they deflect when they are subjected to external loads or inertial forces. Particularly at high speed, a linkage may undergo severe elastic deformation due to its own inertia, to an extent that it can no longer perform its intended function in actual operation. With the speed of machinery constantly increasing, an improved mathematical model of machine dynamics is needed to predict system response to transient inputs. Several authors [l]-[9] have recently introduced models of planar mechanical systems, treating flexible members rather than the traditional rigid links. Winfry [l] originally proposed an analysis method in which the effect of member flexibility is modeled by applying a structural dynamics stiffness technique, using the assumption of superposition (uncoupling) of gross rigid-body motion and a small elastic deformation. Later [2] he utilized a reduction of coordinates technique to determine a particular deflection in a mechanism, based on the same method and assumptions as in [l]. Erdman and co-workers [3] introduced the idea of kineto-elastodynamics, which is the study of the motion of mechanisms consisting of elements that deflect due to external loads or internal body forces. Their method is based on a flexibility matrix approach of structural analysis. They assume linear superposition of small elastic deformation due the inertia forces arising in rigid body motion of the system. They model a planar mechanism as combinations of cantilever beams, two force members, and simply supported beams with end moments. Subsequently, Imam and co-authors [4] presented a general method of deflection analysis for planar linkages, including multi-loop mechanisms, extending the method for kineto-elastodynamic analysis of mechanisms [3] and introducing the rate of change of eigenvalues. Sadler and Sandor [S] have employed lumped parameter models for simulating planar motion of mechanism components that are considered as simply supported beams subject to plane bending in kineto-elastodynamic analysis of mechanisms. Longitudinal deflections are neglected and the assumption of small transverse elastic deformations is made, in order that linear Euler-Bernoulli beam theory would apply. Finite difference formulas are applied to