Meek. Mach. T/wory Vol. 28, No. 2, pp. 193-206, 1993 0094-I 14X/93 $5.00 + 0.00 Printed in Great Britain. All rishta reserved Copyright ~ 1993 Pergamon Press Ltd A THREE-NODE FINITE BEAM ELEMENT FOR DYNAMIC ANALYSIS OF PLANAR MANIPULATORS WITH FLEXIBLE JOINTS AHMAD A. SMAILI Department of Mechanical Engineering. Mississippi State University, Mississippi State. MS 39762. U.S.A. (Received 23 December 1991; received for publication 3 April 1992) AIBtract--This paper presents a three-node isoparametric finite beam element to study dynamic behavior of planar manipulators. The characteristic matrices of the finite element are formulated to include the effects of joint compliances, shear deformation and rotary inertia and to account for coupling effects of nonlinear gross motion of the manipulator links with their distributed flexibility and mass properties. Instantaneous steady-state static response, modal analysis and transient response are obtained. A 2R planar manipulator with rigid and compliant joints is analyzed. I. INTRODUCTION The efficiency of a robot manipulator performing complex tasks at high speed is highly dependent on its dynamic behavior. For a robot manipulator to have acceptable level of dynamic performance its mechanical structure must be capable of carrying payloads as required, has small weight to payload, possess sufficient stiffness and damping to allow fast performance without large structural deformations and excessive vibrations and additionally operate with high accuracy and repeatability. To adequately design the mechanical system for a robot manipulator, its dynamic behavior must be thoroughly investigated. Because of the serial connection of links and drive units of a manipulator via its joint axes produces a complex, nonhomogeneous, continuous and flexible system, it is impossible to comprehend its dynamic performance by closed-form analysis without over-simplifying its dynamic model. Although finite element techniques do not reveal the effect of various parameters on the dynamic behavior of a robot manipulator, they offer a tool by which a nonlinear comprehensive dynamic model of the manipulator can be devised to include all parameters that influence its dynamic response. This paper presents an isoparametric, three-node beam element. While the formulations of the element matrices concentrate on modeling planar manipulators with revolute joints, it can be used for the analysis of linkage mechanism and transmission shafts as well. The methodology presented here can be easily extended to model spatial mechanical systems having various kinematic pairs. The finite element is formulated to include the influence of: (a) Shear deformations and rotary inertia of the links. (b) Interaction of distributed flexibility and mass of the manipulator components with the nonlinear gross motion of its links. (c) Joint compliances, mass of drive units and payload and external damping if present. Although several investigators have used finite element techniques to model manipulators, they invariably failed to include all the influences mentioned above at once. Noted here are the works by Sunada and Dubowsky [I], Bagci and Dado [2], Thompson and Sung [3], Usoro et ai. [4], Smaili and Bagci [5], Naganathan and Soni [6], Liou and Erdman [7], Sadler and Yang [8] and Jonker [9]. A survey of the literature on the subject is presented by Gaultier and Cleghorn [10]. 193