A Note on Equivalent Linkages of Direct-Contact Mechanisms

Wen-Tung Chang,Dung-Yan Yang

doi:10.3390/robotics9020038

Abstract

In this paper, the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis is discussed. Kinematic analyses for three classical types of direct-contact mechanisms consisting of: (a) higher pairs with permanently invariant curvature centers, (b) higher pairs with suddenly changed curvature, and (c) higher pairs with continuously varying curvature are performed, respectively, through their representative case studies. The analyzed results show that the equivalent four-bar linkage cannot give a correct value of jerk for most situations in the three case studies. Subsequently, the concept of “equivalent six-bar linkage” for direct-contact mechanisms is proposed in order to discuss the infeasibility of the equivalent four-bar linkage for jerk analysis. It is found that the suddenly changed or continuously varying curvature of the higher pairs is not considered in sudden or continuous link-length variations of the equivalent four-bar linkage, which further leads to inconsistency between the angular accelerations of the coupler and the contact normal, and finally results in the infeasibility of the equivalent four-bar linkage for jerk analysis of most direct-contact mechanisms. It is also found that the concept of equivalent six-bar linkage could be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms. The presented case studies and discussion can give demonstrations for understanding the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in the aspect of jerk analysis.

Highlights

IntroductionThe simplest direct-contact mechanism consists of a frame, a driving link, and a driven link, in which the driving and driven links are in contact with each other to transmit motion and force
Direct-contact mechanisms, such as gear and cam mechanisms, are mechanisms with higher pairs.The simplest direct-contact mechanism consists of a frame, a driving link, and a driven link, in which the driving and driven links are in contact with each other to transmit motion and force
It is found that the equivalent four-bar linkage of a planar gear mechanism with a pair of involute spur gears is not able to give a correct value of jerk, unless the angular velocity ratio between the two mating gears is exactly negative one (−1)

Summary

Introduction

The simplest direct-contact mechanism consists of a frame, a driving link, and a driven link, in which the driving and driven links are in contact with each other to transmit motion and force Kinematic analysis for such a three-link direct-contact mechanism can sometimes be replaced by its equivalent four-bar linkage in order to calculate the velocity and acceleration at a particular instant [1,2,3,4,5]. When the physical properties for the members of the existing higher pair (such as their masses, centers of masses, and mass moments of inertia) are known, dynamic force analysis for the existing direct-contact mechanism can be performed, for which the indirectly obtained accelerations for the existing higher pair can be used for calculating their inertia forces and moments. The shaking forces and moments [5] in the existing direct-contact mechanism can be evaluated

Objectives

Discussion

Conclusion