Gordon R. Pennock ASME Fellow Associate Professor Edward C. Kinzel Research Assistant School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907-2088 Path Curvature of the Single Flier Eight-Bar Linkage This paper presents a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier eight-bar linkage. The first step is to locate the pole for the instantaneous motion of the coupler link; i.e., the point in the fixed plane coincident with the absolute instant center of the coupler link. Since the single flier is an indeterminate linkage, comprised of one four-bar and two five-bar chains, then the Aronhold-Kennedy theorem cannot locate this instant center. The paper presents a novel graphical technique which can locate this instant center in a direct manner. Then the paper focuses on a graphical method to locate the center of curvature of the path traced by the coupler point. The method locates six equivalent four-bar linkages for the two five-bar chains, investigates six kinematic inversions and obtains a four-bar linkage from each inversion. This systematic procedure produces a four-bar linkage with a cou- pler link whose motion is equivalent up to, and including, the second-order properties of motion of the single flier coupler link. The radius of curvature and the center of curvature of the path traced by the coupler point can then be obtained in a straightforward manner from the Euler-Savary equation. DOI: 10.1115/1.1731298 1 Introduction The planar four-bar linkage is the simplest single degree of freedom linkage and many graphical techniques exist for investi- gating the properties of a coupler curve 1. However, because of the limited number of links this linkage cannot generate certain complex coupler curve shapes 2. In fact, the four-bar cannot satisfy a wide variety of design requirements, for example, a point on the coupler link cannot dwell during a continuous input, and a link cannot move with straight translation. When increased de- mands are imposed on the design then more links must be em- ployed. The next number of links, for a linkage with a mobility of one, is the six-bar linkage and then the eight-bar linkage. These carefully designed linkages are not only more versatile than the four-bar linkage but can provide the required performance while maintaining economy and reliability without resorting to elec- tronic controls and actuation. The linkage investigated in this pa- per is a planar, single-degree-of-freedom, eight-bar linkage, com- prised of one four-bar and two five-bar chains. The linkage is commonly referred to in the kinematics literature as the single flier eight-bar linkage 3. The focus of the research is a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier linkage. The initial step in the procedure is to locate the pole of the coupler link, which is coincident with the absolute instantaneous center of zero velocity henceforth abbre- viated as instant center. However, it is important to note that this instant center cannot be obtained directly from the Aronhold- Kennedy theorem and the linkage is referred to as indeterminate 4. Dijksman 5 presented a geometric approach to determine the instant centers of an indeterminate linkage by applying first-order reduction through joint-joining. Foster and Pennock 6 proposed a more direct approach based on an iterative procedure. The tech- nique was illustrated by Pennock and Sankaranayananan 7 in a study of the path curvature of a geared seven-bar mechanism. The method will also be used in this paper since it is a simple geomet- ric approach, unlike most techniques where velocity information is required 3,8. The second step is to locate the center of curvature of the cou- pler curve for a specified position of the single flier linkage. Dijks- man 9 presented a geometric approach to this problem, which he termed second-order joint-joining, and is an extension of his ear- lier paper on first-order reduction through joint-joining. The tech- nique presented in this paper uses kinematic inversion to deter- mine ten virtual links which can be used to form a series of six four-bar chains. These chains are then manipulated to create an equivalent four-bar linkage for the motion of the coupler link. For the purposes of this paper, an equivalent linkage is defined as having the same kinematic properties of the original linkage up to, and including, the second-order. This stipulation also applies to the four-bar chains formed by the ten virtual links. The inflection circle and the center of curvature of the coupler path can then be obtained in a straightforward manner from the Euler-Savary equa- tion 10. This equation has proved to be important in the study of curvature relationships for the analysis and synthesis of planar, single-degree-of-freedom mechanisms 1,11. The important con- tribution of this paper is that the techniques presented here are purely geometric; i.e., the methods do not require the velocity and acceleration of points fixed in links. The paper is arranged as follows. Section 2 presents the graphi- cal technique to locate the velocity pole for the coupler link of the single flier linkage. Then Sec. 3 presents the graphical technique to locate the center of curvature and determine the radius of cur- vature of the path traced by a coupler point. Section 4 presents a numerical example to illustrate the graphical techniques and com- pares the result for the radius of curvature of the coupler curve with an analytical method presented by Pennock and Kinzel 12. Finally, Sec. 5 presents some conclusions and suggestions for fu- ture research. 2 Graphical Technique to Locate the Pole A schematic drawing of the single flier eight-bar linkage is shown in Fig. 1. The ground is denoted as link 1, the input link is denoted as link 2 and the coupler link is denoted as link 8. Link 2 is pinned to the ground at O 2 and link 4 is pinned to the ground at O 4 . Links 1, 5, 6 and 7 are binary links and links 2, 3, 4 and 8 are ternary links. The revolute joints connecting the moving links are denoted as A, B, C, D, E, F, G and H. For purposes of generality, pin A is not coincident with pin F on link 2, pin E is not coincident with pins F or G on link 3, pin G is not coincident with pin H on link 4, and pins B, C and D are separated by finite distances on the coupler link. The coupler point will be denoted as point Q and is an arbitrary point fixed in the coupler link. The instant center for Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 2003; revised October 2003. Associate Editor: G. Ananthasuresh. 470 Õ Vol. 126, MAY 2004 Copyright © 2004 by ASME Transactions of the ASME Downloaded 02 Feb 2013 to 131.151.67.206. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm