The fourbar linkage is one of the first mechanisms that a student encounters in a machine kinematics or mechanism design course and teaching the position analysis of the fourbar has always presented a challenge to instructors. Position analysis of the fourbar linkage has a long history, dating from the 1800s to the present day. Here position analysis is taken to mean 1) finding the two remaining unknown angles on the linkage with an input angle given and 2) finding the path of a point on the linkage once all angles are known. The efficiency of position analysis has taken on increasing importance in recent years with the widespread use of path optimization software for robotic and mechanism design applications.
Kinematicians have developed a variety of methods for conducting position analysis, but the solutions presented in the literature fall into two general families:
1. The angle between the coupler and the rocker is found using the law of cosines. Once this is known, the coupler and rocker angles are found using some combination of the laws of sines and cosines.
2. A vector loop equation is written around the linkage, and then half-angle tangent identities are used to solve for the two unknown angles.
Two widely-used mechanical design textbooks use method 2, whose derivation is lengthy and whose final results permit no simple geometric interpretation. Method 1 has a much simpler derivation but is difficult to implement in software owing to a lack of four-quadrant functions for sine and cosine.
With this in mind, we have developed a more efficient method for obtaining the position solution for the fourbar linkage that is well-suited to educational settings as well as for design optimization: the projection method. Because the final formulas have an elegant geometric interpretation, we have found that this method is easier for mechanical engineering students to understand and could therefore become a new standard method for mechanical design textbooks. In addition, the final position formula uses the tangent function, which has widely-available four-quadrant implementations. The projection method is easily extended to other common linkages, including the inverted slider-crank, the geared fivebar linkage, and four of the five types of single degree-of-freedom sixbar linkages. This method has been used to develop an educational website, www.mechdes.net, that contains simulations of several common linkages and mechanisms. This paper presents a comparison of the two traditional methods and the projection method, and pseudocode algorithms for each method are given at the end.
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