Polynomial equations for the loci of the acceleration pole of a slider crank mechanism

This paper presents polynomial equations in implicit form for the loci of the acceleration pole of a slider crank mechanism operating at a steady state. The acceleration angle for any point on the coupler is a function of the geometric configuration of the mechanism with a constant crank speed. Accordingly, a geometric approach is applied to formulate three trigonometric equations relating to the location of the acceleration pole for any position of the mechanism. The Cayley's statement of the Bezout's method is then applied to eliminate two variables, the angular displacement of the crank and linear displacement of the slider, in the three equations derived. This approach leads to two polynomial equations for the loci of the acceleration pole on both fixed and coupler planes of the mechanism. The degrees of the two polynomial equations are 24 and 28, respectively. Furthermore, both polynomial equations have full circularity of 12 and 14, respectively. Since each polynomial equation consists of two branches of the mechanism, both loci are symmetrical curves with respect to the ordinate in the fixed plane and abscissa in the coupler plane. Two numerical examples are provided for deriving the corresponding equations.