Journal of the Society of Motion Picture and Television Engineers (1950-1954)

226 RECOMMENDATIONS February It will be obvious from a study of the nature of the action of the tooth that the drive sprocket is the most critical with respect to the shape of the tooth. Epicycloid Curve. The most logical starting place for the analysis of the shape of the tooth is the curve generated by a point on the film relative to the sprocket when the film moves along its path without slipping on the root circle of the sprocket. Since we have assumed that the path is an arc of a circle, the curve so generated is an epicycloid. In the case of a straight path, the resulting curve is an involute and can be treated as a special case of the epicycloid in which the generating circle has an infinite radius. The generated curve is an epicycloid whether the generating circle curves away from the sprocket or curves toward the sprocket and actually encloses it. (Equations to aid in plotting the epicycloid are given in the complete paper.) Since the epicycloid is generated by a circle rolling on the root circle, it is the locus, relative to the sprocket, of a point on the film, such as the edge of a perforation, provided there is no slippage between the film and the sprocket. The epicycloid curve is a valuable reference from which the desired shape of the tooth can be deduced by proper allowance for the amount of film shrinkage to be accommodated. It is obvious that the reference epicycloid to consider is the one that corresponds to the most limiting condition, namely, the film path that curves away from the sprocket along the arc with the minimum radius. Allowance for Shrinkage at Tip of Tooth. Figure 3 shows on a large scale the film and one tooth of a 1 2-tooth sprocket for 1 6-mm film. If the tooth is moving to the left and the film tension is to the right, we have a drive sprocket. The circular pitch of the sprocket is greater than the pitch of the film. If the shape of the tooth is such as to guide the film along the epicycloid (Fig. 3, curve 1) the film will not slip on the sprocket until after it leaves the tip of the tooth, whereupon it will jump to the right and will stop suddenly when the next perforation engages the next tooth to the right. A tooth shape falling to the right of the epicycloid will let the film slip gradually and thus accommodate part of, or all, the shrinkage differential before the film reaches the tip of the tooth. The optimum condition is reached when the tooth just allows accommodation of the maximum shrinkage differential when the film is ready to leave the tip of the tooth. If the film is shrunk less, there will be full accommodation earlier and the film will leave the tooth before it reaches the top. D= 1.1400 INCHES K= 0.0804 INCH B=O.OI11 INCH FOR 1% SHRINKAGE B= 0.0146 INCH FOR 1.5% v SHRINKAGE ^CURVEI EPICYCLOID \ SUPERIMPOSED CIRCULAR ARC 1 ALTERNATIVE TOOTH SHAPE TOOTH FOR 1.5% SHRINKAGE TOOTH FOR I °/o SHRINKAGE 0.004 0.008 0.012 0.016 0.020 0.024 0.028 X , INCHES Fig. 3. Tooth Shape for 12-Tooth, 16mm Sprocket. The shrinkage differential with which we are concerned is that for one pitch length of film. For example, if the shrinkage range is from 0 to 1.5 per cent, the circular pitch of the sprocket is chosen to match unshrunk film (0.300 in.) and the maximum pitch differential is 1.5 per cent of 0.300 in., or 0.0045 in. The proposed tolerances for the pitch of the drive sprocket are plus 0.0003 in., minus 0.0000. Therefore, an additional allowance of approximately 0.0003 in. is made in establishing the location of the tip of the tooth.