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Aug., 1941] CONSIDERATIONS IN SPROCKET DESIGN 167
as the curves show, but owing to sharper peaks in Case //, the maximum instantaneous flutters are in the ratio of 5.20 to 1. Furthermore, the design of Case // will not work well as a drive sprocket (by reversal of rotation), since the film will tend to ride off or produce excessive friction against the guides.
It will be noted that the sprocket of Case / will serve as either a "hold-back" or a "drive" sprocket. If used as a "hold-back" sprocket, a guide above the film or additional balanced film tension is required to move the film down the tooth face. This may be avoided by using Case // for "hold-back" sprockets, but more unsteadiness of film movement results.
Hence, for flutter considerations, most, if not all, of the pitch range should be accommodated by the method of Case /. The tooth pitch should be equal to or just a little less than the maximum film pitch contemplated.
The first step in the sprocket design is to determine the base circle radius that will make the sprocket pitch equal to maximum film pitch for the number of teeth desired. The film thickness must be taken into consideration.
BASIC TOOTH SHAPE
For the second step in the design it is necessary to find the curve generated by a point on the film path when this path is rolled without slipping on the base circle. This is the well known epicycloid (or involute, in case the film path is a straight line). It is recommended that this curve be plotted from the parametric equations given in Appendix / for the three possible cases. The graphical method requires an awkwardly large scale to obtain sufficient accuracy.
The parameter in each case is the angle rolled through on the base circle. When angle 0 is multiplied by a suitable constant, it gives the time required for the film to travel over the path from point (x, y) to P (Figs. 3, 4, and 5).
It is evident that one degree of freedom at the designer's command is the radius of curvature of the film path. In plotting the above curves it will be discovered that the second case of Appendix / gives the longest time of film engagement for a given tooth height. This is desirable in reducing flutter.
The value of a = 0.5c for this. case has been found to be a good basic assumption. A much larger value reduces the engagement