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174 J. S. CHANDLER [j. s. M. P. E.
FRICTION FORCES
The term "theoretical flutter" was used in the preceding section because it is recognized that other sources of flutter exist and the calculated values may not be attained in practice. Two additional sources of flutter will be considered here: (1) friction of film against tooth and (2) friction of film against guide shoe. Fig. 9 (a) shows a portion of the film as it is just beginning tooth engagement at the top of the tooth of Fig. 6. If we assume a coefficient of friction of 0.25, the forces of tooth against film and of shoe against film make an angle of tan"1 0.25 with their respective normals. The magnitudes of these forces relative to the film force, F, are determined by the force triangle of Fig. 9(6).
If such a force analysis is made for the different film positions it will be found that the tooth force remains substantially equal to F, while the shoe force varies from 0.480 .F to 0.4347^. As the tooth face slopes more to the left with a decrease in the ratio of a to c (Appendix /), the film shoe force increases.
The force analysis also shows that the film guide must be above the film to hold it down. If the sprocket is to act as a drive sprocket, the tooth and shoe forces fall on the other side of their respective normals, and the guide must be below the film to support it.
In calculating the force of the film against the shoe, it is well to note that there is an additional radial force toward the center owing to the curvature of the film path. Neglecting the stiffness of the film this radial force = F/r per unit length of film, where F is the film tension and r is the radius of curvature. For the 24-tooth sprocket example, the radial force = 0.17 oF for a film length equal to the pitch.
It is the varying nature of friction which imposes a varying load on the sprocket or causes the film to proceed by jerks and thus introduces flutter.
SPROCKET-TOOTH SHAPER
The usual method of hobbing sprocket teeth leaves tool marks across the face of the tooth. This may cause the film to catch as it moves up or down the tooth. For some experimental work, we have used a tooth shaper in which the cutting stroke is downward from the top of the tooth. With this machine a very smooth tooth surface can be obtained, and the remaining tool marks offer the least resistance to film movement.