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Suppose the natural radius of curvature of the curling film in the longitudinal direction is constant for the considered length of film and denoted by rn, counted positive if the film tends to curl around drum 1. Since in this case the first of the two Eqs. (1) modifies to
El I
Ml
the relation (25) for the parameter tv changes to
(27)
The loop Eq. (4) remains unchanged. This can be visualized from its physical meaning, namely, being an equilibrium condition for the moments acting on the film. If a piece of film is considered between the contact point CPi and any other point of the loop, it is seen that the two opposing bending moments at the ends of this piece are proportional to the change of curvature which the film has undergone when being deformed from its natural state. The difference between these two bending moments is balanced
by a force couple which originates from the relative parallel shift of the opposing force vectors P at the considered points (see Fig. 1). Since only the difference between the two bending moments enters the equilibrium condition, one need consider only the difference of the changes of curvature at the two points. But in this difference the original curvature of the curled film drops out as it was assumed constant over the whole length of film. Thus the loop Eq. (4) and all ensuing relations, including the final result (26), are independent of rn. In other words, the effect of film curling reflects only in the value of the parameter ti as per Eq. (27). This value in connection with Eq. (26) yields the correct length of slack film.
References
1. E. D. Cook, "The technical aspects of the high-fidelity reproducer," Jour. SMPE, vol. 25, pp. 289-313, Oct. 1935.
2. W. J. Albersheim and D. MacKenzie, "Analysis of sound-film drives," Jour. SMPE, vol. 37, pp. 452-479, Nov. 1941.
3. E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves, Dover Publications, New York, Reprint 1945.
Gerhard Schwesinger: Compliance of Film Loops
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