730 BELL SYSTEM TECHNICAL JOURNAL 



axis, by the force per unit length F{AR) working through the distance 

 AR where AR = R2 - Ry. That is 





F(AR)dAR = \eI 



AR 



R,{R, + AR) 



(16) 



Differentiating both sides with respect to AR gives F{AR) for the left- 

 hand side, and after simplification 



^(^«' = f (sr^3- (17) 



Substituting for AR its value R^ — Ri gives 



F{AR) = EI{R2 - Ri)/RiRoK (18) 



The equivalent force per unit length measured along the surface of the 

 arbor must be larger than this value in the ratio of i?2/^2 since the 

 same total force is here distributed over a shorter length. This latter 

 force is/o; hence 



/o = ^ F(AR) = EI ^^' lb. per in. (19) 



The value of /o calculated from this equation in terms of the constants 

 of the spring material and the dimensions of the spring may be used in 

 equations (4) and (8) to calculate the free torque and the slipping 

 torque of the clutch. 



Experimental Check of the Free-Torque Relation 

 In order to check the validity of the relation for the free torque, 

 equation (4), and that for the radial force on the arbor, equation (19), 

 the free torque of a given spring on arbors of various diameters as well 

 as the torque for different numbers of turns on the same arbor was 

 measured. The spring of 0.0085 X 0.022-in. phosphor-bronze ribbon 

 was attached to a short vertical shaft suspended by a torsion fiber of 

 measured torsional stiffness. The free end of the spring was placed 

 over a vertical arbor capable of rotation. The arbor was rotated and 

 the angle of twist of the torsion fiber was measured thus giving a 

 measure of the slipping torque. The precision of the measurements of 

 torque was about 0.5 per cent although the sensitivity to small changes 

 was about 0.2 per cent. No measurable increase of torque occurred 

 by increasing the number of turns on the arbor beyond six. This is 

 to be expected if the coefficient of friction exceeds about 0.12. 



For all succeeding measurements seven to eight turns were used. 

 As a further check on the independence of the torque and the coefficient 



