THE SPRING CLUTCH 727 



As compression exists in the spring wire at any point when the arbor 

 is turned to make the spring unwind, there will be a radial force sub- 

 tracting from /o at every point. This subtracting force is Plr^. The 

 increase of compression in the wire along the length of the line of 

 contact due to friction is 



dP = m(/o - PlrMl (1) 



which upon integration gives 



/ = - (r^/M) In [/o - (P/r2)]C, 



where C is a constant of integration equal to I//0 since P = at ^ = 0. 

 Hence 



P = rMl - e-^'/'-^). (2) 



Since / = I-kYiN, 



P = rMl - e-2-^''). (3) 



Since the torque is equal to Pr2 



T = r^Joil - e-2-^M) (in. -lb.). (4) 



It will be observed that for any but fractional values of N/j. the ex- 

 ponential term becomes very small and 



T = r2% (A^M > 1). (5) 



If Nn =1.0 this expression is in error by only 0.2 per cent. It can 

 thus be seen that provided N/j. does not become too small, variations 

 in A^^ or M do not affect the torque exerted. The torque will depend only 

 on the radius of the arbor and on the force /o which is controlled 

 entirely by the dimensions and the elastic properties of the spring. 



Torque of Spring Clutch in the Gripping Direction 

 If the torque is applied to the clutch in the direction to wind up the 

 spring, instead of unwind it as in the previous case, the force P'Jr2 

 due to the tension P' in the spring wire adds to the inward force /o and 

 the relation corresponding to equation (1) is 



dP' = m(/o + P'lr2)dl. . (6) 



From which by the same method as before 



P' = r2/o(e2-VM _ 1). (7) 



The corresponding torque is 



T' = rif,{e''^^'>^ - 1) (in.-lb.). (8) 



