324 The Ohio Journal of Science [Vol. XX, No. 8, 



his "Mechanics appHed to Engineering," Goodman, neglecting^ 

 the angularity of the helix, gives for the maximum fiber stress- 

 and the total deflection 



(9)f = ^ (10)A = .8"™' 



Trd^ ^ ' Gd^ 



In the above D equals the mean diameter of the coil, d the 

 diameter of the wire, n the number of turns, G the coefficient 

 of rigidity, and the other quantities as before. A moment's con- 

 sideration shows that we will have for the length of the wire 

 and the total weight 



L = xnD and W= (l/4)7r2nwd2D 



From Equations 9 and 10 we find 



A/f2 = x2nd2D/(8PG) = W/(2wPG) 



From this it follows that 

 W = 2wPG(A/f2) 



From this it follows that the weight of the spring depends 

 only upon the material, the applied load, the coefficient of 

 rigidity, the maximum deflection and the allowable fiber stress 

 in shear; and is entirely independent of the size of wire, diameter 

 of the helix and number of turns. The forrnulas for design 

 may now be easily derived: 



(a'") nd2D = 8PGA/(7r2f2) 

 (b"0 dVD = SP/(xf) 

 (c"0 A = SnPDVGdO 



As before, we have two independent equations containing 

 the three unknowns, n, d, and D, so that we may assign to any 

 one of them an arbitrary value and then compute the others 

 and use the third equation as a check. 



As an illustration, let us assume P = 100 lbs., G = 12,000,000, 

 f= 50,000 and A = i". We will then find that nd^D^ 0.1945, 

 and d^/D = 0.00509. If then we assume that the spring is to 

 have twenty turns of wire, we find from the above that 

 D = 0.513", d = 0.1377", and if the spring is to be wound close, 

 its length will be 2.8". The following formula is useful in 

 computing the length of the helix (L), L' = n (d + s) where s 

 is the width of the open space, which is small in the springs 

 considered. 



The Emerson McMillin Observatory, Ohio State University, 

 May 15th, 1920. 



I 



