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BELL SYSTEM TECHNICAL JOURNAL 



li = distance (I A) between spring support points when suspended article 

 is in equilibrium position, 



/ = projection of /» on plane A BCD, 



f = -C minus length (between hooks) of unstretched spring, 



k = spring rate of each spring. 



Consider, first, the action of one pair of springs, say EM and GO of Fig. 

 1.7.1, independent of the remainder of the suspension. Since EM and GO 

 lie in parallel vertical planes and the points M and remain in the initial 

 planes of their respective springs during a vertical displacement, the two 

 springs may be considered to lie in the same plane, and to be translated hori- 

 zontally in this plane so that their outer ends are separated by a distance 2i. 

 Hence Fig. 1.7.3 may be used to represent the independent action of this 

 pair of springs and it is required to find the force Q' needed to transform Fig. 

 1.7.3(a) to Fig. 1.7.3(b). Initially there are two springs, each of length / — / 



f f 



OVJlAJJJLJLtP CK. 



itojuuii 



a b 



Fig. 1.7.3— Diagram used in discussion of tension spring package. 



and spring constant ^, with no initial tension in them. One end of one spring 

 is fixed at point E and one end of the other spring is fixed at a point G 

 distant U from E. The springs are then stretched so that the two initially 

 free ends are located at a point Y equidistant from E and G and distant x-i 

 from line EG. The axis of each spring makes an angle a. with £G, where 



sm a = 



OCi 



^e + xf 



In this state the axial force F in each spring is 



F = kWf^^^ - t+f] 



(1.7.2) 



(1.7.3) 



and the force Q\ required to equilibrate the two forces F is 2F sin a. Con- 

 sidering the force Q' as a function of the displacement Xo , we write 



Q'{4) = 



V7T 



=^ [Vf + x? -i+f] (1.7.4) 



X2 



