DYNAMICS OF PACKAGE CUSHIONING 

 (1 - b)z' 



where 



Q'(z') = 2H \z' 

 .^•2 



+ 



2 = 



Vl +2 



;] 



373 



(1.7.5) 



Consider, next, the configuration shown in Fig. 1.7.4(a), where one end of 

 each of four springs is fixed at a corner of a rectangle of length It and width 

 2.ro . Each spring is again of length i — j. The four free ends of the springs 

 are drawn together at a common point X at the center of the rectangle (see 

 Fig. 1.7.4(b)). The system is in equilibrium in this position. A force Q 

 is then applied at A^ in the plane of the rectangle and normal to the side It. 



f f 



uuuii) 



/ — -V 



f ■ f 

 a 



2X, 



ckflMAJLl/p •- 





^, 



Fig. 1.7.4 — Action of springs in a tension spring package. 



The common point X is displaced a distance X2 to X' (see Fig. 1.7.4(c)). 

 Writing z = xz/t, a = Xo/t, we observe that 



Q(z) = Q'iz + a) + (2'(0 - a), (1.7.6) 



or, from equation (1.7.5), 



Q{z) = 2u\2z-{\-}A / ^ "^ ^ - + - ^-^^^ ^=^11 (17 7) 

 ^^' I ^ iVl + (z + a)2 ^ Vl + (2 - a)0/ ^ '^ 



The standard tension spring package has two sets of four springs so that 

 the force P required to displace the common point X a distance Xi. is 



Piz) = 2Q(z). (1.7.8) 



If X2 is small in comparison with t (i.e., z is small in comparison with 

 unity), equation (1.7.8) may be written approximately as 



P(z) = Akchz - (1 - h)z{2-i)\ 



Even when x^ becomes almost as large as t, equation (1.7.9) has been found, 



e.xperimentally, to be remarkably accurate. 



Writing 



K 



fl- ^-M 



L (1 + a2)3/2j 



m\ - 



(1.7.10) 



