388 BELL SYSTEM TECHNICAL JOURNAL 



1.13 Cushioning with Hyperbolic Tangent Elasticity (Class E) 



In the preceding sections, there have been considered four types of elas- 

 ticity (hnear, cubic, tangent and bihnear) that fit the load-displacement 

 characteristics of the more common cushioning materials and devices. 

 There now remains the problem of finding more nearly ideal shapes of 

 elasticity. By "more nearly ideal" is meant a shape which will result in a 

 smaller maximum displacement for a given maximum acceleration. This is 

 important in the packaging of very delicate articles if shipping space is 

 limited. 



It may be observed (equation (1.2.15)) that the total area under the 

 load-displacement curve is equal to the maximum energy of the system. 

 The maximum ordinate of the enclosed area is proportional to the maxi- 

 mum acceleration. Hence, if we wish to (1) limit the maximum acceleration 

 (2) accomodate a given kinetic energy and (3) have as small a displace- 

 ment as possible, the best shape for the load displacement function is P = 

 constant, where the constant is the product of the supported mass and the 

 maximum allowable acceleration. 



It is not practical to obtain this ideal shape exactly, for there will always 

 be a finite initial spring rate and a rounding off of the load-displacement 

 curve to the limiting maximum load. A function which represents this 

 practical condition (and also includes the ideal case) is the hyperbolic 

 tangent function mentioned in Section 1.4: 



P = Potanh^'. (1.13.1) 



The formulas for maximum acceleration and displacement are found in 

 the same way as for the other classes of cushioning with the results: 



Po -1 (WohkA 

 dm = r cosh exp I o j (1.13.2) 



or 



and 



or 



doPo ,-1 fWlG} 



"^^ = w^cr'^'" '-'n^iT^ ^'-''-'^ 



Gm = ~ tanh ^' (1.13.4) 



W 2 -i 



G„.=i^y^l-exp(-"3p) (U3.5) 



