384 BELL SYSTEM TECHNICAL JOURNAL 



all high-frequency elements of the structure. This subject is treated in 

 detail in Parts II and III. 



The maximum displacement dm , in the case of tangent elasticity, may be 

 calculated from equation (1.9.2) or, in terms of dh/dn , from 



d,„ 2 ^1 

 — - = - cos exp 



db TT 



8 W 



(1.9.5) 



The ratio d,„/db is plotted against db/do in Fig. 1.9.2. 



The use of Fig. 1.9.2 can be illustrated with the example already calcu- 

 lated, in which db/do = 1.15/1.44 = 0.8. Entering the abscissa of Fig. 

 1.9.2 with db/do = 0.8 we find dm/db = 0.915. Hence the maximum dis- 

 placement will be 0.915 X 1.15 = 1.05 inches. 



1.10 Optimum Shape or Load-Displacement Curve for Tangent 



Elasticity 



It is possible to choose the best shape for the load-displacement curve 

 of the cushioning from those represented in Fig. 1.4.7. This will be, of 

 course, not the best of all possible curves, but only the best among "tangent 

 elasticity" curves. The best shape is defined as the one that yields the 

 smallest maximum acceleration (Gm) for a given weight (W2), height of drop 

 (//) and available space (db). This leaves the initial spring rate (^0) as the 

 only remaining variable. To find its optimum value (say ^0), set equal to 

 zero the derivative of Gm (equation (1.9.4)) with respect to ko , remembering 

 that Go and do are functions of ko . The result is 



tt'-^IFs/A /tt^IFs// 



from which 



'-i4*rj-plM*r'-'^°' <^-"'-" 



3AW2h 

 ko = --^. (1.10.2 



Substituting (1.10.2) in (1.9.4) we find the minimum value (G„0 of maxi- 

 mum acceleration to be 



3.9A 

 GL=~-~. (1.10.3) 



db 



To illustrate. the application of equations (1.10.2) and (1.10.3), consider 

 again the case of the 20-pound article dropped from a height of three feet. 

 We found that a linear spring, with a spring constant of 694 lbs/in, would 

 limit the maximum acceleration to 50g if 1.44 inches of displacement were 

 available. If only 1.15 inches of displacement are available, and the initial 

 spring rate is kept at 694 lbs/in, we found the maximum acceleration to be 



