406 



BELL SYSTEM TECHNICAL JOURNAL 



Substituting (2.8.1) in (1.2.13) and performing the indicated integration, 

 we find 



.2 ^7 «0 2 T A 

 Xo — Igll — .T^ — .To . 



m-i 2ni2 



Remembering that Xo = dxi/dt, we solve (2.8.2) for dt: 



dxt 



dt = 



\/2gh-^xl-^x\ 

 y 1112 2nu 



(2.8.2) 



(2.8.3) 



10 12 14 16 18 20 



Fig. 2.8.1 — Duration of acceleration pulse for cushioning with cubic elasticity. See 



equation (2.8.10). 



Then, with the initial condition .r2 = when / = 0, the integral of (2.8.3) is 

 dxo f"^'- dx2 



JQ 



r""- dxo /■•' 



•'O X2 Jn 



J\Jlgh- ^x\- ^^x\ 



(2.8.4) 



;w2 " 2w2 



To integrate (2.8.4), let 



Z = 

 where 



X2 



Vk'-(xi - d„,y + di ' 



k- = hii - 



Vl + J5 



(2.8.5) 



(2.8.6) 



