DYNAMICS OF PACKAGE CUSHIONING 



413 



i.e., do is the displacement that would have been reached if the spring rate 

 remained constant. 



The velocity of W2 at time ts is 



[x2\t^t. = \/2pcoscoo/, = A/lghfl -^y (2.10.6) 



If y/lgh > coods, the displacement will exceed ds and the equation of 

 motion becomes 



m2X2 + kbX2 — (^6 — kQ)ds = 0, X2 ^ d, 

 The solution of (2.10.7), with initial conditions 



[X2\t^t, = dg 



L^2J<=<, 



VW^- 



(2.10.7) 



(2.10.8) 



is 



X2 = 



where 



kndi 





— COft/g) 



(2.10.9) 



+ 1 - r W. 



X2 ^ d. 



tan" a = 



e-) 



^6 ( To — 



^6 

 W6 = 4/ — . 

 m2 



(2.10.10) 



By dififerentiating (2.10.3) and (2.10.9) twice with respect to t, the 

 accelerations for the two regions are found to be 



X2 = — Gogsinuot, ^ X2 ^ ds , (2.10.11) 



X2 > d,, 



(2.10.12) 



where 



Go = 



_ /2hko 

 ~ V W2 



(2.10.13) 



Typical shapes of the acceleration pulse represented by equations (2.10.11) 

 and (2.10.12) are shown in Fig. 2.10.1. The curves are drawn for da/do = 



