DYNAMICS OF PACKAGE CUSHIONING 425 



^max with the "static response" i.e. with the value (.t.,«) that x would have 

 if the acceleration x^ reached the same maximum value (co2\/2g/0 in a very 

 long time. The resulting value may be found from (3.2.4) by omitting 

 the transient term jhix. Then 



Xst = C02 



or 



C02 / 



Xsi = ~2 V2gh. (3.2.8) 



The amplification factor for the interval < / < t/co-z is then 



^^ ^ x^. ^ _jo^ ^^ _2nT_ ^ (o = t^ ^M. (3.2.9) 



Xst ^ _ 1 ^ _1_ 1 



CO2 CO2 



It should be observed that Am depends only on the frequency ratio coi/co2 . 

 That is, since coi/co2 = T2/T1 , the amplification factor depends only on the 

 ratio of the pulse duration to the half period of vibration of the element. 



Thus far we have studied only the motion in the interval ^ / ^ 7r/aj2 . 

 We must not, however, overlook the possibiHty of larger displacements of nti 

 with respect to m^ occurring after rebound. In fact, examination of (3.2.6) 

 reveals that x has no maximum in the interval ^ t ^ 7r/co2 whencoi < aj2 . 

 It is very likely, then, that larger values will occur at later times. 



After rebound, nii executes free vibrations with respect to W2 . We have 

 to compare the magnitude of .Tmax , in the interval ^ / ^ t/(j02 , with the 

 amplitude of the free vibration. Calling the relative displacement during 

 free vibration x' and measuring a time coordinate /' from the instant the 

 package leaves the floor, we have 



mix' + kix' = 0, (3.2.10) 



with initial conditions 



[X J<'=0 — [x\t^T/oi2 J 



(3.2.11) 



[X Jr=0 — [X\t=r/U2 • 



The solution of (3.2.10) with initial conditions (3.2.11) is 



4/^' + -=-/).„(„,,+^^^), 



aji(aj^ - CO?) \ 2aj2/ .^ 2 12) 



^ > — • 



0)2 



