DYNAMICS OF PACKAGE CUSHIONING 



401 



The acceleration solution of (2.5.2), with the initial conditions of the drop 

 test (see (2.2.2) and (2,2.3)) is 



where 



i2 = - '^^i^M. .-^-^-' COS (co../vr^^^ + 7) 



V 1 - ,8i 



X-i 



tan 7 = 



2/31- 1 

 2/32 Vl - 



(2.5.5) 

 (2.5.6) 



Gog 



Fig. 2.5.2 — Acceleration-time curves for linear cushioning with various amounts of 

 damping (no rebound). See equation (2.5.5). 



The acceleration is thus a damped sinusoid with an abruptly reached 

 initial value whose magnitude depends upon the amount of damping. For 

 small damping, the initial acceleration is small and then the acceleration 

 increases, but never reaches the value that would be reached without any 

 damping. For high damping 032 > 0.5) the initial value is greater than 

 without any damping and falls off thereafter. Figure 2.5.2 shows the shapes 

 of the acceleration time curves for several values of 0i . All of the curves 

 are for no rebound. It may be seen, from equation (2.5.5) and Fig. 2.5.2 

 that the addition of damping changes the shape of the acceleration-time 

 relation in three ways. First, a damped sinusoid replaces the pure sinusoid; 

 second, the frequency is reduced; and, third, the initial phase is changed. 



It is useful to consider in detail the effect of damping on maximum accel- 

 eration. Let 



Gm = maximum number of g's with damping 



'2^2 



-/ 



W-, 



= maximum number of g's without damping. 



