400 BELL SYSTEM TECHNICAL JOURNAL 



The complete acceleration history of a rebounding package with un- 

 damped Hnear cushioning is thus a half sine wave pulse of amplitude Gm = 

 ■s/lhki/Wi and duration 7r/w2 followed by an oscillating acceleration of 

 amplitude given by (2.4.11) and frequency given by (2.4.6). Such a wave 

 shape is shown in Fig. 2.4.1. 



2.5 Influence of Damping on Acceleration 



The presence of damping in cushioning is always desirable to prevent the 

 building up of large amplitudes as a result of periodic disturbances. How- 

 ever, damping also has an effect on the maximum acceleration that is at- 

 tained in a drop test. From the latter point of view there is an optimum 

 amount of damping and an amount that should not be exceeded if the maxi- 

 mum undamped acceleration is not to be exceeded. 



We shall consider the case of a linear cushion with damping proportional 

 to velocity. The system is represented in Fig. 2.5.1. With the addition 



4 



5 h 



Fig. 2.5.1^Idealization of linear cushioning with velocity damping. 



of the damping term the equation of motion of m^ , during contact of the 

 package with the floor, is 



miX2 -f C2X2 + ^2.^•2 = 0, (2.5.1) 



in which C2 is the damping coefficient of the cushioning. Equation (2.5.1) 

 is more conveniently expressed as 



X2 -h 2/320)2X2 + W2X2 = 0, (2.5.2) 



where 





co2 = 4/^, (2.5.3) 



2w2a;2 



W2 is the undamped circular frequency of vibration of W2 on its spring and ^2 

 is the fraction of critical damping. ^2 = means no damping and 182 = 1 

 means just enough damping so that there will be no oscillation if the pack- 

 aged article is displaced and released. 



