DYNAMICS OF PACKAGE CUSHIONING 405 



do and Go are the maximum displacement and acceleration that would obtain 

 if no friction were present. From (2.7.2) the maximum displacement with 

 friction is 



Hence, the presence of friction decreases the maximum displacement since 



dm < do . 



From (2.7.3) the acceleration is 



- = - yCl + ^^Y sin (co2f + a), (2.7.5) 



so that the maximum acceleration is 



G^=^Gl + {0. (2.7.6) 



which is greater than the maximum acceleration without friction. 



It would appear, at first glance, that cushioning with friction always 

 gives a greater acceleration than the corresponding cushioning without 

 friction. However, the reverse is actually true provided we allow the same 

 displacement in both cases. This may be done, as may be seen from (2.7.4), 

 by decreasing the spring rate in the cushioning with friction to 



k, =. k2-^. (2.7.7) 



do 



The maximum acceleration in the cushioning with friction is then, from 

 (2.7.6), 



That is, for the same maximum displacement, the maximum acceleration 

 is reduced by the addition of dry friction. 



2.8 Acceleration-Time Relation for Cubic Elasticity 



As an example of the effect of nonhnearity of the cushioning on the shape 

 of the acceleration-time function, the case of cubic elasticity (Class B) 

 will be analyzed. The system to be considered is illustrated in Fig. 1.2.1, 

 and the load-displacement relation for the cushioning is given by 



P = koX2 + rxl . (2.8.1) 



