DYNAMICS OF PACKAGE Ci'SIIIOXIXG 



389 



where, as before 



k = |/ 





Go 



/2hko 



V 1F2 • 



Equations (1.13.3) and (1.13.5) are plotted, in Figs. 1.13.1 and 1.13.2, 

 against the dimensionless parameter Po/WoGo . The latter is the ratio of 

 the maximum force, that the hyperbolic tangent cushioning will transmit, 



.|.:..|....i;::-|:;::|;.;.i;;;:|:. 



.|.;..(-,.,|.:„t^ 



3 



2.0 



1.5 





1.0 H— 



■■ 





1.0 



2.0 



2.5 



30 



1.5 



Pq 

 W2G0 



Fig. 1.13.1 — ^Maximum displacement for cushioning with hyperbolic tangent elasticity. 



See equation (1.13.3). 



to the force that linear cushioning would transmit under the conditions 

 specified. 



To find the value of ^0 which yields the minimum value of acceleration 

 for a given maximum displacement, differentiate (1.13.4) with respect to ko 

 and set the result equal to zero: 



•2 ^0 U„i, 



sech" 



Po 



0. 



(1.13.6) 



This is satisfied by ^0 — ^ '^- , which represents the rectangular load dis- 

 placement curve and confirms the conclusion reached from energy 

 considerations. 



