408 BELL SYSTEM TECHNICAL JOURNAL 



Then (2.8.4) becomes 



1 r^ dZ 



dZ 

 k \/(l - Z2)(l - yfe2Z2)' 



in which the integral is the elliptic integral of the first kind (see Hancock 

 "Elliptic Integrals," John Wiley and Sons, New York, 1917). In (2.8.7), 



CO, = coo (1 + B)"\ (2.8.8) 



where coo = 's/ka/m^ is the radian frequency that would obtain if the cushion- 

 it. g were Unear with spring rate ^o • The motion for the linear case has a 

 half period, or pulse duration ro = tt/coo . The half-period (72) of the motion 

 with cubic elasticity is twice the time required for X2 to increase from 

 to dm • 



From 2.8.5 



[Z]x,=o = 0, 



(2.8.9) 



Hence, from (2.8.7), the half-period is 



2 r^ dZ 2K 



"'^'^ci V(l - Z2)(l - ^) = ^ ^^'^-^^^ 



where K is the complete eUiptic integral of the first kind. The duration of 

 the acceleration pulse is therefore 2K/uc ■ We can define a radian frequency 

 of the acceleration by 



TT TTOic 7rCOo(l + -S) /T Q 1 1 \ 



"^ = 7. = 2^ = 2K • ^^-^'^^^ 



The ratio a3o/co2 (i.e., t^/tq) is plotted in Fig. 2.8.1 which illustrates how the 

 pulse duration decreases as the parameter B increases. Hence, for a given 

 cushioning with cubic elasticity, the pulse duration decreases as the height 

 of drop increases. This is in contrast to the linear case in which the dura- 

 tion is independent of the height of drop. 



To find the acceleration X2 , we return to (2.8.7) and write it in the form 

 of an elliptic function: 



sncoc/ = Z. (2.8.12) 



Substituting the expression for Z given in (2.8.5) and solving for X2 , we find 



X2 = dmCn{oict - K). (2.8.13) 



Finally, differentiating (2.8.13) twice with respect to /, we find the accelera- 

 tion to be 



X2 = Jcdm[2kHn\uict - K) -l]c«(co.^ - K). (2.8.14) 



