DYNAMICS OF PACKAGE CUSHIONING 409 



The ratio —Xi/Gog is plotted in Fig. 2.8.2 against a radian coordinate 

 (woO for several values of B. It may be seen that, as B increases, the maxi- 

 mum acceleration increases, the duration of the pulse decreases (see Fig. 

 2.8.1) and the acceleration-time curve becomes bell shaped. For reference, 

 the sinusoid for the linear case {B = 0) is plotted in the figure. 



Figure 2.8.2 is plotted for perfect rebound. If rebound does not occur, 

 the curves continue, mirrored in the time axis, so as to form a periodic 

 vibration of period 2x2 . 



2.9 Acceleration-Time Relation for Tangent Elasticity 



In this section the effect of tangent elasticity on the shape of the accelera- 

 tion-time relation will be studied. The shape of the load displacement 

 curve is given by 



2kodb TTXi Ci A 1\ 



P = tan—-. (1.4.3) 



■K Mb 



The system considered is again that shown in Fig. 1.2.1. Referring to the 

 energy equation (1.2.13): 



Pdxo = m2gh, (1.2.13) 



2 "^ in 



we substitute the above value of P and perform the indicated integration to 

 obtain, for the velocity. 



^2 = y^ 



X2= A/lgh + ^-^hogcos'^. (2.9.1) 



W2 TT^ 2db 



Then, as in Section 2.8, 



'^ dx2 C"^ dx: 



t 



r^ dx2 r^ dXi 



= = / «W2 (2.9.2) 



Jo X2 Jo ^ /n 1 t ^kodi, irX2 



\/ 2gh H log cos -^ 



y m2ir 2db 



and the half-period (72) of the motion is again twice the time required 

 for X2 to increase from to dm . Hence 



r<im (1X2 



where, from Section 1.9, 



Hm 



^'cos-^exp| -^(7) |. (1.9.5) 



