DYNAMICS OF PACKAGE CUSHIONING 



363 



Class B — Ciibic Elasticity. This includes cushioning which does not bot- 

 tom in the anticipated range of use, but the slope of the load-displacement 

 function generally increases with increasing displacement as in the curved 

 full line of Fig. 1.4.2. A suitable load-displacement function is 



P = ko X2 + rxo 



(1.4.2) 



ko is the initial spring rate of the cushioning, as shown by the slope of the 

 dashed straight line in Fig. 1.4.2, and r determines the rate of increase of the 

 spring rate. The same function can be used if the slope of the curve de- 

 creases gradually with increasing load as shown by the curved dashed line 

 in Fig. 1.4.2. In this case the parameter r is negative. 



n ^2 



Fig. 1.4.2 



Fig. 1.4.2 — Cubic elasticity. 

 Fig. 1.4.3 



2 "b '^2 



Fig. 1.4.3 



Class B. 

 Tangent elasticity. Class C. 



Class C^Tangent Elasticity. Cushioning that bottoms, but not very 

 abruptly, can be represented by the load-displacement function 



_, 2^0 db ^ Ttxt 

 P = tan — - 



TT Mb 



(1.4.3) 



Referring to Fig. 1.4.3, ^o is the initial spring rate and dh is the maximum 

 available displacement. The figure shows hov the stiffness of the cushion- 

 ing (i.e., the slope of the curve) increases as the displacement approaches 

 the maximum available {db) at hard bottoming. The shape of the curve 

 is typical of load-displacement curves for a great variety of packages with 

 distributed cushioning. 



Figure 1.4.7 illustrates the wide variety of shapes of non-linear cushioning 

 characteristics that can be obtained with the single function given by equa- 

 tion (1.4.3) simply by varying the parameter Uq; and a similar set is given by 

 each value of db . Although these families of curves do not include all pos- 

 sible shapes, one of them can usually be found to fit a practical shape for 

 cushioning of this class over the anticipated range of use. 



Class D — Bi-linear Elasticity. This is characterized by a load-displace- 



