DYXAMICS OF PACKAGE CUSHIONIXG 361 



and, from (1.2.14), 



G„. = ^\ (1.2.16) 



where P,,, is the maximum value of P. If /"(.to) is a monotonic function, 

 P„, may be obtained from (1.2.2) by substituting (/,„ for .vo: 



P,n-P{(L). (1.2.17) 



In the unusual case where P(.V2) is not monotonic, the maximum value of P 

 in the interval < .T2 < dm must be chosen instead of equation (1.2.17). 

 The general procedure is to calculate dm from (1.2.15), Pm front (1.2.17) 

 and then Gm from (1.2.16). If P can be expressed analytically in terms of x^ 

 and if the integral in (1.2.15) can be evaluated in terms of elementary func- 

 tions, simple formulas can be found for dm and G,„ . If this is not possible, 

 then the integration can be performed graphically or numerically. Both 

 of these procedures will be illustrated. In either case the maximum accel- 

 eration and displacement are obtained in terms of the weight of the pack- 

 aged item, the height of drop and parameters descriptive of the load-dis- 

 placement characteristics of the cushioning. 



1.3 Linear Elasticity 



For cushioning with a linear load-displacement relation, equation (1.2.1) 

 applies. Substituting this value of P in (1.2.15), and performing the in- 

 tegration, we find 



2^^ (1.3.1) 



From (1.3.1) and (1.2.17), 



Pm = V2hW2h, (1.3.2) 



and, from (1.3.2) and (1.2.16), 



Grn=y^^- (1.3.3) 



Notice that equation (1.3.3) holds only if there is space available for a 

 displacement dm and if the cushioning is linear and capable of transmitting 

 aforceP„j. Also, from (1.3.3) and (1.3.1), 



2h 



(1.3.4) 



V 



and 



Gn 





