DYNAMICS OF PACKAGE CUSHIONING 357 



dead load stresses (obtained in the usual manner) by the ratio of the maxi- 

 mum acceleration to the acceleration of gravity. If the criteria for the use 

 of maximum acceleration alone are not satisfied, then Parts II and III will 

 supply a numerical factor (the Amplification Factor) by which the maximum 

 acceleration should be multiplied, and the remainder of the procedure is the 

 same as before. 



The determination of the maximum acceleration is founded on a knowledge 

 of the load-displacement characteristics of the cushioning. When the cush- 

 ioning system is simple enough, the load-displacement relation may be found 

 or designed by purely analytical procedures. The tension spring package, 

 discussed in Sections 1.7 and 1.8, is an example where such a treatment is 

 possible. In many instances, as with distributed cushioning, the load- 

 displacement relation is more easily found by test. 



A load-displacement test is made by applying successively increasing 

 forces, with weights or in a load testing machine, to the packaged item 

 completely assembled in its package, and measuring the corresponding 

 displacements. The force is applied usually by means of a rod inserted 

 in a hole cut through the oiiter container and the cushioning to the packaged 

 item. It is convenient to use a low loading rate in the test, and, in doing so, 

 the effect of resisting forces that depend on velocity is lost. These forces 

 are often of little importance but, in certain designs, it is necessary to con- 

 sider them. This is done for velocity damping in Sections 2.5, 2.6, 3.2 and 

 3.5. 



Most of Part I is concerned with cushioning having non-linear load-dis- 

 placement characteristics. Linear cushioning is rarely encountered, but 

 it will be treated first because of its simplicity and because it will be con- 

 venient later to express the maximum acceleration in non-linear cases in 

 terms of the maximum acceleration in a hypothetical linear case. 



1.2 Derivation of Equations of Motion 



To introduce the method of analysis that will be used in Part I, the sim- 

 plest possible system is considered first. The mi system is omitted entirely, 

 the mass of the outer container (m^) is neglected, and the cushioning is 

 assumed to have no damping or friction. There remain only the mass 

 W2 (the mass of the packaged item alone) and the supporting spring, as 

 shown in Fig. 1.2.1. If the spring is linear its displacement is proportional 

 to the applied load throughout the range of use (see Fig. 1.4.1). The spring 

 rate (^2) of a linear spring is a constant usually expressed in terms of pounds 

 per inch. The force (P) transmitted through a linear spring is therefore 

 given by 



P ^ koxo, (1.2.1) 



