DYNAMICS OF PACKAGE CUSHIONING 



447 



nomena involved. The treatment is by no means complete, but a more 

 detailed investigation is beyond the scope of this paper. 



4.2 Effect of Distributed Mass and Elasticity of Cushioning 

 ON Acceleration of Packaged Article 



Referring to Fig. 4.2.1, we consider the packaged article, of mass W2 , 

 to be supported by distributed cushioning of mass mc and depth ^. The 

 cushioning may be a pad, say of rubber, in which case / is the pad thickness, 

 or it may be a helical metal spring, in which case f is the coil length. The 

 package is dropped vertically from a height h and has attained a velocity v 

 at the instant of contact (/ = 0) of the outer container and the floor. The 

 outer container is assumed to be heavy enough so that there is no rebound. 

 A horizontal plane in the cushioning is located by a coordinate x measured 

 from the end of the cushioning attached to the outer container. The vertical 



-V 



i 1 1 1 1 1 ; 



/ (cushion) 



•: 



Floor 



////////////////////////// 



Fig. 4.2.1 — Packaged article of mass tni, supported on distributed cushioning of depth t 

 and mass vie, depicted at the instant of first contact of the outer container {m^) and 

 the floor. 



displacement of the plane x is designated by u. The undamped motion of 

 the cushioning after contact is governed by the one-dimensional wave 

 equation: 



b U 



d U 



a/2 ^ a^2' 



(4.2.1) 



in which a is the velocity of propagation of longitudinal waves in the cushion- 

 ing. If the cushioning is continuous. 



2 E 



a = — . 



(4.2.2) 



where E is the modulus of elasticity and p is the density of the cushioning. 

 If the cushioning is a helical spring, 



2 kf 



a = — , 

 where k is the spring rate. 



(4.2.3) 



