DYNAMICS OF PACKAGEXUSHIONING 



397 



Two pairs of equations are necessary to describe the action of the system; 

 one pair appHes during the time of contact of m^ with the floor and the 

 second pair appKes if rebound occurs. 



The mass W3 is assumed to be inelastic (see Section 0,2) so that, during the 

 interval of its contact with the floor, the equation of motion for m^ will be 

 the same as before (2.2.1). In addition, there w ill be an equation of equili- 

 brium for the mass W3 : 



R = kiXi + nizg , 

 where R is the upward force exerted by the floor on mz . 



(2.3.1) 



h 



/////>///////// 



Fig. 2.3.1 — Two-mass system representing packaged ^^ticle, linear cushioning and 



outer container. 



Equations (2.3.1) and (2.2.1) will hold as long as i? is positive. To find out 

 when R > 0, solve (2.2.1) for ^2-^*2 and substitute in (2.3.1): 



R = W2 +Ws- W2X2 



(2.3.2) 



That is, a necessary condition for rebound is that the mass of the cushioned 

 article, multipHed by its maximum acceleration, exceeds the total weight 

 of the package. The condition for rebound may be written 



Gm > 



W2 + W3 

 W2 



(2.3.3) 



This is a necessary, but not a sufiicient, condition for rebound because there 

 will be energy losses as a result of damping and permanent deformation. 

 Gm will generally have to be considerably greater than the right hand side 

 of (2.3.3) for rebound to occur. 



If rebound does not occur, equation (2.2.9) continues to apply, except for 

 damping which will be considered in Section 2.5. 



