DYNAMICS OF PACKAGE CUSHIONING 395 



study, in Part III, of its influence on the response of elements of the packaged 

 item. 



The first case to be considered will be the simple single mass and linear 

 spring example described in Sections 1.2 and 1.3. Following this the 

 phenomenon of rebound of the package will be considered. The influence 

 of velocity damping and dry friction will be studied; and, finally, the effects 

 of non-linearity of the cushion elasticity on the acceleration-time relation 

 will be investigated. 



2.2 Acceleration-time Relation for Linear Elasticity 



Returning to the elementary example studied in Sections 1.2 and 1.3, we 

 first write the equation of motion for the mass W2 , on its linear spring of 

 spring rate ki (see Fig. 1.2.1.). Equation (1.2.3) becomes 



ni2X-i + ^2-V2 — ^Wog. (2.2.1) 



Using the initial conditions 



N.=o = 0, (2.2.2) 



[:v-2],=o = V2^, (2.2.3) 



the solution of (2.2.1) is 



Xo = ^^-^ — —^ — --^- sm (co2/ — a) + ^ 



C02 



(2.2.4) 



or 



where 



and 



W2 



^ + dl, sin (co./ - a) + ^', (2.2.5) 



= 4 A = 2^2 = ^ (2.2.6) 



a = tan ^ 7^=7 = tan ^ rY' (2.2.7) 



W2 V 2gh ko dm 



0)2 is the circular frequency, /> is the frequency and To is the period of vibra- 

 tion of the mass W2 on its spring; d,,, has the same definition as in Section 1.3 

 (equation (1.3.1)). 



Now, Wi/ki is the static displacement of the mass nio on its spring. This 

 is usually very small in comparison with the maximum displacement {dm) 

 during impact. Hence 1^2/^2 will be neglected, and (2.2.5) becomes 



.r2 = dm sin oiot- (2.2.8) 



