DYNAMICS OF PACKAGE CUSHIONING 453 



ir/o}2 • We shall therefore treat separately, as in Part III, the motion 

 during contact 



^ z=: = T 



< / < — 



0)2 



and after rebound 



/ > -■. 



During the first interval, the initial and boundary conditions are 



[hUo = 0, (4.3.6) 



[a. 



= -V, (4.3.7) 



u]i^o = — sin Wo/, (4.3.8) 



C02 



[-1 



= 0. (4.3.9) 



The first and second conditions state that, at the instant of contact, all 

 points in the bar are moving with the approach velocity v, without relative 

 displacement. The third condition prescribes the half -sine wave motion 

 of the end of the bar that is attached to W2 • The fourth condition states 

 that the strain at the free end of the bar is always zero. 



By the usual methods, a solution of (4.3.4) satisfying conditions (4.3.6) 

 to (4.3.9) is found to be 



V COS —{I - x) sm W2 / „ . 00 sm -— sm — y- 



« = -, + ^ Z. ~P7 v^ n (4.3.10) 



co2t- TT-a n=i,3,5-- 9 ( uray 



The displacement is seen to be a forced vibration at the frequency (002) 

 of the applied acceleration, on which are superposed the free vibrations of 

 the bar given by the series expression. The frequency of the fundamental 

 mode of vibration of the bar isxa/2f and the frequencies of the higher modes 

 are the odd integral multiples of the fundamental. 



