DYNAMICS OF PACKAGE CUSHIONING 433 



^ = ""lYy {e-^'-^'lA sin Ut J^y - 8)+Bsm ic,[t - 7 - f)] 



/W2'«Ji 



- e-^^^^'lA sin (ojo^ + 7 - 5)- 5 sin (wW + 7 + f)]} (3.5.1.) 



where 



Wi 



W2 = 



12 

 W2 



COl' = coiVl — /3i 



C02 = C02 'x/l — /3f 



1//1 = VGS2C02 - /3io;i)2 + (coi' - wO- 



\/B = \/(^2C02 - i8lCOl)2 + (cOi' + CO02 



t- ^ = 2/32 VI - /31 

 / / 



OJi — COg 



tan d = 



tan f = 



/32C02 — iSlCOi 

 COl + 0)2 



/32C<J2 — /?lCOi 



The relative displacement of nti with respect to W2 is seen to consist of a 

 forced, damped vibration (w2 , ^2) on which is superposed the free damped 

 oscillations (wi , /3i) of Wi . 

 The amplification factor 



2 

 ^0 = -^ = z;^ 3.5.2 



is plotted in Figs. 3.5.2 to 3.5.7 for six values of /3i and six values of ^2 ■ 

 These curves were obtained by direct solution of the differential equation 

 on the Westinghouse Mechanical Transients Analyzer. The amplifica- 

 tion factor in this case includes the effect of damping; i.e., the reference 

 acceleration is not the maximum acceleration of W2 , but is the maximum 

 acceleration that W2 would reach if the damping 182 were zero. Conse- 

 quently, the amplification factors for large values of coi/w2 do not approach 



* See footnote, Section 3.2. Only enough data were obtained with the analyzer to find 

 the general shapes of the curves, so that the fine structure is not revealed. Checks on 

 the analyzer results were made by computing Aq from equations (3.5.1) and (3.5.2) for 

 wi/w2 = 1, /3i = /Sj; aii/w2 = 0; 0)1/032 — > 00 . 



