DYNAMICS OF PACKAGE CUSHIOXIXG 449 



where 



o;o = — (4.2.13) 



and 



2 t/^c/ml co^„fA 



B„ = ' '"•' v;-2 ^ ^^ / (4 2.14) 



W2 Wo a^ 



The acceleration of Wo is, therefore, a sum of sinusoids of frequency ojn 

 and ampHtude vuoBn . Now, iwo is the maximum acceleration that niz 

 would attain if the mass of the cushioning were negligible. Calling G„ 

 the maximum acceleration in the n^^ mode and Go the maximum accelera- 

 tion neglecting the mass of the cushioning, as in Part I, we have 



^ = ^n. (4.2.15) 



But Bn depends only on the ratio mdmi , as may be seen from equations 

 (4.2.9) and (4.2.14). Similarly the ratio of the frequency (w„) of any mode 

 to the frequency (ojo) with massless cushioning depends only on mdmi , 

 as may be seen from equations (4.2.3), (4.2.9) and (4.2.13). Hence, both 

 the amplitude and frequency ratios for the acceleration in any mode depend 

 only on the ratio of the mass of the cushioning to the mass of the packaged article. 

 The ratios Gn/Go andco„/coo are plotted against mdmi in Figs. 4.2.2 and 4.2.3 

 for the first five modes. It may he seen from these figures that the accelerations 

 in the higher modes can he very important. For example, if the cushioning 

 weighs half as much as the packaged article the maximum acceleration in 

 the second mode is about 40% of the acceleration in the first mode and the 

 latter is about the same as found by the elementary method of Part I. 

 This could have a disastrous effect on an element of the packaged article if 

 the latter had a fundamental frequency near that of the second mode of the 

 cushioning, the latter being found, from Fig. 4.2.3, to be about five times 

 the fundamental frequency of the package. 



It must be remembered that damping has been neglected in the above 

 investigation and damping in the cushioning will serve to mitigate the se- 

 verity of the higher mode accelerations to a great extent. However, the 

 danger is always present at the start of a design and the possibilities of un- 

 favorable combinations should be studied in every case. 



