DYNAMICS OF PACKAGE CUSHIONING 455 



k = m when — < / < -^ ■ — ~ 



a a 



w = 0, 1 , 2, 3 • • • . 



The expression (4.3.14) is simple enough so that the maximum value (e^,) 

 of the strain at the attached end can be obtained without difficulty for any 

 ratio of the fundamental frequency (wi = ira/lf) of the bar to the frequency 

 (coa) of the disturbing acceleration. The amplilication factor 



■Am — 



(4.3. I5) 



vuzp^ iraj2 V 



may then be calculated. The results of these calculations are plotted in 

 Fig. 4.3.2. The important feature of this curve is that the amplification 

 factor is everywhere less than the corresponding amplification factor for the 

 one-degree-of -freedom system (Fig. 3.2.2, /3i = 0). Hence the assumption 

 of lumped parameters is on the side of safety as regards amplification factor. 

 It is interesting to observe that the curve of A^ vs. coi/w2 , for this case, 

 is a straight line between ui/002 = and a;i/co2 = 1. This arises from the 

 fact that, for wi/w2 < 1, equation (4.3.14) reduces to 



f= cos .,*-!, (^51). (4.3.16) 



Hence, when the duration of shock is less than the half period of the funda- 

 mental mode of vibration, the maximum value of strain occurs at the end of 

 impact and is equal to twice the ratio of the approach velocity to the velocity 

 of wave propagation in the bar. 



The whole solution of the problem is not yet completed; for, although it is 

 fairly evident from the fact that there is at least one maximum in the inter- 

 val < / < ir/co-i for all values of coi/co2 , it must be verified that the maxi- 

 mum strain (and, therefore, the amphfication factor) is never greater after 

 / = ir/a)2 than before. Defining a new time coordinate 



t' = t - -, (4.3.17) 



aj2 



we have, for the initial and boundary conditions of equation (4.3.4) for 

 t ^ 7r/w2 , 



