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BELL SYSTEM TECHXICAL JOURNAL 



The amplification factor, in this case, will be taken as the ratio of .Tmax 

 to the relative displacement (xst) resulting from a slow apphcation of the 

 maximum value of .fo . From (3.6.1), 



X2 



Gmg 



2 



2 ' 



where Gm is given by equation (1.5.6). Then 



(3.6.4) 



■Am. — 



•''max -^max '*'! 



Xst 



Gmg 



(3.6.5: 



Fig. 3.6.2 — Amplification factors for undamped cushioning with cubic elasticit}'. 

 rebound. See ecjuation (3.6.6). 



or 



Am = 



Wi 





X2(X) sin coi(X - t„) d\. 



Perfect 



(3.6.6) 



Am was evaluated, mostly by graphical methods, for four values of B 

 (0, 2, 20 and cc) and the results are plotted in Fig. 3.6.2. Observing that 

 B = corresponds to hnear cushioning, it may be noted that cubic non- 

 linearity in the cushioning does not change the amplification factor by 

 more than 35% even in the most extreme case (B -^ oc). The severity 

 of the shock, however, may be much greater for the cubic cushioning than 

 for linear cushioning with a spring rate equal to the initial spring rate (^o) 

 of the cubic cushioning. This is because A,n is multiplied by Gm to obtain 

 Ge and, for large values of B, Gm may be much larger than the maximum 

 acceleration for the linear case. In other wordfe, in comparing Class B 



