416 BELL SYSTEM TECHNICAL JOURNAL 



in Section 2.9. The system considered is that shown in Fig. 1.2.1 and the 

 load displacement curve of the cushioning is given by 



P=Potanh^^ 



Substituting the above expression for P in the energy equation (1.2.13), we 

 find the velocity to be 



^2 = A/lgh - ^ log cosh ^-^ . (2.11.1) 



Then, as before, 



r^ dx2 

 t = — (2.11.2) 



and the half period (72) of the motion is twice the time required for X2 to 



increase from to dm , or 



C^"* dx2 

 T2 = 2 — , (2.11.3) 



Jo ^2 



where, from Section 1.13, 



The radian frequency of the acceleration is defined as 



IT 



032 = — 



T2 



an J this is to be compared with the frequency 



TT 



coo = — = 



To Y W2 



that would obtain if the cushioning had a constant spring rate equal to the 

 initial spring rate (^0) of the hyperboUc tangent cushioning. The ratio 

 wo/aJ2 (or T2/T0) is plotted, in Fig. 2.11.1, against the dimensionless param- 

 eter P0/W2G0 (see Section 1.13). It may be observed that the pulse 

 duration becomes very long when P0/W2G0 is small, i.e., when the horizontal 

 portion of the load displacement curve (Fig. 1.4.5) comes into play. The 

 influence on the shape of the acceleration-time curve is illustrated in Fig. 

 2.11.2. The curve marked Po/W^Go ^ 00 is the sinusoid for the Unear 

 case. For small values of P0/W2G0 the curve approaches a square wave. 



