68 



54 



2.0 



i.e 



1.6 



1.2 



1.0 



0.8 



0.6 



0.8 



0.6 



0.4 



•02 



0.1 



0.2 0.3 



0.4 0.5 



0.6 



0.7 



08 09 



1.0 



Figure 28 - Plot of the Dynamic Response Factor or Load Factor N 

 for a Harmonic System under Exponential Forcing 



N represents the ratio of the maximum deflection of a harmonic system of natural frequency v, when 

 acted on by a suddenly applied force f, «" . to its static deflection under a steady force f„. Here 

 I is the time and a and F„ are constants. The plot serves also for a proportionally constrained dia- 

 phragm whose swing time is T,, when acted on by an exponential pressure wave of time constant T^ - l/a; 

 that is, the incident pressure is p^ = p^« ""'where ( is the time and p,^ and a are constants. 



For the circular diaphragm already considered, z^o can be calcu- 

 lated either from formulas already given for M, M,, F^, and T, , or directly. 

 For a pressure equal to p^, Fg = 7ro^p^/2 by Equation [42b]. The curvature 

 of the diaphragm Is given nearly enough by the approximate formula for a 

 sphere, 2zja'^ for small z^ ; hence the stress force per unit area normal to 

 the plane of the diaphragm is ^ = - ^ah.zja'-, on the assumption of equal hy- 

 drostatic pressures on the front and back. Thus by Equations [31b] and [40b], 



« 



= - ^^4^ / (l - \) 2r dr = - iTxohz, 



and k = 2-r:a)i. Hence, for small 2^„ , 





[93] 



lis. 



If the accurate formula for the curvature is used, or C = g V' i. 

 a quadratic equation must be solved for z^„ . 



Detailed formulas have been given here for only one type of wave, 

 the idealized shock wave of exponential form. The waves emitted during re- 

 compression of the gas globe can be approximated roughly by superposing sev- 

 eral exponential terms, but simple final formulas are not obtainable; see 

 Reference (l6). 



