183 



An liilementary Approximation to the Solution 



In case the effect of the constraint forces, that is, 

 acceleration of the material in the central flat region, is small throughout 

 most of the notion, so that it inay be neglected, the equations (D) take on 

 a particularly simple form, especially if the material is one '.";hic!i does not 

 work harden. Then the order of the set of differential relations becomes 

 two, so that only tvio initial conditions, those on H and on H, can be applied. 

 Although it is not necessary, it is also convenient to suppose that O' is 

 a very large number, compared to unity, which is cectainly true for many 

 cases of interest. For the typical values listed in a previous section 



cr 



^_- 80 . 



Let us denote the constant CT by c3~. Then, neglecting high 



P 



PV ^ . 



powers of v, the solution of the equations may be written 

 c 



so that 



(29) 



These equations tell us that the radial speed U is a small 

 constant at the bending wave, and that the speed of the bending wave from 

 the edge to the center of the diaphragm is a constant, ir.denendont of the 

 initially imposed norrial velocity v. The thickness distribi^tion in the 

 deformed diaohragm given by equation (29) (c), Hhows a dimpling tendency at 

 the center J in fact at the last inonent the thic] ness becomes zero at the 

 very center - no doubt a consequence of the ideali2ation3 and approximations. 



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