This should be expected, because when the damping of the hanging mass be- 

 comes very large, the mass cannot move very much, and we have again the 

 fixed end spring with resonant frequencies at n, 2tt, 3tt, etc. , and infinite am- 

 plitudes. To put it another way, if a very large amount of energy is dissipated 

 by the hanging mass, the energy propagating along the spring must be very 

 large, since the spring is the only link between source and load. 



Two remarks are in order here. First, as Figure 3 shows, there 

 are resonances (some of them highly sharp depending on the values of u ) at 



uj' s: n or uj = ^ . For c ^ 10,000 ft/sec and L ^ 20,000 ft, this corre- 

 sponds to frequencies as small as 1.5 rad/sec or periods as high as four sec- 

 onds . Since such periods can be found easily in a realistic sea, it is not cor- 

 rect to consider the cable nonflexible when an accurate estimation of the stres- 

 ses is desired, unless such frequencies are filtered out by the vessel very ef- 

 fectively. Also, for small values of B, there are resonances at values of m' 

 much smaller than tt, corresponding to periods much larger than four seconds- 

 which are quite frequent and prominent in a rough sea. Second, an idealized 

 system, such as the free end or fixed end flexible cable, will give portions of 

 the curves of Figure 3 rather accurately. However, this does not render the 

 present analysis, which includes the dynamics of the array, superfluous. Our 

 objective here is not to display a rough picture of the phenomena, which could 

 be done by the free end or fixed end cable, but rather to find as accurately as 

 possible the dependence of the maximum stress on the various quantities . In 

 terms of a realistic and sound design of the cable, which hinges on the magni- 

 tude of the maximum stress, simple models --such as the nonflexible and the 

 free end or fixed end (flexible) cables --are useless, and a complicated model, 

 like the present one, is indispensable. 



In order to demonstrate how these curves can be used, let us con- 

 sider a cable and array as specified below: 



L = 20,000 ft M = 600,000 1b 



a 



E = 20xl0^psi A = 3,000ft'^ 



S = 3 in^ a = 1.2 



m = 10.6 lb/ft 



c = 13, 600 ft/ sec 



19 



Arthur a.IlUtIc Jnr. 



S-7001-0307 



