C. PARAMETRIC ANALYSIS OF THE MAGNITUDE OF THE 

 MAXIMUM STRESS FOR SINUSOIDAL INPUTS 



The magnitude of the normalized amplitude of the maximum stress, 

 as given by Equation 20, is obviously a function of only three variables, namely 

 to' , u , and 8 . It is plotted in Figure 3 versus m' with u and B as parameters .* 

 As it is stated at the end of Section IVBl the maximum stress as given by Equa- 

 tion 20 occurs in the cable if 2 «)' + ^ ^ - . When this condition cannot be 



fulfilled, which can happen only for small values of tii' , the stress at the top of 

 the cable (Equation 22) is used in Figure 3, because it is greater than the stress 

 anywhere else in the cable. Figure 3 then shows the dependence of the maximum 

 stress on the input characteristics (u) and Uq), the characteristics of the cable 

 (c, E, and S), the length of the cable(L), and the weight and shape of the array. 



Let us see whether Figure 3 is compatible with the well known results 

 of simpler systems. For 8 = 0, there is no damping in the system. In this 

 case the resonant frequencies of the system are the roots of the equation 

 tan uu' = M-/ID'. The smallest of these roots lies between and tt/2, the next 

 between tt and 3 tt/2, and so on. At these resonances, the reflections of the on- 

 coming stress waves by the hanging mass have the proper phase so that their 

 individual contributions result in an infinite total stress. As the hanging mass 

 is decreased indefinitely (u -•=°), the resonant frequencies approach the values 

 n/2, 3 tt/2, etc. , and we have the case of the free end spring. As the hanging 

 mass is increased indefinitely (u -» 0), the resonant frequencies approach the 

 values TT, 2Tr, etc. , and we have the case of the fixed end spring. On the other 

 hand, when 8 7^ 0, energy is dissipated by the hanging mass, and the amplitudes 

 of the reflected waves are diminished. Resonance can still occur but with finite 

 amplitude . The actual value of B (amount of damping) should have a small ef- 

 fect on the values of the resonant frequencies and a very profound effect on the 

 amplitudes at resonance. One could expect that the amplitude at resonance will, 

 be decreased as the damping is increased. 



Figure 3 shows quite clearly all these expected trends, except one; 

 that is, for the resonance occurring near uj' = n (and 2rr, 3tt, etc., as well), 

 the amplitude increases when the damping is increased beyond a certain value. 



*In computing these curves, the values of I E | and 1 El have also been 



recorded. These values can be of use in the design of the joints at the top and 

 bottom of the cable. 



17 



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