example, there is little difference between \- = 36-^^ and \^ = ■» so that the first mode is nearly inex- 

 tensible. For X^ = <», equation (B6) reduces to 



tan 



(BID 



which is also plotted in Fig. B2. With this frequency equation the symmetric natural frequencies are 

 again well ordered and alternate with antisymmetric frequencies, but there is a shift of between 0.937r 

 and V in these symmetric mode frequencies with respect to the taut string symmetric mode frequen- 

 cies. 



The modes shapes are affected by the frequency crossover in a very interesting manner. A sym- 

 metric mode must possess an even number of nodes. The mode shape acquires two additional nodes in 

 crossing over, thus altering its form while preserving symmetry. The transition is smooth as shown in 

 Fig. B3 which is adapted from a related numerical study by West, Geschwindner and Suhoski (B5). A 

 dashed line which corresponds to the example A.^ = 367r^ is included in Fig. B3a. 



Some recent NRL experiments (B2) demonstrated this behavior as shown in Fig. B4. To the 

 right of the crossover the mode shapes were identifiable with the « = 1 (0) and n = 2 (n) mode 

 shapes of a taut cable, although the natural frequencies of the n = 1 curve deviate from those of a taut 

 string. To the left of the crossover the n = 2 mode was essentially unchanged while the mode shape 

 for the other curve (•) acquired two additional nodes and thus resembled an « = 3 mode of a taut 

 cable. The observed modal transitions were of the form predicted by Irvine and Caughey and are indi- 

 cated in Fig. B3(b). According to the theory the first crossover occurs at X^ = 4tt^ and this provides a 

 means for comparison with the data. Since the product EA is not a simple material constant for cables, 

 particularly at low tensions, a range of values for EA was found from curves of elongation versus ten- 

 sion measured at NRL. The crossover range computed in this way is included in Fig. B4 and is in good 

 agreement with the experimental observations. The in-water tests with the same cable afforded another 

 opportunity for comparison. The same range of EA was employed, but the linear weight was adjusted 

 to reflect the buoyancy. The result is shown in Fig. B5 and again there is good agreement with the 

 theory. 



126 



