|l^ L increases, say doubles, due to an increase In the cable length of L to twice 

 L, the appropriate nondimensional frequency, co', at the corresponding value of ji 

 is not twice the co' at the shorter length. This results in a lower circular frequency, 

 60, at the longer cable length. This applies to the larger values of [^qx | correspond- 

 ing to these lengths. A similar argument applies at cable lengths lower than 200 feet. 

 For intermediate cable lengths a relatively small variation in I ^qxI results in a 

 significant increase in co', resulting in an increase in the circular frequency, OJ. 

 This applies to a range of co' from 0.477 to 0.877 — i.e., to values of |^ax| larger 

 than those corresponding to the first peak in Figures 2 to 13 and 15 to 22, but less 

 than those associated with the peak responses occurring at oo' approximately equal 

 to 3. 142. As co' values tend toward this value, significant changes in | ^qxI result 

 in larger variations in circular frequency for different cable lengths due to the influ- 

 ence of the c/L ratio in computing the latter. Hence the variation of cable stress 

 with cable length depends upon the shapes of the computed curves relating j^Lavl 

 to CO'. 



The physical behavior of the cable assembly may be described by reviewing 

 the significant results obtained by Little' and by Whicker.'^ The latter is a rather 

 simplified (i.e., no damping) theoretical analysis of the effect of ship motion on 

 mooring cables in deep water. On Figure 33, values of co' corresponding to 77/2 and 

 77 are indicated, as well as the roots of the equations tan co' = fi/co' and tan co' = -fi/co'. 

 The case of the fixed-ended spring for zero damping (/S = 0.0) is shown by Whicker 

 and by Little to have resonant frequencies of to' - 77, 277, ... n 77 when the relative 

 mass, ^i, of the cable and payload Is decreased infinitely (I.e., the payload mass 

 increases Indefinitely). The case of the free-ended spring for zero damping (j3 = 0.0) 

 is represented by: 



1. Values of co' equal to Tr/2, 377/2. . .n77/2 when the relative mass is increased 

 Indefinitely. When the damping jS = 0. 0, it is shown by Little that values of 

 co' equal to 77/2, 377/2. .. n 77/2 result in Infinite dynamic stress for Infinitely 

 large values of /i. 



2. Values of co' corresponding to the roots of the equation tan co' = -ju/co'. 

 Whicker shows that for finite values of fl, the least root of the above equa- 

 tion results in infinite total stress. 



In addition. Little shows that for finite values of M/ and for zero damping. 

 Infinite total stress will result when the roots of the equation tan co' = jJi/u)' are 

 satisfied. This Is in contrast to the results given by Whicker, yet both formulations 

 appear to be correct. Thus, It would appear that the Little and Whicker results are 

 compatible for long cable lengths, but that the Whicker results are not applicable 

 for short lengths, since all of recorded data supports Little's conclusions. In any 

 event, the least of the roots of the above equation lies between and 77/2. As /i Is 

 Increased indefinitely, the roots of the above equation approach 77/2, 3t7/2. , .n77/2. 

 There is no comparable analogy with the well-known results of simpler systems. 



17 



