Pa'idoussis 



two are associated with the zeroth mode, and the other two with the 

 first mode and its mirror image about the [im(to)] -axis. 



Surprisingly, the correspondence of the rigid-body frequencies 

 to those of the flexible body, for the apparently arbitrary value of 

 u = 0, 7 persists for other values of fg) as shown in Table 2, This 

 value of u = 0,7 can be explained as follows. We have defined the 

 dinaensionless frequency of the rigid body by oirb~ ^L/U. On the 

 other hand, the dimensionless frequency of the flexible body was 

 defined as tO|w= [ (M+m)/El] J2L , which may be rewritten as 

 <*5f b ~ [ (M +m)/M] ''^^u^L/U, where u is the dimensionless flow 

 velocity (§5, 1). The a§.sumptions made in the theory require that 

 m = M, so that to^|j = V2 ufiL/U, Now, if the dimensional frequency, 

 SI, of the rigid body^nd of the flexible body are identical, we may re- 

 write this as w^[, = V2uw^|j, and we can see thai identity of the dimen- 

 sionless frequencies will occur when u = 1/V2 ~ 0,707, 



Calculations were also conducted to pin-point the thresholds of 

 yawing and oscillatory instability in terms of f2t A etc, , and to 

 compare with the existing stability diagrams, e.g. Figs, 6 and 7. 



In the case of a rigid body with parameters corresponding to 

 those of Fig. 6, it was found that oscillatory instability exists for 

 — fg— 1 a-iid that yawing occurs for f2 > 0.5. Correspondingly, for 

 the case of Fig. 7 it was found that yawing persists throughout, and 



Upon examination, the stc jle branch of the zeroth mode as given in [26] 

 was found to be in error; the locus moves away from the [ Re (co)] - 

 a:xis much faster than shown in Fig. 3, [ 26] , The corrected value 

 for u = 0,7 is given here. 



1006 



