Ramberg's investigation was that the Independence Principle is not generally valid for stationary, yawed 

 cylinders. It was found that the variation of the shedding frequency of the vortices deviated substan- 

 tially from the Independence Principle for yaw angles greater than /3 = 25° to 30°, and that the vortex 

 shedding from stationary cylinders was strongly influenced by the end conditions. 



A special case (one of only two) where the Independence Principle was found to apply is for the 

 condition of lock-on between the vortex and the vibration frequencies. A typical example of the results 

 obtained by Ramberg is shown in Fig. 2.19. The bounds for lock-on during cross flow oscillations are 

 plotted for yaw angles up to 50° from normal incidence. The yawed cylinder results are in excellent 

 agreement with comparable experiments performed with a cylinder positioned normal to the incident 

 flow. The region inside the dashed lines and data points corresponds to the lock-on regime. The 

 findings from Ramberg's investigation imply that the various methodologies for predicting vortex- 

 excited oscillations at normal incidence can be applied with resonable confidence to a cylinder at an 

 angle of inclination to the flow. King's findings further suggest that such an extension of the results 

 obtained at normal incidence is valid for flexible cylindrical structures in flowing water. 



2.6 Surface Roughness Effects. Another factor influencing the hydrodynamic forces that result 

 from vortex-excited oscillations is the surface roughness of the cable or cylinder. Sarpkaya (30) has 

 measured the unsteady hydrodynamic forces on sand-roughened cylinders forced to vibrate and has 

 compared his measurements to similar experiments with a smooth cylinder. Some typical results for 

 the total hydrodynamic force coefficient Ct,max are plotted in Fig. 2.20. Substantial increases are 

 apparent in the unsteady hydrodynamic force coefficient for the rough cylinder, though additional study 

 is necessary to determine which components of the total force are amplified by the roughness. The 

 resonant effect of lock-on is apparent, however, when the coefficient Cj is plotted against the displace- 

 ment amplitude Y/D as in Fig. 2.21. Once again the peak value of Ct is obtained at 2Y/D = 1 for the 

 diff'erent reduced velocities in the lock-on regime. The increase in Cj for the roughened cylinder as 

 compared to a smooth cylinder is readily apparent in Fig. 2.21. The Reynolds number based on the 



21 



