Yeff.max = yMAxIi"' I \^> (2) Ia/^a- = YMAxhi (2.3a) 



where the modal scaling factor /, is defined by 



. ilif (z) dz I, / -.1 



J>/(z)</z 

 Equation (2.3b) was obtained by Iwan (8) and Skop and Griffin (9) in the course of developing a 



"wake-oscillator" formulation for predicting the vortex-excited oscillations of structures (see Appendix 



D). Typical values of /, are tabulated in references 3, 8 and 9, and in Table El. 



Experimental measurements of Yeff.max as a function of i[,J^l are plotted in Fig. 2.2. These 

 results encompass a wide range of single cylinders of various configurations at Reynolds numbers from 

 Re = 300 to 10^. All available experiments to date indicate that the limiting unsteady displacement for 

 an elastically-mounted rigid cylinder is about IYeff.max ~ 2 to 3, as shown in the figure. This result 

 has been obtained both in air and in water, even though the mass ratios of vibrating cylinders in the 

 two media differ by as much as two orders of magnitude. For cylinders and cables vibrating in water 



the mass ratio — ^ varies from slightly greater that 1 to about 8; in air typical values of the mass ratios 



corresponding to the figure cover a range j = 25 to 500. The experimental data in Fig. 2.2 indicate 



pD' 



that the reduced damping can increase by a factor of nearly ^/fy {ijp. = 0.01 to 0.5) while the peak- 

 to-peak displacement amplitude decreases by a factor of only about two or three (2-3 diameters to 1 

 diameter). In that range of the figure the hydrodynamic forces predominate and the light, flexible 

 structures that are typically employed in water (cables, cantilevered beams) can undergo large cross flow 

 oscillations. 



A fairly wide variety of structural elements is included in the data. For example, the experiments 

 of Vickery and Watkins (7) were conducted with a pivoted rigid cylinder in incident flows of both water 

 and air. As noted previously. Dean, Milligan and Wootton (5) measured the cross flow response in 

 water of both spring-mounted cylinders and flexible cylindrical beams. King's results (3, 10, 11) were 



