The equation of motion of a rigid, spring-mounted cylinder is 



YID + 2Cw„ K/Z) + uilYlD = (p V^L/lm) Cl = lJi(^sCL (D4) 



where, as noted earlier in the report, fi = pD^liiT^St^m. The natural frequency aj„ and damping ratio t, 



are taken for the purpose of modeling as those values measured in a stationary fluid having the same 



physical properties as the flowing fluid.t The solution to the above equations in the entrainment region 



where resonant oscillations occur is sought as 



YID = A sinwr, (D5a) 



CJClo = 5 sin (o)/ + 0). (D5b) 



Here, w is the entrainment frequency and the conditions w/a>„ ~ 1 and w/w^ = 1 are implicitly 



assumed; A and B are, respectively, amplification factors for the cylinder displacement amplitude and 



the fluctuating lift. 



When equations (D5) are substituted into equations (Dl) and (D4), the entrained response is 

 found to be 



>( = (fiQo/5c)/(8^ + 4)'/^ (D6a) 



B^=\- (F/GSgCIo) [8/(62 + 4)], (D6b) 



= arctan(-2/8), (D6c) 

 where 8 must satisfy the cubic equation 



8 3 - AS 2 + (4 - HF/C GSg) 8 - M^ - F/H Sg) = 0. (D7) 



In these equations, the detunings 8 and A are defined by 



. 8= (2/C)Wa)„- 1), (D8a) 



A= (2/n(a'>„- 1), (D8b) 



and the response parameter Sq = t,lix. 



It has been shown (D3,D9) that the above equations yield an accurate representation of the 

 entrained resonant response of spring-mounted rigid cylinders when the empirical parameters G, H and 

 F are appropriately selected. Based on an analysis of several diverse sets of experimental data for 



tMore recent studies (D7,D8) have demonstrated that, for oscillations of bluff cylinders in water, this simplified specification of 

 the damping is not appropriate. This point is discussed in more detail in Appendix C, and the presentation here is meant only to 

 demonstrate the general approach to a "wake-oscillator" type of formulation. For most cylinders in air, the structural damping 

 and the still fluid damping are virtually identical. 



154 



