qj = 0, J ^ i (D15b) 



Qi/Cio = Bi sin {wt + cp/) (D15c) 



Qj = 0,J^i. (D15d) 



Here again Aj and 5, are modal amplification factors for the cylinder displacement and fluctuating lift, 



and </), is the phase of the fluctuating lift relative to the cylinder displacement. The conditions w/w„ , = 



1 and w/ois = I imposed on the response frequency w must be satisfied for the assumed solution to be 



valid. 



The entrained pure mode response of the system is found to be 



Bl = (1/4,) {1 - {FjG;SGjClo)[hJ{h} + 4)]}, 

 4>j = arctan (—2/8,), 

 where 8, must satisfy the cubic equation 



(D16a) 

 (D16b) 

 (D16c) 



bj - A,8,2 + (4 - //,./;./{, G,Sg,,)8,-4(A, - Fjl^iScj) = 0, 



(D17) 



and where 



8,. = (2/C,) (ft>/w„,, - 1), 

 A, = (2/C,) {cos/co„j - 1), 



(D18a) 



(D18b) 

 (D18c) 

 These equations for the /''' pure mode response are identical to the equations for the vortex-excited 



response of a rigid, spring-mounted cylinder except for the multiplicative factor of 4,'^^ appearing in 



the equation for B,. Hence, the behavior of the /''' modal response as a function of A,, or oislf^n.h is 



similar to the behavior of the rigid, spring-mounted cylinder, complete discussions of which of can be 



found elsewhere (D9). 



The vortex-excited oscillation of an elastic cylinder under the condition w^ = w„ , is 



Y{x,t)/D = AiipiU) sin cot. 

 The maximum oscillation amplitude Yj/^^xM in the /''' pure mode entrainment region is then 



yi.MAx(x)/D = A,,MAx\^i(x)\ 



157 



