where, as before, the /"^ natural frequency w„, and the /"^ damping ratio Ci are those values measured 

 in a stationary fluid having the same physical properties as the flowing fluid (see previous footnote). 

 Here m, is the effective mass of the /"^ mode of the vibration. 



Finally, let the fluctuating lift coefficient Q in the shedding region be developed as 



Q = i:fi(')'/''(^)' (D12) 



where the Q,(/) are the lift response factors. When equation (D12) is substituted into equation (Dl), 



and the latter is multiplied by \pjix) and integrated over the length of the structure, the lift equation 

 for the /"^ lift response factor reduces to 



+ EZZ lijJ^sG^QjQkQi + (Gi/cos)QjQkQi 



- oilHiQjQi^Q, - H,QjQkQi\ = (OsFiqjD; j,kj^ i simultaneously (D13) 



where 





This equation of motion for the structure reduces to 



qjD + li,(^„jqjD + wl^qj D = oiliiaQ,. 

 where 



is a constant and m is the virtual mass per unit length of the structural member. 



(D14a) 



(D14b) 



(D14c) 



The undamped linear equivalents to the governing equations show that if the system is responding 

 at a frequency w then q, <x Q, /[I — (a)/aj„,)^] and Q, oc q^/[\ — (w/ws)^]. Thus, for a>/a>„, ~ 1 and 

 (o/ws ~ 1 simultaneously, the equivalent linear solution reveals that the /"' vibration mode undergoes a 

 resonant type behavior while the remaining vibration modes remain small. Hence, for the flow condi- 

 tion (os ~ o)„j, the solution to the equation (D14) is sought as the pure mode form 



qi/D = Ai sin lot (D15a) 



156 



