y + 2Csy+y = fj- 



(Q - Q). 



(E.l.l) 



where the dot (•) notation denotes differentiation with respect to time. Here y = y/D, t = co„ t, the 

 mass ratio fx = — -^ — 5 — and i^ 'S the structural damping ratio. The fluid force coefficients are 



Lift: Q 



Reaction: C^ 



MlpV^D 



Q sin (ax + 0), a = , 



Cn sin (ar + <j>x), a = 



(E.1.2a) 

 (E.1.2b) 



yipV^D 



as suggested by Griffin and Koopmann (El), Griffin (E2) and Chen (E3), where </> and 01 are the 

 phase angles between the lift and the displacement and between the reaction and the acceleration, 

 respectively. The two forces represented by Q, and Q are orthogonal (E2). 



If the cross flow displacement y is written as 



y = Fsin ar , a = , 



with Y = Y/D, then the equation of motion separates into 



sinar: —a^Y+ Y — fi 



cosar: 2Cs Y — n 



(Ci cos <f> — Cn cos 1) = 



(Cl sin0 — Cr sin 0i) = 



(E.1.3a) 

 (E.1.3b) 



when the coefficients of sin o t and cos a t are grouped appropriately. The various force components are 

 identified as follows: 



STRUCTURAL 

 INERTIA AND STIFFNESS 



-a^Y + Y 



FLUID 

 INERTIA AND ADDED MASS 



(Q COS0 — Cr cos0i) (E.1.4a) 



162 



