Besch and Liu 



by assuming a lumped parameter representation for the strut, with 

 elastic and inertial properties lumped at discrete points along a 

 straight elastic axis. This procedure is well established as an accur- 

 ate means of predicting vibration mode shapes and frequencies of 

 elongated structures in air. The hydrodynamic forces on the strut 

 were also lumped at stations along the axis. Values of structural 

 parameters and hydrodynamic forces at spanwise stations were assign- 

 ed by dividing the strut into strips normal to the elastic axis. A nu- 

 merically converged solution was obtained when 10 strips were used. 



Displacements were assumed to occur in bending normal to 

 the plane of the strut and in torsion about the strut elastic axis. The 

 equations of motion for the entire system written in matrix form are 



h. J r , lh.| _ _ lh. 



M 



e. 



[c] L 1 + < 1+ Js> [ K ] k = { F i} 



The hydrodynamic force F was expressed in terms of the physical 

 displacements and their time derivatives, permitting the structural 

 and hydrodynamic expressions to be combined. Further simplificat- 

 ion is achieved by representing strut motion as a series of standing 

 waves in the form 



h. = H. e St and 0. =0. e St 



li li 



The resulting system of linear equations for the hydroelastic system 

 is 



(s 2 [M-] + s [C-] ♦ [K-] ) I" 1 ) = |0} 



Solutions to the above equations are the complex eigenvalues of 

 which may be written 



I 2" 1 



s = - f w + j \ i - r <* 



in terms of the damping ratio f and the undamped natural frequency 

 u) . Each eigenvalue of s corresponds to a mode of oscillation of the 

 strut-fluid system. Flutter occurs at the lowest speed for which the 

 real part of one of the eigenvalues becomes zero. 



352 



