Theodorsen Analysis" described on p. 37 of Reference ik by McGoldrick and 



Jewell. 



For greater generality, allowance is made for possible additional 



damping due to other causes by adding a force - c,v acting on the foil along 



the same line as F and positive in the same direction, and also a moment 



- CpQ about the 6 axis. Without too great a complexity, the more general 



expressions - c^^v - c,„e and - c„ v - c 6 could be used but this was 

 11 12 21 22 



not thought worthwhile for the present purpose. 



Equations of motion may now be written for the foil. Let the effective 

 center of the foil be at a distance h ahead of the e-axLs , h being positive 

 toward the approaching stream. The total effective mass of the foil 

 (including virtual mass) will be denoted by m, and its total effective 

 moment of inertia about an axis drawn through its center of mass and 

 parallel to 9-axis by I . The displacement of its center of mass is 

 V + he and the total upward force on the foil is - F - c-,v + Fj , - F being 

 due to the attachment structure that exerts the force F on the beam. The 

 total moment about a line drawn through the center of mass parallel to the 

 e axis positive in the same direction as 6 is , similarly: 



-G-ce + M„-h(-F- c,v + F^ ) 

 2 6 1 L 



Hence 



m (v + he) = - F - c,v + F 



I^e = - G - C2e + Mg - h (- F - c V + Fj^) 



A simpler solution, however, is to add h times the first equation to the 

 second equation, thus obtaining for the second equation: 



See p. 78 of Reference 3, for example, c,, , c^p an<i c,p, c^-, are direct 

 and cross damping constants, respectively. 



hi 



