m and I are further defined later, 

 o 



There remain then a total resultant lift force F and a total moment 



ij 



** 



M about the 6 axis whose magnitudes can be written: 



o 



2 • • r, , 



F^ = Bs e - BSv + cse [U5a] 



M„ = LBS^e - LBSv - E^Se [li5b] 



t) Li 



Here B, C, L, £ind E^ are constants (defined below), L having the 



dimension of length. F^ and M^ are positive in the respective directions 

 L a 



of V and 9; v = dv/dt and 9 = d9/dt. Since M denotes the total moment 



9 



about the 9 axis, F may be supposed (for concreteness of thought) to act 

 Li 



throu^ this axis, as drawn in Figure 3 (i.e., Theodorsen gives the total 

 force F and total moment M about a certain axis . Hence , we cause no 

 error if we avbitvavily assume F to act through the axis. ) 

 According to Theodorsen 's calculations; 



B = Trpb£^ ; C = B(^b + e); L = lb-e; E =B(-b + e) 

 ^ f ' '2 2 L 2 



in terms of the density p of the surrounding fluid, the half-chord length 



b and length £„ of the foil, and the distance e that the axis of rotation 

 f 



or the 9 axis lies ahead of the midchord line (here ahead means toward 

 the approaching stream). The first two terms of M^, may be regarded as 

 arising from a force equal to the first two terms of F acting at the 

 foivard quarter-chord point. With these values inserted. Equations [U5a,b] 

 might be called the low-speed approximation to Theodorsen 's expressions. 

 For a derivation of the approximation, see Reference h. In this reference 

 it is shown that if e = b/2. Equations [H5a,b] agree with the "Modified 



** These values of F^ and M. agree with those of P and Vi^ given, 

 respectively, on pp. SUl and 8^2 of Reference U. 



U6 



