displacement v , associated with a force F on the heaon and -F on the 

 o 



foil; let V, v , and F be positive in the same direction. In terms of 

 o 



elastic constants k and k of the connecting structure: 



F=k, (v-v); G=k„(e-e) 

 1 o do 



Figure 3 represents a section in the principal plane of the beam 



and shows positive values except where the associated symbol is preceded 



by a minus sign (as in -F). A heavy line represents a section through 



the foil plane; the foil itself, of width 2b, being perhaps reduced to a 



connecting structure here. The axis A in the foil plane and the axis A 



o 



through the principal axis of the beam coincide when foil and beam are 

 un deflected. 



Let beam and foil be immersed in a stream of fluid approaching at 

 \miform speed S, as shown in Figure 3. Expressions for the resulting lift 

 forces on the foil are adapted from Theodorsen's formulas for a uniform 

 foil of infinite length vibrating harmonically. An approximation closer 

 than the common steady -motion approximation is used (see p. 8^+1 of 

 Reference ^4 (Appendix H)), the additional complexity being only moderate. 

 In the case of ship rudders or control foils, Theodorsen's parameter l/k 

 or S/b(i), in which b is the half- chord* length and w the circular frequency, 

 is less than 1 (or at least < 2); then Theodorsen's functions "F" and 

 "G" may be replaced without great error by their values at S = or 1/2 

 and 0, respectively (see pp. 8UO-8U1 of Reference U). The effects of 

 certain other terms may be included in the effective mass m and the 

 effective moment of inertia I (taken about the effective center of mass); 



* Not to be confused with external force and moment respectively on beam. 



UI4 



