G act at an intermediate point instead of at an end of the uniform beam; 

 shear warping is included but rotary inertia and damping are ignored. 



5. In Section 5, the analysis is further extended to include the 

 effects of both external (Rayleigh) and internal damping, the latter being 

 associated with bending strains; shear warping is included but rotary 

 inertia is ignored. It is shown that the equations derived are valid 

 even if the external force F and the moment G are not vibrating in phase. 

 Formulations are developed from which the final calculation of the co- 

 efficients can be made. The procedure for making these calculations is 

 clearly indicated. 



6. In Section 6, flutter of a rigid foil flexibly attached to a 

 uniform beam is considered, the entire system being immersed in a stream 

 of fluid approaching at uniform speed. Expressions for the resulting 

 lift forces and moment on the foil are adapted from Theodorsen's equations 

 for a uniform foil of infinite length vibrating harmonically. An 

 approximation closer than the common steady-motion approximation is used. 

 For greater generality, allowance is made for still-water damping 

 (structural plus fluid) as well as for hydrodynamic damping. Equations of 

 motion for the foil are related (or coupled) to the equations previously 

 derived for the response of the beam subject to harmonic loads; i.e., the 

 coefficients previously derived for the beam and Hooke's law for the 

 flexible connecting structure serve to couple the motions of the foil to 

 the harmonic loads imposed upon it by the motions of the beam. A method 

 is described for finding the critical flutter speed and frequency for this 

 system, at which a steady vibration of the foil and beam is possible in 



90 



