2. HARMONIC END FORCING OF UNDAMPED UNIFORM BEAM WITH SHEAR WARPING 



Consider a uniform besiin that is maintEiined in vibration by a trans- 

 verse force F and a couple G, acting in a principal plane containing the 

 principal axes of all beam cross sections; see p. l88 of Reference 11. 

 It will be assuTKd that F and G are both harmonic functions of the time; 

 nonharmonic forcing is much more complicated. 



In developing the theory, F and G will be assumed to act at one end 

 of the beam, taken as x = ; generalization for other positions will be 

 given later. Furthermore, f and G will be assumed to vary at the same 

 frequency and in phase, so that 



F = A sin cot; G = B sin wt [la,b] 



in terms of constants A and B. The resulting formulas can then be 

 applied to a case in which the frequencies of F and G differ by first 

 putting B = 0, then A = 0, and adding the two motions thus obtained. If, 

 on the other hand, F and G have the same frequency but differ in phase 

 (F = A sin ojt, G = B sin((jjt + 4))), it is readily seen in the same way that 

 the general formulas are all valid as they stand (hence, for instance, in 

 Equations [l2a,b] v and 6 are intermediate in phase between F and G). 



Let v(x, t) denote the displacement of the beam, positive in the 

 same direction as F, and let G be positive in the same direction as 

 positive 3v/8x. F and G are assumed to be applied in such a way that 

 significant distortion of the end of the beajn, near x = 0, is avoided 

 except for the normal distortions of cross sections accompanying bending 

 and shear. Let the other end of the beajn, at x = Jl , be entirely free. 



