a = tan q£ + 1. The effect of the small omitted term 1/C is chiefly to 

 shift the position of the infinite values very slightly along the ql axis, 

 by Aqjj. = + l/C, since, when qZ << 1, cos qj, - (l/C) = cos [ql + (l/C)]. 



At the free-free frequencies defined by aC = 1, resonance may be 

 said to occur, since the beam can vibrate at these frequencies even if 

 F = G = 0. In the one-dimensional cases, an attempt to force at a reso- 

 nant frequency necessarily results in an infinite amplitude. 'In a two- 

 dimensional (or two degrees of freedom, (v,e)) case like the case under 

 discussion, however, and presumably in any multidimensional case, finite 

 amplitudes will occur, provided the applied reactions are in certain 

 ratios to each other. 



Conversely, at the built-in frequencies defined by aC = -1, there is 

 "anti resonance. " In a one-dimensional case of this kind, the forced 

 amplitude cannot be budged from zero at the forcing point; however, in 

 similar miilti dimensional cases, the amplitudes of displacement or 

 rotation at the forcing point merely stand necessary in certain fixed 

 ratios to each other, these ratios being independent of the ratios of 

 the applied forces or moments. 



The detailed discussion of individual cases is greatly simplified 

 when shear warping is neglected because then, in the fonnulas , only the 

 single parameter q varies with frequency. If shear warping is included, 

 the parameter C also varies and the variation of the coefficients 

 becomes more complicated. It may reasonably be surmised, however, that 

 in this case, also, the a's and b's will vary widely with frequency. In 

 rou^ly cyclic fashion, the infinite a's will occur at the natural 



18 



