The general harmonic case of a transverse force and couple acting 

 on one end of the beam can be resolved into two sets of forces and couples 

 like the set jvist described, with the two sets acting in perpendicular 

 principal planes. 



Although nonharmonic forcing is much more complicated, solutions to 

 such forcing can be found using Fourier Series or Fourier Integral methods. 



Basic equations for the harmonic forcing are developed, including 

 the shear-warping effect; however, rotary inertia is ignored because of 

 its small effect in practice (see p. 18T of Reference ll). 



Let M denote the moment acting across any cross section on material 



lying toward x = 0, taken positive in the same direction as G or 6, where 



6 denotes 3v/8x for convenience, P the corresponding shear force taken 



positive with F and v, and y the mass of the beam per unit length. Then 



the appropriate differential equations are as follows (see Equations [2.12. 



to [2.15] in Reference 11, where V = -P because V wa^ taken positive 



toward negative v, and the I term is to be omitted) ; the reader can also 



find the basis for these equations in many texts on vibration theory: 



9^v _ 8P 3M , ^- 



^ "~2 - 3"!^ ' ^= -P [2a,b] 



3t 



3V 3y I^ r 1 



e = — = Y + 2aP; -^ = — [2c, d] 



3x ' ' 3x EI 



Here — - is written for the constant usually denoted by KAG, the shear 

 2o 



rigidity, and this parameter as well as y and EI are assumed uniform 

 along the beam. 



