For V and 6 and also for M, expressions have been found that satisfy 

 Equations [2Toi] and [2801] and, hence, also Equations [27] and [28]. But in 

 Equations [27] and [28], all coefficients are real. Consequently, the 

 real parts of v, G, and M by themselves must satisfy these equations and 

 must also be harmonic solutions of Equations [2U] and [25]. 



The real parts of the end displacements v and 6 must, therefore, 



represent v and 6 as functions of the time in a possible forced 



vibration. Keeping only the real parts in both members of the last two 



formulas for v and 6 gives, since e = cos cot + i sin ojt : 



V = aT T A cos ojt - aVi A sin ut 

 o 11 11 



+ a' B cos wt - a" B sin art [^3a] 



12 12 



Sq = a' A cos wt - a" A sin wt 



+ a' B cos ojt - a" B sin ut [^3b] 



22 22 



Selecting real parts in Equations [29e,f] gives as the corresponding real 

 external force and moment: 



F = A cos uit ; G = B cos art 



Thus the displacement v and slope 6 produced by force actions 

 proportional to cos art alone contain terms proportional to both cos wt and 

 sin (jjt when damping is present. Equations [U3a,b] might be accepted as 

 the final expressions for end displacement and slope produced by harmonic 

 end forcing. 



If, however, interactions of the beam with attached structures are to 

 be considered, it may be preferable to avoid explicit mention of sin wt and 

 cos wt. 



39 



