F and v are taken positive in the same direction, and G is positive in the 

 direction of 3v/3x viewed as a rotation 6. Then, the boimdary conditions 

 for a beam of length Z are: 



atx=0: M=-G;|^=-P=F [26a,b] 



at X = I: M=0;|^=-P=0 [26c, d] 



The final results of the investigation will be relations between v 



and e , the values of v and of 6 or 3v/3x at x = , and F and G. 

 o 



(a) Harmonic motion. This problem is solved only for the case in 

 which all time-dependent variables vary harmonically with time at 

 circular frequency w. In developing the solution, complex quantities are 

 employed becaiose of their algebraic compactness. Complex quantities are 

 distinguished from real ones by adding a bar over the symbol. All time- 

 dependent complex variables are assumed to vary with time in proportion 



, iwt / . _ I r-v 



to e (i = /- Ij . 



The complex analogs of Equations [2U], [25], [26a,b,c,d] thus 

 become: 



M = EI 



gx 3t3x / 3x 



? — — 2 — 



3'^v '^' 3v ^ 3 M „ roHi 



St"^ -^ Sx"^ 



at X = 0: M = - G; ^ = F [29a,b] 



e.t x= I: M=0; |^=0 [29c, d] 

 3x 



29 



