Note that : 



, . ^ 1 dF r, ■ ^ 1 dG 



A sin ojt = ; B sin ut = 



0) dt w dt 



Thus Equations ['i3a,b] can also be written in the form: 



— ^ I 



11 dF ^12 dG , , , , 



a' F + -^i — + a' G + — [UUa 



^ 11 w dt 12 (^ d* 



^21 dF , ^22 dG ri i.i 



e = a' F + ^ + a' G + -— TT [l^Ub] 



o 21 0) dt 22 u) Qi. 



The eirfit coefficients a' , a" /lo, etc. , may be regarded as an extended 



11 11 



set of influence coefficients for the damped beam under harmonic end 



forcing, corresponding to the four coefficients a , a , a , and a 



^' 11 12 21 22 



for the undamped beam. As an alternative, inertia coefficients corres- 

 ponding to b , b , b , and b for the undamped beam could be 

 calculated. 



The following extension of the results is perhaps obvious. F and G 

 have been assumed to vibrate in phase, but this restriction is easily 

 removed. Let B = 0. Then the motion of the beam is excited only by F, 

 and all formulas will obviously 'hold if art is replaced by (wt + a) in 

 which a is an arbitrary phase angle. Similarly, if A = 0, only G is 

 active and wt may be replaced by (oat + S), in which 6 is another arbitrary 

 phase angle. Since the equations of motion are linear, these two motions 

 and the exciting force actions can be superposed. Hence, Equations 

 [^3a,b] will remain valid if the external force and moment are assumed to 

 be 



F = A cos (ujt + a) ; G = B cos (wt + 6) 



UO 



