If, at the w chosen in calculating the parameters, e b - b e 7^ 0, 

 determine S from Equation [55] using the second form of the left member; 

 and, if S > 0, test this value by substituting it into Equation [5^a] if 

 b^ or into Equation [5'+b] if e, ^ 0. If the chosen one of Equations 

 [5^a,b] is satisfied, simple reasoning from Equation [55], using the 

 first form of the left member, shows that the other one of Equations 

 [5^a,b] is also satisfied, so that S is a common root. If this procedure 

 cannot be used, one of the following procedures is available. 



If e,bp - b-,ep = 0, then e-,bo - ti-,eo = also, to satisfy Equation 



[55]; otherwise there is no common root. If b-j^ ^ and e-^ y 0, it follows 



that e = (e A) ) b and e^ = (e /b ) b^, and, of course, e, = (e /b ) b . 

 2112 3113 1111 



Thus the two Equations [5^a,b] are proportional and have the same roots, 

 which may be found by solving either equation provided it contains S. If 

 either b = or e =0 but not both, one of Equations [5^a,b] reduces to 

 0=0 and the other must be solved for S. 



Finally, if b-, = e-, = 0, Equations [5^a,b] are, at most, linear in S; 

 and S determined from one of these equations may be tested in the other 

 equation. 



In the very rare case that b =b =b =e =e= e=0, steady 

 vibration of the system may occur at the assumed u with any value of S. 



If one of these procedures does not yield a common root for Equations 

 [5^a,b], then there is no common root and steady vibration cannot occur at 

 the chosen w. Furthermore, for our present purpose any negative S is to 



be rejected. Usually no acceptable S will be found; by repeated trials, 



2 

 using interpolation wherever useful, the rare values of w at which an 



5U 



