different ratio, or with either F or G alone, will result in infinite 

 amplitudes of vibration. With F = G = 0, however, the beam may execute a 

 free vibration with 



^= -^=- 0.982 q 

 V b -y -i. 



o 12 



(i) At ql = 2.55^ = 4.870: a^^ = - 8.11, a^^ = 1 -^^ , a^^ = - 5.89, 

 and A = - 1.22. As qii increases further, the a's continue to decrease 

 numerically. 



(j) Second sliding-end frequency: Just beyond q_l = l.T5Tr or 

 5.50, tan q£ = - tanh qi and a = - 2.02; a.^^ = 1.01, a = 0, and 

 A = - 1.02. The general situation is as in Case (b). 



(k) At qi = 2-n = 6.283: a^^ = - I.OOI+, a^^ = , a^g = l.OOU, and 

 A = - 1.008. 



During each further 2tt range of qJi (2it < qil 4 U, Utt < q«, <^ 6-n , etc.), 

 a cycle of zero and infinite values of a. . occurs similar to that in the 

 range < qJl <^ 2tt except that during the first quadrant of each 2Tr 

 range, two additional points occur that have no analog in the first 2ir 

 range. Successive points of interest are now spaced almost exactly n/4 

 apart, and the peaks in the curve that contains the infinities become 

 extremely narrow. 



The two additional points in the fifth quadrant and two other 

 points illustrating the concentrated occurrence of large a's are as 

 follows : 



16 



