of Beams and the Whirling of Shafts. 89 



and KZ 3 must lie between it and — approximately. A few 



trials show that KZ 3 = 202° is a close solution. 

 When /i = / 3 , the equation becomes 



01 + 02 = + ^2 



or <t>i + <pi = ^2 \ 



and 0(K:Z 1 )+0(iKZ 2 )=OJ ' 



The first equation corresponds to symmetrical vibrations. 



The second equation corresponds to skew symmetrical 

 vibrations with a node at the centre o£ the middle bay, being 

 the same as that for two bays of lengths Z 1? -JZ 2 . 



These are special cases of a general theorem which is 

 given in § 11. 



For three bays, the symmetrical case gives the lower 

 whirling speed. 



§ 10. The equation for four bays simply supported is 



(0 1+ </) 2 )((/) 2 + (/)3)(^3 + ^)-t2^3+^)-t3 2 (01+*2)=O^ 



When Zj, Z 2 , Z 3 , Z 4 are unequal, it appears that this equation 

 could be solved by a somewhat tedious process of tabulation, 

 in the form 



01 + 02 03 + 04 



unless (0i + 02)j (03 + 04) can vanish simultaneously. 



If, however, the beam is symmetrical about its middle 

 point, so that Z l = Z 4 and Z 2 = Z 3 , the equation reduces to the 

 two equations 



01 + 08 = 



and (0! + (f> 2 ) </> 2 — ^ 2 2 = . 



The first of these equations is that for two bays simply 

 supported, and the second is that for two bays fixed in 

 direction at one extremity. This is a particular case of a 

 general theorem which will now be given. 



§ 11. Consider the case of 2u bays symmetrical a hour the 

 middle point, so that Z 1 = ?2?i, h — hn-ij etc. Then by a 



