of Beams and the Whirling of Shafts. 91 



in the same form as a string would vibrate. It will now be 

 shown that all the other solutions are greater than — . 



The period equation reduces to 



= 



yfr n sin not 



sm a ' 



where cos a = —• . 



Hence other solutions are given by sin nu-=0, sin. a=£0,, 

 leading- to 



sir 



<f)=lkC0S — , 



' n 



where * is an integer, not zero or a multiple of 71. 



When the argument is less than 7r, ~ is always numerically 

 greater than unity. Thus these solutions are all greater 

 than y. 



When the argument lies between ir and the point near 

 *— where <fi = 0, y is negative and decreases numerically 

 from unity to zero. Hence the next gravest period is given 



by 



d>= — i/r cos — . 



r n 



Thus for three bays 



(£>== — -^ cos ;r = —-h^, 

 o 



or B^(KZ)=^iKZ). 



This solution is about 204°. 



These solutions tend to the limit it when n is large. 



§ 13. The question what is the best distribution of supports 

 so as to make the whirling speed of a shaft of given length 

 as great as possible is of special interest to engineers. It 

 will be shown that when all the supports are simple the 

 whirling speed is a maximum when the supports are 

 equidistant. 



Consider the function 



/(K) = An si n K^i sin K/ 2 ... sin K7„. 

 The determinant An is of the second degree in <f>' and yv. 



