668 On the Whirling Sjieeds of a Loaded Shaft. 



12-6EI 

 Substituting f i = "531, we find co 2 = — -= y\t 



Ul — ^2)t 



This determines the speed o£ a simple oscillatory vibration. 

 For such a vibration to be possible it is necessary that I 2 be 

 greater than Ii, or for a solid circular cylinder, that l\ is 

 greater than 0'866d\. 



The normal whirling speed in this case is determined from 



Putting ^ = '5 31 we find 



EI 

 mi 6 

 • We thus have a simple normal whirling speed corresponding 



.EI 



to co 2 =66'2 — jx, and a simple oscillatory vibration corre- 



EI 



sponding to co 2 = 12'6 



(i 2 -liV 



The ratio of oscillatory speed to normal whirling speed is 



v. 



5-25(1,-1 



§ 23. For a uniformly loaded uniform shaft supported in 

 three bearings the general solution is well known. It is 

 given by 



cot yjrl -f cot yjrV = coth yjrl -f coth yjrl' ; 



and particular solutions by yjrl — air and yjrl' = hir simul- 

 taneously, where a and b are integers. ^r 4 =— — where m is 

 the mass per unit length. 



The following Table gives the values of yjrl corresponding 

 to the first and second whirling speeds, and singular solutions, 

 for different ratios of lengths of span. 







I. 



*JBLE 



III 









V 

 7 = 



o-oi 





o-i 





0*5 



0-8 



10 



First ijfl= 



3'92 





3-8L 





3-56 



3-38 



3-94 



Second x}/l= 



71 





6-88 





7-41 



455 



7-08 



Particular ^1~ 







314 





6-28 



15-71 



3-14 

 G-28 



