654 Prof. H. H. Jeffcott on the Whirling Speeds of a 



the sum of the numerical quantities along the principal 

 diagonal of the' determinant, is 17'152. From physical 

 considerations we expect all the eight roots to be positive. 

 Hence the sum of the remaining six roots lies between 2*152 

 and *652. Thus the two roots obtained are the first and 

 second whirling speeds. 



Taking ^ = 13 to 14 and 2 to 2*5, and substituting the 

 values of E, I, I, g~, we finally obtain the whirling speeds in 

 revolutions per minute : — 



First whirling speed 1440 to 1500 r.p.m., 

 Second whirling speed 3410 to 3820 r.p.m. 



It is to be noted that in this calculation the portions of 

 the shaft of increased thickness are assumed to act fully in 

 supporting the bending moment quite up to the section 

 change point. Some allowance, however, should be made 

 for this end effect, and at the outset virtual change points of 

 section should be chosen, making the virtual lengths of the 

 thick portions of the shaft slightly shorter. Thus the shaft 

 is somewhat less stiff than assumed above, and when the 

 virtual values of the shaft lengths are used in the calculation, 

 the whirling speeds are found to be somewhat lower. 



§ 13. As we have obtained the deflexion equations in 

 §§10 and 12, we may readily apply Dunkerley's method to ; 

 those examples. ^ -. -< 



Dunkerley's rule is -=-% = — 2 H 1 + , where Ox is 



ill (Oi &>2 



the first whirling speed of the loaded shaft, and a> 1} co 2 , ... r 

 are the whirling speeds of the several loads acting alone. 

 Applying this rule to § 10, we have 



EI 539C> 1 . ft9 o r ,™? 3 



= 188416 M ' ° r a^ 02866 ]^ 



income" 

 Also ~ =-00682^. 



0) 2 2 El 



1 mP 



Hence ~ ='03548^. 



IV EI 



' H 1 2 =28-2x ] ^. 



ml 6 



It will be noted that this value of the first whirling speed 

 is too low. 



Correcting Dunkerley's rule by taking into account the- 



