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



be the distance from of a point in the right span, z being' 

 reckoned positive towards the right. Let z be the distance 

 from O of a point in the left span, z' being reckoned positive 

 when measured towards the left. Let u be the deflexion of 

 the point z in the right span, u being reckoned positive when 

 upwards in the figure. And in like manner let v be the 

 deflexion of the point z' in the left span, v being positive 

 when downwards in the figure. 



Let the shaft have changes of section at 0, A, B, 0, D, E* 

 Let the moment of inertia of the area of the cross-section 

 about a diameter between and A be I 1? between A and B 

 be I 2 , between B and C be I 3 , and between. C and L be I 4 . 

 Also let the moments of inertia between and 1), D and E, 

 E and L' be I/, I 2 ', and L/ respectively. Let 0A = #,. 

 OB = b, 00 = c, OL = Z, OB = d, OE = e, QL' = Z'. 



Let the force X/ and couple M : act at P, and the 

 force Xi' and couple M/ act at P\ Let 0P = £, OPW. 

 Let Y, Y' be the reactions at the bearings L, L'. Let E 

 denote the value of Young's Modulus of elasticity for the 

 material of the shaft. 



§ 3. Following the ordinary Bernoulli-Eulerian theory of 

 beams, we have, from to A, 



-EI 1 g=Y(/-,)-X 1 (c 1 -.-)+M„ 



••• -EL ~ =Y(fe-fc»)-X 1 (* 1 *-i*«) + M l5 + C 1 . 



.-. -BI 1 u=T(iJ»»-i*»)-X 1 (t« 1 *"r-i*») + JM 1 s«+C,*+G t '. 

 From A to B, 



-EI 2 g = Y(/-; 0-X.h - «)+.Mi, 



.'. -EI, ~ = Y(7:-^)-X I (z I --4; 2 ) + M 1 ; + C 2 . 



.-. -EI ! » = Y(W-J:')-X,(|:,r-|: 3 ) + |M 1 :UC 2; + G, 



Now u l = u^ and l-j-) ~ (jz) 5 when r=w. Hence we 



obtain C 2 , G 2 in terms of 0^ G,. 

 Thus 



-EIj^j =Y(la-±a 2 )-X,(z l a-ia 2 ).+ M 1 a-tC 1 , 

 and - Ejj^) =Y(la-ia 2 ) -X^^a-^ + M^ + C 2 . 



