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



§ 15. When each span of the shaft is of uniform section 

 the algebra is simplified. 



Referring to fig. 1, we have, from to P, 



-EI^=YG-^/-X 1 ( 2l -2)+M 1 . 



.-. -El~=Y(lz-^)-X l (z l z-^) +M 1 « + 0, 

 and -Mu = Y{ik'-iz»)-X 1 (iz 1 z 1 -iz , >) + iM,2» + 0* + a. 

 Now m = when c = 0. .-. G=0. 

 From P to L, -EI~ =Y(l-z). 



-Elf = Y(L— *--») + K, 

 and -EIm = Y(^ 2 — j l- 3 ) + K^ + H. 



Now w = when 2 = Z, hence H is determined. 

 EIk = i Y(2/ 3 - 3?£ 2 + - 3 ) + K(Z - e) . 

 Thus for the span I we have, from to P, 



2EI^ = X 1 .:(2.: 1 -;) - Y~(2Z- ~) -2M^-2C, 



and from P to L, 



6EIu=Y(l-z){2l 2 + 2lz-z 2 ) + 6K{l-z), 



2EIp = -Yz(2l-z)-2K. 



7 



Now at P, where z - c 1} the values of u and — are equal 

 in OP and PL. Thus we get dz 



6EIu 1 = 2X 1 z 1 *-Yz 1 2 ('dl-z 1 )-m l z 1 2 -6Cz 1 



=Y(/-: 1 )(2/ 2 + 2/, 1 -, 1 2 ) + 6K(/-, 1 ), 

 and 



2El(gy=X 1 c 1 2 -Y, 1 (2/-, 1 )-2M 1 , 1 ~2C, 



= -Y,: 1 (2/-.: l )-2K. 



Hence K = C-iX 1 .- 1 2 + M r ; : , 



and 6Cl=X 1 z 1 2 (3l-z 1 )-2Yl s -m 1 z 1 (2l-z 1 ). 



