Loaded Shaft supported in Three Bearings. 657 



These equations determine C and K. We thus obtain; 

 from to P, 



6EIlu=X 1 z{-3k i (z 1 -z)+z 1 s -lz 2 }+Ylz(2l*-Zlz + z 2 ) 



+ 3M 1 ,:{/(2; 1 -3)-. 1 2 }, 

 and from P to L, 



6EIlu=-X l z l ^l-z)+Ylz{l-z)(2l-z)+m i z ] ; 2 (l l -z). 

 Also from to P, 



6EI/^=X ] {3/c(2: 1 -,c)-: 1 2 (3/-: 1 )}+Y/{2/ 2 -6^ + 3^ 2 } 



+ 3M 1 {2^ 1 -r)-e 1 2 }. 



Corresponding expressions hold for the span V . 

 Thus from to P', 



6Er/'r = X 1 'y{-3rr 1 'ti'-^)- r ~' 2 + -i /3 } 



+Y'l'z'(2l ,2 -3l'z'+z'*) + m l 'zV'{W-z')-Zi% 

 and 



6ET7'^-, =X/{3ZV(2r 1 , -y)-^ 1 ' 3 (3Z'-- 1 ')} 



" +Y / l / (2l" 2 -6rz' + dz' 2 )+m i '{2l , (z 1 '-z')-z 1 ' 2 }. 



Now introduce the fact that the shaft is continuous at the 



origin, i.e. l-^) = 1-^7) for OP and OP'; and write 

 I' = »I. \dz/ Q \az/ 



Then for ,: — 0, putting l~-\ ==( ,p) j we obtain 



M X/{ -z^M-Zi) } + 2^YIH' + SfjMJ'z&l-Z!) 



=x x n{-z^\?>v -z^)} + 2Ynv*+nuiz{(2i' -z?). 



Also, by taking moments about the origin, 



YZ + Yr-X^-XV + Mi + Mx'-O. 



These two equations give us Y and Y' in terms of the 

 loading. We thus find 



2Y?Z / 0*Z + Z')=XA 1 {2ZZ , .+ 3/ife 1 ^^ 1 8 } 



^X 1 'lz l '{2l' 2 -3l\: 1 '+z l ,2 }~M i r{^: l (2l-: ] ) + 2ir} 



+ M 1 7{3: 1 '(2/'-: 1 ')-^' 2 }- 

 2Y7/ /2 ^ + / / ) = X 1 /x/ , : 1 {2/ 2 -3/: 1 + : 1 2 } 



+ X^:/{2^/ / + 3/':/-:r 2 } + M 1 /'{3 / a: 1 (2/-:0--V ? } 



-M 1 7[3: 1 '(2/-: 1 ') + 2 A J/'}. 



