Loaded Shaft supported in Three Bearings. 643 



§ 8. We have now got general expressions for the de- 

 flexions and slopes at any point of each span due to the 

 system of loads. In the problem of whirling speeds the 

 loads are due to centrifugal action. j 



Thus X = mw 2 H ; and M= (L — LW-t-, as will be shown 



. dz o du 



later, or approximately for a disk-shaped load M=m^V-j-, 



where mk 2 is the moment of inertia of the load about a 

 diameter perpendicular to the plane of bending. 



Substitute these values of X and M in the general equations 

 for deflexion and slope, and put z s (and zj) in turn equal to 

 the co-ordinate of each of the load positions. We thus 

 obtain a series of equations of the form 



cf)u 1 = a Y u x -f a 2 u 2 -t- . . . + ttyt'i + . . . + 



-'(a^'Si— '©,+•■• 



<j>u 2 — biit-i + b 2 u 2 + , . . + Mi + ■ • • + 



v© 1+ v(a-K.. + ^) i+ ... 



etc., etc., etc., 



<t>Vl = flUl +/ 2 ^2+ • • • +frVl + . . . + 



nih^(S), + --- + A&\ 



+ 

 i 



+ 



' \az /i 

 etc., etc., etc.. 



4>[j;J = h^i + k^2 + •-..+ l r vi + . . . + 



^(£\^ , (5).+"-+v(S) 



etc., etc., etc., 



* w?) ~^ lMl +p* u i + ■ • • +A v i + ••• + 



, /du\ ,(du\ ,(dv\ 



etc. j etc., etc., 



where a, £>, /, etc., are known or calculable constants, and <fi 

 is proportional to 



