PERIOD OF VIBRATION OF STEAM VESSELS. 131 



The period of a beam supported at the ends and uniformly loaded can be ob- 

 tained as follows: — 



Let zv be the load per foot run, and L the length of the beam. The bending 

 moment at any point B will be 



M = y2wL{y2L - x) - yiwiy^L - xf 



and 



w 



IVx^ x' 



•' 2Ei\ 8 12 



Making x ^ ^L, the elevation of the point of support above the tangent at mid 

 length (equal deflection at mid length) will be 



5^^^ (49) 



r. 



384^/ 



Applying formula (47), we have 



^o WEI \20El ^^ 



since W = J/^wL = load between mid length and end. 



This gives the deflection of the center of gravity of the load in neutral posi- 

 tion below the points of support. 



We have — 



w{y,-yfdx ^2u^ (51) 



''"^ — W^ — ^-Wp "" .00008542 



2 r 8 



|r p X .00008542 ^^. 



1= ':^U_ =:e7X-°^°25 (52) 



i2o£7 



Expressed as a fraction of the maximum deflection, we find 



-J = .']%']2y^ and I = .643;, 



Since | varies directly as the load and becomes o when the load is o, it is the 

 



deflection to be used in the formula for the period 



By comparing this result with the period for a uniformly loaded cantilever of 

 half the length, it will be found that the period of the cantilever is about 70 per cent 

 as great. 



