128 PERIOD OF VIBRATION OF STEAM VESSELS. 



tangent to the curve at B with the horizontal axis. For relatively small deflection, 



i = A. (34) 



di d^y . ^ ^ ^. . M (^c\ 



ana -j- —-j-^= reciprocal of radius of curvature =-Ey v.)o/ 



Applying the above formulae to the cantilever in question, we find that, at any 

 point B 



M= W{l-x) 



and 



By integration, 



d^^^ _ W{l-x) 

 dx dx^ EI 



dy 

 t = 



wilx-^\ 



And by further integration. 



dx EI 



\2 6 

 ^-^ £1 



By making x = I, we find 



S = ^ (36) 



2EI ^^ ^ 



Since by (36) the deflection varies directly as the load, the accelerating force 

 when the weight is raised and released will vary directly as the distance from the 

 neutral position, and will equal the weight when the weight is raised to the level, a 

 distance = 8.. The period of vibration therefore will be given by the formula (33). 



7 = ^v^ (37) 



CANTILEVER OF UNIFORM SECTION UNIFORMLY LOADED. 



Fig. 7, Plate 59. 



Let the weight per unit length be w. The bending moment at any point B 

 will be 



M= w{l—x) I ' ^ ' 



Applying formula (35), integrating and making x = I, we find 



8EI 



Deflection of end of cantilever = — — - (38) 



