PERIOD OF VIBRATION OF STEAM VESSELS. 129 



This deflection is evidently not the deflection of the total weight. The deflec- 

 tion to be used in calculating the work is the deflection of the center of gravity of the 

 total weight, and that is found as follows: — 



Let wdx be the weight on length dx at any point B of the cantilever, whose 

 deflection is y. Then wdx.y will be the moment of this weight about the axis X, 



and J wydx will be the total moment about that axis. If this be divided by wl, 



the total weight, the result will be the deflection of the center of gravity of the load, 

 or 



^' wydx 



^-f 



wl 

 From this we find — 



(39) 



In this case, however, where the load is not concentrated, while wlxh repre- 

 sents the total work, including both the work done in deflecting the beam to neutral 

 position from position of no load, and also the work done in lowering the weight, 

 it does not represent the kinetic energy of the loaded cantilever, because all parts of 

 the weight have not the same velocity. The velocity of any point is proportional to 

 the deflection at that point. The virtual center of mass would be obtained as in the 

 case of a compound pendulum, by determining the square of the radius of gyration, 

 using the deflection of each point, however, instead of the distance of the point 

 from the pivot. If k^ be the square of radius of gyration so obtained, the equiva- 

 lent pendulum will have a length -j instead of h, just as in the case of the compound 



o 



pendulum. 



The period of vibration of the uniformly loaded cantilever will then be — 





vAh (41) 



where h is the deflection of the center of gravity of the weight, and 1^ is the square 

 of radius of gyration. 



The radius of gyration is obtained by the formula — 



wy^dx 



^-f 



wl ("t') 



Solving, we find 



^^=-13!^ (43) 



3240^/^ ^^^^ 



and from (40), 



k'^ \lwl'' 



h t62 EI 



(44) 



