130 PERIOD OF VIBRATION OF STEAM VESSELS. 



Expressed as a fraction of the maximum deflection, we find 



J = .6^2yi and S = .43/1 



CANTILEVER OF UNIFORM SECTION, LOADED IRREGULARLY. 



The deflection of the center of gravity of the load would be given by the for- 

 mula 



« = >-4- (45) 



I wax 



in which w is a variable. ° 



The total work consists of two parts. One part is due to the flexure of the beam 

 against its resilience, and the other part is due to the lowering of the center of grav- 

 ity of the weight against the resistance offered by the resilience of the beam. In 

 the case in question, where the support is considered rigid, these two parts are equal. 

 The formula for the total work as set forth in books on mechanics may be written — 



u=wh = y2 fMdi ^ywh (46) 



Since di - —^rjr, formula (46) may be changed to read — 

 EI 



^-S 



' M'dx 



o Wei (47) 



in which W represents the total weight. 



This may prove more convenient for use than formula (45). 



No formula for determining k^ in terms of M direct seems practicable, so that 

 k' would be obtained by first determining y, and then using the formula — 



I w/dx 



f 



(48) 



wdx 



BEAM SUPPORTED AT ENDS. 

 Fig. 8, Plate 59. 



From the consideration that the period of vibration depends on the deflection 

 of the center of gravity of a concentrated load from the position where the weight 

 of the load is entirely unbalanced, it will be found that the period of vibration of a 

 beam resting on two points of support and loaded at the center will be the same as 

 a cantilever of half the length loaded at the end with half the load. 



