PERIOD OF VIBRATION OF STEAM VESSELS. 127 



and from (28) 



V 



= -|j5,'-5^ (31) 



It will be seen that the character of the motion is exactly the same as in a simple 

 pendulum, and the period can be obtained in the same manner. The time occupied 

 by the weight in moving from A to O, with variable velocity, will be the same as 

 the time taken in traversing a quadrant of radius AO^Si at the velocity Va of the 

 weight when passing O. 



S,^g ^% 



^ K o 1^ ~ 2^'i (32) 



The conditions after passing O are the reverse of the conditions of the approach, 

 the kinetic energy being converted into work, and the same formulae hold. The time 

 of a double vibration from A to Z and back to A again will therefore be 



7'=4/=27lJ?=;^^ (33) 



o o 



By comparing this with equation (17), it will be noticed that the period is the 

 same as that of a pendulum of short swing having a length equal to 6, the total 

 extension of the rod to position of equilibrium. The period of the rod in tension 

 is not dependent on the amount of the vibration so long as the tension is not entirely 

 relieved, and so long as the extension of the rod is proportional to the force acting 

 on it. 



A further consideration of the subject will indicate that when the accelerating 

 force is directly proportional to the distance of a body from neutral position, the 

 period of vibration will vary as the square root of the distance from neutral position 

 to point where accelerating force is equal to the weight of the body. The period is 

 that of a short swing pendulum having a length equal to the above-mentioned 

 distance. 



CANTILEVER OF UNIFORM SECTION LOADED AT END. 

 Fig. 6, Plate 59. 



Let the cantilever be assumed of uniform section and without weight, of length 

 /, and loaded at the free end with weight W. Before the weight is applied, the free 

 end is assumed at A in same level line as the fixed end C. When weight is applied 

 and brought to rest, the free end will be at O, the position of neutral equilibrium, and 

 the deflection will be some length 8. Let x and y be the coordinates of any point 

 B on the cantilever when in neutral equilibrium, and let i be the angle made by a 



