PERIOD OF VIBRATION OF STEAM VESSELS. 133 



MOTION IN A HORIZONTAL PLANE. 



If a load W be applied at the end of a cantilever by small increments, a steady 

 deflection 8 will be produced. The work done may be expressed as j/2 PV8, and this 

 is the measure of the resilience of the cantilever when deflected by an amount equal 

 to S. If a load 2W be similarly applied, the deflection will be 25, and the work done 

 will be 5^ X 2W X 2d = 2W8. This is the measure of the resilience of the can- 

 tilever when deflected by an amount equal to 25. Now let the load W be applied at 

 once at the end of the cantilever. The work done at deflection 5 will be PV 8, of 

 which YzW 8 will be stored in the cantilever as resilience, and the remaining y2W8 

 will represent the energy of motion of the weight W as it passes the point of steady 

 deflection. It will be found that when the cantilever has deflected 25, and the re- 

 silience equals 2W8 as shown above, the work done by falling of the load W is also 

 2W8, and there is no energy of motion and the weight is momentarily at rest. But 

 the weight W is not sufficient to hold the deflection at 25, and the resilience causes 

 the weight to rise. As it passes the deflection 5, the energy exerted by the canti- 

 lever is 2WI8 — y2W8 = iy2W8. The weight has been raised a distance 

 25 — 5 =5 and the work done is W8. The remaining YzlVb represents the 

 energy of motion of the weight, which, it will be noticed, is the same as in the down- 

 ward motion at the same point. It will be found that when the deflection is o, the 

 energy exerted will be 2W8, and the work done in raising the weight will be 

 2W 8 also, and the load will be again momentarily at rest. 



As has been shown previously, the period of this vibration will be the same as 

 that of a simple pendulum of length 5. The period will remain unchanged when 

 the vibration is reduced as the relation of velocity to length of travel of the weight 

 is constant. 



If the cantilever is turned on its side, and the load W is deflected by hand 

 or some independent force to a horizontal distance 5 to the right, the resilience 

 will be YiW 8 as before when the deflection was in a vertical plane. If the weight 

 be released, the energy exerted by the resilience of the beam when the weight 

 passes the neutral position will be Yz IVS, and as the weight is supposed to move 

 in a horizontal plane, this will represent the energy of motion. When the weight 

 has moved a distance 5 to the left of neutral position, this energy will have been 

 all expended, as the energy of resilience is again YIV 8, and the weight will be 

 momentarily at rest. It will be noticed that the weight moves through a distance 

 25, and has an energy of motion at mid vibration of YW8, corresponding ex- 

 actly to the motion in a vertical plane. The period in the two cases will therefore 

 be identical. 



DIPPING MOTION. 



The amount of the dipping motion of a vibrating vessel can be obtained by 

 considering that in the case of the "Rod in Tension," 



2\^ 



(32) 



