144 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



traveling at speed the depth of the water in which the vessel is running has a distinct bearing 

 on its trim. A ship running at speed in very shallow water will drag down at the stern more 

 than when running at the same speed in a greater depth. 



Also, in most previous cases of observation of vibration, the period of vibration has* been 

 observed while the ship was made fast to a dock with the machinery running, as in the case 

 of a dock trial. 



Would not the method of securing the ship to the dock, and also' the position of the 

 helm, have a distinct bearing on the stresses existing in the vessel, and therefore a bearing on 

 the period of vibration? 



These two observations are submitted merely with a view that, in the discussion of this 

 paper, they may have some bearing on the subject. 



Rear Admiral C. W. Dyson, U. S. Navy (Communicated) : — I have read through 

 the paper on "Vibration of Beams of Variable Cross-Sections," prepared by Mr. Akimofif, 

 but am sorry to say that my time is so fully occupied that I have not been able to make such a 

 study of the paper as its character justifies. 



I can say, however, that for some time back we have been balancing turbines, and in 

 some cases propellers, for our vessels, following in general the method pursued by Mr. 

 Akimoff and at times employing Mr. Akimofif to conduct the balancing test. In all vessels 

 where this balancing has been accomplished and the vessels have been tried in setvice, the 

 lack of vibration experienced has been very noticeable as compared with previous vessels. 



Mr. F. Hymans (Communicated) : — In the science of dynamics there is no more fas- 

 cinating subject, nor one of greater importance for the engineer, than that of the motion of 

 elastic systems. According thereto, an elastic system moves as if it were perfectly rigid, 

 superposed by a motion of its constituent parts relative to each other. 



Take for example a wheel rolling on a rail. Each of its material points tends to describe 

 cycloidal paths, with velocities, which within each revolution continually change in magnitude 

 as well as in direction. These changes are only then possible when each material point is 

 subject to certain forces exerted upon it by the surrounding matter, and we recognize that, 

 while the body rolls, the stresses in each of its material points must vary to correspond to 

 the motion impressed upon it. 



But a stress is always accompanied by a deformation, and hence, while a wheel rolls, 

 there occurs a continual change of figure of the wheel as a whole, which we call a vibration. 



How many who have looked upon a rolling wheel have been alive to the fact that there 

 is internally a continuous process of readjustment going on, in the endeavor of the wheel 

 actually to roll? 



And so it is with most machinery in motion. Superficially the appearance is that of 

 equilibrium, yet in reality equilibrium is seldom the fact; and upon closer investigation we 

 see that ever-changing conditions occur which set up internal stresses and become manifest 

 in vibrations if these changes occur rapidly enough. 



In the majority of cases, as in the example of the rolling wheel, the vibrations in ma- 

 chines are set up by inertia forces ; that is to say, they are due to certain motions impressed 

 on its material points. If so, the forces corresponding to the impressed motions pass within 

 given intervals through one and the same cycle, and are therefore periodic. Now we know 

 that if the frequency of these forces is the same as the frequency of the free vibration of the 

 system, the vibrations set up by them will be extremely severe. 



