ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 143 



steaming along at a speed of about 15 knots. The minute the blade dropped off a tremendous 

 vibration was set up, and, judging from the difficulty experienced in standing perfectly rigid 

 on the bridge, the amplitude of the vibration was at least one inch. I also found, from prac- 

 tical experience, that the vibration is always in the vertical plane and never in the horizontal 

 plane, no matter what the cause of this vibration may be. 



I have noticed, in some of our cruisers, that the reciprocating generators in action will 

 cause considerable vibration while the ship is dead in water. It seems that at certain times the 

 vibration of the machine itself would get into synchronism with the vibration of the ship 

 and accelerate it to such an extent that it would be felt in the after end of the ship and in the 

 forward end, but would not be very noticeable amidships. This is the case in all ships whether 

 steaming or at anchor, that is to say, the vibration is excessive at the extreme ends, but 

 very moderate amidships. 



I can personally recall hundreds of instances where the above has occurred on board ship 

 during my career in the service. 



Prof. S. E. Slocum (Communicated) : — Mr. Akimoff's article on "Vibrations of Beams 

 of Variable Cross-Section" affords an interesting summary of the methods applicable to prob- 

 lems of this type. There is one general method, however, which does not seem tO' be men- 

 tioned, but which is of such importance that it should certainly be included in the summary, 

 namely, the application of Fourier's integral or Fourier's series. 



The method generally followed in mathematical physics is to state the physical condi- 

 tions underlying a given problem in the form of a partial differential equation. Such a differ- 

 ential equation is of course simply a symbolic statement of the problem and can usually be 

 written down without difficulty. Its importance lies in the fact that when once formulated 

 it presents the conditions governing the problem in a functional form and therefore involves 

 implicitly not only the known facts from which it was obtained but also the unknown cir- 

 cumstances attending the phenomena. 



It is usually possible to find by trial, or otherwise, a set of particular solutions satisfy- 

 ing the given differential equation. From these there may be selected a set which also satisfy 

 the given boundary conditions of the problem. In the case of a vibrating rod, these boundary 

 conditions would be the end conditions of the rod as determined by its restraints, etc. 



Now it is evidently possible to formi a series of terms from these particular solutions 

 such that this series will satisfy the original differential equation and also the boundary con- 

 ditions of the problem. Also, as in the case of Ritz's method, the more terms this series in- 

 cludes the more nearly will the solution it represents approach the general solution of the 

 problem. An infinite series of such terms, provided it is convergent, can thus be made to 

 define a function which is the required general solution of the original differential equation. 

 Now Fourier's integral is an expedient for determining the function defined by such an in- 

 finite series of terms, and therefore affords a method for determining the function represent- 

 ing the general solution of the given partial differential equation. The use of Fourier's in- 

 tegral, which is a limiting form of Fourier's series, thus provides a powerful general method 

 which has a wide application to problems of the type here considered. 



Lieut. W. L. Kraemer, U. S. N. (Commimicated) : — In reading over an advance copy 

 of the paper on "Vibrations of Beams of Variable Cross-Sections," by Mr. N. W. Akimoff, 

 a few points that might be relative to the discussion are mentioned, that is, that with a ship 



