146 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



1. That the axis of the bar, when without stress, is straight. 



2. That a cross-section of the bar is not sensibly altered in its internal arrangement in 

 the bending. 



3. That the bending consists of a rotation of each cross-section around an axis passing 

 through its center of gravity. 



4. That the kinetic energy due to the rotation of the material points of each cross-sec- 

 tion around the above axis may be neglected. 



5. That the potential energy of shear is negligible. 



6. That the deflections are very small. 



In the application to vibration of ships, the following of the above conditions are surely 

 not verified. 



1. The axis, being the line joining the centers of gravity of the cross-sections, is not 

 straight. 



2. In a beam, consisting of cross-sections as we have in a ship, shear must play an im- 

 portant part, neither can it be assumed that the cross-sections remain plane without sensible 

 alteration of their figure. 



The above statements are made for the single purpose that we should not expect too 

 much of an agreement between calculation and actual results. But even so, it surely is a 

 remarkable achievement to be able tO' determine the periods of free vibration of as compli- 

 cated a structure as a ship if only with a fair degree of accuracy. 



As to the propelling machinery for ship of average size, the contending systems to-day 

 are the steam turbine and the Diesel engine, the latter in the writer's opinion a sure winner 

 in the near future. 



Thanks to Mr. Akimoff's paper and enterprise, turbine rotors are to-day balanced stati- 

 cally as well as dynamically. While therefore initially we start off with ideal conditions, it 

 nevertheless remains true that wear of the turbine blades takes place and so gradually de- 

 stroys the perfect balance. With Diesel engines, on the other hand, periodic impulses to the 

 ship's hull may be minimized with good design, but cannot altogether be avoided, no matter 

 what we do. 



In an unbalanced turbine rotor the horizontal and vertical components of the forces set 

 up by its rotation are equal in magnitude, and each of them is capable to excite vibrations. 

 With this type of propulsion, therefore, it is necessary to ascertain the periods of the free 

 transverse vibrations of the hull in the horizontal as well as in the vertical plane. With 

 Diesel engines the horizontal component of the inertia forces is small and can be taken care 

 of, and we are therefore mainly interested in the period of the vibrations in the vertical plane. 



The President : — I think the thanks of the Society are due to Mr. Akimoff for his 

 paper, and if he will accept the same without a formal motion, we will be glad to tender our 

 thanks to him. 



Rear Admiral Francis T. Bowles, Past President : — I trust, Mr. President, that you 

 and the members of the Society will pardon a momentary interruption of the regular proceed- 

 ings. I have received to-day a copy of a report made by the Shipping Committee to the National 

 Foreign Trade Council on November 8. This report is signed by Mr. James A. Farrell, Mr. 

 P. A. S. Franklin, and Mr. Robert Dollar, and it presents so clearly and so thoroughly the 

 great problem facing this country in the utilization of the 11,000,000 gross tons of merchant 



