ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 141 



no vibration due to these two speeds, so that the vibration could not be due to the recipro- 

 cating forces of the engine. 



The measurement of vibrations on the Minnesota was brought about by the fact that 

 considerable trouble had l>een experienced with shaft breakages on battleships of this class, 

 which it was thought were fundamentally due to hull vibrations and lack of rigidity in the 

 engine and shaft-bearing foundations. 



I was a member of a board which conducted tests both on the Minnesota and the Kansas 

 to ascertain, if possible, the causes for the shaft breakages. The results were very interest- 

 ing and will no doubt be of interest to the Society in connection with the general subject of 

 ship vibration. On both ships at engine speeds of 80 to 81 revolutions per minute there were 

 found to be marked vertical hull vibrations with a frequency of 320. As the engines were 

 triple-expansion engines with two low-pressure cylinders, that is, four pistons and cranks, the 

 vibrations' could possibly have been due to the unbalanced forces of the engines ; but as the 

 period at which the trouble was experienced did not agree with any natural period which 

 could be expected from the hull structure, an investigation was made of the possibility of tor- 

 sional vibrations in the shaft. 



The results were very interesting. At the critical speed — that is, about 80 or 81 revo- 

 lutions — there was found, by means of a torsion meter which would record the torsional 

 stress in the shaft during a complete revolution, that, when the shaft was transmitting a torque 

 corresponding to a mean maximum sheering stress of 3,000 pounds per square inch, the shaft 

 was in torsional vibration sO' that the real stress in it varied from 3,000 pounds negative to 

 12,000 pounds positive. In other words, there was a complete reversal of stress, and the 

 maximum positive stress was four times that which could be expected from the mean load 

 transmitted. This reversal stress occurred 320 times a minute and readily accounted for the 

 breaking of the shafts. 



This torsional stress, when reacting through the crank pins on the connecting rod and 

 pistons, resolved itself into bending stresses on the crank shaft. This, transmitted through 

 the engine frame and foundations, resulted in the forced hull vibrations which were, of 

 course, caused by the shaft. Instead of the shaft breakages being due to hull vibrations they 

 were due to torsional vibrations of the shaft, and the hull vibrations were a symptom, not 

 the cause of the disease. 



This torsional stress on the shaft, as I say, reacted through the crank pins and connect- 

 ing rods and resolved itself into fore-and-aft bending moments on the crank shaft. 



This, in turn, was transmitted to the hull and caused the hull to vibrate at the same 

 period as the torsional vibration of the shaft. 



The President: — Is there anyone else who has anything to add to this discussion? 



Mr. E. H. Rigg, Member: — Mr. Chairman and gentlemen, the Society is to be congratu- 

 lated on Mr. Akimoff's paper, because it is one which treats of an extremely difficult subject. 

 Most previous investigations have been confined to the balancing of the engines, that being a 

 field in which a great deal of work has been done. The Transactions of this Society and of 

 kindred societies contain much interesting literature on the subject of balancing engines, and 

 the art is thoroughly well understood at the present time. 



Commander McEntee referred to some interesting investigations; perhaps, as a matter 

 of record, I may say that the results of those or similar investigations were published in 1912, 



