140 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



free space between propeller tips and the hull ; in fact, everything but plain static 

 and dynamic unbalance. The latter may be considerable, and it is not clear why one 

 should look for any other causes before this, most immediate, source of trouble, so 

 easy to detect and to- remove, has been looked into. 



DISCUSSION. 



The President: — This paper, No. 8, entitled "On Vibrations of Beams of Variable 

 Cross-Section," is now open for discussion. Is there any member who desires to say any- 

 thing in connection with this subject? 



Commander William McEntee, C. C, U. S. N., Member: — As I received a copy of 

 this paper only two days ago, I have not been able to give it the careful study which it merits. 

 From experience I have had in measuring vibrations on naval vessels I can heartily endorse 

 the statement made by the author on page 113 that the hull of a ship vibrates as a beam 

 and not as a string. 



In connection with the general subject, I would like tO' inquire of the author whether 

 he has read the valuable papers on the subject of ship vibrations by Dr. Giimbel published 

 in the Jahrbuch der Schiffbautechnischen Gesellschaft in 1902 and published in the Transac- 

 tions of the Institution of Naval Architects in 1912. In those papers Dr. Giimbel gave two 

 very valuable rules connecting the frequency of vibrations and the lengths of similar ships. 

 These are as follows: — (1) The frequency of the natural periods of vibration for similar 

 ships are inversely propoTtional to the linear dimensions. (2) The product of the frequency 

 and the length of the ship are constant for similar vessels. Dr. Giimbel states that, in the 

 measurements made by Schlick on the German cruiser Vineta, the number of first order ver- 

 tical vibrations per minute multiplied by the length of the ship in feet was 45,300. 



In measurements made on the battleships Kansas and Arizona, I found on the Kansas the 

 first period vertical vibration had a frequency of 101. As the ship's length between perpen- 

 diculars is 450 feet, the product is 45,450. On the Arizona, the length of which is 600 feet 

 between perpendiculars, the first-period vertical vibration had a frequency of 76, the product 

 being 45,600. This agreement, with Schlick's results on the Vineta, is remarkably close. 



For second-period vertical vibrations I found on the Kansas, and on her sister ship, the 

 Minnesota, the frequency to be 220. The Minnesota was fitted with a four-bladed propeller^ 

 The second-order vertical hull vibrations occurred when the engines were making 55 revolu- 

 tions per minute, whereas on the Kansas, which was fitted with three-bladed propellers, the 

 same vibrations appeared when the engines were making 73 revolutions per minute, thus in- 

 dicating clearly that the vibrations were due to the propeller blades. 



These vibrations were not due to the unbalanced forces of the engines themselves, for 

 when the Minnesota's engines were run with propeller uncoupled, no vibrations appeared 

 either at 55 or at 73 revolutions. It is interesting to note that the ratio between the first 

 and the second period vibrations on the Minnesota was 2.2. This may be compared with the 

 figures given by the author for the vibrations of a free bar of uniform cross-section. 



We disconnected the propellers in the Minnesota to run the engines uncoupled, and found 



