ON VIBRATIONS OF HEAMS OF VARIABLE CROSS-SECTION. 113 



much more complicated law. Yet Messrs. Pollard and Dudebout quote one authority 

 who reports that, in a certain destroyer, the speeds at which the hull felt most sensi- 

 tive were at 120, 240 and 360 revolutions per minute. They also quote Mr. Yarrow, 

 who noticed that in a small boat such critical speeds were 200, 40c, 600 and 800 rev- 

 olutions per minute; while at 300, 500 and 700 revolutions per minute all vibra- 

 tions ceased. Hence their conclusion that the theory of vibrating strings can be 

 made to answer the required problem of finding such critical speeds, or at least the 

 fundamental tone. They proposed the formula — 



4/ ^ 1.026 \I Jm \w/ 



vibrations per second, where — 



/ = length of the hull. 

 PF = the total volume of the hull. 

 w =the displacement volume of the hull (unloaded). 



( — r ) = the average value, on half-length of ship, of the quotient ° _ . ^• 



\ I / m moment of mertia 



H=tht extreme total height of hull considered as beam; all units are metric. 



Without the slightest desire to disparage the authority of the great teachers in 

 question, the writer regrets being quite unable to agree with such analysis, especially 

 as regards the very premises from which they started. A hull is a beam, not a string, 

 no matter how analyzed; and being a beam of variable cross-section, it is naturally 

 quite difficult of handling mathematically, as will be shown later in this paper. But, 

 and here lies the most important point, all the experiments so far quoted were made 

 in connection with engine-driven ships, years before the advent of the steam turbine. 

 Now the action of a reciprocating engine is in itself very rich in harmonics, which 

 follow the series 1:2:3: etc. ; and where the third harmonic is distinctly felt, as is 

 well known from automobile practice, for instance. So that, after all, it is not im- 

 possible that some of the vibrations were felt simply through the harmonics of the 

 reciprocating mechanism coming into step with this or that harmonic of the free 

 vibration of the hull. In order to justify the vibrating string theory, it would be 

 necessary to provide a large, variable speed electric motor, with a purposely unbal- 

 anced rotor, and fed from the shore. Such a motor could be placed in various posi- 

 tions and various effects could be properly recorded. But, as a matter of specula- 

 tion, the variable-section beam theoi-y is much more satisfactory, as the writer hopes 

 to show later on. 



3. Free-Free Bar. — The gravest frequency (of transverse vibration) can be 

 found from the following rough formula (steel) : — 



n =- 60,000 r/f, 



where n is the frequency per second, r the radius of gyration, and / the length, both 

 in feet. The frequencies of higher harmonics will be found from the series, 



3^ : 5^ : 7== : Q'' : . . . . ' 



