112 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



gravity of the rod. In this connection mention may be made of an interesting prob- 

 lem recently suggested to the writer by Admiral G. E. Burd, of the Brooklyn Navy 

 Yard : — A rod 4.9 feet long, swinging about one of its ends, will have a frequency of 

 60 beats (half-oscillations) per minute. Swinging about its center of gravity, it 

 will have a frequency = o ; somewhere between these two points the frequency has 

 a maximum value. It is required to determine such a point and the maximum value 

 of frequency. This problem is capable of immediate reduction to a very easy case 

 of finding a maximum of a simple expression ; the answer is that the required point 

 is not quite 17 inches above the middle of the rod, and the great^t frequency is not 

 quite 64.5 beats (single oscilladons) per minute. 



It is well to remember, in connection with all problems of pendular motion, that 

 the period is not materially affected by damping unless the latter is excessive. 



2. Vibrating Strings. — It will be remembered that here the displacement of any 

 point is characterized rjot by the time alone, but also by the location of the point. 

 Hence, instead of an ordinary we have a partial differential equation, very well 

 worked out, 



d^y _ f^zd'^y 



of which the solution is usually given in two forms, trigonometric, or by arbitrary 

 functions (see any book on differential equations). The string can be excited by 

 either plucking or striking it with a small hammer, by bowing it, or, finally, by 

 sounding another body near it of corresponding pitch. From one viewpoint we are 

 interested only in the latter method, corresponding to the stationary mode of vibra- 

 tion, with permanent nodes (or positions of rest) and antinodes. Such a string, 

 rigidly fixed at both ends, will have its gravest tone when it vibrates as a whole; 

 besides this it can also, be made to vibrate (either by proper mechanical excitation 

 or by means of resonance) in any whole number of loops, with frequencies 2, 3, 4, 

 times that of the fundamental tone ; these tones are called overtones, or har- 

 monics, first, second, etc. (In more rigid investigation it can be shown that over- 

 tones and harmonics do not always mean exactly the same thing, owing to the rigidity 

 of the string.) The first harmonic will mean one node, in the middle; to the second 

 harmonic will correspond two nodes, dividing the string into thirds, etc. As a rule a 

 string, plucked or struck in an arbitrary manner, will not vibrate in any of such 

 normal modes, but the general vibration will consist of a mixture of these tones. But 

 in engineering investigations it is perfectly proper to consider these normal modes 

 as actually taking place separately. 



It is of interest to note, in this connection, the explanation of the hull vibra- 

 tions as proposed by Messrs. Pollard and Dudebout, in Vol. IV of their well-known 

 treatise, "Theorie du Navire" (Paris, 1894). According to their views, actual ex- 

 perience tends to show that the hull should not be considered as a vibrating rod or 

 beam, but as a vibrating string. It is known (as will be mentioned later) that in a 

 vibrating rod the frequencies do not follow the simple series 1:2:3: etc., but a 



