114 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



So that, for instance, the frequency of the first harmonic will be ^ that of the 



9 

 fundamental, etc. To the fundamental tone will correspond two nodes, 0.224 / from 



each end ; the ends and the middle will, of course, be antinodes. The first harmonic 

 (or octave) will be characterized by three nodes, one in the middle and one 0.13 / 

 from each end. The next harmonic will have four nodes, etc. The ends will, of 

 course, be antinodes under all circumstances. All this applies only to the rod of con- 

 stant cross-section, and for this reason cannot be used for finding frequencies of so 

 complex a structure as a hull. 



4. Clamped-Free Bar. — The ' approximate formula for finding the gravest fre- 

 quency (per second) of such a bar is: — 



n = 9,320 r/f 



(for steel), where n, r and / denote the frequency, the radius of gyration and the 

 length (both in feet). The frequency of the next harmonic will be 6.25 times greater ; 

 that of the following harmonic will be 17.6 times greater than that of the funda- 

 mental, etc. 



Vibrating in the fundamental tone, the rod will, of course, have one node, where 

 clamped, and the free end will be antinode. In the next harmonic there will be an 

 additional node, 0.226 / from the free end. In the next harmonic there will be two 

 such nodes, one 0.132 / away from the free end and the other almost exactly at the 

 middle; etc. 



5. Supported-Supported Bar. — The fundamental frequency can be given by 

 the formula: — 



n = 26,400 r//^ 



the notations being the same as before. 



The frequency of the first harmonic is four times and of the second nine times 

 higher than that of the fundamental. 



Vibrating in the gravest node, the rod has no nodes between the ends ; to the 

 first harmonic there will correspond one node, in the middle ; the second harmonic 

 will be characterized by two nodes, dividing the length into thirds, etc. 



6. Clamped-Clamped Bar. — Its frequencies are the same as those of the free- 

 free bar. 



General remark. — The frequencies (and therefore the periods) and the nodes 

 of all these bars are found in a perfectly uniform manner from the same partial dif- 

 ferential equation: — 



— Z. J- a — - = o. 

 dt' dx' 



The methods of solution will be found in Lord Rayleigh's "Theory of Sound," or 

 in St. Venant's "L'Elasticite des Corps Solides," and it will be at once apparent that 

 the difficulty lies not in the work of finding the general solution itself, but in adjust- 

 ing the arbitrary constants to fit the special conditions of each problem, such as 

 tension, bending moment, or shear at the ends, etc. 



