ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 139 



The displacement will then be — 



D = ^^ — ^-^ — 1(1- -0000279 ^^dx = about s,ioo tons. 



2240 ^o 



We first calculate the coefficients : — 



a22 = 6.46x 10-"; (811 = 123; 1812 = 36.6; (822=104.7 

 Then 



A = 6.97X io~*. 

 And, if 



E = 4.32xio«|^, 



the frequency-constant will he p = 23.5, so that the corresponding number of revo- 

 lutions will be (about) — 



27t 



This example is given, of course, only for the purpose of illustrating the work- 

 ing of the method; it involves integration by parts of the easiest kind possible. 



In actual practice neither the load curve nor the curve of moments of inertia is 

 parabolic or symmetrical. It becomes necessary, therefore, to resort to graphical 

 methods, perfectly evident, in working out the expressions of a's and /8's. It will be 

 remembered, of course, that in integrating the product of two curves it is necessary 

 to plot an additional curve, of which every ordinate is the product of the correspond- 

 ing ordinates of the two curves ; then ordinary methods can be applied, planimeter, 

 Simpson's rule, etc. 



After the a's and jS's have been found, the problem is completed exactly as 

 above. 



The values of the hyperbolic sine and cosine will be found in Smithsonian Tables 

 or in Mellor's "Higher Mathematics," where also an easy exposition is given of the 

 elements of the general theory of hyperbolic functions and their applications. 



5. Remarks. — The importance of a practical method whereby to find an approx- 

 imate value of the fundamental frequency cannot be overestimated; anyone inter- 

 ested in designing a ship, in its construction or equipment, would naturally give 

 almost anything to know, beforehand, the rotative speeds, from which to keep away, 

 in specifying or installing all machinery. 



In reviewing various causes to which forced vibrations may be due one is some- 

 what surprised to see that the effect of skin friction has been rather overlooked ; yet 

 it is not unlikely that this may start longitudinal vibrations, in much the same man- 

 ner as a violin bow in a steel rod clamped in the middle. 



Another point which is mostly overlooked is the question of balance. Dr. 

 Schlick and others explain vibrations by everything under the sun — the transverse 

 couple, the unbalanced forces or couples of the reciprocating mechanism ; the slight 

 difference of pitch of the propeller blades ; the wrong number of blades ; too little 



