ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 121 



Instead of finding the frequency we shall try the next approximation by substituting 

 the last two equations into (4), as usual, 



+ .0832''-. 04 ~ +.001275 jijdz 



= •^^^^,y' (s.2H05^'x' -7.17687 Px'+i.488o9x' 



-.5y+.oo966^j. 

 Integrating twice, 



-y= •'^y,y^"(.75074^'-i-09782/^x + .687/^x''-.35884/^x^ 



x^ x^ \ 



+ .02657 x^ - .00772 — + .00007 — 1, 



hence the end amplitude, through 



11 = 10.9812^, 



and its frequency — 



» = -^aII 



_ 3-3] 



27tP \ 7 



A third approximation carried out precisely in the same manner would result in the 

 end amplitude through the equation — 



-~ = 11-063—-, 



Ji yi 



and its frequency — 



27iP y y 



where, again, the breadth of the bar does not enter. 



6. Clamped-Free Bar of Varying Depth (h = Ax), the breadth, b, being con- 

 stant. 



The exact form of the equation, says Professor Morrow, first assumed for the 

 type of vibration, is immaterial to the final result ; but, by choosing a suitable equation, 

 the labor involved in obtaining this result to any required degree of accuracy is 

 reduced. . -- 



In the present case Professor Morrow recommends, instead of (5), the form — 



/, 2X , x^\ 



