x« 



120 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



Integrating twice, we have 



" / 5 



-:y = ^■2'^. I (■ 122619 /*-. 173469 /^x + .o83 x*-. 03 4" + • 000850 ^ 

 En \ LI 



Hence the ampHtude of the free end {x = o) is given by 



-v" FA' 



-21 = 4.07767^ 



In order to find a better approximation, both these equations are substituted into 

 (41) in the usual manner, with the result — 



- S = ^^1^ ^'" (•°^°436 1^ X- - .014455 /V 



x' x^" \ 



+ .001984 x" -. 000595 — + .000007 -Tj- ] 



or, integrating twice, 



-y= '^^j/°i y y^' (• 240053 /^ - . 3408 /' X + . 1 70304 /* x' 



- .072279 /^ X* + .003543 X - 000827 ~r + 000006 ~ 



whence the amplitude of the free end, by — 



v" FP 



and its frequency — 



2.063^ \E 



In other words, all clamped-free bars of which the depth is constant and the 

 breadth proportional to the distance from the free end will vibrate with a period 

 quite independent of the breadth of the fixed end (hence of the value of ^). 



5. Clamped-Free Bar of Varying Breadth (b = Ax^), the depth being constant 

 and = h. Here — 



-^=^^'r^n^-^)3'.«'^ (4u) 



ax tPx' yi-i, 



= g(.^_.8f..o;,4a8i^), 

 (by substituting the tentative curve from (5)). Integrating twice, 



" / 5 8 \ 



- y = ^2\ • 098928 /^ -. 143537 /'x+. 083 x^-. 04 ^+.001275 ^j, 



whence the expression of the end amplitude — 



-v" FP 



->^= 10.1083^. 



yi yi 



