ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 119 



The value of G is found from the consideration that, since the total force on the 

 bar vanishes, we have — 



ay^Cydx = o (8) 



In other words, 



r (>-! - — Gl'x + ^ GPx'- 3 Glx''+ Gx') dx = o 



'^o 2 2 



so that — 



^ _ 28 J, 



therefore the equation assumed for 3; becomes — 



, 14 X 70 x*" o x^ 28 x^ 



Substituting this for y in equation (2), 



dx^ EJ y, 

 we have 



d'^y ay " C'r . ^4 ■s' , 7° ^^ ^o s'^ 28 s'^X . s , 





^^3' .5^'-. 7 — + . 7:^-.6— -.16 — 



Integrating twice and evaluating the second constant by reference to (8) we have — 



-y = ^y y[ ( . 001993 /" - . 00925 P x^-. 0416 ^r" - . 038 ~ 

 hj I 



x^ x^ x'^X 



+ • 0138-^ - . 00925 — + . 00185 -^ J 



whence the frequency — 



22.4 r IB 



which is very near the value given in Chapter I. 



In this paper we are especially interested in beams of variable cross-section. For 

 this reason the following three examples, worked out by Professor Morrow, should 

 be studied with greatest care: — 



4. Clafnped-Free Bar of Varying Breadth (h^Ax). 



Let the depth h be constant. Our equation (4) will then become 



A A3 



or, by virtue of (5) and considering that a^= Ahz; also that / 



d^y _2yyl /„2_^ 

 dx' Eh' ' 



/ 21 /V' 



12 



