122 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION, 



which, inserted in — 



gives- 



dx" EbA^x^' y^' 



d^y _ yyl ( ^ 2x .x^ 



dx^ EA" V l^'^ P 



x^ ^ x^ 



integrating twice, we have — 



-3' = J^2 ( •483^'-i-2/x + x'-.3y + .o5 



and the equation of the end amplitude {x^o) — 



yl ^EA^ 



y\ yl 



In the second approximation, carried out in the usual manner, this becomes- 



yl EA"" 



-— = 2.3i2^r^, 

 yx ll 



and in the third approximation — 



y[ EA" 



with the frequency — 



I.S3I2 A 

 271 I y y 



A IE 



so that here the period is independent of the breadth, but the influence of the depth 

 is exhibited by means of the constant A. 



7. Review. — It will thus be seen that Professor Morrow's method is not nearly 

 so difficult as it appears ; the integrations to be performed are of the most elementary 

 nature, and the method itself is perfectly uniform. The order of procedure is as 

 follows : — 



(a) Bearing in mind the obvious relations (i) and (2), form the equation 



(4). 



(b) In this substitute a tentative equation of the .form (5), satis fymg the end 



conditions, not neglecting to change x for :s, when placing (5) under the integral 

 sign {x being here not a variable but a limit of integration). 



(c) Multiply by dx and integrate ; multiply the result by dx and integrate 

 again, finding each of the constants of integration through the end conditions. 



(d) The result will be the approximate vibration curves. Make x = o, in it, 

 and the result will be an equation of the vibration of the end, of the usual kind, 

 y" = — a'y ; hence the frequency n = a/2.n. 



(£•) For the next approximation, substitute the last two equations, in (4), in 

 the following manner : — Place y, just found, for y^ under the integral sign, and the 

 value y ly = — o\ before the integral sign and perform the integration. 



