138 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



can be written simply as — 



2 ai{aij-'^(iy) =o 



The first two roots of the transcendental equation — 



tan k + tanh ^ = o 

 are — 



ki = o and ki = 2.365. 

 So that — 



2 cos /^2 cosh — ^ - cosh ^2 cos -j- 

 1^1 = const. ; 1^2"= -^ • 



^ Vcos^ ^2 + cosh^ ^2 



To the first term of the series will thus correspond a bodily displacement of the 

 bar (no bending) ; the second term will mean a certain elastic curve corresponding 

 to the fundamental tone of a free- free bar. 



If ^1 = I /y/~2^, the values of a and (3 will be as follows: an = o ; ai2 = o ; 



a22 = f (i - bo^){^P,'fdx = -3ii^ ( I - .087 ^/^) ; 

 -/ ^ 



■ /3ii = / (i-.333f/'); ft2 = . 297^/^1 i322 = /( i - . 481 c/"); 



and our maximum-minimum conditions will be : — 



«i ( ttii - ^ |8ii ) -I- ^2 ( "21 - ^ 1821 ) = o ; 



«i(ai2-^/3i2) + Ch.ia'n-'^M = o- 



In order that ai and 02 may not be identically = o, the determinant must 

 vanish; hence, since an = ai2 = o, 



The first root ( X = o) means a non-vibratory displacement; the second root, 



^ _ a22 I 



^22 . ?: 



12 



will determine the frequency-constant of the fundamental tone through — 



In Professor Timoshenko's treatise, from which this problem has been adapted, 



the following practical illustration is given (in round figures) : — 



Let the characteristics of a 5,000-ton vessel be: — 



lbs 

 2.1 = 328 ft.; /o = 2,316 ft.*; Foy= 1,437^-^ ; ft = c = 2.79 X lo-^; 



