Rotatory Inertia on the Vibrations of Bars. 39 



•'• d{ - 7i 2 cos 1\ 1 '- J 2 cosh y 2 V} 



+ d'-\ -yr sin 7iZ / -7,y 2 sinh y/J =0. (14) 



•'• ^{(-^ 4 7i + 7i 3 ) sin 7i r + (-y 2 X 4 - 72 3 ) sinli y/} 



+ ^ , {(X 4 y 1 -y 1 S)cosy i r+(-y 1 X 4 -yiy 2 2 jcoshy 2 /'} = 0.(15) 



In virtue of relations (9), (14) and (15) become re- 

 spectively 



d | - cos y/ - ^ 2 cosh y 2 Z' | + d f j - sin y-Jf - ^ sinh y/ 1-0, 

 rf { sin y/_ £ sinh y/ i + d' ( -cos y/- ^cosh y/ \ =0. 



* 7s ' * 72 £ 



Eliminating d and d' between tliese, we have 



2 + y ' 4 t 7 / cos y ± r cosh y/- r yi "'~ 72 sin y 1 Z / sinh y/ = 0. 

 7i72^ 7i72 " 



Or, again using (9), 



2 + (2 + A. 4 ) cos y x V cosh y 2 V -2\ 2 sin y/ sinh y 2 Z' = 0. . (B) 



On neglecting X 2 and putting yi=y 2 = A, there results 



cos \l' cosh \Z' + 1 = 0. 



The ordinary equation obtained by neglecting the rotatory 

 inertia first given by Euler *. 



2. Correction for Rotatory Inertia in a Clamped-free Bar. 



To obtain the roots of (B) we proceed as follows ; — 

 By (9) we have 



yr^tLv/A^+^ + X 4 ], 



72 2 = i[ v /A?T4X 4 -X 4 ]. 

 Expanding these and rejecting all terms of order A 5 , 



w{i- ¥♦£}:} " 6 ' 



A itself is a small quantity but X- is not small. 



K 



* 1740 circa. 



