Rotatory Inertia on the Vibrations of Bars. 43 



2. Correction for Rotatory Inertia in a Clamped-cIamped ] JSar. 

 Substituting in (C), 



7l =\[i-f|], 72 =x[i-V| 



and calling, as before, \— ^z\l' = n, we have 

 1 — cos n 1 1 + — ) cosh n ( 1 — — J — -~ sin n sinh n— 0. 



if *•" ■ I f i *•" • i 1 V • • T, r> 



1 — < cos n — n-j sin n > \ cosh >z — n — smh » V — -x sm n Sinn n = U, 



1 — cos?z cosh n + n-r- (cos ?z sinh ?i + sin n cosh h) — ^-sin nsmhn = 0. 



Put »=^o + 7 7> 



where v? is now a root of 



cos n cosh n =l, (25) 



and 7) is small. 



1— (cos n — 7) sin ?? )(cosh n -\-r) sinh?? ) 



+ j X 2 (cos n sinh n + sin rc cosh n Q ) — - sin n sinh rc o = 0» 



Therefore, remembering 1— cos n cosh « = 0, 



77(003 ?? sinh 7? — sin rc cosh n ) 



n X 2 



= -°X 2 (cos ?? sinh >? + sin rc cosh w ) — - sin » sinh rc =(X 



"o - r cos n o snin "o + s i n n o cosn "o~~ ~ sin n sinh ;? | 



•'• ^ = 4 X | ; : ^ 



L_ cos n sinh n — sin n cosh n - 1 



o 

 ?? 2 1 tan n + tanh ;? G — — tan n tanh » 



- ~4 - ^ ; 



L. tan ?? — tanh n Q — ' 



or, transforming by means of (25), 



"o 9* 2 f L9^n 2 . n n ~] 



or "o o« 2 r . ,»„ . 2 l <>A 



-4°V ? ltan^'-r--tan F »), 



according as >? lies in the 1st or 4th quadrant. 



