﻿242 Dr. J. Morrow on the Lateral Vibration of Bars 



Section VI. Loaded Massless Bar Clamped at each End. 

 § II. Here for x<a, 



-■Blg=sPy-«y.f« + M 1 +(M,-M 1 )*, 



where M x and M 2 are the terminal couples. 

 This may be written 



dhi ., . , 

 —x- -\- n" y = lix — k. 

 dx 2, J 



where 2 _ Z_ ; _ m h>> \ M 2 — M t _ M x 



n ~Er mi~~ mi ' c ~m' 



The solution is 



y=— z (hx — h) + D sin na? — C cos nx. . . . (19) 

 The conditions at the origin give 



C=-4, D=~4- 



For A>a we must add my a (x—a) to the right-hand side 

 of (19). Then, if 



_m^a (M 2 -M 2 ) , ■my ff a~M 1 



~ EI I "" E1Z ~'~ ' S ~ EI ' 



the solution is 



h' (s + K'l . 7 hf , } . 



y=—w— { s— sm n/ + — o cos m > sm n&' 



/A' . 7 5 + A7 A s , rt/v , 



+ -^ sm nZ =— cos w/ } cos wa' -f -9 • • (20 ) 



\n s w J rr ' 



Equating the two values of -j- and of y a we get, 



after some reductions, 



M l = my a u M 2 — M x = my a y. 



Where 



a -f b cos nZ — Z cos nb H (sin 726 — sin wZ + sin na) 



n v ' 



2(1 — cos ?i/) — raZ sin nl 



and 



_ (Z>— a) (1 — cos n l)-\-l (cos nb — cos rca) 



' 2(1 — cos nl) — nl sin rcZ 



