38 Mr. J. H. C. Searle on the- Effect of 



(ii.) At a pivoted (supported) end v = and M = 0. 



,-. / p = 0and (4) g'=0. . . . (11) 



(iii.) At a free end M = and 8 = 0. 



... (3)4(4) ^ = 0, and mK ^=EA K ^. 



v J v J dx- dxdi z dx° 



Thug, remembering $} = Kfj 9 



^ ; i (i2> 



and A 4 '|?-+^f =0. I 



The latter equation contains the small term X 4 -fl- due to 

 the rotatory inertia. * 



Six combinations present themselves : — 

 («) Clamped-free, 

 (A) Clamped-clamped, 



(c) Free-free, 



(d) Pivoted-clamped, 



(e) Pivoted-free, 

 (/) Pivoted-pivoted. 



III. 1. Clamped-free Bar. 

 We have, dropping suffixes, 

 ^dcos.yxf+d' siii y^+d 7 ' cosh 7 S f+d" / sinh7,e : . (13) 



Taking origin at clamped end, 



/=0 when f = 0. 



^-=0whenf=0. 

 .-. y ± d= +y 2 d'". 



.-. /=^(cos7 l f— cosli7 2 f) +rf'(sin -yjf — — sinh y 2 f h 

 At the free end, 



jh=0, when f=-=7', say, 



