90 Mr. C. Chree on Bars and 



Thus, for a free-free bar, R = when 2 = and when 

 z = l l +l 27 .-. B 1 = 0, and 



— A 2 sin ot 2 (Ji + h) + B 2 cos cc 2 (h + k) = ; 

 whence, after a simple reduction, we get 



(m 1 + ri 1 )a 1 tan (a^i) + (m 2 + n 2 )ot 2 tan (a 2 £ 2 ) =0. . (45) 

 For a fixed-free bar, w = when e=0, and R=0 when 

 z = li + l 2 , and we get 



tan( ai 4)tan(« 2 y-^±^^ = 0. . . (46) 



"*2 ~> "2 a 2 



For a fixed-fixed bar, w — when £ = and when z = li + l 2 , 

 whence we get 



(m 2 + n 2 )u 2 tan (a^) + (m x + raj oc 1 tan (a 2 £ 2 ) =0. . (47) 



we see 



Remembering that a 1 = k\ / ~ and a, 2 = k* / ^~, 



these are the equations giving the frequency in the three 

 standard forms of longitudinal vibration. Putting m 1 = m 2 

 &c, we deduce the ordinary results for a uniform bar. 



We would specially remark that from (45) and (47) it 

 follows that the frequency of the corresponding vibrations in 

 the free-free and fixed-fixed conditions are no longer the 

 same ; while from (46) it follows that the frequency of the 

 fixed-free vibrations depends on which end is fixed. Thus, if 

 a fixed- free bar be reversed, the pitch is altered. This might 

 prove a delicate test of the uniformity of material in a bar of 

 strictly uniform cross section. 



From (46) we see that if (m 1 + n 1 )a 1 < (?n 2 + n 2 ) a 2j then 



cos (»& + *&) >0, and .-. *[^/j£ + l *\/ ]f\ < p 



while if (nil + ^i)a x > (m 2 + n 2 )cc 2y then 



*[VI+n/l>f- 



Thus the arrangement giving the highest pitch consists in 

 fixing that end for which (m + n)\/ ^ is greatest. Thus, if 



the materials differ only in density, the pitch is highest when 



n 



the densest material is at the fixed end. Since M = — (3m— n), 



m \ n 



we may infer, as a general rule, that if the two materials have 

 the same density, the highest pitch is obtained when the 

 material which possesses the largest elastic constants, or 

 greatest elasticity, is at the fixed end. 



