52 Mr. J. H. C. Searle on the Effect 0/ 



Hence y l l' = i7r . . (L) 



(45) becomes 



f=d' sin™ £ = d' sin™ w (M) 



2. Correction for Rotatory Inertia in a Pivoted-pivoted Bar. 

 :\(l + ^)in (L), 



Taking ./. \ 2 \ . 



7i 



and putting I 



K 



X 2 ' 



^ 2 ) (46). 



1 K. .„ 



iir[ 1 



Tn the fundamental i = l. 



1 -j- 



2 I 2 



V : P0=1+ c>l2 iV 



= 1 + 4*834:802 — . 



Clebsch *, in investigating the effect of stiffness on the 

 vibrations of a string, has worked oat the above case in a 

 more general form, viz., by putting on a longitudinal 

 tension. He obtains an equation corresponding to (M) above, 

 but does not enter into numerical values. It will be seen 

 that the case of a pivoted-pivoted bar is distinct from those 

 in which both ends are not pivoted. 



The foregoing represents a fairly complete investigation of 

 the effect of rotatory inertia on the vibrations of straight 

 uniform bars under various terminal conditions. 



It will be seen that the effect is small and, in practice, 

 negligible. It increases rapidly with the order of the tone, 

 but as in experiment we are concerned mainly with the 

 fundamental this is again of small importance. 



It has seemed desirable, however, to retain the rotatory 

 inertia term in the differential equation and to commence 

 approximations only when compelled to do so, i. e., in the 



* Clebsch, Theorie der Elasticitdt der fester Korper, § 61. 



