1915-16.] Torsional Vibration of Beams of Commercial Section. 41 
will be the same as that of a similar beam having a load at the free end, 
the eccentricity of which is the same as that of the uniform load, and 
having a mass moment of inertia equal to half that of the distributed load. 
VIII. Beam of Non-Uniform Section. 
If the beam is not of uniform section throughout, but is made up of 
separate sections as in fig. 4, the lengths of which and the effective 
1 
k h >! 
L ^ 
! J 
J' L 
p 
J/ 
\ 
i 
Fig. 4. 
polar moments of inertia of which are respectively l v l 2 , l 3 , etc., and 
J\, J' 2 , J' 3 , etc., the frequency of vibration of the loaded system may be 
obtained as follows : — 
Writing equation (5) in the form 
we have for a composite “ fixed-free ” beam, assuming C constant throughout, 
i.e. the moment of inertia J" of a uniform beam of given length which has 
the same frequency of vibration as the composite beam is given by 
If the section changes uniformly throughout the length of the beam, J' 
will be a function of l, and the summation in equation (13) can be carried 
out by integration. 
The significance of the above investigation in relation to the vibration 
of structures will be best understood by the consideration of a specific 
example. Thus suppose we take the case of an encastre I-section beam 
8" deep by 4" flange, and covering a span of, say, 20 feet. Suppose this 
