1915-16.] Torsional Vibration of Beams of Commercial Section. 43 
being noted. The effective polar moments of inertia of the beams were 
previously determined by measuring the angle of twist on a measured 
length corresponding to a known torque. 
The results of the tests are tabulated below, the frequency as calculated 
from the geometrical polar moment of inertia being tabulated for com- 
parison with the actual frequency of vibration. 
Table III. 
I-section beam.4-78" deep x 1‘75" flange, span 57-8". 
Moment of inertia of eccentric load in 
inch lb. units. 
37-3 
52-7 
67-2 
83-1 
Calculated frequency per minute, using 
geometrical polar moment of inertia J. 
542-5 
457-5 
406-1 
364-0 
Calculated frequency per minute, using 
actual polar moment of inertia J'. 
171-9 
145-0 
128-2 
115-5 
Observed frequency per minute 
165-2 
139-2 
123-0 
113-0 
Table IY. 
I-section beam S'Ol" deep x 3-00" flange, span 57 , 8 // . 
Moment of inertia of eccentric load in 
575 
74-1 
90-7 
107-0 
inch lb. units. 
Calculated frequency per minute, using 
366-0 
321-5 
291-2 
268-5 
geometrical polar moment of inertia J. 
Calculated frequency per minute, using 
actual polar moment of inertia J'. 
Observed frequency per minute 
181-2 
159-5 
144-3 
132-6 
176-5 
155-5 
140-5 
129-2 
Tables III and IV indicate that the observed frequency of vibration 
is in close agreement with the frequency as calculated from equation (5). 
Comparison between the actual frequency and the erroneous frequency 
as obtained by using the geometrical polar moment of inertia is instructive, 
as showing the large discrepancy that exists between these two. 
The above demonstration has been based upon the supposition that 
the relationship between torsional stress and torsional strain is a linear 
one, i.e. that stress is proportional to strain. Investigation shows, how- 
ever, that this law only holds for a small range of stresses, and that strain 
increases more rapidly than stress. Nevertheless, the above formulae 
indicate the frequency of application of a periodically applied force which 
would initiate vibrations in an eccentrically loaded beam. 
The investigation leads to the conclusion that beams of commercial 
section, when loaded non-centrally, may have a period of vibration 
commensurate with that of machinery in operation. 
{Issued separately May 8, 1916.) 
