42 
Proceedings of the Royal Society of Edinburgh. [Sess. 
beam is subjected to a load of 300 lbs. at its middle point. If the load 
is central, the natural frequency of tranverse vibration is given by the 
expression * 
_ 13^84 /Eg 
1 2?r V wl*' 
Assuming a value for E = 30'0 x 10 6 inch lb. units, this gives 
n Y = 27*6 per second 
= 1656 per minute. 
If the load has an eccentricity equal to r, besides the above transverse 
vibration, the system will have a natural frequency of torsional vibration 
the magnitude of which is obtained from equation (5), making the necessary 
correction for a “ fixed-fixed ” beam, viz. 
1 /CJV 
^ 7j- \ wrH ' 
A 2 
The value of J' as obtained from Table II is ^r, which, for the 
60 
dimensions of a standard commercial 8" x 4" I section, is equal to *468 
inch 4 units. 
Hence, assuming a value for C = 12 x 10 6 inch lb. units we obtain 
55-0 , 
n 2 = per second 
r 
3300 
= per minute. 
r 
This shows that the frequency of torsional vibration diminishes as r 
increases. When the eccentricity is 2 - 0", the frequency is approximately 
the same as that of the transverse vibrations. 
Values of n 2 for various values of r are tabulated below. 
Eccentricity r in inches . 
2" 
4" 
6" 
8" 
10" 
12" 
Frequency in vibrations per 
minute. 
1650-0 
825-0 
550-0 
412-5 
330-0 
275-0 
The above table shows that for quite moderate eccentricities the 
natural frequency of torsional vibration is considerably greater than the 
frequency of transverse vibration for the same load applied centrally. 
In order to verify the above formulae, experiments were carried out 
on two eccentrically loaded I-section beams. These were supported 
between the centres of a six-foot lathe, one end being secured in the chuck, 
while the eccentric load was applied at the free end. The system was 
set vibrating, the number of vibrations over a measured interval of time 
* See Morley, Strength of Materials , p. 398. 
