1915-16.] Torsional Vibration of Beams of Commercial Section. 39 
fact that when the system is vibrating the frequency of vibration at C must 
be the same as the frequency at D. 
Let the node be at point E between C and D and distant a from 
point C. Then, since no motion ensues at E, the portions AE and BE of the 
beam will behave as “fixed-fixed” beams of length and — 
loaded at C and D respectively. The position of the node is determined 
by equating the frequencies of these two “ fixed-fixed ” beams. 
The frequency of the beam AE is given by 
n = 
or 
V ‘‘U+A 1 
and that of the beam BE by 
n = 
CJ' 
Equating these we have 
T C3 a ) I 
J al l 
1 (« + / 1 ) 2 (l 3 + l 2 -a) 
= Io 
k(h ~ a ) 
(8) 
(9) 
(10) 
Equation (10) has two real roots corresponding to two possible positions 
of the node, giving in general a higher and a lower frequency of vibration. 
Fig. 3. 
The value of a as obtained from equation (10), when substituted either in 
equation (8) or equation (9), gives the frequency of vibration of the beam. 
When a beam is loaded at more than two points the motion becomes 
more and more complex as the number of loads increases. 
If the beam is loaded at three points C, D, and E, as in fig. 3, a 
possible mode of vibration is that in which nodes are formed at F and G, 
under which conditions the three portions AF, FG, and GB of the beam 
behave as “ fixed-fixed ” beams. Expressing the condition that the 
frequency is the same at C, D, and E, we have 
L 
Z, 
(h 
x) = u 
\X + y) + y) 
h 
(h - v) 
(ii) 
^1 + ^2-*)' 
which gives us two equations for the determination of the unknowns x 
