38 
Proceedings of the Royal Society of Edinburgh. [Sess. 
VI. Beam Loaded at More than One Point. 
When the beam is loaded at more than one point, the nature of the 
vibration depends upon the relation of the moments of inertia of the applied 
loads and the position of these relative to the fixed ends. Suppose the 
beam AB (fig. 2) is fixed at each end and is acted upon by two loads at C 
and D. Under these circumstances, the system may vibrate in one of two 
distinct modes depending upon the relationship between I 1 and I 2 and 
between l x and l 2 . 
The lowest frequency of vibration will occur when the frequency 
amplitude and phase of vibration are the same at the points C and D. 
Under these conditions, the portion CD of the beam will simply oscillate as 
a whole about the axis of vibration, the parts AC and DB behaving as 
“ fixed-free” beams of lengths l x and l 2 respectively, and having loads at 
their free ends whose moments of inertia are J 1 and I 2 respectively. 
If the frequencies at C and D are to be equal, we have from equation 
(5) 
Vi ~ 
Also, if the amplitude at C is to be the same as the amplitude at D, 
from equation (1) 
TA = T2^2* 
If these two conditions are to hold simultaneously, then clearly r 1 = r 2 , 
i,e. the eccentricity of w 1 must equal that of w 2 . 
If the above conditions are fulfilled, and if the phase of vibration at C 
is the same as that at D, the beam will vibrate with a frequency given by 
Li 1 19L 
w 27 r \ I A 27 r\ I 2 / 2 
(?) 
If the above conditions do not hold, a node will be formed at some 
point between C and D, the position of which may be determined from the 
