Page 411 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

398 Classical Methods Chap. 12

Figure 12.7-2.
Figure 12.7-2 shows the iih section, from which the following equations can
be written:
V, - tn^oi^i y^ + c,<Pi) (12.7-1)
^ + \
i
M ,,, = M , - (12.7-2)
'i+i T, + (12.7-3)

(12.7-4)

^/+i = 2 1 7 ),. '*■^ '+ > (£ 7 ),-
i P \ ( P
y,+i = y,' + + K+i t f i + ^i+i 2 El (12.7-5)
3EI
(12.7-6)
,

P
<Pi+ i = < + T,+ ^hi
where T = the torque
h = the torsional influence coefficient = l/GIp
ip = the torsional rotation of elastic axis
For free-ended beams, we have the following boundary conditions to start
the computation:
K, = M, = r , = 0
e^ = e y\ = 1-0 <P =
\
Here again, the quantities of interest at any station are linearly related to and
and can be be expressed in the form
a + bo ^ (12.7-7)
Natural frequencies are established by the satisfaction of the boundary conditions
at the other end. Often, for symmetric beams, such as the airplane wing, only
one-half the beam need be considered. The satisfaction of the boundary conditions