Page 187 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

174 Properties of Vibrating Systems Chap. 6

moment method/ the deflection at the various stations is equal to the moment of the
M/EI area about the position in question. For example, the value of iZ2i ^ ^12
found from Fig. 6.1-3 as follov/s:
1 11 i i
^ 2 (2 /rx 3
£7 3 El
The other values (determined as before) are

- U l L n i l
~ 3 El “21 “ ‘ 3 El
8 2.5 /3
l
"22 ~ 3 El (I'yri — C 3 El
- ill - Ill
“33 3 El “i3 “ “31 - 3 El
The flexibility matrix can now be written as

P 27 14 4
3 El 14 8 2.5
4 2.5 1
and the symmetry about the diagonal should be noted.
Example 6.1-4
The flexibility influence coefficients can be used to set up the equations of a flexible
shaft supported by a rigid bearing at one end with a force P and a moment M at the
other end, as shown in Fig. 6.1-4.

Figure 6.1-4.

The deflection and slope at the free end is
y = -f-
(6 .1-1)
6 = a2\P +
which can be expressed by the matrix equation
yi ^ r«ii «12] i P
o f - [«21 «22j\M ( 6.1-2)
The influence coefficients in this equation are
P P J _
«11 3£/- «12 «21 2E r El (6.1-3)

E. P. Popov, Introduction to Mechanics of Solids (Englewood Cliffs, NJ: Prentice-Hall, 1968),
p. 411.