Page 197 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

184 Properties of Vibrating Systems Chap. 6

Figure 6.5-2.

Solution: Compared to the previous Example 6.4-1, we now have an additional coordinate
^3, which results in a 4 X 4 matrix. To the three configurations of the previous
problem, we add the fourth configuration, as shown in Fig. 6.5-2. The new 4 x 4
stiffness is easily determined and is given as

' F ^ ■ 24 6 6 1 6 ■ 'vF
/2 / / I /
1
1
M, 6 8 0
E l / 2 1 1
' ' 'T 6 1 <
2 8 1 2 02
M 2 T
1
1
6
M 2 7 0 2 1 4 0,
J 1 \
which we partition by the dotted lines and relabel as
1^,2
(6.5-1)
a:2, 1K 22
Note here that is the stiffness matrix of the previous problem. Multiplying out the
new matrix, we obtain

SOI = K 2 , V + K 22& (6.5-2)
Because for the pinned end, the moment is zero, we let = 0 and solve for 0 in
terms of the other coordinates, thus reducing the size of the 4 x 4 matrix to a 3 X 3
matrix.
0 = -K22'K2,V (6.5-3)

Substituting this into the first equation, we have
d = {K,,-K,2K22'K2,)V (6.5-4)

Because the first term of this equation is that of the previous example, we need only