Page 190 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.3 Stiffness Influence Coefficients 177

ration = 0, X2 = 1.0, and JC3 = 0 are /c,2, /C22, and /C32 in the second column.
Thus, the general rule for establishing the stiffness elements of any column is to set
the displacement corresponding to that column to unity with all other displace­
ments equal to zero and measure the forces required at each station.
Example 6.3-1
Figure 6.3-1 shows a 3-DOF system. Determine the stiffness matrix and write its
equation of motion.
A ky ^2 ^3 ^4 t
i-AVW- my m2 ^3 —VWV-|

X2 Figure 6.3-1.

Solution: Let Xj = 1.0 and X 2 = X3 = 0. The forces required at 1, 2, and 3, considering
forces to the right as positive, are

/ l = /Ci + /C2 = /C ,i
f l ^ ^ ^21

/.I ^ 0 = /C31
Repeat with X2 = 1, and Xj = X3 = 0. The forces are now

f \ ~ ~ ^2 ~ ^ \ 2
/ z = /C2 + /C3 = /C22
/3 ” “ ^3 ^32
For the last column of /c’s, let X3 = 1 and x, = x^ = 0. The forces are
/l - 0 = ^,3
/2 ^ ~^3 ^ ^23
/3 ~ ^3 + ^4 ^33
The stiffness matrix can now be written as
(^ 1 + ^ 2) -^ 2 0
K = ~^2 (^2 + ^3) -^ 3
/
0 —C3
and its equation of motion becomes
my 0 0 ' P‘l "(^1 + ^ 2) -k2 0
0 m2 0 -^ 2 (^2 + ^3) -^ 3 x, \ = I F2
0 0 m3 0 -k j {k, + k,)_
^3 1^,1
Example 6.3-2
Consider the four-story building with rigid floors shown in Fig. 6.3-2. Show diagramat-
ically the significance of the terms of the stiffness matrix.