Page 185 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

172 Properties of Vibrating Systems Chap. 6

6.1 FLEXIBILITY INFLUENCE COEFFICIENTS
The flexibility matrix written in terms of its coefficients a^j is
^12 « 13 '
«21 ( 6.1-1)
x A = ^22 ^23 r
x ^ j «31 ^32 ^33 1/3
The flexibility influence coefficient a^j is defined as the displacement at i due
to a unit force applied at j with all other forces equal to zero. Thus, the first
column of the foregoing matrix represents the displacements corresponding to
/, = 1 and /2 = /3 = 0. The second column is equal to the displacements for
/2 = 1 and /j = /3 = 0, and so on.

Example 6.1-1
Determine the flexibility matrix for the three-spring system of Fig. 6.1-1.
Solution: By applying a unit force ]\ = 1 at (1) with /2 = = 0, the displacements, X|,
^2, and X3, are found for the first column of the flexibility matrix
l/ki 0 o' f/, = l
l/kt 0 0
l/k, 0 0
0
Here springs k2 and are unstretched and are displaced equally with station (1).
Next, apply forces /j = 0, /2 = 1, and = 0 to obtain
“
i \
0 ^ 0
1
•^2 > == -1— - f
L )
1 1
X3 0 7— + T- 0

V )
In this case, the unit force is transmitted through k^ and ^2, and k^ is unstretched.
In a similar manner, for /j = 0, f 2 = 0, and /3 = 1, we have
0 0

0 1 1
^2 0 1- + -T-

0 0
V

I—VNM^ (1) (2) ^AAA/— (3)

I— > 3 Figure 6.1-1.