Page 192 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.4 Stiffness Matrix of Beam Elements 179

of Appendix C. The determinant of [a] is found from the minors using the first
column.
1.5 1.5 0.5 0.5 0.5 0.5
kl= ^{0.5 0.5 + 0.5
1.5 2.5 1.5 2.5 1.5 1.5
= ¥ { 1 -5 -0 .5 + 0 } = ^
For the adjoint matrix, we have (see Appendix C)
1.5 -0.5 0
adj [a -0.5 1.0 -0.5
0 -0.5 0.5
Thus, the inverse of [a] is
1.5 -0.5 0 3 - 1 O'
[a]-' = [k] = 2k -0.5 1.0 -0.5 = k - 1 2 - 1
0 -0.5 0.5 0 - 1 1
which is the stiffness matrix.
Example 6.3-4
By using the stiffness matrix developed in Example 6.3-3, determine the equation of
motion, its characteristic determinant, and the characteristic equation.
Solution: The equation of motion for the normal modes is
-2 0 01 ■ 3 - 1 O’

— (o^m 0 1 0 + k - 1 2 - 1
0 0 1 0 —1 1
from which the characteristic determinant with A = (o^m/k
( 3 - 2A) - 1 0
1 (2 - -A) - 1 =
0 - 1 (1 -A )
The characteristic equation from this determinant is
- 4.5A^ -f 5A - 1 = 0

6.4 STIFFNESS MATRIX OF BEAM ELEMENTS

Engineering structures are generally composed of beam elements. If the ends of
the elements are rigidly connected to the adjoining structure instead of being
pinned, the element will act like a beam with moments and lateral forces acting at
the ends. For the most part, the relative axial displacements will be small
compared to the lateral displacements of the beam and we will assume them to be
zero for now.
Figure 6.4-1 shows a uniform beam with arbitrary end displacements, v^,6^
and V2,02, taken in the positive sense. These displacements can be considered in