Page 331 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

318 Introduction to the Finite Element Method Chap. 10

(a)
Í/2 l/Z
c,t

(2)
ct 4) Figure 10.5-1. Uniform cantilever
( 2) (3) beam.
Element b: Element b is the same as element a except for the displacement
vector, which is

\e.

With the global coordinates coinciding with the beam axis, the assembly of the
system matrix is simply that of superimposing the preceding matrices for elements a
and b into a 6 X 6 matrix. That is, for the stiffness matrix, we have
r _
1 1

1Element a 1
1
1 1
1 ' ^’2
<
L j 1 02
1 Element b \
1
1 _i 03
V 'V
Because = 6^ = 0 due to the constraint of the wall, the first two columns can be
ignored. Also, we are not concerned with the force and moment, Ej and A/|,
respectively, in the vibration problem. We can, therefore, strike out the first two rows
as well as the first two columns, leaving the equation
1 ■2
312 0 1 6.5/

ml 0 2/2 1 6.5/ -0.75/2 »2
>
840 54 6.5/ 1 156 -11/ i ^3

-6.5/ -0.75/2 1 /2 n
\ .^ 4
- 'L'
24 0 1 -12 3/ ^2^
ISEI\ 0 2/2 1 -31 0.5/‘ 1 < Ml
1 _
l / ■ ) -12 -31 \ 12 -31 V

'
3/ 0.5/2 1 -3 / /2 \ 0 iM,
To solve for the free vibration of the beam, the force vector is made equal to
zero and the acceleration vector is replaced by times the displacement.