Page 333 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

320 Introduction to the Finite Element Method Chap. 10

i/2 t/2
Figure 10.5-2. Two-element dis-
mi ml erete mass model of a uniform ean-
2 4 etilever beam.
Thus, the original 4 x 4 equation has been reduced to a 2 X 2 equation, the final
form being
m^ 0
16 - 5 ‘
0 m^ \ r / 1 7/-’ f - 5 2
An acceptable discrete mass distribution is one in which the mass of each
element is divided into half at each end of the clement. Thus, if the total mass of the
uniform beam of length / is ml, the mass of each element is m l / 2 and = 2{ml/4)
= m l / 2 and m^ = m l / 4, as shown in Fig. 10.5-2.
The equation of motion and solution then becomes
2 0 16 ~ 5 ] '

—A A 1
0 1 - 5 2j
where
rml 11- . 7 f ml^
A = -
4 48 T7 192 [ El
A, = 0 . 3 6 3 2 o;, = 3.516 exact value = 3.516
A . = 9 .6 3 7 ia. = 22.033 exact value = 22.034
0.327 1.5271
^1 = 1.000 ^2 = 1.000
Example 10.5-3
Determine the equation of free vibration of the portal frame with identical elements.
Solution: By labeling the joints as shown in Fig. 10.5-3, the stiffness and mass for each
element arc available from Eqs. (10.4-7) and (10.4-8). Because joints 0 and 3 have
zero displacements, we write only the terms corresponding to joints 1 and 2.

Element 0-1, a = 90°, c = 0, x = 1:
1 “ 0 - 6 / “
1 () - R 0
1 6/ 0 2 / “
k = ^ ,3 1
'^0-1
1 12 0 6/ î'E
1 0 R 0
1 6 / 0 4 l \
-
5 4 0 1 3 / '
0 0
ml - 1 3 / 0 - 3 / “
420
156 0 2 2 /
0 0 0
2 2 / 0 4 / “