Page 335 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

322 Introduction to the Finite Element Method Chap. 10

Assembling these matrices, we have
(12 + R) 0 61 \ ~R 0 0" Ml
0 (12 + R) 61 \ 0 -12 61
El 6/ 61 8/2 1 0 -61 11"
(a)
/■’ - R 0 0 1 (12 + R) 0 61 Ü2
0 -12 -6 / 1 0 (12 + R) -61 ^2
0 61 2/2 ! 61 -61 8/2 _ Ô2
~(156 + N) 0 22/ 1 \N 0 o '
0 156 22/ 1 0 54 -13/
ml 111 22/ 8/2 ! 1 0 13/ -11 r2
42Ô 0 0 1 (156 + N) 0 111 (b)
0 54 13/ 1 0 (156 + N) -111
1
0 13/ -3/2 ! 111 -111 SI r2
We next note that r, = ¿s = 0, which eliminates columns 2 and 5 as well as rows 2
and 5. The equation for free vibration with N = 140 substituted then becomes
"296 22/ 1 70 0 “
^ml 111 8/2 1 1 0 -3/2 / ë,
420
IQ 0 T 296 111 ^2
1
0 -3/2 1 111 8/2_ < ^2,
■(12 + /?) 61 1 o' M l 'o'
El 6/ 8/2 ! 0 2/2 0

-|- —r- 1 < > = < (c)
/2 -R 0 (12 + /?) 61 M2 0
0 2/2 1 6/ 8/2_ k ^2 ^ .0.
Example 10.5-4
Figure 10.5-4 shows the lowest antisymmetric and the lowest symmetric modes of free
vibration for the portal frame. Determine the natural frequencies for the given
modes.

Figure 10.5-4.