Page 210 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.12 Unrestrained (Degenerate) Systems 197

to (f)^. However, we find that (/>, and (¡>2 are not orthogonal to 4)^2-
1 0 0 - h'

1 1) 0 2 0 = 4 ^ 0
0 0 1 1 + h

1 0 0

cf>lM<f>,2 = 0 1) 0 2 0 1 ) = 2 b ^ 0
0 0 1 1 + /? i

6.12 UNRESTRAINED (DEGENERATE) SYSTEMS
A vibrational system that is unrestrained is free to move as a rigid body as well as
vibrate. An airplane in flight or a moving train is such an unrestrained system. The
equation of motion for such a system will generally include rigid-body modes as
well as vibrational modes, and its characteristic equation will contain zero frequen­
cies corresponding to the rigid-body modes.
Example 6.12-1
Figure 6.12-1 shows a three-mass torsional system that is unrestrained to rotate freely
in bearings. Its equation of motion is
/ \
0 0 0 », ^0^
"

0 0 < ^ 2 >-t- -K, (K,+K,) - K 2 < » 2 > = < 0

0 0 0 ~K, 0

t -V ^ 2 \ -V V /
We will here assume that 7, ^ J?, ^ J ^nd = K2 = K, and let A = co^J/K, in

which case, the preceding equation reduces to
1 0 o' ■ 1 -1 ()]■ 0

-A 0 1 0 -1 2 -1 0
0 0 1 0 -1 1J ,0
The characteristic determinant for the system is
(1 - A) -1 0
-1 ( 2 - A ) -1 = 0
0 -1 (1 - A)
which when multiplied out becomes
A(1 - A)(A - 3) = 0

Figure 6.12-1.