Page 248 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.1 Root Solving 235

8.1 ROOT SOLVING
Figure 8.1-1 shows a 3-DOF system for which the normal modes and natural
frequencies are desired. The equation of motion for this system is
"2 3 -1 0 “ f-^i' fo]
m 1 1 ^2 / ^ -1 2 -1 X2 » = 0 .
1 0 -1 1 loj
or
‘2 3 -1 O'
-A 1 + -1 2 -1 -^9 =
1 0 -1 1
where A = oj^m/k.

The eigenvalues of the system are found from the characteristic determinant
equated to zero:
(3 - 2A) -1 0
-1 ( 2 - A ) - 1 = 0
0 - 1 ( 1 - A )
This determinant reduces to a third-degree algebraic equation. Using the method
of minors (see Appendix C) and choosing the elements of the first column as
pivots, we have
( 2 - A ) - 1 - 1 0
( 3 - 2 A ) + 1 = 0
- 1 ( 1 - A ) - 1 ( 1 - A )
and the characteristic equation becomes
A^ - 4.50A^ + 5A - 1 = 0
There is no simple equation to find the roots of this equation. However, it is
a simple matter to plot it as a function of A and find its zero crossing. There are,

however, a number of computer programs that will solve for the roots (eigenvalues)
of the polynomial equation. The procedure is quite straightforward and based on
the following idea.
By letting the A-degree equation be expressed as
/(A) = A^ + CiA"-^ + + • • • +c, = 0
'

and by assuming a number for A and substituting it into this equation, a value is
k k
2mI 2m—AAAAr- m - v w v - m

^3 Figure 8.1-1.