Page 145 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

132 Systems with Two or More Degrees of Freedom Chap. 5

which are satisfied for any /I, and A2 if the determinant of the above equations is
zero.
(2k — co^m) —k
= 0 (5.1-4)
—k (2k —2ci)^m)

.
Letting 0)^ = X and multiplying out, the foregoing determinant results in a second-
degree algebraic equation that is called the characteristic equation.
A^_(3A]A + | i A ] L o (5.1-5)
\ m ) 2 \ mI
The two roots A, and A2 of this equation are the eigenvalues of the system:

A| j 2 2 ' r \ / 3 ] — = 0.634- }
j m
.
A ,= i | + 1 ^ ) A = 2 .3 6 6 ^ (5.1-6)
\2 2 j m m
and the natural frequencies of the system are

'0.634-^
‘>^1 = = \A m
o>. = \ ' f - = ^ J f 3 ^66^
.
3
m
From Eq. (5.1-3), two expressions for the ratio of the amplitudes are found:
A. k 2k —2(o^m

(5.1-7)
^2 2k —
Substitution of the natural frequencies in either of these equations leads to the ratio
of the amplitudes. For = 0.634A/m, we obtain
1 = 0.731

'2k —(o]m 0.634
which is the amplitude ratio corresponding to the first natural frequency.
Similarly, using to? = 2.366/c/m, we obtain
1
= -2.73
2k —(o\m 2 - 2.366

for the amplitude ratio correspionding to the second natural frequency. Equation
(5.1-7) enables us to find only the ratio of the amplitudes and not their absolute
values, which are arbitrary.
If one of the amplitudes is ehosen equal to 1 or any other number, we say that

the amplitude ratio is normalized to that number. The normalized amplitude ratio is
then ealled the normal mode and is designated by (fyfx).
The two normal modes of this example, which we can now call eigenvectors, are