Page 156 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 156
Sec. 5.4 Forced Harmonic Vibration 143
5.4 FORCED HARMONIC VIBRATION
Consider here a system excited by a harmonic force F, sin cot expressed by the
matrix equation
m, 0 ■ l i l t ^11 ^12
0 m2 ^ 2 1 ^22 X- 0 sm cot (5.4-1)
Because the system is undamped, the solution can be assumed as
^X 1^ i X t
sm cot
Substituting this into the differential equation, we obtain
(^11 m^co 1
(5.4-2)
( ^ 22 2 ^ )
or, in simpler notation.
[Z(o.)]
2T,
Premultiplying by [Z(co)] \ we obtain (see Appendix C)
X ) IF \
(5.4-3)
X 0 |Z(o>)|
By referring to Eq. (5.4-2), the determinant |Z(w)| can be expressed as
|Z(w)| = m|/?i2(a)| — —(o^) (5.4-4)
where oij and W2 ihe normal mode frequencies. Thus, Eq. (5.4-3) becomes
X, (A:22 “ m2(o^) -A, F,
(5.4-5)
|Z(< -A . (All -
or
(A22 -
=
2(ca(-a,^)(o.^-a,^)
(5.4-6)
- k n F,
X2 =
