5.8 Free Vibration of Damped Two Degrees of Freedom System
175
The roots of Eq. (a) are
λ 11 = − 1.309 + 0.951 i
λ 12 = −1.309 − 0.951 i
(d)
λ 21 = − 0.191 + 0.588 i
λ 22 = − 0.191 − 0.588 i
Therefore,
A 11
B 11
= γ 11 =
λ 11 + 1
λ
2
11 + 2 λ 11 + 2
=
−0.309 + 0.951 i
0.191 − 0.588 i
(e)
Multiplying both the numerator and dominator by the complex conjugate of the
dominator yields
γ 11 = − 1.618
(f)
where a small imaginary part remains as a round-off error. This negligible imaginary
value suggests that the phase difference between the modes is negligible, that is, the
masses vibrate as if no damping is present in the system.
Similarly, we have γ 12 = −1.618, γ 21 = 0.618, γ 22 = 0.618.
We have
a 1 = 1.309, p d 1 = 0.951, a 2 = 0.191, p d 2 = 0.588
and
α 1 = − 1.618, α 2 = 0.618, β 1 = 0, β 2 = 0
These parameters when substituted into Eq. (5.53) yield (Fig. 5.13)
x 1 (t) = e
− 1.309 t
C 1 (1.618) − C 2 β 1
C 1
C 1 cos 0.951 t
+
C 1 (β 1 ) + C 2 (− 1.618)
C 2
C 2 sin 0.951 t
+ e
− 0.191 t
C 3 (− 0.618 ) − C 4 (β 2 )
C 3
C 3 cos 0.588 t
+
C 3 (β 2 ) + C 4 (0.618 t)
C 4
C 4 sin 0.588 t