5.8 Free Vibration of Damped Two Degrees of Freedom System
173
Note that C 1 and C 2 are real numbers
where
C 1 = B 11 + B 12 , C 2 = i ( B 11 − B 12 )
(5.48)
Similarly,
γ 11 = α 1 + i β 1 , γ 12 = α 1 − i β 1
(5.49)
Then
γ 11 B 11 e
λ 11 t
+ γ 12 B 12 e
λ 12 t
= e
− a 1 t
(C 1 α 1 − C 2 β 1 ) cos p d 1 t
+ (C 1 β 1 − C 2 α 1 ) sin p d 1 t
(5.50)
Similar treatment is meted out to second pair of complex conjugate.
C 3 = B 21 + B 22 ,
C 4 = i ( B 21 − B 22 )
(5.51)
γ 21 = α 2 + i β 2 ,
γ 21 = α 2 − i β 2
(5.52)
Therefore, Eq. (5.46) can be written as
x 1 ( t) = e
− a 1 t
γ 1 C 1 cos p d 1 t + γ
1 C 2 sin p d 1 t
+ e
− a 2 t
γ 2 C 3 cos p d 2 t + γ
2 C 4 sin p d 2 t
x 2 ( t) = e
− a 1 t
C 1 cos p d 1 t + C 2 sin p d 1 t
+ e
− a 2 t
C 3 cos p d 2 t + C 4 sin p d 2 t
(5.53)
where the amplitude ratios are given by
γ 1 =
C 1 α 1 − C 2 β 1
C 1
,
γ
1 =
C 1 β 1 + C 2 α 1
C 2
γ 2 =
C 3 α 2 − C 4 β 2
C 3
,
γ
2 =
C 2 β 2 + C 4 α 2
C 4
(5.54)
Equation (5.53) denotes the damped response of two masses, which can be
compared with Eq. (5.10) for the undamped case. Equation (5.53) can be written
as
x 1 ( t) = B
1 e
− a 1 t cos
p d 1 t − ϕ
d 1
+ B
2 e
− a 2 t cos
p d 2 t − ϕ
d 2
x 2 ( t) = B 1 e
− a 1 t cos
p d 1 t − ϕ
d 1
+ B 2 e
− a 2 t cos
p d 2 t − ϕ
d 2
(5.55)