172
5 Vibration of Two Degrees of Freedom System
Two complex conjugate pairs are assumed of the following form:
λ 11 = − a 1 + i p d 1
λ 12 = − a 1 − i p d 1
λ 21 = − a 2 + i p d 2
λ 22 = − a 2 − i p d 2
(5.42)
where damped natural frequencies are p d i = p i
1 − ζ 2 and a i are positive
numbers.
Substituting each root given by Eq. (5.42) in turn into Eq. (5.40) yields
m 1 λ
2
i j + ( c 1 + c 2 ) λ i j + ( k 1 + k 2 )
− c 2 λ i j − k 2
− c 2 λ i j − k 2
m 2 λ
2
i j + c 2 λ i j + k 2
A i j
B i j
=
0
0
(5.43)
From Eq. (5.43), we get four possible ratios
A i j
B i j
= γ i j =
c 2 λ i j + k 2
m 1 λ
2
i j + ( c 1 + c 2 ) λ i j + ( k 1 + k 2 )
=
m 2 λ
2
i j + c 2 λ i j + k 2
c 2 λ i j + k 2
(5.44)
i = 1, 2 and j = 1, 2.
The ratios γ 11 , γ 12 , γ 21 and γ 22 are modal conjugate pairs that relate the modal
amplitudes which are generally of complex modes.
A i j = γ i j B i j
(5.45)
The complete solution of the equation is
x 1 (t ) = γ 11 B 11 e
λ 11 t
+ γ 12 B 12 e
λ 12 t
+ γ 21 B 21 e
λ 21 t
+ γ 22 B 22 e
λ 22 t
x 2 (t ) = B 11 e
λ 11 t
+ B 12 e
λ 12 t
+ B 21 e
λ 21 t
+ B 22 e
λ 22 t
(5.46)
( B 11 , B 12 ) and (B 21 , B 22 ) are complex conjugate pairs that are to be determined
from initial conditions.
Now,
B 11 e
λ 11 t
+ B 12 e
λ 12 t
= e
− a 1 t
C 1 cos p d 1 t + C 2 sin p d 1 t
(5.47)