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the uncoupled differential equations appear as
+
=
+
=
=
...
M n d q
K n d q
P
or
M i q i
K i q i
P i
i
1 , 2 ,
,n d
(10.72)
= Φ T P ,M i = φ i M φ i ,K i = φ i K φ i , and P i = φ i P .
It is apparent that the coupled governing equations for our n d DOF system have been
replaced by n d equations [Eq. (10.72)], each of which has the same form as the governing
equations for a single-DOF system. From the theory of ordinary differential equations, we
know that the solution to these equations is
in which P
q
=
Aq
(
0
) +
B
q
˙
(
0
) +
F
(10.73a)
where
q 1
(
0
)
V 10
V 20
V n d 0
q 2
(
0
)
R − 1
Φ − 1 V
Φ − 1 V 0
q
(
0
) =
=
=
(
0
) =
(10.73b)
.
q n d (
0
)
V 10
V 20
.
V n d 0
q 1
˙
(
0
)
q 2
˙
(
0
)
V 0
V
R − 1
Φ − 1
Φ − 1
˙
q
(
0
) =
=
=
(
0
) =
(10.73c)
.
q n d (
˙
0
)
sin
ω 1 t
ω 1
0
ω
cos
1 t
0
sin
ω 2 t
ω 2
ω
cos
2 t
A
=
B
=
. . .
. . .
sin
ω n d t
ω n d
0
cos
ω n d t
0
and
F 1
0
t
F 2
1
M i ω i
F
=
with
F i =
P i (τ )
sin
ω i (
t
− τ)
d
τ
. . .
0
0
F n d
, V n d 0 are the prescribed initial conditions (di sp lace-
ments a n d velocities) of the system which are represented by the vectors V
V 10 ,
V 20 ,
and V 10 ,V 20 ,
...
,V n d 0 , and
...
(
0
) =
V 0 and
) = V 0 .
In scalar form, the solution to Eq. (10.72) is
V
(
0
t
sin
ω i t
ω i +
1
M i ω i
q i =
q i (
0
)
cos
ω i t
+˙
q i (
0
)
P i (τ )
sin
ω i (
t
− τ)
d
τ
i
=
1 , 2 ,
...
,n d
(10.74a)
0
n d
ij V k 0 M i
n d
MV 0
M i =
φ i
q i
(
0
) =
m jk
φ
j
k
(10.74b)
n d
M i
M V 0
M i =
n d
m jk φ ij V k 0
) = φ i
q i (
˙
0
j
k
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