Lecture 15: Forced vibrations, 2DOF systems
0.1
Matrix methods for 2DOF system with forcing
We have seen how matrix methods simplify analysis for free vibration
of 2DOF. Now we use those to enable analysis of forced systems. We
will see how the use of diagonalization discussed in the previous lecture
enables us to solve such, relatively complex problems.
Consider the vector
x
= (
x
1
, x
2
) of positions for a linear vibrat
ing system with 2DOF. Let the force on the system be expressed as
Bu
(
t
) where
B
is a 2
×
2 matrix and
u
(
t
) the forcing vector
u
(
t
) =
(
u
1
(
t
)
, u
2
(
t
))
T
. Equations of motion can be written in matrix form as:
M
¨
x
=

Kx
+
Bu
(
t
)
.
We premultiply by
M

1
and rewrite this as
¨
x
=

M

1
Kx
+
M

1
Bu
(
t
)
.
From the previous lecture, we know that we can simplify this system by
using the change of voordinates
y
=
V

1
x
where
V
is the eigenvector
matrix. We obtain
¨
y
=
Dy
+
V

1
M

1
Bu
(
t
)
.
The expression
V

1
M

1
Bu
(
t
) =
v
(
t
) is a column vector that we can
write in components as
v
(
t
) = (
v
1
(
t
)
, v
2
(
t
)).
Thus the equations for
y
1
, y
2
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 Winter '08
 Mezic,I
 Complex number, k2, mass m1, vibration absorber, response X1

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