Dissipated energy.
Forced damped vibration of a linear massdamperspring system con
sists of two parts: the exponetialy decaying part and an oscillatory
part. The exponentially decaying part is the homogeneous solution to
the forced damped vibration equation of motion, stemming from New
ton’s law. If there was no forcing the system would end up ultimately
(when
t
=
∞
) an equilibrium position. In the presence of forcing, the
system ends up oscillating permanently, at the same frequency as the
forcing, but with a changed phase of the response. The oscillatory part
of the solution is
x
=
Xsin
(
ωt

φ
)
where
X
is the amplitude of the response to forcing and
φ
is the phase
of the response. We would like to ﬁnd out how much energy is neces
sary to put into the system to keep it oscillating with that response.
We will ﬁnd that by considering the work of the dissipation force over
a cycle of the forced oscillation. Note that
˙
x
=
ωXcos
(
ωt

φ
) =
±
ωX
p
1

sin
2
(
ωt

φ
) =
±
ωX
r
1

x
2
X
2
=
±
ω
√
X
2

x
2
and the dissipative force is
F
d
=
c
˙
x
If we integrate it over a cycle of oscillations, we get:
W
d
=
I
F
d
dx
=
I
c
˙
xdx
=
I
c
dx
dt
dx
dt
dt
=
I
c
˙
x
2
dt
=
cω
2
X
2
Z
2
π
ω
0
cos
2
(
ωt

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 Winter '08
 Mezic,I
 Damper, Resonator, equivalent viscous damping

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