Lecture 6
Harmonically forced vibrations
In this section we study one of the most important phenomena in sci
ence and technology: that or resonance. If it was not for resonance,
you would not have stadium rock concerts (might be for the better, I
prefer smaller venues myself). Resonance can also cause a lot of trou
ble, as you could see in the Millenium bridge video shown in class, and
of course the Tacoma bridge disaster that I mentioned in the introduc
tory lecture. I will explain the physics of the Tacoma bridge disaster
towards the end of this lecture. Bit first, let us derive the equation of
motion for the system shown in figure (1).
Figure 1: A massspringdamper system acted upon by a timedependent
force
F
(
t
).
Let us consider the case when the external force is harmonic, with
zero phase.
Remark 1
There is no loss of generality when we do this: remember
that harmonic force can be written as
F
(
t
) =
F
0
sin(
ωt
+
ϕ
)
,
for some constant phase
φ
and constant force amplitude
F
0
. If we now
let
τ
=
t
+
ϕ/ω
, then the harmonic force becomes
F
(
τ
) =
F
0
sin(
ωτ
)
,
and since
dτ
=
dt
, we have
dx/dt
=
dx/dτ
and
d
2
/dt
2
=
d
2
/dτ
2
.
What we have done with this transformation is equivalent to looking
at the process at an initial time
t
0
=
ϕ/ω
instead of
t
= 0
.
1
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Taking into account the spring, damping and external harmonic forc
ing, the equation of motion reads
m
¨
x
+
c
˙
x
+
kx
=
F
0
sin(
ωt
)
(1)
From lecture 4, we know the solution to the homogeneous part
m
¨
x
+
c
˙
x
+
kx
= 0
,
is
x
h
(
t
) =
x
(
t
) =
e

c
2
m
t
sin(
ω
d
t
+
φ
)
,
and that solution decays exponentially fast in time to equilibrium
x
h
=
0.
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 Winter '08
 Mezic,I
 Complex number, Fundamental physics concepts

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