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◦
This gives
δ
→
180
◦
(
tan
δ
approaching
0
from below)
◦
The mass lags
180
◦
out of phase, ie. it moves in the opposite direction to the driving force
◦
This system is inertia dominated
◦
The amplitude still falls off at high
ω
like
ω

2
but it is not due to the damping [see loglog plot shown in class]
3. Velocity Response
•
There is also a resonance in velocity – mass moves faster than forcedend of spring
•
This actually occurs at the natural frequency of the system
˙
X
(
t
) =

ωA
(
ω
) sin(
ωt

δ
)
•
The velocity amplitude is:
V
(
ω
) =
a
0
ωω
2
0
p
(
ω
2
0

ω
2
)
2
+ (
γω
)
2
=
a
0
ω
2
0
p
(
ω
2
0

ω
2
)
2
/ω
2
+
γ
2
•
This is maximised when the denominator is minimised:
d
dω
(
(
ω
2
0

ω
2
)
2
ω
2
+
γ
2
) = 0
•
But the
γ
2
term just vanishes from the derivative
•
Taking derivative of a ratio:
d
dx
(
A
B
) =
A
0
B

B
0
A
B
2
gives:
2(
ω
2
0

ω
2
)2
ω
(
ω
2
)

(2
ω
)(
ω
2
0

ω
2
)
2
= 0
•
Gathering terms together this gives:
(
ω
2
0

ω
2
)[2
ω
](2
ω
2

ω
2
0

ω
2
) = 0
•
There are three different families of roots here:
ω
= [0
,
±
ω
0
,
±
iω
0
]
•
Of these only
ω
=
ω
0
is positive and real
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 Fall '07
 PierreSavaria
 mechanics, Force, Inertia, Mass, Statistical Mechanics

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