Environmental Engineering Reference
In-Depth Information
|
Γ
(
)
|
Letting
d
t
m
=
0 leads to two extreme points,
d
t
=
t
m
1
0
(6.236)
and
ω
m
+
φ
m
1
⎡
⎣
ζ
⎤
ψ
m
m
ζ
m
+
ζ
−
−
m
−
1
ζ
1
m
⎦
,
t
m
2
=
−
ω
m
ln
(6.237)
ψ
m
m
m
m
2
ζ
−
1
ζ
m
+
ζ
−
1
ω
m
+
φ
m
(
ζ
m
−
ζ
−
1
)
with Max1
[
|
Γ
m
(
t
)
|
]=
|
φ
m
|
and Max2
[
|
Γ
m
(
t
)
|
]=
|
Γ
m
(
t
m
2
)
|
, respectively. Therefore,
|
Γ
m
(
t
)
|
decreases monotonically from
t
=
0when
t
m
2
=
0 (very like in classical heat
conduction). When
t
m
2
>
0, however,
|
Γ
m
(
t
)
|
first increases from
|
φ
m
|
to a maxi-
mal value Max2
[
|
Γ
m
(
t
)
|
]
as
t
increases from 0 to
t
m
2
and then decreases for
t
≥
t
m
(Fig.6.4). There is no oscillation if
ζ
m
>
1.
When
τ
T
>
τ
0
,
+
ατ
T
m
2
2
+
ατ
0
m
2
2
π
π
2
√
ατ
0
m
l
.
1
>
1
≥
l
2
l
2
This, with Eq. (6.220), yields
m
2
2
π
1
+
τ
T
α
l
2
2
√
ατ
0
m
l
ζ
=
>
1
.
m
Fig. 6.4
|
Γ
(
t
)
|
at the overdamped oscillation and t
m
2
>
0 (after Xu and Wang 2002)
m
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