Environmental Engineering Reference
In-Depth Information
While
Γ
m
(
t
)
is oscillatory, it is not periodic because of the decaying amplitude.
Γ
m
(
t
)
oscillates in time with a fixed damped period
T
dm
given by
π
ω
dm
.
2
T
dm
=
(6.229)
Critically-Damped Oscillation
For this case,
ζ
m
=
1. This requires, by Eq. (6.220),
1
−
τ
T
τ
0
1
±
=
.
ω
(6.230)
m
τ
T
Therefore, critically damped oscillation appears only when
0
. When the sys-
tem is in critically damped oscillation, there are two equal numbers
τ
≤
τ
T
λ
1
and
λ
2
.
Therefore,
a
m
e
−
ω
m
t
b
m
t
e
−
ω
m
t
Γ
(
t
)=
+
,
m
which becomes, after determining the integration constants
a
m
and
b
m
by the initial
conditions [Eq. (6.214)],
e
−
ω
m
t
Γ
m
(
t
)=
[
φ
m
+(
ψ
m
+
ω
m
φ
m
)
t
]
.
(6.231)
d
2
d
|
Γ
m
(
t
)
|
|
Γ
m
(
t
)
|
By letting
=
0 and analyzing the sign of
, Xu and Wang (2002)
d
t
2
d
t
obtain the maximal value of
|
Γ
(
t
)
|
m
ψ
m
+
ω
m
φ
m
φ
+
ψ
ψ
m
m
ω
m
e
−
Max
[
|
Γ
(
t
)
|
]=
(6.232)
m
m
at
ψ
m
ω
m
(
ψ
m
+
ω
m
φ
m
)
,
t
m
=
(6.233)
that is positive if
2
m
ψ
> −
ω
m
φ
m
ψ
m
.
This clearly requires that
ψ
m
=
0. Therefore,
|
Γ
m
(
t
)
|
decreases monotonically as
t
increases from 0 when
2
m
ψ
≤−
ω
m
φ
m
ψ
m
.
This is very similar to the classical heat-conduction equation. When
2
m
ψ
> −
ω
m
φ
m
ψ
m
,
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