Environmental Engineering Reference
In-Depth Information
series,
∞
m
=
1
Γ
m
(
t
)
sin
(
β
m
x
)
,
T
(
x
,
t
)=
(6.240)
∞
m
=
1
D
m
sin
(
β
m
x
)
,
g
(
x
)=
(6.241)
where
Γ
m
(
t
)
and
β
m
are defined in Section 6.10.1, and
l
2
l
D
m
=
g
(
x
)
sin
(
β
m
x
)
d
x
.
(6.242)
0
in Eq. (6.240) automatically satisfies the boundary conditions in
Eq. (6.209). Substituting Eqs. (6.240) and (6.241) into the equation of (6.209) and
making use of the orthogonality of the set
T
(
x
,
t
)
{
sin
(
β
m
x
)
}
yields
QD
m
α
k
¨
ζ
m
ω
m
˙
2
m
e
iΩ
t
Γ
m
(
t
)+
2
Γ
m
(
t
)+
ω
Γ
m
(
t
)=
(
1
+
i
Ωτ
q
)
,
(6.243)
τ
q
whose solution is readily obtained as
B
m
e
(
Ω
t
+
ϕ
m
)
i
Γ
(
t
)=
.
(6.244)
m
Here,
QD
m
α
B
m
=
B
Ω
m
,
(6.245)
k
ω
m
Ω
m
η
m
+
i
B
Ω
m
=
Ω
∗
m
,
(6.246)
−
Ω
∗
m
)
2
m
(
1
+
4
ζ
ζ
m
Ω
m
2
tan
−
1
(
ϕ
m
)=
−
−
Ω
∗
m
,
(6.247)
1
1
√
ατ
η
m
=
m
,
(6.248)
β
q
Ω
m
=
Ω
ω
m
.
(6.249)
For resonance,
B
Ω
m
2
reaches its maximum value. Note that, by Eq. (6.246),
B
Ω
m
+
Ω
∗
m
2
m
η
2
=
Ω
∗
m
.
(6.250)
(
1
−
Ω
∗
m
)
2
+
4
ζ
m
Therefore, resonance requires
∂
B
Ω
m
2
∂Ω
∗
m
=
0
,
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