Civil Engineering Reference
In-Depth Information
-
ˆ
ˆ
½
2
G
=
1
G
= −
1
0.3042
=
0.9526
°
11
22
°
°
°
ˆ
ˆ
(
)
ω
=
0.7
G
=
0.3042
G
=
0.3042
−
0.0926 0.3042
⋅
0.9526
=
0.2898
®
¾
2
21
32
°
°
ˆ
G
0.0926
ˆ
=
2
2
G
1
0.0926
0.2898
0.9526
°
=−
−
=
°
31
¯
¿
33
1
0
0
ª
º
«
»
ˆ
(
)
0.7
0.3042
0.9526
0
G
xx
ω
=
=
«
2
»
«
»
0.0926
0.2898
0.9526
¬
¼
-
ˆ
ˆ
½
2
G
1
G
1
0.1541
0.9881
=
= −
=
°
11
22
°
°
°
ˆ
ˆ
(
)
ω
=
1.1
G
=
0.1541
G
=
0.1541
−
0.0238 0.1541
⋅
0.9881
=
0.1522
®
¾
3
21
32
°
°
ˆ
G
=
0.0238
ˆ
2
2
G
1
0.0238
0.1522
0.9881
°
=−
−
=
°
31
¯
33
¿
1
0
0
ª
º
«
»
ˆ
(
)
1.1
0.1541
0.9881
0
G
ω
=
«
xx
1
»
«
0.0238
0.1522
0.9881
»
¬
¼
Denoting
ª
(
)
º
2
S
ω
=⋅Δ
ª
0.5
ω
240.4
º
⋅⋅
a
a
1.79
ªº
x
1
ª º
«
»
«
1
2
»
«»
« »
«
»
(
)
=
2
S
ω
=
0.7
⋅Δ =
ω
«
2 7.6 0.4
⋅ ⋅
»
≈
2.46
1.55
«»
« »
x
2
«
»
«
»
«»
a
« »
«
»
230.4
¬¼
(
)
⋅⋅
¬ ¼
«
»
3
2
S
ω
=⋅Δ
¬
1.1
ω
¼
«
»
x
3
¬
¼
then the three time series are given by (see Eq. A.19)
13
ˆ
() () (
)
()
¦¦
xt
=
G
ω
⋅
2
S
ω ω
Δ ⋅
cos
ω ψ
t
+
1
1
nj xj
j j
nj
11
==
ˆ
ˆ
ˆ
() (
)
( )
(
)
()
(
)
Ga
cos
t Ga
cos
t Ga
cos
t
=
ω
ω ψ
+
+
ω
ω ψ
+
+
ω
⋅ ⋅
ω ψ
+
11
1
1
1
11
11
2
2
2
12
11
3
3
3
13
(
)
(
)
(
)
=
1.79 cos 0.3
⋅
t
+
ψ
+
2.46 cos 0.7
⋅
t
+
ψ
+
1.55 cos 1.1
⋅
t
+
ψ
11
12
13
23
() () (
)
ˆ
()
¦¦
xt G
2
S
cos
t
=
ω
⋅
ω ω
Δ ⋅
ω ψ
+
2
2
nj xj
j j
nj
11
==
ˆ
ˆ
ˆ
() (
)
( )
(
)
()
(
)
Ga
cos
t Ga
cos
t Ga
cos
t
=
ω
ω ψ
+
+
ω
ω ψ
+
+
ω
⋅ ⋅
ω ψ
+
21
1
1
1
11
21
2
2
2
12
21
3
3
3
13
ˆ
ˆ
ˆ
() (
)
( )
(
)
( )
(
)
Ga
cos
t Ga
cos
t Ga
cos
t
+
ω
ω ψ
+
+
ω
ω ψ
+
+
ω
⋅ ⋅
ω ψ
+
22
1
1
1
21
22
2
2
2
22
22
3
3
3
23
(
)
(
)
(
)
1.075 cos 0.3
t
0.748 cos 0.7
t
0.239
cos 1.1
t
=
⋅
+
ψ
+
⋅
+
ψ
+
⋅
+
ψ
11
12
13
(
)
(
)
(
)
+
1.431 cos 0.3
⋅
t
+
ψ
+
2.343 cos 0.7
⋅
t
+
ψ
+
1.532 cos 1.1
⋅
t
+
ψ
21
22
23
