Civil Engineering Reference
In-Depth Information
mDLM
nyz
θ
=
, or
, or
-
®
()
lim
s
1
→∞
Φ=
(D.7)
mn
=
s
¯
Therefore, an exponential type of function with a limiting value of unity is usually
chosen for the representation of indicial functions interpreted from experimental results
(see Salvatori & Borri [31]). It is convenient to express these functions by the non-
dimensional relative time parameter
ˆ
ssVB
=
. Thus,
N
j
mDLM
nyz
θ
, or
, or
=
-
®
ˆ
b e
−
cs
j
Φ=−
¦
()
ˆ
s
1
(D.8)
mn
j
=
¯
j
1
=
where the constants
b
and
c
may be determined from experiments or, as shown
below, from the aerodynamic derivatives (see Chapter 5.2).
The current motion induced load
()
ae
t
q
may then be obtained by history integration
−∞ to the present time
t
)
(theoretically from
()
()
()
()
t
ª
d
r
τ
d
r
τ
º
()
q
t
³
C
s
K
s
d
=
+
τ
(D.9)
«
»
ae
ae
ae
d
d
τ
τ
¬
¼
−∞
Integration by parts
()
t
ds
ds
C
τ
τ
=
=−∞
t
ae
()
() ()
()
t
s
³
d
q
=
ª
C
⋅
r
τ
º
−
⋅ ⋅
r
τ τ
¬
¼
ae
ae
ds
d
τ
−∞
(D.10)
()
t
ds
ds
K
=
=−∞
t
τ
τ
() ()
ae
()
³
+
ª
Kr
s
⋅
τ
º
−
⋅ ⋅
r
τ τ
d
¬
¼
ae
ds
d
τ
−∞
and assuming negligible initial displacement and velocity conditions, then
t
t
() ()()
()()
()()
()()
³
³
q
t
=
C
0
r
t
+
C
′
s
r
ττ
d
+
K
0
r
t
+
K
′
s
r
ττ
d
(D.11)
ae
ae
ae
ae
ae
−∞
−∞
()
()
ds
C
ds
K
ae
ae
()
()
′
s
′
s
where
C
=
and
K
=
. Introducing a harmonic motion
ae
ae
ds
ds
a
t ae
ªº
«»
y
()
it
it
ω
ω
i
e
i
r
=
⋅
= ⋅
a
(D.12)
«»
«
¬¼
z
a
θ
