Civil Engineering Reference
In-Depth Information
while
r
r
+
renders
k
1
k
1
+
−
1
(
)
2
(9.167)
r
≈⋅
r
−+
r
r
k
k
+
1
k
k
−
1
2
t
Δ
Dynamic equilibrium at
t
is given by
k
Mr
+
C
r
+
K
r
=
R
(9.168)
net k
net k
dyn
k
Introducing
r
and
r
from Eqs. 9.166 and 9.167
1
1
(
)
(
)
Mr
⋅
−
2
r
+
r
+
C
⋅
r
−
r
+
K
r
=
R
(9.169)
k
1
kk
1
et
k
1
k
1
et k
k
+
−
+
−
2
2
t
Δ
Δ
t
and solving for
r
k
+
1
-1
t
⎡
t
⎤
⎛
Δ
⎞
(
)
⎛
Δ
⎞
2
2
t
2
t
r
=+
MC
Δ
R
+ −Δ
MKr
−−
MCr
(9.170)
⎜
⎟
⎢
⎜
⎟
⎥
k
+
1
net
dyn
net
k
net
k
−
1
2
k
2
⎝
⎠
⎝
⎠
⎣
⎦
Thus, it is seen that
r
may be estimated based on knowledge about the load and
response quantise at
t
and
k
+
1
t
.
For the establishment of initial conditions at
−
1
t
=
0
before the iteration procedure can
r
. Dynamic equilibrium at
start it is necessary to define (choose)
r
and
t
=
0
will
then render the corresponding acceleration
(
)
−
=⋅
1
rM R
−
CrKr
−
(9.171)
0
dyn
net
0
net
0
0
while eliminating
r
from (see Eqs. 9.166 and 9.167)
1
2
⎫
(
)
r
≈⋅
r
−
r
⎪
0
1
−
1
t
Δ
⎪
⎪
⎬
⎪
⎪
(9.172)
1
(
)
2
r
≈⋅
r
−+
⎪
r
r
0
1
0
−
1
2
t
Δ
⎭
renders
2
t
Δ
r
=−Δ⋅
r
t
r
+
r
(9.173)
10
0
0
−
2
