Civil Engineering Reference
In-Depth Information
Then, Eq. (3.19) becomes
z
k
+1
=
F
d
z
k
+
H
d
a
k
ð3
:
21Þ
where
F
d
is the 2
n
×2
n
state transition matrix in discretized form and
H
d
is the 2
n
× 3 ground
acceleration transition matrix in discretized form. Note that by representing the earthquake
ground acceleration
a
(
s
) as a delta forcing function in Eq. (3.18), Eq. (3.21) becomes an explicit
solution procedure.
3.1.3 Solution Procedure
The state transition matrix
F
d
=
e
A
Δ
t
requires that an exponential power of a matrix be per-
formed. For a single degree of freedom (SDOF) system, it has a closed-form solution:
2
3
ζ
1−
ζ
1
cos
ω
d
Δ
t
+
p
sin
ω
d
Δ
t
p
1−
ζ
sin
ω
d
Δ
t
4
5
2
2
ω
e
A
Δ
t
=
e
−
ζωΔ
t
ð3
:
22Þ
ω
1−
ζ
ζ
1−
ζ
p
p
−
sin
ω
d
Δ
t
cos
ω
d
Δ
t
−
sin
ω
d
Δ
t
2
2
where
ζ
is the damping ratio,
ω
is the natural frequency, and
ω
d
is the damped natural frequency.
These quantities can be computed for a SDOF system with mass
M
, damping
C
, and stiffness
K
as:
s
,
K
M
q
1−
ζ
C
2
ω
M
,
2
ω
=
ζ
=
ω
d
=
ω
ð3
:
23Þ
However, for multi-degree-of-freedom system, the state transition matrix has no closed-form
solution and therefore it generally needs to be evaluated numerically. One good method is to
use the Taylor series expansion, i.e.
2
3
e
A
Δ
t
=
I
+
A
Δ
t
+
A
Δ
t
ð
Þ
+
A
Δ
t
ð
Þ
+…
ð3
:
24Þ
2
!
3
!
However, if some entries in
A
Δ
t
are much larger than 1.0, squaring and cubing this matrix can
result in an even larger matrix that requires summing many terms before the factorial in the
denominator takes over for convergence. This may also lead to numerical error in the evalu-
ation due to multiplication, addition, and then division of large numbers. An improved method
called the scaling and squaring method may be used to enhance the numerical accuracy of the
calculation and increase the speed of convergence:
"
#
p
p
=
I
+
A
Δ
t
p
2
3
+
A
Δ
t
=
p
ð
Þ
+
A
Δ
t
=
p
ð
Þ
e
A
Δ
t
=
e
A
Δ
t
=
p
+…
ð3
:
25Þ
2
!
3
!
