Civil Engineering Reference
In-Depth Information
of dynamic response calculations. Today, linear dynamic analysis for multi-degree-of-
freedom (MDOF) systems is a well-established subject, and it has been discussed in detail
in many structural dynamic textbooks (e.g., Chopra 2011). Therefore, the goal of this
section is not to focus on the general linear dynamic analysis, but rather to focus on a specific
dynamic analysis tool - the state space method - that works together harmoniously with the
force analogy method on solving nonlinear dynamic problems. To date, only one textbook
talks about the state space method for earthquake engineering purposes (Hart and Wong
2000), though the method is widely used in control engineering. Therefore, it is worthwhile
for this chapter to begin with the discussion on the state space method for solving linear
dynamic problems.
3.1.1 Equation of Motion
One important aspect of performing a dynamic analysis is that the displacement response
changes with respect to time. The time variable
t
is typically inserted to the displacement vector
in the form
x
(
t
) to denote that the displacement vector is a function of time. When the displace-
ment is differentiated with respect to time, velocity
x
(
t
) is obtained, i.e.
x
ðÞ=
d
x
ðÞ
dt
ð3
:
1Þ
where a dot above the variable represents differentiation of the variable with respect to time.
Differentiating the velocity with respect to time once again gives the acceleration
x
(
t
):
=
d
2
x
ðÞ
dt
2
x
ðÞ=
d
x
ðÞ
dt
ð3
:
2Þ
where, for an
n
-degree-of-freedom (
n
-DOF) system, these
n
× 1 displacement, velocity, and
acceleration vectors are of the form:
<
=
<
=
<
=
x
1
ðÞ
x
2
ðÞ
.
x
n
ðÞ
x
1
ðÞ
x
2
ðÞ
.
x
n
ðÞ
x
1
ðÞ
x
2
ðÞ
.
x
n
ðÞ
x
ðÞ=
,
x
ðÞ=
,
x
ðÞ=
ð3
:
3Þ
:
;
:
;
:
;
Consider the equation of motion for an
n
-DOF system subjected to the
n
× 1 earthquake
ground acceleration vector
g
(
t
):
Mx
ðÞ+
Cx
ðÞ+
Kx
ðÞ= −
Mg
ðÞ
ð3
:
4Þ
where
M
is the
n
×
n
mass matrix,
C
is the
n
×
n
damping matrix, and
K
is the
n
×
n
stiffness
matrix. The stiffness matrix
K
for an
n
-DOF system is the same as what is assembled in the
classical stiffness method of analysis, similar to what was obtained in Section 1.3 and through-
out Chapter 2. The earthquake ground acceleration vector
g
(
t
) corresponds to the effect of
ground motion at each degree of freedom (DOF).
