Civil Engineering Reference
In-Depth Information
10.12 PRACTICAL CONSIDERATIONS
The major problem inherent to dynamic structural analysis is the time-consuming
and costly amount of computation required. In a finite difference technique, such
as that represented by Equation 10.163, the system of equations must be solved at
every time step over the time interval of interest. For convergence, the time step
is generally quite small, so the amount of computation required is huge. In modal
analysis, the burden is in computing natural frequencies and mode shapes. As
practical finite element models can contain tens of thousands of degrees of free-
dom, the time and expense of computing all of the frequencies and mode shapes
is prohibitive. Fortunately, to obtain reasonable approximations of dynamic
response, it is seldom necessary to solve the full eigenvalue problem. Two practi-
cal arguments underlie the preceding statement. First, the lower-valued frequen-
cies and corresponding mode shapes are more important in describing structural
behavior. This is because the higher-valued frequencies most often represent
vibration of individual elements and do not contribute significantly to overall
structural response. Second, when structures are subjected to time-dependent
forcing functions, the range of forcing frequencies to be experienced is reason-
ably predictable. Therefore, only system natural frequencies around that range are
of concern in examining resonance possibilities.
Based on these arguments, many techniques have been developed that allow
the computation (approximately) of a subset of natural frequencies and mode
shapes of a structural system modeled by finite elements. While a complete dis-
cussion of the details is beyond the scope of this text, the following discussion
explains the basic premises. (See Bathe [6] for a very good, rigorous description
of the various techniques.) Using our notation, the eigenvalue problem that must
be solved to obtain natural frequencies and mode shapes is written as
2
[
M
]
[
K
]
{
A
}=
{
A
}
(10.179)
The problem represented by Equation 10.179 is reduced in complexity by
static
condensation
(or, more often,
Guyan reduction
[11]) using the assumption that
all the structural mass can be lumped (concentrated) at some specific degrees
of freedom without significantly affecting the frequencies and mode shapes of
interest. Using the subscript
a
(active) to represent degrees of freedom of inter-
est and subscript
c
(constrained) to denote all other degrees of freedom Equa-
tion 10.179 can be partitioned into
[
K
aa
]
K
ac
]
[
K
ca
]
K
cc
]
{
A
a
}
{
2
[
M
aa
]
0
]
[0]
{
A
a
}
{
=
(10.180)
A
c
}
[0]
A
c
}
In Equation 10.180, [
M
aa
] is a diagonal matrix, so the mass has been lumped at
the degrees of freedom of interest. The “constrained” degrees of freedom are
constrained only in the sense that we assign zero mass to those degrees. The
lower partition of Equation 10.180 is
[
K
ca
]
{
A
a
}+
[
K
cc
]
{
A
c
}={
0
}
(10.181)
