Civil Engineering Reference
In-Depth Information
If we were to apply the equivalent damping given by these values to the entire frequency
spectrum of a structure, the effective damping ratio for any mode would be given by
2
i
−
0
.
0375
+
0
.
0135
i
=
2
i
If the values of
and
are applied to a multiple degrees-of-freedom system, the damp-
ing ratio for each frequency is different. To illustrate the variation, Figure 10.16 depicts
the modal damping ratio as a function of frequency. The plot shows that, of course, the
ratios for the specified frequencies are exact and the damping ratios vary significantly for
other frequencies.
Rayleigh damping as just described is not the only approach to structural
damping used in finite element analysis. Finite element software packages also
include options for specifying damping as a material-dependent property, as
opposed to a property of the structure, as well as defining specific damping
elements (finite elements) that may be added at any geometric location in the struc-
ture. The last capability allows the finite element analyst to examine the effects of
energy dissipation elements as applied to specific locations.
10.10 TRANSIENT DYNAMIC RESPONSE
In Chapter 7, finite difference methods for direct numerical integration of finite
element models of heat transfer problems are introduced. In those applications,
we deal with a scalar field variable, temperature, and first-order governing equa-
tions. Therefore, we need only to develop finite difference approximations to first
derivatives. For structural dynamic systems, we have a set of second-order dif-
ferential equations
{
¨
}+
{
˙
}+
(10.154)
representing the assembled finite element model of a structure subjected to gen-
eral (nonharmonic) forcing functions. In applying finite difference methods to
Equation 10.154, we assume that the state of the system is known at time
t
and
we wish to compute the displacements at time
t
+
t
;
that is, we wish to solve
[
M
]
[
M
]
[
C
]
[
K
]
{}={
F
(
t
)
}
{
¨
{
˙
(
t
+
t
)
}+
[
C
]
(
t
+
t
)
}+
[
K
]
{
(
t
+
t
)
}={
F
(
t
+
t
)
}
(10.155)
for
{
(
t
+
t
)
}
.
Many finite difference techniques exist for solving the system of equations
represented by Equation 10.155. Here, we describe Newmark's method [7] also
referred to as the
constant acceleration method.
In the Newmark method, it is as-
sumed that the acceleration during an integration time step
t
is constant and an
average value. For constant acceleration, we can write the kinematic relations
t
2
2
+
¨
av
+
˙
(
t
+
t
)
=
(
t
)
(
t
)
t
(10.156)
˙
=
˙
+
¨
av
(
t
+
t
)
(
t
)
t
(10.157)
