Civil Engineering Reference
In-Depth Information
where in the context of solid mechanics, [
k
m
] is the element elastic stiffness and [
m
m
]the
element mass. In equation (3.117) a time dependent forcing term
has been included
on the right hand side. In addition to these elastic and inertial forces, solids in motion
experience a third type of force whose action is to dissipate energy. For example, the solid
may deform so much that plastic strains result, or may be subjected to internal or external
friction. Although these phenomena are non-linear in character and can be treated by the
non-linear analysis techniques given in Chapter 6, it has been common to linearise the
dissipative forces, for example by assuming that they are proportional to velocity. This
allows (3.117) to be modified to
{
f
(t)
}
[
c
m
]
d
u
d
t
[
m
m
]
d
2
u
[
k
m
]
{
u
} +
+
= {
f
(t)
}
(3.118)
d
t
2
where [
c
m
] is assumed to be a constant element damping matrix.
Although in principle [
c
m
] could be independently measured or assessed, it is common
practice to assume that [
c
m
] is taken to be a linear combination of [
m
m
] and [
k
m
], where
[
c
m
]
=
f
m
[
m
m
]
+
f
k
[
k
m
]
(3.119)
where
f
m
and
f
k
are scalars, the so-called “Rayleigh” damping coefficients. They can be
related to the more usual “damping ratio”
ζ
(Timoshenko
et al
., 1974) by means of
f
m
+
f
k
ω
2
2
ω
ζ
=
(3.120)
where
ω
is the natural (usually fundamental) frequency of vibration.
The most generally applicable technique for integrating (3.118) with respect to time is
“direct integration” in an analogous way to that previously described for first order prob-
lems. Two of the simplest popular implicit methods are described in subsequent sections,
where the solution is advanced by one time interval
t
, the values of the displacement
and its derivatives at one instant in time being sufficient to determine these values at the
subsequent instant by means of recurrence relations. Both preserve unconditional stabil-
ity, and examples that utilise both element assembly and element-by-element strategies are
presented in Chapters 11 and 12.
Attention is first focused however on the “modal superposition” method.
3.13.1 Modal superposition
This method has as its basis the free undamped part of (3.118), that is when [
c
m
]and
{
f
}
are zero. The reduced equation in assembled form is
[
M
m
]
d
2
U
d
t
2
[
K
m
]
{
U
} +
= {
0
}
(3.121)
which can of course be converted into an eigenproblem by the assumption of harmonic
motion
{
}
=
{
}
U
A
sin
(ωt
+
ψ)
(3.122)
