Civil Engineering Reference
In-Depth Information
c
1
2
k
Figure 10.15
A model of a
bar element with damping.
decay of the motion.) The answer to the damping question is complex. For
example, structures are subjected to the atmosphere, so that air resistance is a
factor. Air resistance is, in general, proportional to velocity squared, so this effect
is nonlinear. Fortunately, air resistance in most cases is negligible. On the other
hand, the internal friction of a material is not negligible and must be considered.
If we incorporate the concepts of damping as applied to the simple harmonic
oscillator, the equations of motion of a finite element model of a structure become
(10.132)
where [
C
] is the system viscous damping matrix assembled by the usual rules. For
example, a bar element with damping is mathematically modeled as a linear
spring and a dashpot connected in parallel to the element nodes as in Figure 10.15.
The element damping matrix is
c
(
e
)
=
[
M
]
{¨
q
}+
[
C
]
{˙
q
}+
[
K
]
{
q
}={
F
(
t
)
}
c
−
c
(10.133)
−
cc
and the element equations of motion are
m
(
e
)
f
(
e
)
(10.134)
The element damping matrix is symmetric and singular, and the individual terms
are assigned to the global damping matrix in the same manner as the mass and
stiffness matrices. Assembly of the global equations of motion for a finite ele-
ment model of a damped structure is simple. Determination of the effective vis-
cous damping coefficients for structural elements is not so simple.
Damping due to internal friction is known as
structural damping,
and exper-
iments on many different elastic materials have shown that the energy loss per
motion cycle in structural damping is proportional to the material stiffness and
the square of displacement amplitude [2]. That is,
c
(
e
)
k
(
e
)
{¨
u
}+
{˙
u
}+
{
u
}=
kX
2
(10.135)
where
is a dimensionless structural damping coefficient,
k
is the material stiff-
ness, and
X
is the displacement amplitude. By equating the energy loss per cycle
to the energy loss per cycle in viscous damping, an equivalent viscous damping
coefficient is obtained:
U
cycle
=
c
eq
=
k
(10.136)
