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system, which is not always feasible, at least not
in a satisfactory and economic way. Many appli-
cations of such dynamic improvements may be
found in the literature; see for example Chopra
(2000, pp. 767-777) or Bozorgnia and Bertero
2004, pp. 10-11 - 10-28). Other approaches deal
with changes to the dynamic system by introduc-
ing new components. They may be defined by
ing mass and stiffness cause strong decreases in
the amplitude
u
1
.
Figure 3d compares the amplitudes
u
1
and
u
2
for some given stiffness
k
2
. We realise that
the amplitude
u
2
increases, but
u
1
decreases as
m
2
increases down to a minimum of
u
1
. Further
augmentation of
m
2
reduces the effect of the
compensator,
u
1
increases again, but
u
2
essen-
tially does not decrease. Figure 3e demonstrates
that the use of passive compensators measured
as the value
u
1
with compensator compared to
u
1
without compensator is only efficient if the
damping
c
2
is relatively small. As soon as the
damping is too large, the damper
c
2
acts more like
a stiff link, increasing the mass
m
1
by
m
2
but not
allowing significant relative displacement. So an
independent oscillation of
m
2
is suppressed, and
the compensator is not efficient.
1. Uncoupling the mass from the ground mo-
tion
u
g
(
t
) by introducing a base isolation by
an elastic and damping layer between sur-
rounding ground and the base of the edifice
(Chopra 2000, pp. 741-766).
2. Introducing an additional mass-stiffness-
damping element at the mass
m
1
to modify
the Eigen frequencies (Figure 3a). There may
be passive masses or active or semi-active
systems using e.g. control to accelerate
the mass
m
2
to minimise the deflection of
the mass
m
1
. Such mass-stiffness-damping
systems are called compensators, absorbers
or tuned mass dampers.
Multi-Mass Compensator Systems
The idea of a passive compensating device is
easily expanded to problems with many degrees
of freedom. For the sake of simplicity, we are
restraining ourselves here to one-dimensional
chains of masses, stiffness and dampers excited
at one end of the chain as indicated in Figure 4a.
For the following examples we again use data
like that proposed in Equation (11).
The corresponding system is given by
For base isolation and controlled compensa-
tor systems, more detailed information about the
advantages and problems are given by Bozorgnia
and Bertero (2004, pp. 11-1 - 11-17) and Xu et
al. (2004).
The influence of a compensator on the dynamic
response of the initial mass may be learned from
Figure 3b. The amplitudes “
u
1
for different
k
2
”
are significantly smaller at the Eigen frequency,
but have 2 maxima of a significant fraction of
the initial maximum amplitude. The maximum
amplitude of the “best
u
1
” proposal one in Figure
3b depends strongly on the damping in the system
(Figure 3e). There may be problems since we need
to provide space for the corresponding deflections
of mass
m
2
. Figure 3c depicts the dependence of
the relative amplitude
u
1
compared to the one
without compensator on the compensator's mass
and stiffness. Even relatively small values of damp-
Mu Cu Ku
+
+
=
(
t
F
(26)
where
m
0
..
..
0
1
0
m
0
M
=
2
(27)
:
:
::
:
0
0
.
..
m
n
denotes the mass matrix which may or may not
be diagonal.
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