Geology Reference
In-Depth Information
formulations are derived by the famous topic
written by Den Hartog (1956, pp. 87-121). Reso-
nances that would cause inacceptable vibrations
are removed by the elastic coupling of additional
masses which shift the Eigen frequencies and
the amplitudes to regions and values that do not
affect the usability of the system. The improve-
ment of the dynamics of bridges and chimneys by
compensators is well-known (Nawrotzki 2005).
For high buildings passive compensators play an
important role as well (Eddy 2005).
m
k
c
u
F
=
=
=
=
1
1
1
1
0 002
.
.... .
0 05
(11)
1
1
g
,max
,max
=
1
1
so all the other values may be considered as rela-
tive values related to this basic data. One result
of the ODE (9) is the Eigen frequency of the not
damped oscillator
One Mass Oscillator and Compensator
k
m
ω
0
=
(12)
1
From the classical approach of dynamics, it is a
good idea to start with a single mass oscillator.
It is defined by
the
mass
m
1
,
the
stiffness
k
1
and
the
damping
c
1
(Figure 3a). Stiffness and damping
are attached to some base which may be fixed in
time (
u
g
= 0) or has defined displacements (
u
g
=
u
g
(
t
)). The system is excited by either the ground
motion
u
g
(
t
) or a force
F
1
(
t
) acting on the mass
m
1
. The resulting displacement history
u
1
(
t
) may
be found by integrating the ODE
1
1. We usually discuss 3 aspects of the ODE
(9):
Assuming that no external excitation is
acting (
F
1
(
t
) = 0 and
u
g
(
t
) = 0) yields the
Eigen problem
mu
1
+
cu
+
ku
=
0
(13)
1
1
=
( )
(9)
mu
1
+
cu
+
ku
F t
and the Eigen frequency following Equation
(12) for the not damped and
1
1
1
where
F
1
(
t
) is either the external force or caused
by the displacement
u
g
(
t
)
k
m
c
m
2
2
2
ω
=
1
−
1
=
ω
−
γ
d
0
1
2
4
(14)
(
)
1
1
F t
( )
=
k u t
( )
−
u t
( )
(10)
1
1
g
1
We shall use the complete form of Equation
(9) in contradiction to other authors who prefer to
divide Equation (9) by the mass and find a dimen-
sionless damping often labelled ζ. The proposal of
a viscous damping proportional to the velocity is
convenient even if we often do not know exactly
which values to use for
c
.
For the following examples we use a simple
dynamic system where
where
γ
=
c
m
1
(15)
2
1
1
for the damped system. The resulting dis-
placement-time history is of the type
i
ω
t
−
γ
t
u t
( )
=
u e
e
(16)
d
1
0
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