Geology Reference
In-Depth Information
Ω
application of viscous or frictional damping de-
vices to increase the structural damping suffers
from small storey drifts, TLCGD should be pre-
ferred. Assuming that each floor can be repre-
sented by a rigid diaphragm with three degrees
of freedom (e.g. by applying static condensation
of a finite element discretization), its motion is
given by the horizontal displacements of its cen-
ter of mass
w
and
v
together with a rotation
θ
about the vertical axis. For the subsequent section
the choice of the coordinate system is in accordance
with literature: the vertical axis is denoted
x
, the
horizontal axes are
y
and
z
, respectively. Recent
research on the application of TLCGD on mul-
tiple purpose buildings has revealed that it is fa-
vorable to distinguish between moderate and
strong asymmetry. In terms of modal floor dis-
placements strong asymmetry is observed if the
modal center of velocity lies within the floor plan
but distinct from the center of mass,
C
(
)
(
)
M
*
=
M
1
+
µ
1
−
κκ
,
Ω
*
=
,
(
)
1
+
µ
1
−
κκ
m
M
κκ
κκ
f
µ
*
=
=
µ
)
<
µ
(
*
1
+
µ
1
−
κκ
(8)
All properties of the equivalent TMD system
are denoted by a star
*
. Optimal tuning for a two
DOF system is classically done by applying the
Den Hartog criterion (Hartog 1956),
ω
*
*
1
3
8 1
µ
A opt
,
*
*
δ
=
=
,
ζ
=
(
)
*
*
opt
Ω
1
+
µ
*
+
µ
(9)
that minimizes the dynamic magnification factor
of absolute floor acceleration in case of time
harmonic base excitation. The same parameters
apply for minimizing the displacement magnifica-
tion factor under the condition of time harmonic
forcing, Warburton (1981). Since the TLCGD is
fully described by the equivalent TMD system
their natural frequencies and damping ratios are
identical,
ω
=
, , 0
. In this case the rotation of
the floor dominates and, e.g., is excited by the
horizontal ground acceleration of an earthquake.
In a perfect symmetric structure modal center of
stiffness,
C
y
z
M
C
C
M
M
=
=
*
and can be
calculated evaluating Eq.(9), with Eq.(8) substi-
tuted,
*
=
ω
,
ζ
ζ
, , 0
, and center of mass
coincide and the rotational mode is seismically
not forced. If the modal center of velocity lies
outside the geometrically regular floor plan,
translation dominates and the building is consid-
ered moderately asymmetric. Given the modal
floor displacement with respect to the center of
mass by the (modal) displacement vector, mode
number
j
understood,
φφ=
y
z
A opt
,
A opt
,
opt
opt
K
C
C
K
K
(
)
1
+
µ
1
−
κκ
ω
3
8 1
κκµ
µ
A opt
,
δ
=
=
,
ζ
=
(
)
opt
1
opt
Ω
+
µ
+
(10)
, ,
the coor-
dinates of the modal center of velocity
C
φ φ φ
y
z
u
T
APPLICATION TO MULTIPLE
STOREY ASYMMETRIC BUILDINGS
=
y
,
z
, 0
for small rotations become, see
V
C
C
V
V
Figure 2a,
The damping of excessive vibrations of asym-
metric plan multiple purpose buildings is a com-
mon task in civil engineering. Even discomfort
of people living in lightly damped tall buildings
is observed under wind excitation. Since the direct
φ
φ
φ
φ
r
y
=
y
−
,
z
z S
C
C
C
V
M
V
u
T
r
y S
=
z
+
,
φ
≠
0
(11)
C
u
M
T
u
Search WWH ::
Custom Search