C
ALIFORNIA
I
NSTITUTE OF
T
ECHNOLOGY
E
ARTHQUAKE
E
NGINEERING
R
ESEARCH
L
ABORATORY
I
NTERPRETATION OF
M
ILLIKAN
L
IBRARY
’
S
V
IBRATING
M
ODES
U
SING A
M
AGNETO
C
OIL TO
M
EASURE
P
HASE
S
HIFTS
BY
M
ING
H
EI
C
HENG
,
T
HOMAS
H.
H
EATON AND
M
ONICA
D.
K
OHLER
R
EPORT
N
O
.
EERL
2014-02
P
ASADENA
,
C
ALIFORNIA
D
ECEMBER
2014
Page
1
of
19
Interpretation of Millikan Library’s V
ibrating
Modes
Using a Magneto Coil to Measure Phase Shifts
Ming Hei Cheng
, Thomas H. Heaton, and Monica D. Kohler
Department of Mechanical and Civil Engineering, California Institute of Technology
Abstract
A new set of natural frequencies for the 9
-story
reinforced concrete Millikan Library building
on the Caltech campus is computed
using the observed phase shift between
the
driving force
of a
shaker installed on the building’s roof
and structural response at
resonance. The phase of the
shaker’s output force was recorded by a magneto coil and magnet attached to the shaker
’s rotating
mechanism
, and the phase of the structural response was obtained from acceleration time series
recorded by an accelerometer on the
roof. These new results refute
previous studies’ identification
of the 3
rd
EW and 2
nd
torsional modes which used
spectral analysis of forced and free vibrations
,
but did not consider the phase shift
. In addition, the newly identified 3
rd
EW mode shape is inde-
pendent of the other EW mode shapes, unlike previous findings. This new interpretation is com-
patible with results from subspace system identification
based on two sets of earthquake records
.
Keywords
: Millikan Libr
ary; natural frequency;
phase shift;
magneto coil; subspace system iden-
tification
1. Introduction
Phase
shift
s relate to the time lag
between
an input excitation and a
vibrating mechanical sys-
tem’s response
, and
can be
effective in
identify
ing
the natural frequencies of a dynamic system.
Techniques using measured phase shifts
have been adopted in various engineering applications
such as
the design of beam resonator
s in Microelectromechanical
System
s (MEMS) (e.g. Piekarski
et al., 2001; Scheible
and Blick, 2004) and the design of Electromagnetic Suspension systems
(EMS) in maglev train
s (e.g. Wang et al., 2008; Zhou et al.
, 2011). The
se studies
motivate
the
work presented here in which we use phase shift measurements between a driving force and the
resulting
structural response to identify the normal modes of the Millikan Library building on the
Caltech campus
.
Millikan Library (Fig. 1) is a reinforced
concrete building with nine stories and one level of
basement. It has a stiff core
wall that encases the elevators and a stairwell. In addition, there are
30-
cm thick shear walls on the east and west sides of the building which make the building very
stiff in the NS direction (the first NS modal frequency is 1.7 Hz). EW vibrations are resisted by a
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2
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19
combination of the core wall and massive moment frames (the first EW frequency is 1.
15
Hz). A
Kinem
etrics
VG
-1 shaker
with counter
-rotating weight baskets
has been installed on the roof for
the past 40 years and it has been used frequently for a variety of research and class projects. Since
Millikan Library was constructed
in 1966, its natural frequencies have been monitored. The NS,
EW and torsional natural frequencies of the first two modes obtained from forced vibration tests,
ambient vibration tests, and several earthquakes are well documented (e.g. Udwadia and Trifunac,
1974; Foutch and Jennings, 1978; Luco et al., 1987). Todorovska (2009a, b) estimated the 1
st
mode
natural frequencies
using the spectral
peak of the
transfer function between the roof response and
the ground response for
four
earthquakes. Michel and Gueguen (2010) adopted a pseudo-
Wigner
-
Ville method to address the variation of 1
st
mode natural frequencies
during earthquakes.
Bradford
et al. (2004) and
Clinton et al. (2006) estimated the higher mode natural frequencies through
forced
vibration experiment
s. The frequencies were identified based on the spectral
peak
s of roof dis-
placement records
.
The previously proposed 3
rd
EW mode for Millikan Library is problematic because the corre-
sponding mode shape is not independent of the other EW modes. The 1
st
mode has no zero-
cross-
ing
s, the 2
nd
mode has one zero
-crossing,
but
the 3
rd
mode
also
has
only
one zero-
crossing. These
previously proposed mode shapes are shown in Fig. 2.
In this study, a new set of natural frequen-
cies for the
Millikan Library building
is proposed using the observed phase shift between the driv-
ing force and the building’s
response
at resonance,
a parameter that was not considered in the
previous studies. The results presented here show that the newly identified 3
rd
EW mode
shape
is
independent of the other EW modes.
In this study, the phase shift
s were
measured by recording the voltage from a magneto coil and
magnet that was attached to the shaker producing the forced vibrations. The results from the phase
shift experiment can serve
as an independent measure of natural frequencies. Th
e new interpreta-
tion
of natural frequencies presented here are
compatible with the results from deterministic
-sto-
chastic
subspace identification. S
ubspace system identification using both input and output data
has been used
extensively
to
extract modal parameters, including natural frequencies, damp
ing
ratios and mode shapes from civil engineering structures (e.g. Overschee, 1994; Ljung, 1999; Guy-
ader and Mevel, 2003; Moaveni and Asgarieh, 2012)
.
2. Using p
hase s
hifts to
identify m
odal f
requencies
Measurements of the spectral
amplitude of the roof motions of Millikan Library
have been
used to identify natural frequencies
by other researchers
. While this methodology is clearly effec-
tive for identifying the 1
st
NS, EW, and torsional modes of the building (Li et al., 2002; Arakawa
and Yamamoto, 2004; Bradford et al., 2004), identifying the higher modes from spectral peaks is
ambiguous. Therefore, we use a phase shift experiment to assist in the identification
of
higher
modes.
Assuming Rayleigh damping
, the time
-dependent
response o
f a single building mode is
Page
3
of
19
identical to that of a single
-degree
-of-freedom damped linear oscillator
(Taylor, 2005;
Chopra,
2007)
. Consider a single
-degree
-of-freedom damped oscillator that is driven by a sinusoidal driv-
ing force. The phase shift
δ
that describes the lag (in degrees) of the oscillator’s motion behind the
driving force is
훿훿
=
푎푎푎푎푎푎푎푎푎푎푎푎
(
2훽훽훽훽
훽훽
0
2
−훽훽
2
)
(1)
where
β
is the damping constant
,
휔휔
0
is the natural frequency of the system, and
ω
is the driving
frequency.
When
휔휔 ≪휔휔
0
, the oscillator’s response
is in phase with the driving force and
δ
is zero or
negligible. As
ω
increases,
δ
also increases. When
휔휔
=
휔휔
0
, the argument of the arctan function in
Eq.
1 is infinity and the oscillator
lags 90
o
behind the driving force; if the forci
ng function is
sinusoidal, then the response is cosinusoidal. When
휔휔
>
휔휔
0
, the argument of the arctan function
is negative and becomes zero as ω increases until the os
cillator
becomes 180
o
out of phase with
the driving force. As the
damping
of the system
is decreased, the rate of change of the phase be-
comes larger at the resonant frequency. However, at the resonant frequency, the phase lag is always
90
o
(Fig.
3).
For a building with multiple
independent
vibrating modes, the phase lag will always
be 90
o
at each of its natural frequencies. Even w
ithout the assistance
of mode shapes, vibrating
modes of a building can be easily identified by the observed phase shift between the driving force
and the structural response.
Forced
vibration experiments have been performed
for decades using the shaker
installed on
the roof of Millikan L
ibrary. However, no previously published work has compared the phase lag
between the shaker and the building
’s acceleration
response
. To measure the shaker’s cyclic mo-
tion,
in light of the problem with the orthogonality of mode shapes for previously proposed modes,
we constructed a simple magneto device to record
shaker phase data (
Fig.
4). The magneto coil
was affixed to the cage of the shaker and a tiny magnet was attached
to the bucket of the shaker.
A sharp impulse voltage was generated when the magnet passed under the magneto device. This
occurred every time the two counter
-rotating buckets overlapped,
generating
the maximum input
force from the shaker.
For a single
-degree
-of-freedom system, the phase difference between the harmonic input driv-
ing force and the system response increases
from 0
o
to 180
o
as the driving
frequency increases
from just below to just above the natural frequency. It is not as easy to identify such phase shifts
from 0
o
to 180
o
for the higher modes of a multi
-degree
-of-freedom system, especially when there
are several modes with similar frequencies. Since we are really only interested in identifying modal
frequencies, we are only interested in find
ing cases when the building motion is 90
o
out of phase
with the shaker force.
Page
4
of
19
A
shaker
frequency sweep experiment was conducted on July 31, 2011
for Millikan Library
.
No
lead weight
s (i.e.,
the buckets that hold the lead weights were empty
) were
used
in the shaker
during the experiment. The driving frequency was increased from 0.9
Hz to 9.7
Hz (the maximum
allowable)
, incrementally by 0.05
Hz. The magneto device time series voltage data was
recorded
on a 24-
bit datalogger which uses GPS
-based
time sync
hronization
via an external
antenna
. The
recorded data was then compared to the building roof response which was extracted from 24
-bit
accelerometer data (station “
MIK
” provided by the Southern California Earthquake Data Center
archive
).
The time history plots from the experiment are shown
in
Fig
s. 5- 10. The acceleration records
are filtered using a second
-order Butterworth bandpass filter with a half
-width of 0.15
Hz centered
on the excitation frequency. The vertical dotted lines indicate
the time at which the shaker gener-
ates the maximum force, i.e. the peaks of the sinusoidal output. At resonance, the sharp voltage
spike
occurs when the building response has zero amplitude; i.e. the building and shaker are 90
o
out of phase. Note that the
figures are plotted with different time scales because building response
takes less time to complete one cycle at higher frequencies than at lower frequencies.
Fig
. 5
shows the recorded building acceleration from the roof (station “MIK”) when the shaker
is driven at 1.15 Hz, the frequency of the 1
st
EW mode. Fig.
5(
a) and 5(c)
display the EW and NS
acceleration
time series
respectively, recorded during EW forcing by
the shaker at maximum ex-
citation of that
mode
. Fig.
5(b)
and 5(d) similarly display the EW and NS acceleration time series
for
NS
forcing by
the shaker, which is nearly orthogonal to the EW mode. As expected, driving
the shaker at the building’s 1
st
EW mod
e frequency in the EW direction results in a clear 90
o
phase
shif
t between the magneto output and the building motion. All other phase shifts are zero, except
for the NS building response during the EW forcing by the shaker. This illustrates
that the building
does not oscillate in a pure
ly EW direction
; it exhibits a mi
nor component of
NS motion
.
Similarly, the data
for
the 1
st
NS mode at 1.7
Hz
(Fig.
6) shows that the building does not
oscillate
in pure NS motion, but also exhibits a minor component of
EW
motion.
Fig.
7 shows the
1
st
torsional
mode
that
occurs at 2.4
Hz.
The shaker is located
approximately
6.1
m
south
of the
center of mass of the building, as shown in Fig.
1.
Due to the off
-center location of the shaker, it
is more likely to trigger the torsional response of the building during the EW forcing than duri
ng
NS forcing
. Since the driving force is not perfectly aligned in any one direction, an eccentric force
is generated in the EW direction when the shaker is operated in the NS direction, and vice versa.
Both the NS and EW building response
s during the EW excitation of the shaker at the 1
st
torsional
modal frequency show a clear 90
o
phase shift relative
to the magneto output. No phase shift occurs
during the NS forcing of the shaker because the eccentric force is exciting the building at a no
n-
resonance frequency in the translational NS direction. Fig
s. 8- 10
show
the
2
nd
EW mode that oc-
curs at 4.7 Hz, the 2
nd
NS mode at 7.2 Hz, and the 2
nd
torsional mode at 7.7 Hz. T
he phase shift
experiment results are summarized in Table 1
. For simplicity, a
phase shift of 0
o
is assigned for all
Page
5
of
19
the non-
resonant situations (i.e. when the two oscillations are either 0
o
or 180
o
out
-of-phase) (
Table
1).
3. Deterministic
-stochastic s
ubspace i
dentification
We now verify our newly identified natural frequencies of Millikan Library
with
a commonly
used input
-output system identification method. This is to provide an independent measure of
building properties in order to validate the results using the phase shift method. D
eterministic
-
stochastic subspace identification assumes that
the d
ynamic behavior of a building can be modeled
by an n
th
-degree
-of-freedom spring
-mass system
as
푀푀푈푈
̈
(
푎푎
)
+
퐶퐶푈푈
̇
(
푎푎
)
+
퐾퐾푈푈
(
푎푎
)
=
퐹퐹
(
푎푎
)
(2)
where
푀푀푀푀ℝ
푛푛푛푛푛푛
is
the
mass matrix
;
퐶퐶푀푀 ℝ
푛푛푛푛푛푛
is the
damping matrix
;
퐾퐾푀푀 ℝ
푛푛푛푛푛푛
is the
stiffness ma-
trix
;
푈푈
(
푎푎
)
푀푀ℝ
푛푛푛푛1
is the
displacement vector
; and
퐹퐹
(
푎푎
)
푀푀ℝ
푛푛푛푛1
is the
excitation force as a function
of continuous time
t
.
Eq.
2 describes a model in continuous time, whereas measurements are always ob
tained as
discrete time samples
. By applying di
scretization and
including
modeling noise
, the linear time
-
invariant structural model can be converted to a discrete-
time subspace model a
s
푥푥
푘푘+1
=
퐴퐴푥푥
푘푘
+
퐵퐵푢푢
푘푘
+
푤푤
푘푘
(3)
푦푦
푘푘
=
퐶퐶푥푥
푘푘
+
푣푣
푘푘
(4)
where
푥푥
푘푘
푀푀ℝ
푛푛푛푛1
is
the
discrete state vector
;
푦푦
푘푘
푀푀ℝ
푁푁푛푛 1
is
the
sampled output vector
;
푢푢
푘푘
푀푀ℝ
푁푁푛푛 1
is
the sampled input vector
;
푤푤
푘푘
푀푀ℝ
푛푛푛푛1
is the process noise vector
( mainly due to modeling inaccura-
cies
);
푣푣
푘푘
푀푀ℝ
푁푁푛푛 1
is
the
measurement noise vector
(mainly
due to sensor inaccuracies
);
퐴퐴푀푀 ℝ
푛푛푛푛푛푛
,
퐵퐵푀푀 ℝ
푛푛푛푛푛푛
, and
퐶퐶푀푀 ℝ
푁푁푛푛푛푛
are
the
system
matrices;
k
is the time sampling
instant;
n
is the system order;
and
N
is the number of input/
output
sample
s. The
state
noise
is assumed to be zero
-mean Gaussian
white
noise
. Detailed derivations of the mathematical setup
can be found in Overschee and Moor
(1994),
Ljung
( 1999), and
Guyader a
nd Mevel
(2003).
Theoretically, the properties of a system with
n
degree
s-of-freedom can be modeled by a
sub-
space model with order
N=2n
. In practice, determination of the most probable model order is not
trivial due to noise present
in the system, including modeling inaccuracies and meas
urement noise
.
A common practice is to use
a stab
ilization plot to distinguish stable modes
of the system
from
spurious
modes
(Peeters and Roeck, 2001; Altunisik et al., 2011; Döhler et al., 2011).
An
N
th
order
subspace system contains
N
numbers of modes
. Modal parameters of the system for the
i
th
mode
can be obtained from Eq
s. 5-
7.
The
modes
of each system order
will be compared to those with a
Page
6
of
19
lower
system
order. If the frequency, damping ratio
, and associated mode shape (usually calculated
as modal assurance criterion
values
) differences are within certain limits, the particular
mode
is
lab
eled as stable
. Modal parameters (frequency
f
i
, damping ratio
ζ
i
, and mode shape
Φ
i
) can be
obtained from the eigenvalue
λ
i
and eigenvector
φ
i
of system matrix
A
as
푓푓
푖푖
=
|
푙푙푛푛
(
휆휆
푖푖
)
|
2휋휋
훥훥훥훥
(5)
휁휁
푖푖
=
−2휋휋
푅푅푅푅
[
|
푙푙푛푛
(
휆휆
푖푖
)
|
]
푓푓
푖푖
훥훥훥훥
(6)
훷훷
푖푖
=
퐶퐶휑휑
푖푖
(7)
Seismic records of Millikan Library
during
two
earthquakes
, the M4.4 August 9, 2007 Chats-
worth
CA
earthquake and the M7.2
April
4, 2010
El Mayor Cucapah (Baja California)
earthquake,
were selected for system identification
. The data were recorded by Kinemet
rics FBA
-11 sensors
on a 16-
bit real
-time datalogger monitoring system (RTMS) of Digitexx Data Systems, Inc. The
channels
in both the NS and EW directions
at each floor
were used
. The channels at the basement
level
were
used as sample input
, while
the
other channels from the 1
st
floor to the roof
were used
as sample output. The data were downsampled from 200 to 100 samples per second incorporating
a Chebyshev Type I filter.
Stabilization plots with system order from 5
to
80
for the selected earthquakes are shown in
Fig
s. 11-
12.
A mode is called stable in this study if the frequency
difference between two succes-
sive orders is less than
1%
, the damping ratio difference is less than
5%, and the mode shape
difference is less than
2% using the
modal assurance criterion
. A mode that remains stable for six
successive system orders is considered a physical mode for Millikan Library
. The identified modes
for
the 2007 Chatsworth
CA
earthquake are 1.16
Hz
, 1.75
Hz
and
4.76
Hz
, 7.03
Hz and 10.0
Hz.
The identified modes for the 2010 El Mayor Cucapah earthquake are 1.09
Hz, 1.66
Hz, 4.62
Hz
,
6.98
Hz
and 10.5
Hz
. The associated damping ratio for each mode identified by the subspace
method is less than 5%
, which is within the expected range for this type
of building
. Results are
summarized in Table 2.
4. Discussion
Table
2 compares the previously proposed (Bradford, 2006) and the newly identified natural
frequencies for Millikan Library. Since the frequency sweep experiment was performed at i
ncre-
ment
s of 0.05
Hz, the result
s from phase shift analysis are
limited
to the resolution of
this frequency
increment
, but they can be used to confirm or refute
the previously reported natural frequencies
.
The
1
st
mode
frequencies
from
the
phase shift experiment (~1.15 Hz for EW, ~1.7 Hz for NS, and
~2.4 Hz for torsion) and subspace system identification (1.09-
1.16
Hz for EW and 1.66-
1.75
Hz
for NS
; 1st torsion was not identified
) match well with the previously proposed frequencies
. Sim-
ilarl
y, t
he 2
nd
mode
frequencies
in translational motion from the phase shift experiment (~4.7
Hz
Page
7
of
19
for EW and ~7.2
Hz for NS) agree with
the results
from subspace system identification (4.62-
4.76
Hz for EW and 6.98-
7.04
Hz for NS).
The
result
s validate
the previously identified natural fre-
quencies for the 1
st
mode (NS, EW, and torsion) as well as
the 2
nd
NS and EW mode
s. The phase
shift results refute
the previously identified 3
rd
EW mode and 2
nd
torsional mode, because these do
not generate a 90
o
phase
shift. The
2
nd
torsional mode of ~7.7
Hz
identified
in this study by the
phase shift experiment was previously misinterpreted as the 3
rd
EW mode.
Finally, a 3
rd
EW mode
at 10.0-
10.5 Hz
is suggested by the subspace system identification.
In this newly defined set of natural frequencies, the mode shapes are
shown to be
independent
of
each
other
(Fig
s. 13-
15)
. The newly defined EW mode shape
s have no zero-
crossing
s for the
1
st
mode
, one zero-
crossing
for
the 2
nd
mode
and two zero-
crossings
for
the 3
rd
mode
. This
finding
also
supports the implication that the previous study
misinterpreted the 2
nd
torsional and 3
rd
EW
mode
s. Note that
this frequency (10
.0-10.5 Hz)
exceeds the safe operation of the shaker and cannot
be verified by the phase shift analysis
. There is a minor variation in the identified natural frequen-
cies
between the two proposed methods, but this is
not uncommon.
Previous studies show that the
natural frequencies of Millikan Library during forced vibrations
are 4% lower than those during
ambient vibration
(Bradford, 2006; Clinton et al.
, 2006)
. Also, lower natural frequenc
ies
are
de-
tected when a larger shak
ing
force
is used in the forced vibrations
. In this study, earthquake data
are
used in the subspace sys
tem identification and forced vibration data
are
used in the phase shift
analysis.
Clinton et al. (2006) found that a
seismic event may cause a drop
of up to 30%
in natural
frequenc
y.
In a separate study
(Cheng and Heaton, 2013)
, a simple
Timoshenko beam (combined effect
of shearing and bending) with rocking
-type soil
-structure interaction is used
to estimate first
-order
building response
using the frequency ratios
. Unlike the translational modes of Millikan
Library
which
appear
to cause global bending as well as shearing, the torsional modes of a structure pri-
marily involve shearing along the vertical axis
; thus
the frequency ratios should theoretically ap-
proach that of a shear beam. That is, for a uniform shear beam, if
ω
0
is t
he frequency of the 1
st
mode, then the frequency of the 2
nd
mode is 3
ω
0
and the 3
rd
mode is 5
ω
0
. The ratio of the
first two
torsional frequencies
(2.4
Hz and 7.7
Hz) of Millikan Library is 3.2
Hz
, which is close to the
behavior of a uniform shear beam. The
third torsional mode is expected at 5
ω
0
, or 12.5
Hz
. Solely
based on
the
1
st
and 2
nd
natural frequencies of Millikan Library, the 3
rd
natural frequency is esti-
mated to be 15.3
Hz for the
NS direction and 10.0
Hz for the EW direction using this simple
Timoshenko beam
approach
. The
predict
ion
in EW direction agrees
with
the
result
from
the sub-
space system identification
(Table 2)
. There are some stable modes identified by the subspace
method at around 15
Hz, but they do not appear in a consistent way to be quantified as a physical
mode.
Frequencies above 10
.0 Hz are beyond the safe operation of the shaker
, so
future
investiga-
tion is needed for conclusive verification of
the 3
rd
vibrati
onal
modes
.
Page
8
of
19
5. Conclusion
s
The natural frequencies of
the 9
-story
reinforced concrete
Millikan Library
building
have been
revisited
using
a simple and robust phase shift analysis. This method can successfully identify
higher modes of the building and it complements
the traditional subspace system identification
which doe
s not use phase shift measurements. To meas
ure the shaker’s cyclic motion, a simple
magneto device wa
s constructed to record
shaker phase data. A magneto coil was affixed to the
cage of the shaker and a tiny magnet was attached to the bucket (for holding lead weights) of the
shaker. The phase shift between the building’s response and the driving force was measured from
a voltage jump generated each time the magnet passed under the magneto device when the counter
-
rotating buckets overlapped.
A new set of natural frequencies for Millikan Library is proposed,
and
the
associated
mode
shapes are shown to be independent of each other
. The new frequencies are 1.15 Hz
(1
st
EW), 1.7
Hz (1
st
NS),
2.4
Hz (1
st
torsion),
4.7
Hz (2
nd
EW)
, 7.2
Hz (2
nd
NS)
and 7.7
Hz (2
nd
torsion)
. The
newly identified 2
nd
torsional mode was misinterpreted in previous studies as the 3
rd
EW mode.
This interpretation is compatible with the results from subspace system identification based on two
sets of earthquake
records.
Th
e subspace method also suggests that the 3
rd
EW mode may occur
in
the
range of 10
.0-
10.5 Hz.
Acknowledgements
This wor
k is partially supported by the
National Science Foundation (EAR
-1027790),
George
Housner Earthquake Engineering Research Endowment (EAS
-41212), Fred L. Hartley Family
Foundation, and Croucher Foundation. Their support is gratefully acknowledged.