introduction to vibration monitoring [compatibility mode]
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Vibration monitoringVibration monitoring
. . .Istanbul Tecnical University
undesbakir ahoo.com
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http://faculty.uml.edu/pavitabile/22.515/ME22515_PDF_downloads.htm
Safak E., Structural monitoring, what is it, why is it done, how is it done, and what is
, , -2007, Istanbul, Turkey
Celebi M. Seismic instrumentation of buildings, USGS Open-File Report 00-157,.
Heylen W., Lammens S. And Sas P., Modal Analysis Theory and Testing, KatholiekeUniversiteit Leuven, 1997.
Ewins D.J., Modal Testing, Theory, Practice, and Application (MechanicalEngineering Research Studies Engineering Design Series), Research Studies Pre; 2edition (August 2001) ISBN-13: 978-0863802188
Maia, N. M. M. and Silva, J. M. M.Theoretical and Experimental Modal AnalysisResearch Studies Press Ltd,, Hertfordshire, 1997, 488 pp.,ISBN 0863802087
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e a m o v ra on mon or ng s o escr e a s ruc ure n erms o s
modal parameters which are the frequency, damping and mode shapes.
If we explain modal analysis in terms of the modes of vibration of a simpleplate:
Suppose we apply a sinusoidal force. We will change the rate of oscillation
of the frequency but the peak force will always be the same. We will alsomeasure e response o e p a e ue o e exc a on w an
accelerometer attached to one corner of the plate.
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we measure e response o e
plate, we will notice that the
amplitude changes as we changethe rate of oscillation of the in utforce. There will be increases aswell as decreases in amplitude atdifferent points as we sweep inme.
The response amplifies as weapp y a orce w a ra e ooscillation that gets closer andcloser to the natural frequency (or
and reaches a maximum when therate of oscillation is at theresonant frequency of the system.
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e t me ata prov es very use u n ormat on. ut we ta e t e
time data and transform it to the frequency domain using the Fast
Fourier Transform then we can compute something called the
frequency response function .
Now, there are some very interesting items to note. We see that
frequencies of the system. Now, we notice that these peaks occur atfrequencies where the time response was observed to have
input excitation.
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ow, we over ay t e t me trace w t t e requency trace w at we
will notice is that the frequency of oscillation at the time at which the
time trace reaches its maximum value corresponds to the frequency
where peaks in the frequency response function reach a maximum.
the frequency at which the maximum amplitude increases occur or
the frequency response function to determine where these natural
requenc es occur. ear y e requency response unc on s eas er
to evaluate.
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The figure shows the deformation patterns that will result when the
excitation coincides with one of the natural frequencies of the
system.
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We see that when we dwell at the first natural frequency, there is a first
bending deformation pattern in the plate shown in blue. When we dwell at
the second natural frequency, there is a first twisting deformation pattern in
the plate shown in red.
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When we dwell at the third and fourth natural fre uencies, the second bendin and
second twisting deformation patterns are seen in green and magenta, respectively.
These deformation patterns are referred to as the mode shapes of the structure.
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How many points are enough for a vibrationHow many points are enough for a vibration
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measurement?measurement?
or a tota o measurement po nts, we can see t at t ere are
sufficient number of points to describe the mode shape for each
mode.
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How man oints are enou h for a vibrationHow man oints are enou h for a vibration
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measurement?measurement?
For a total of 5 measurement oints alon one ed e of the late if
we compare mode 1 and 3, we see that there are not enough points
to adequately describe the mode shape for each mode. The same.
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measurement?measurement?
If we increase the number of measurement points to 15, we see
that the modes can be measured well only if the measurement.
figure, then it will be very hard to distinguish between modes 1 and
3. The mode shapes look almost the same.
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measurement?measurement?
If we only take measurements along the front and back edges of theplate, then it would be very hard to distinguish between the first rigid
body mode and the first flexural mode.
From all these simple examples above, it becomes obvious that weneed a distribution of points located appropriately such that each
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measurement?measurement?
If I am only interested in characterizing modes 1 and 2, then
possibly I could get a fairly good decription with only 6 points as
shown but fewer oints than that would be difficult es eciall if we
needed to distingish the flexible modes from the rigid body modes.
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and the frequency domain and the modaland the frequency domain and the modal
First lets consider a simple
that the beam is excited by a
pulse at the tip of the beam.
The response at the tip of the
beam will contain the response
o a e mo es o e sys em
(shown in the black time
response plot); notice that therev
different frequencies.
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and the frequency domain and the modaland the frequency domain and the modal
We know that the cantilever beam
of vibration. At each of these
natural frequencies, the structural
definite pattern, called a modeshape. For this beam, we see that
shown in blue, a second bending
mode shown in red, and a third
bendin mode shown in reen.
Of course there are also other
higher modes not shown but only
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and the frequency domain and the modaland the frequency domain and the modal
Now, the physical beam could
also be evaluated using an
finite element model (shown in
black in the upper right part of the
.
This model will generally beevaluated using some set of
interrelationship, or coupling,
between the different points, or
de rees of freedom used to modelthe structure. This means you pull
on one of the dofs in the model,
the other dofs are also affected
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and also move.
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and the frequency domain and the modaland the frequency domain and the modal
This coupling means that the
equations are more complicated in
system behaves. As the number
of equations used to describe the
,
complication in the equationsbecome more involved. We often
use matrices to hel or anize all
of the equations of motion
describing how the system
behaves which looks like:
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and the frequency domain and the modaland the frequency domain and the modal
Mathematically, we perform
and use the modal transformation
equation to convert these coupled
single degree of freedom systemsdescribed by diagonal matrices of
,
modal stiffness in a new
coordinate system called modal
s ace described as:
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and the frequency domain and the modaland the frequency domain and the modal
We can see that the
space to modal space using the
modal transformation equation is a
complicated set of coupledphysical equations into a set of
systems.
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and the frequency domain and the modaland the frequency domain and the modal
And we see in the figure that the
down into a set of single dof
systems where the single dof
blue, mode 2 is shown in red andmode 3 is shown in green.
Modal space allows us to describe
the system easily using simple
y .
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and the frequency domain and the modaland the frequency domain and the modal
Now lets go back to the time and
frequency responses shown in.
response can be obtained from
the contribution of each of the
.
in black comes from thesummation of the effects of the
res onse of the model shown in
blue for mode 1, red for mode 2,
and green for mode 3. This
applies whether I describe thesystem in the time domain or the
frequency domain. Each domain is
equivalent and just presents the
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data from a different viewpoint.
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and the frequency domain and the modaland the frequency domain and the modal
Please note that we have only
shown the magnitude part of the,
complex which is correctly
displayed using both magnitude
parts of the FRF. Since we can break the analytical
systems, we could determine the
FRF for each of the single dof
s stems as shown with mode 1 inblue, mode 2 in red, and mode 3
in green.
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and the frequency domain and the modaland the frequency domain and the modal
We could also determine the time
response for each of these single
Or we could simply inverse
Fourier transform the FRF for
.
Or we could also measure theresponse of the beam at the tip
response of each modes of the
system, and we we would see the
the system with mode 1 in blue,
mode 2 in red, and mode 3 in
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and the frequency domain and the modaland the frequency domain and the modal
As a result, we see that there is no
difference between the time, ,
space and physical space. Each
domain is just a convenient way
.
However, sometimes one domainis much easier to see things than
. ,
total time response does not
clearly identify how many modes
there are contributin to theresponse of the beam.
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Single degree of freedomSingle degree of freedom
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functionfunction The force equilibrium for a viscously damped SDOF structure:
)()()()( tftKxtxCtxM =++ &&&
Transforming this time domain equation into the Laplace domain:
2 =
FXZ
or
=
where Z is the dynamic stiffness. Inverting Z gives the transfer function:
)/()/(
/1
)(
)()(
2MKpMCppF
pXpH
++==
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frequencies, damping ratiosfrequencies, damping ratios e enom na or o e equa on
/1)(
)( 2MpX
pH ==
is referred to as the system characteristic equation. Its roots are called the
)/()2/()2/( 22,1 MKMCMC =
t ere s no amp ng, t e system un er cons erat on s a conservat ve
system (C=0).The undamped natural frequency (rad/s) is then defined
as:
1=
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,,
frequencies, damping ratiosfrequencies, damping ratios Depending on the value of the damping ratio, the systems are classified as
overdamped (1>1), critically damped (1=1) or underdamped (1
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,,frequencies, damping ratiosfrequencies, damping ratios The equation
2,1 KCC =
111111 * jj =+=
Where 1 is the damping factor and 1 is the damped natural frequency
1
2
111 )1( += j
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With the knowledge of the equation
111111 * jj =+=
/1)()(
2
MpXpH ==
becomes:
))((
/1)(*
1 1
=pp
MpH
Applying the theory of partial fraction expansion yields:
*
A
In this formula A and A * are the residues.
1
1*
11 2with
)()()( 1
jA
pppH =
+
=
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The previous section discussed the relation between input (force) andoutput (displacement) of a single degree of freedom system in the Laplace
domain.
This relation can also be expressed in the frequency domain. The transfer
function evaluated along the frequency axis (j) is called the frequency
response function (FRF).
)()()()(
*
1
*
1
1 1
+
==
= jA
jAHpH
jp
The FRF is a subset of the transfer function. The contribution of the
complex conjugate part (or negative frequency part) is negligible around
. ,
)( 1
= A
H
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)()( *
*
1 1
+===
AA
jHpH jp
yields the expression in the time domain: the impulse response function.
***
The residue A1 is the real part of the pole which defines the initial
11 11 eeeeet +=+=
, 1and 1 is the frequency of oscillation.
e mpu se response o a sys em s e sys em response o a rac
impulse at time t=0.
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Multi degree of freedomMulti degree of freedom
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functionfunction e equa on o mo on s:
{ } { } { } { }fxKxCxM =++ ][][][ &&&
If we transform this time domain equation into the Laplace domain (variable
p), assuming the initial displacements and velocities are zero yields:
[ ] [ ] [ ]{ } { })()()(2
FXZpFpXKCpMp
==++
where [Z(p)] is the dynamic stiffness matrix. The inverse of [Z(p)] is [H(p)]
)()()( pFpHpX =
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functionfunction
Standard calculus proves that the inverse of a matrix can be calculated
from its adjoint matrix:
[ ] [ ] [ ]
)(
)()()(
1
pZ
pZadjpZpH ==
Where adj([Z(p)]) is the adjoint matrix of [Z(p)] which can be expressed as.
columnandrowwithout)],([oftdeterminanthe:
][)])(([
jipZZ
ZpZadj
ji
t
jiij=
)]([oftdeterminanthe:)(
oddisif-1even;isif,1
pZpZ
jijiij +=+=
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functionfunction The frequency response function can be written as:
*m AA
)()( *1 kkkjp
jjp
+
==
=
=
)()()(
*
*
1 k
ijkijkm
k
ijja
jajh
+
=
=
hij() means a particular output response at point i due to an input force at pointj.Since [M], [C], [K] are symmetric, [H(j)] is also symmetric. This implies that
= impacting point i and measuring the response at pointj and get exactly the sameFRF as impacting pointj and measuring the response at point i. This is what ismeant by reciprocity.
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ResiduesResidues The residues are directly related to mode shapes and a scaling factor as:
This shows that the frequency response function can be written in terms ofres ues.
When written as a mode sha e, then it becomes ver clear that if the valueof the mode shape at the reference point is zero (or almost zero) then thatmode will not be seen in the frequency response function.
Always select a reference point where all the modes can be seen all thetime from that reference point.
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ResiduesResidues Never select the reference point at the node of a mode!
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FRFs can be generated from residues and poles. The residues are directlyof the system.
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First let's start with an analyticalelement model shown. Basically,
we use the FEM to approximate a
interconnected by springs to
represent the physical system.
Since the analytical approximationis described in terms of a force
described in the system, we end up
with one equation for each mass (or
approximate the system.
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Since we need many small littledescribe the system, I end up with
many equation and unknowns.
Right away, it becomes convenient
to describe all these equations
us ng ma r ces. ow once ave
assembled all these equations, amathematical routine called an
the system in simpler terms - the
system's frequencies and mode
.finite element process.
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I can take those same equationsLaplace domain.
Now in the Laplace domain, we
have, [B(s)], the system equation
and its inverse,[Hs)], the system
rans er unc on. ow we now a
this inverse is the adjoint of thesystem matrix (or the cofactors of
determinant of the system matrix.
This inverse is described in all
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Now another im ortant relationshi
is the Frequency Response
Function, FRF. This is the system
transfer function evaluated alongthe jaxis. The FRF is actually a
matrix of terms, [H(j)].
Well, since we are dealing with a
matrix, it is convenient to identify
in ut-out ut measurements with a
subscript. So a particular output
response at point 'i' due to an input
force at point 'j' is called hij(j). .
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Now what we need to realize is that
those FRFs that were generated
(synthesized) contain information
relative to the systemcharacteristics.
generated from residues and poles.
And that the residues are directly
related to the mode sha es and the
poles are the frequency and
damping of the system.
.
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If there were some other wa to
estimate those FRFs without assuming
physical properties then I could employ
the modal parameter estimationtechniques to extract the desired
information. This is where modal
testing comes in.
Basically, my structure is excited with
some measured force. The responseof the system due to the applied force
is measured along with the force. Now
this time data is transformed to the
frequency domain using the FFT and
basically a ratio of output response to
input force is computed to form an
approximation of the FRF.
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So we could measure one in ut-out ut
FRF based on this approach. If we
used a shaker to excite the structure
and move the accelerometer to manypoints then we could measure a
column of the FRF matrix. So the big
advantage of making measurements is
that I measure the response of the
system due to the applied force Idon't ever make any assumptions as to
e mass, amp ng an s ness o e
system - and I avoid any erroneous
approximations I may make. Of
u , umake very good measurements
otherwise I will distort my system
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of structural systemsof structural systems
There are three main approaches to evaluate seismic behavior
.
1. Laboratory Testing
2. Computerised analysis
3. Natural Laboratory of the Earth:Integral to the naturallaboratory approach is the advance instrumentation of selected
earthquakes.
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The methods used in studying structural response records are quite diverse:
a Mathematical modelin finite element models var in from crude to verdetailed, subjected to timehistory, response spectrum or modal analyses).The procedure requires the blueprints of the structures which may not bereadily accessible;
(b) System identification techniques: single input/single output or multiinput/multi output. In these procedures, the parameters of a model are
,
(c) Spectral analyses: response spectra, Fourier amplitude spectra,
, , - ,functions ( ) [using the equation : 2xy (f) = S
2xy (f)/ Sx (f)Sy (f)] and
associated phase angles
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Improve our understanding ofthe behavior and potential fordamage in structures under the
d namic loads of earth uakes.
Emergency response : Adetailed real time hazard analysisn ur an env ronmen s
Improvement in mathematical
program should provide enoughinformation to reconstruct theresponse of the structure in
response predicted bymathematical models and thoseobserved in laboratories.
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Quantify the interaction of the soil and the structure: The nearby free-
field and round-level time histor should be known in order to uantif theinteraction of soil and structure.
Determine the importance of nonlinear behavioron the overall and localresponse o e s ruc ure,
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as the response increases and determine the effect of this nonlinearbehavior on the frequency and damping
Correlate the damage with inelastic behavior
-building response damage
Facilitate decisions to retrofit/strengthen the structural system as
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objective way following big earthquakes and aftershocks
retrofitted in the structure
Evaluating whether the intended benefit from retrofitting is
Determine the maximum interstory drifts in the structure
Providing an early warning system for traffic closure when thebridges are subjected to excessive wind loading
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e most w e y use co e n t e
United States, the Uniform
Building Code (UBC-1997 andprior editions), recommends, for
seismic zones 3 and 4 a minimum
of three accelero ra hs be
placed:
an aggregate floor areas of 5500m2 or
more
in every building over ten stories
regardless of the floor area.
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-illustrated in Figure.
Experiences from pastearthquakes show that the UBCminimum guidelines do not ensuresu c en a a o per orm
meaningful model verifications.
As an example, three horizontalaccelerometers are required todefine the horizontal motion of afloor (two translations andtorsion).
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Rojahn and Matthiesen(1977) concluded that the
high-rise building can bedescribed by the
modes of each of the threesets of modes (two
.
Therefore, a minimum of 12
necessary to record thesemodes.
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If vertical motion and rocking are expected to be significant and
,required at the basement level.
s type o nstrumentat on sc eme s ca e t e ea extens veinstrumentation scheme herein and is illustrated in the Figure.
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center of the diaphragm as well as the edges.
Performance of base-isolated systems and effectiveness of theisolators are best captured by measuring tri-axial motions at top
and bottom of the isolators as well as the rest of the
superstructure.
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Instrumented
1. Structural parameters: the construction material,
, , ,
-.
a. Severit -of-shakin factor to be assi ned to eachstructure on the basis of its closeness to one or more ofthe main faults within the boundaries of the area
. . ,Andreas, Hayward, and Calaveras faults areconsidered).
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Instrumented
b. Probability of a large earthquake (M = 6.5 or 7 occurring on thefault(s) within the next 30 years was obtained. The purpose of this
recording useful data within an approximately useful life of astructure.
c. Expected value of strong shaking at the site, determined as the
product of a and b.
3. Building usage, functionality, occupancy and relevance to life safetyrequirements following damaging earthquakes.
4. Other parameters of interest to owners or public officials.
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Instrumented
Once the particular structure to be instrumented isidentified, the engineering staff
obtains instrumentation permits for selected structures
gathers information relative to the project including
ambient response studies.
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Field Station If physically feasible, it is advisable to include into the
instrumentation scheme, a building specific free-field station.
Such a free-field station is usually deployed at a distance greaterthan 1.5-2 times the height of the nearest/tallest building. This isdue to the desire that motions recorded by a free-field station should
.
In general, free-field and ground-level motions should be known in.
However, data recorded at building specific free-field stations can be
well as ground motion studies including development of attenuationrelationships and quantification of site response transferfunctions and characteristics.
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Tests on Existin tructures to
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Determine Dynamic Characteristics Although it is possible to obtain a satisfactory understanding of a
structure's expected dynamic behavior by preliminary analyticalstudies, an ambient-vibration and/or a forced vibration test on an
frequencies.
recorders at three to five locations that are expected (from analytical
studies or other information) to have maximum amplitudes duringthe first three to four vibrational modes.
Thus, elastic properties of the structure can be determined. If thesub ect structure ex eriences nonlinear behavior durin a stronshaking, it will be much easier to evaluate the nonlinear behavioronce linear behavior is determined before the nonlinear behavioroccurs during the strong shaking.
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Tests on Existin tructures to
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Determine Dynamic Characteristics - , -
more difficult to perform. The required equipment (vibrationgenerator with control consoles, weights, recorders, accelerometers,
and cables is heavier and the test takes lon er than the ambient-vibration test.
State-of-the-art vibration enerators do not necessaril have thecapability to excite to resonance all significant modes of all
structures (elebi and others, 1987).
Dynamic Analysis
A simplified finite-element model can be developed to obtain theelastic dynamic characteristics.
This is performed with any one of the several tested computerprograms available (e.g. SAP2000, ANSYS, and STRUDL).
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instrumentation scheme, anoptimum list of hardware is
develo ed after carefulconsideration of cost and datarequirements.
While developing the
instrumentation scheme within thebudgetary constraints, it is best to
channels for each recordingsystem. Most recording systemshave maximum of 12 or 18channels of recording capability.
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to install seismic instruments:
1. After an instrumentation scheme is
locations are chosen, monitoring teamand the owner's representative reviewthe site to determine exact sensor
satisfactory to both parties.
This is important from viewpoint of long- ,
with the occupant's space, placement ofdata cable runs, and aestheticrequirements of the owner.
Figure exhibits a sample schematicshowing locations of sensors, routing ofcables, location of junction boxes andrecordin units.
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. ext a tec n c an s ou nspect t e ent re structura sc emewith an electrical contractor who will install the data cable,junction boxes at key locations and terminal boxes (if required)
a eac sensor s e. e mo ern recor ng sys ems may norequire terminal boxes as they have internal terminals. Actualcabling by the contractor is monitored by the monitoring team
'as desired and that all building code regulations are followed.
3. The cable-termination box includes data circuits, batteries andbattery charges. This box is normally mounted on the wall abovethe recorder. The recorder location is selected on the basis of
secur y, yp ca y n a e ep one or e ec r ca sw c room, an nsome circumstances is enclosed with separate fencing in anopen area.
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. e ns rumen a on un ergoes a pre m narycalibration in the strong-motion laboratory and is theninstalled in the structure with appropriate testproce ures nc u ng a s a c sens v y es oreach component and determination of direction ofmotion for upward trace deflection on the record.
For modern digital systems, this information is entered
general database.
Other documentation includes precise sensorlocation, period and damping of each unit, location ofcable runs access information and circuit dia rams.
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condition that whether 2 dimensional or 3 dimensional motions of thestructure are going to be monitored.
In 2 dimensions, the degrees of freedom are 2 translations and one rotation.
A typical example to such a structure is a multistorey building with shearwa s an a r g ap ragm.
In order to determine these two translations and one rotation, three.
These three measurements have to satisfy the following conditions:
The measurements have to be taken from two separate locations
The three measurement directions should not be parallel.
The three measurement directions should not intersect each other.
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,3 translations and 3 rotations.
,satisfy the following conditions in order to solve for the 3 rotationsand 3 translations from the dynamic equilibrium equations:
The measurements have to be taken at least from 3 separate locations.
.
The 6 measurement directions should not be parallel.
The 6 measurement directions should not intersect each other.
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n genera t e or er or p ac ng sensors:
Roof
Basement
Any location where stiffness and/or mass changes significantly
Any location where the curvature of the deformed shape is expected
to change.
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. .
2. The second group of sensors should be placed on the top of thefoundations (in the ground floor or basement).
3. The third rou of sensors should be laced at the locations where therigidity and the mass of the structure change.
4. The rest of the sensors should be placed on locations where theamplitudes of the vibration modes of the structure are expected to be
.
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1. Effective Independence Technique
2. Optimum Driving Point Based Method
3. Non-optimum driving point based method
. -
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T hniT hni
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where Q is the Fisher information matrix
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T hniT hni
The best state estimate can be obtained by maximizing Q which results in
. ,that the measurement noise is uncorrelated and possesses identical
statistical properties of each sensor. The Fisher Information Matrix can
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Effective IndependenceEffective Independence
TechniqueTechnique
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TechniqueTechnique
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Optimum driving point based
method
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method
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Effective Indepence Driving
Point Residue Technique
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Point Residue Technique
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order to have a successful program. Unless maintenance arrangementsare made, successful recording of data cannot be accomplished. Therefore,routine maintenance is conducted every 3-12 months if circumstances and
x w.
This maintenance includes the following:
1. Remote calibration of period and damping.
2. Inspection of battery terminals, load voltage, and charge rate (batteries arereplaced every 3 years).
3. Measurement of threshold of triggering system and length of recording cycle.
As a final maintenance procedure, a calibration record is obtained and thenexamined for the desired characteristics. All inspection procedures arerecorded in the permanent station file at the laboratory.
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, , ,biaxial or triaxial) were used to instrument structures.
However, observations of damages during the 1994Northridge and 1995 Kobe earthquakes, have forced
based seismic design methods and to find new
techniques to control drift and displacements.
To verify these developments, sensors directlymeasuring displacements or relative displacements
(transducers, laser devices and GPS units) are nowbeing considered.
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