introduction to vibration monitoring [compatibility mode]

8/12/2019 Introduction to Vibration Monitoring [Compatibility Mode]

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ERASMUS Teaching (2008), Technische Universitt Berl in

Vibration monitoringVibration monitoring

. . .Istanbul Tecnical University

undesbakir ahoo.com


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ERASMUS Teaching (2008), Technische Universitt Berlin

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

P. Gundes Bakir, Vibration based structural health monitoring 2


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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


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



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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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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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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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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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


here.


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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


and also move.


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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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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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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 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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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


data from a different viewpoint.


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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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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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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

ERASMUS Teaching (2008) Technische Universitt Berlin


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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

++==


ERASMUS Teaching (2008) Technische Universitt Berlin


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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


1



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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.

P. Gundes Bakir, Vibration based structural health monitoring 56.



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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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-



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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.


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.


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 h iT h i


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T hniT hni


where Q is the Fisher information matrix


T h iT h i


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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


Effective IndependenceEffective Independence

TechniqueTechnique


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TechniqueTechnique



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Optimum driving point based

method


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method


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.


introduction to vibration monitoring [compatibility mode]

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