vibrations of machine foundationsszepesr/anyagok/oktatas/ngb-se005_4/eloadas/gepalap… ·...

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Vibrations of

Machine Foundations

Richard P. Ray, Ph.D., P.E.Civil and Environmental Engineering

University of South Carolina

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ATST Telescope and FE Model

Fundamentals-Modeling-Properties-Performance

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Summary and Conclusions (Cho, 2005)1.

High fidelity FE models were created2.

Relative mirror motions from zenith to horizon pointing: about 400 μm in translation and 60 μrad in rotation.

3.

Natural frequency changes by 2 Hz as height changes by 10m.4.

Wind buffeting effects caused by dynamic portion (fluctuation) of wind 5.

Modal responses sensitive to stiffness of bearings and drive disks

6.

Soil characteristics were the dominant influences in modal (dynamic) behavior of the telescopes.

7.

Fundamental Frequency (for a lowest soil stiffness):OSS=20.5hz; OSS+base=9.9hz; SS+base+Coude+soil=6.3hz

8.

A seismic analysis was made with a sample PSD9.

ATST structure assembly is adequately designed:1.

Capable of supporting the OSS2.

Dynamically stiff enough to hold the optics stable3.

Not significantly vulnerable to wind loadings


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Topics for Today

FundamentalsModelingPropertiesPerformance

AlapokModellezésTulajdonságokGyakorlati Alkalmazás

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Foundation Movement Alapok Mozgáslehetőségei

X

ZY

θ

ψ

φ


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Design Questions (1/4) Tervezés

How Does It Fail?Static SettlementDynamic Motion Too Large (0.02 mm)Settlements Caused By Dynamic MotionLiquefactionWhat Are Maximum Values of Failure? (Acceleration, Velocity, Displacement)

Hogy rongálódik/megy tönkre?

Statikus süllyedésDinamikus mozgás túl nagy (0,02 mm)Süllyedés dinamikus mozgás következtébenMegfolyósodásRongálódás maximális értékei (gyorsulás, sebesség, eltolódás)

Fundamentals-Modeling-Properties-Design-Performance

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Velocity Requirements Sebesség Követelmények

Massarch

(2004) "Mitigation of Traffic-Induced Ground Vibrations"


0,40

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300 800

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Design Questions

(2/4) Tervezés

What Are Relations Between Loads And Failure Quantities?

Loads -Harmonic, Periodic, RandomLoad→ Structure →Foundation → Soil →Neighboring StructuresModel: Deterministic or Probabilistic

Mi a kapcsolat terhelési és törési mennyiségek között?Terhelések- Harmónikus, Periódikus, VéletlenszerüTerhelés → Épület →Alapozás → Talaj → KözeliÉpületekModel: Determinisztikus és probabilisztikus


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Harmónikus

Periódikus

Véletlenszerű

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Design Questions (3/4) Tervezés

Hogy határozzuk meg a tervezéshez szükséges paramétereket? (How do we measure what is necessary?)

Teljes méretarányú teszt (Full scale Test)Prototípus teszt (Prototype Test)Kis méretű teszt (Small Scale Tests (Centrifuge))Laboratóriumi teszt (Laboratory Tests (Specific Parameters))Számítógépes program (Computer Model)


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Design Questions (4/4)Milyen biztonsági tényezőt használjunk? (What Factor of Safety Do We Use? )

Van a biztonsági tényezőnek értelme? (Does FOS Have Meaning)Mi történik törés után (What Happens After There Is Failure)

Életvesztés (Loss of Life)Tulajdonvesztéd (Loss of Property)Gyártás kihagyás (Loss of Production)

Mi a munka célja, tervezett élettartalma, értéke (Purpose of Project, Design Life, Value)


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r -2 r -2 r -0.5

r -1

r -1

r

Nyíróhullám

Vertical component

Horizontal component

Shear window

Rayleigh wave

Relative amplitude+

+

+

+

- -

+

+

Wave TypeHullám típus

Összes energia százaléka

Rayleigh 67Shear 26

Compression 7

Waves


Compresszió

hullámNyíró ablak

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Alapok Modellezése (Modeling Foundations)

Egyesített Tömb (m,c,k)Lumped Parameter (m,c,k) Block System

Parameters Constant, Layers, Special Ellenállási Függvények Impedance Functions

Function of Frequency (ω), LayersPeremérték Feladatok Boundary Elements (BEM)

Infinite Boundary, Interactions, LayersVéges Elemes (Finite Element/Hybrid (FEM, FEM-BEM))

Complex Geometry, Non-linear Soil


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Lumped Parameter

(Egyesített tömb))sin( tPP o ω=

m

Gk

m

cν ρ

)sin(0 tPkzzczm ω=++ &&&

r


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Egy szabadságfokú (Single Degree of Freedom)

)()(

)(

)()(

0

2

NormmNForceSpringzk

Norsecm

msecNForceDampingzc

NorsecmkgForceInertiazm

zkzczm

z

z

z

zzz

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ −

=

⎟⎠⎞

⎜⎝⎛=

=++

&

&&

&&&

k

m

c

z

Tehetetlenségi erő

Rugó erő

Csillapító Erő

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Egy szabadságfokú Single Degree of Freedom

condependssforsolution

smcsthen

mksetandembydivide

ekcsmswhereconstantezformtakewillsolution

zkzczm

n

nst

st

stzzz

0

0)(.......

0

22

2

2

=++

=

=++

==

=++

ω

ωα

α

αα

&&&

c=0…Undamped

c=2mω…Critically

Damped

c<2mω…Underdamped

osztás

megoldás állandó

ahol

tehát

s megoldása c-től függ

Nem csillapított

Alul csillapított

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Single Degree of Freedom

)cos()0()sin()0()(

)0()0(

)(,)cos()sin()()sin()cos('

.).(,)(

...0

2121

22

tztztz

AzandBz

conditioninitialfBAtBtAtztiteidentitysEuler

condinitfwhereeetz

issmcs

nnn

n

n

nnti

titi

nn

n

nn

ωωω

ω

ωωωω

αααα

ωω

ω

ωω

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

=+=+=

=+=

±==++

−

&

&

undamped

)0(z&

z(0)t

Kiinduló feltétel

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[ ] tn

t

nncrit

n

n

n

n

etztztz

condinitfwhereettzmcsandmccthenif

mc

mcsthen

presentdampingifsmcs

ω

ω

ω

αααα

ωω

ω

ω

−

−

++=

=+=

−=−====

−⎟⎠⎞

⎜⎝⎛±−=

=++

)0()1)(0()(

.).(,)()(2

20

22

0

2121

22

22

&

critical


z(0)

)0(z&

t

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( )( )

⎥⎦

⎤⎢⎣

⎡+

+=

+=

+=+=

±−=−±−=

−===

<<

−

−

−−−−+−

)cos()0()sin()0()0()(

)cos()sin(()(

)(

1

02

2121

22

2

tztDzzetz

tBtAetz

eeeeetz

iDDDsthen

DandratiodampingccDsuppose

thenmcif

DDD

ntD

DDtD

tititDtitDtitD

Dnnnn

nDcrit

n

n

n

DDnDnDn

ωωω

ω

ωω

αααα

ωωωωω

ωω

ω

ω

ω

ωωωωωωω

&

dunderdampe


See Chart

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Single Degree of Freedom)sin(0 tPzkzczm Pzzz ω=++ &&&

k

m

c

)sin( tPP Po ω=

( ) ( ){ }

( )( )

22

2222

0

1

2tan

sin

cos)sin(

⎟⎠⎞⎜

⎝⎛−

⎟⎠⎞⎜

⎝⎛

=−

=

−+−

+

+= −

n

P

n

P

P

P

P

PP

DDDt

D

mkc

tcmk

PtBtAez n

ωω

ωω

ωωφ

φωωω

ωωω

critccD =

mk

n =ω 21 DnD −=ωωkmccrit 2=

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SDOF Átmenti és Állandó Transient

and Steady-State

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( )( )

222

0max

2222

0

21

1

sin)(

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

−+−

=

n

P

n

P

P

PP

DkPz

tcmk

Ptz

ωω

ωω

φωωω

222

21

1

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

n

P

n

P

staticmax

D

zz

ωω

ωω


Dynamic Magnification (Logarithmic)

0.1

1

10

100

0.1 1 10Frequency Ratio (ωP/ωn)

Mag

nific

atio

n

D=0.02D=0.05D=0.10D=0.20D=0.50

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Lumped Parameter System

Kx

Z

ψ

KzCz

Cx

KψCψ

/2 Cψ

/2

X

)sin(0 tPzkzczm Pzzz ω=++ &&&

mIψ


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Rendszer paraméterek Lumped Parameter Values

Mode Vertical

zHorizontal

x Rocking

ψTorsion

θ

Stiffness k

Mass Ratio m

Damping Ratio, D

ν−14Gr

ν−28Gr

)1(38 3

ν−Gr

316 3Gr

5rIρθ

38)2(

rm

ρν−

34)1(

rm

ρν−

2/1ˆ425.0

m 2/1ˆ288.0

m 2/1ˆ)ˆ1(15.0

mm+ m̂2150.0

+

m̂

D=c/ccr

G=Shear Modulus ν=Poisson's Ratio r=Radius ρ=Mass Density Iψ

,Iθ

=Mass Moment of Inertia

58)1(3

rIρ

νψ −


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Design Example 1

(Példa)VERTICAL COMPRESSORUnbalanced Forces

(kiegyensúlyozatlan erők)•Vertical = 45 kN•Horizontal

Primary = 0,5 kN•Operating Speed

= 450 rpm•Wt Machine + Motor = 5 000 kg

Soil PropertiesShear Wave Velocity Vs

= 250 m/secDensity, ρ

= 1600 kg/m3

Shear Modulus, G = 1,0e8 PaPoisson's Ratio, ν

= 0,33

DESIGN CRITERION:Smooth Operation At SpeedVelocity <0,10 in/sec Displacement < 0,002 in <0,05mm

Jump to Chart


függőlegesvízszintesüzemelő seb.

Gép+motor súlya

Talaj jellemzőkTervezési feltétel

Nyíró hullám

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rGrQmm

kQZ

zstatic ×××

=−

=== 800

100,141000)00045(667,0

4)1(05,0 ν

mr 5.105.0

075.0==

( )

mmZZD

Mm

D

mr

m

staticdynamic

z

05,0210,153,0

ˆ425,0

65,055,1160040002367,0

4)1(ˆ 33

==

⎟⎠⎞

⎜⎝⎛≈==

=×

×=

−=

ρν

Try a 3 x 2,5 x 1 foundation block, r = 1,55 m

Mass = 18 000 kg Total Mass = 18 000 + 5 000 = 23 000 kg

Jump to Figure


667,0100,14

)1(4 8 rGrk ×××

=−

=ν

34)1(ˆ

rmm

ρν−

=tömeg

tömbalaptest

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Design Example -

Table Top5m

10m5m

4m

Q0

=1800 N

ψ

m=250 000 kg

Iψ

=1,0 x 107

N-m-sec2

Soil PropertiesShear Wave Velocity Vs

= 200 m/secShear Modulus, G = 6,80x107

PaDensity, γ

= 1700 kg/m3

Poisson's Ratio, ν

= 0,33

DESIGN CRITERION

5.0 mm/sec Horizontal Motion at Machine Centerline

X = 0,04 mm from combined rocking and sliding

Speed = 320 rpm

Slower speeds, X can be larger


X

Emelt (asztal) alap

sebesség

Kisebb sebességnél, x megnőhet

együttes rocking, csúszás

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Horizontal Translation Only

mmkQXMag

mD

rmmmlwrtEquivanlen

xstaticx

37

02/1

3

103,199,3108,6

33,028

18002,141,0ˆ288,0

49,08

2ˆ99,3510

−×=××

−==≈∴==

=−

==×

==ρ

νππ

Rocking About Point "O"

0,25019,029,3)29,31(

15,0ˆ)ˆ1(

15,0

29,3)39,3(1700

100,18

)67,0(38

)1(3ˆ

/4,32100,110054,1

/10054,1)33,01(3

39,31080,68)1(3

8

/5,3332039,33

5103

5

7

5

7

10

10373

43

43

=∴=+

=+

=

=×

=−

=

=××

==

×=−

×××=

−=

===×

==

ψψψ

ψ

ψψ

ψ

ψ

ψ

ρν

ω

ν

ωππ

Magmm

D

rI

m

secradIk

radNGrk

secradrpmmlwrEquivalent

n


Ax = 40x10-3

mm

csak vízszintes mozgás

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mmResonanceAt

mmhXMotionHorizontal

radkMDeflectionAngularStatic

mNMBaseAboutMomentStatic

s

os

33

37

710

0

100,85)104,3(0,25

104,341054.8

1054.810054.1

9000900051800

−−

−−

−

×=×=

×=××=×==

×=×

===

−=×==

ψ

ψ

ψ

ψ


ψ

X

X = 40x10-3

mm

Dynamic Magnification (Linear)

0

5

10

15

20

25

30

0,0 0,5 1,0 1,5 2,0Frequency Ratio (ωP/ωn)

Mag

nific

atio

n

0,02

0,05

0,1

0,2

0,5

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Impedance Methods

Based on Elasto-Dynamic SolutionsCompute Frequency-Dependent Impedance Values (Complex-Valued)Solved By Boundary Integral Methods Require Uniform, Single Layer or Special Soil Property Distribution Solved For Many Foundation Types


Ellenállási függvények

Frekvenciától függő ellenállási értékek

Peremérték integrál módszer

Egyenletes, egy réteg, speciális talajérték eloszlás szükséges

Többfajta alap típusra megoldott

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Impedance Functions

( ))sin()cos( titPePP oti

o ωωω +==

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++×=+== SOILSTATIC

z

zz DKCikKCiK

ARS

ωωωω 2)(

Radiation DampingSoil Damping

Jump Wave

Sz


Energia csillapítás Talaj csillapítás

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Impedance Functions

Luco

and Westmann

(1970)

sVr

Gra ωρω ==0


ψ

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Impedance Functions


ψ

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Boundary Element

Stehmeyer

and Rizos, 2006


Peremérték feladatok

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B-Spline

Impulse Response Approach


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[ ]{ } [ ]{ } { } tie ωpuKuM =+&&


{ } { }[ ] [ ]{ }{ } { }pUMK

Uu=−

=2ω

ω thene ti


G1

,ρ1

,ν1

u1

u2

u7

u8

[ ] ),,( 1111 νρGfnK =

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

8

7

2

1

8,87,82,81,8

8,77,72,71,7

8,27,22,21,2

8,17,12,11,1

uu

uu

kkkkkkkk

kkkkkkkk

[ ] )( 11 ρfnm =

linearlinear

dcxyybxau iii

==

+++=

σε

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[ ]{ } [ ]{ } { } tie ωpuKuM =+&&

tie

ppppp

uuuuu

kkkkkkkkkkkk

kkkkkkk

uuuuu

mm

mm

m

ω

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

5

4

3

2

1

5,54,53,5

5,44,43,42,4

5,34,33,32,31,3

4,23,22,21,2

3,12,11,1

5

4

3

2

1

5

4

3

2

1

&&

&&

&&

&&

&&

{ } { } { } { }[ ] [ ]{ }{ } { } { }[ ] { } valuedcomplexare

forsolvegivenandethenezif titi

−=−

−==

ZKZpZMK

ZzZ

,,2

2

ωω

ω ωω &&

( )22 1221* DiDDGG −+−=

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Dynamic p-y

Curves

Tahghighi

and Tonagi

2007


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Soil PropertiesShear Modulus, G and Damping Ratio, D

Soil TypeConfining StressVoid RatioStrain Level

Field: Cross-Hole, Down-Hole, Surface Analysis of Seismic Waves SASWLaboratory: Resonant Column, TorsionalSimple Shear, Bender Elements


Talaj jellemzők

Hézagtényező

Talajtípus

Nyúlás szint

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Crosshole TestingOscilloscope

PVC-cased Borehole

PVC-cased Borehole

DownholeHammer(Source) Velocity

Transducer(GeophoneReceiver)

Δt

Δx

Shear Wave Velocity:Vs

= Δx/Δt

TestDepth

ASTM D 4428

Pump

packer

Note: Verticality of casingmust be established by

slope inclinometers to correctdistances Δx with depth.

SlopeInclinometer

SlopeInclinometer

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Resonant Column Test

G, D for Different γ


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Torsional

Shear Test

Schematic Stress-Strain


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Hollow Cylinder RC-TOSS


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TOSS Test Results


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Steam Turbine-Generator (Moreschi

and Farzam, 2003)


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Machine Foundation Design Criteria

Deflection criteria: maintain turbine-generator alignment during machine operating conditions

Dynamic criteria: ensure that no resonance condition is encountered during machine operating conditions

Strength criteria: reinforced concrete design


Jump to Resonance

El/kihajlási kritérium: a turbina-generátor szintben maradjon működése alatt

Dinamikus feltétel: nincs rezonancia a gép működése alatt

Erősségi feltétel: előfeszített beton tervezés

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STG Pedestal Structure


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Vibration Properties Evaluation

Identification of the foundation natural frequencies for the dominant modesFrequency exclusion zones for the natural frequencies of the foundation system and individual structural members (±20%)Eigenvalue analysis: natural frequencies, mode shapes, and mass participation factors


Az alap saját frekvenciáinak meghatározása a domináns lengésekre/módokra

Kihagyási frekvencia zónák a természetes frekvenciákra

Eigenérték

elemzés:természetes frekvenciák, lengésmódnál alakok, tömeg együttdolgozása

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XYZ

XYZ

Finite Element Model Structure and Base


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Low Frequency Modes

1st

mode

6.5 Hz95 % m.p.f.

2nd

mode

7.2 Hz76 % m.p.f


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High Frequency Modes

28th

mode

46.3 Hz0.3% m.p.f

42nd

mode

64.6 Hz0.03% m.p.f

Excitation frequency: 50-60 HzFundamentals-Modeling-Properties-Performance

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Local Vibration Modes

Identification of natural frequencies for individual structural members

Quantification of changes on vibration properties due to foundation modifications


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ATST Telescope and FE Model


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Optics Lab mass/Instrument weight = 228 tonsWind mean force = 75 N, RMS = 89 N Ground base excitation PSD = 0.004 g2/hzConcrete Pier

High Strength Concrete (E=3.1×1010 N/m2, ν=0.15)

Soil Stiffness, kFour different values using Arya & O’Neil’s formula based on the site test data (Shear modulus:30~75ksi, Poisson’s ratio:0.35~0.45)

Assumptions in FE analyses


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•

Soil property range: Shear modulus (30~75ksi), Poisson’s ratio (0.35~0.45)•

Pier Footing: Diameter (23.3m)•

“min”

for shear modulus of 30 ksi; “max”

for 75 ksi

Frequency vs

Soil Stiffness

Stiffness min min+33.3% min+66.6% maxKx 1.19E+10 1.83E+10 2.48E+10 3.12E+10Ky 1.19E+10 1.83E+10 2.48E+10 3.12E+10Kz 1.48E+10 2.45E+10 3.41E+10 4.38E+10Krx 1.34E+12 2.21E+12 3.09E+12 3.96E+12Kry 1.34E+12 2.21E+12 3.09E+12 3.96E+12Krz 1.74E+12 2.61E+12 3.49E+12 4.36E+12

6.3 7.0 7.4 7.56.4 7.1 7.5 7.79.4 9.7 9.9 109.4 10.3 11.1 11.810.4 11.9 12.6 13.311.2 13.0 13.6 13.7

456

MODE123

Stiffness units = SI, frequency mode (hz)


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Summary and Conclusions (Cho, 2005)1.

High fidelity FE models were created2.

Relative mirror motions from zenith to horizon pointing: about 400 μm in translation and 60 μrad in rotation.

3.

Natural frequency changes by 2 hz

as height changes by 10m.4.

Wind buffeting effects caused by dynamic portion (fluctuation) of wind 5.

Modal responses sensitive to stiffness of bearings and drive disks6. Soil characteristics were the dominant influences in modal

behavior of the telescopes.7.

Fundamental Frequency (for a lowest soil stiffness):OSS=20.5hz; OSS+base=9.9hz; SS+base+Coude+soil=6.3hz

8.

A seismic analysis was made with a sample PSD9.

ATST structure assembly is adequately designed:1.

Capable of supporting the OSS2.

Dynamically stiff enough to hold the optics stable3.

Not significantly vulnerable to wind loadings


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Free-Field Analytical Solutions

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

RVz C

rHaRVLiru ωρβ

ωθ 2003

0 )(2

)0,,(

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

RVr C

rHaRVMiru ωρβ

ωθ 2103

0 )(2

)0,,(

uruz


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Karlstrom

and Bostrom

2007

Trench Isolation


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Chehab

and Nagger 2003


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Celibi

et al (in press)

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Thank-you

Questions?

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r -2 r -2 r -0.5

r -1

r -1

r

Shear wave

Vertical component

Horizontal component

Shear window

Rayleigh wave

Relative amplitude+

+

+

+

- -

+

+

Wave Type Percentage of Total Energy

Rayleigh 67Shear 26

Compression 7

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Waves

Rayleigh, R Surface

Shear,S Secondary

Compression, P Primary

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Machine Performance ChartPerformance Zones

A=No Faults, New

B=Minor Faults, Good Condition

C = Faulty, Correct In 10 Days To Save $$

D = Failure Is Near, Correct In 2 Days

E = Stop Now

0.002

450

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