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USCUSCUSC
Vibrations of
Machine Foundations
Richard P. Ray, Ph.D., P.E.Civil and Environmental Engineering
University of South Carolina
USCUSCUSC
ATST Telescope and FE Model
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Topics for Today
FundamentalsModelingPropertiesPerformance
AlapokModellezésTulajdonságokGyakorlati Alkalmazás
USCUSCUSC
Foundation Movement Alapok Mozgáslehetőségei
X
ZY
θ
ψ
φ
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
USCUSCUSC
Velocity Requirements Sebesség Követelmények
Massarch
(2004) "Mitigation of Traffic-Induced Ground Vibrations"
Fundamentals-Modeling-Properties-Performance
0,40
USCUSCUSCFundamentals-Modeling-Properties-Performance
300 800
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Harmónikus
Periódikus
Véletlenszerű
USCUSCUSC
USCUSCUSC
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)
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
USCUSCUSC
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)
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
Compresszió
hullámNyíró ablak
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Lumped Parameter
(Egyesített tömb))sin( tPP o ω=
m
Gk
m
cν ρ
)sin(0 tPkzzczm ω=++ &&&
r
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
USCUSCUSC
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
USCUSCUSC
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
USCUSCUSC
[ ] 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
Single Degree of Freedom
z(0)
)0(z&
t
USCUSCUSC
( )( )
⎥⎦
⎤⎢⎣
⎡+
+=
+=
+=+=
±−=−±−=
−===
<<
−
−
−−−−+−
)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
Single Degree of Freedom
See Chart
USCUSCUSC
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=
USCUSCUSC
SDOF Átmenti és Állandó Transient
and Steady-State
USCUSCUSC
( )( )
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
USCUSCUSCFundamentals-Modeling-Properties-Performance
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
USCUSCUSC
Lumped Parameter System
Kx
Z
ψ
KzCz
Cx
KψCψ
/2 Cψ
/2
X
)sin(0 tPzkzczm Pzzz ω=++ &&&
mIψ
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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 −
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
függőlegesvízszintesüzemelő seb.
Gép+motor súlya
Talaj jellemzőkTervezési feltétel
Nyíró hullám
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
667,0100,14
)1(4 8 rGrk ×××
=−
=ν
34)1(ˆ
rmm
ρν−
=tömeg
tömbalaptest
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
X
Emelt (asztal) alap
sebesség
Kisebb sebességnél, x megnőhet
együttes rocking, csúszás
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
Ax = 40x10-3
mm
csak vízszintes mozgás
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
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
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
Ellenállási függvények
Frekvenciától függő ellenállási értékek
Peremérték integrál módszer
Egyenletes, egy réteg, speciális talajérték eloszlás szükséges
Többfajta alap típusra megoldott
USCUSCUSC
Impedance Functions
( ))sin()cos( titPePP oti
o ωωω +==
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛++×=+== SOILSTATIC
z
zz DKCikKCiK
ARS
ωωωω 2)(
Radiation DampingSoil Damping
Jump Wave
Sz
Fundamentals-Modeling-Properties-Performance
Energia csillapítás Talaj csillapítás
USCUSCUSC
Impedance Functions
Luco
and Westmann
(1970)
sVr
Gra ωρω ==0
Fundamentals-Modeling-Properties-Performance
ψ
USCUSCUSC
Impedance Functions
Fundamentals-Modeling-Properties-Performance
ψ
USCUSCUSC
Boundary Element
Stehmeyer
and Rizos, 2006
Fundamentals-Modeling-Properties-Performance
Peremérték feladatok
USCUSCUSC
B-Spline
Impulse Response Approach
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
[ ]{ } [ ]{ } { } tie ωpuKuM =+&&
Fundamentals-Modeling-Properties-Performance
{ } { }[ ] [ ]{ }{ } { }pUMK
Uu=−
=2ω
ω thene ti
USCUSCUSCFundamentals-Modeling-Properties-Performance
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
USCUSCUSC
[ ]{ } [ ]{ } { } 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 −+−=
USCUSCUSC
Dynamic p-y
Curves
Tahghighi
and Tonagi
2007
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
Talaj jellemzők
Hézagtényező
Talajtípus
Nyúlás szint
USCUSCUSC
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
USCUSCUSC
Resonant Column Test
G, D for Different γ
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Torsional
Shear Test
Schematic Stress-Strain
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Hollow Cylinder RC-TOSS
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
TOSS Test Results
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Steam Turbine-Generator (Moreschi
and Farzam, 2003)
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
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
USCUSCUSC
STG Pedestal Structure
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
Fundamentals-Modeling-Properties-Performance
Az alap saját frekvenciáinak meghatározása a domináns lengésekre/módokra
Kihagyási frekvencia zónák a természetes frekvenciákra
Eigenérték
elemzés:természetes frekvenciák, lengésmódnál alakok, tömeg együttdolgozása
USCUSCUSC
XYZ
XYZ
Finite Element Model Structure and Base
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
Low Frequency Modes
1st
mode
6.5 Hz95 % m.p.f.
2nd
mode
7.2 Hz76 % m.p.f
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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
USCUSCUSC
Local Vibration Modes
Identification of natural frequencies for individual structural members
Quantification of changes on vibration properties due to foundation modifications
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
ATST Telescope and FE Model
Fundamentals-Modeling-Properties-Performance
USCUSCUSC
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