Vibrations of Machine Foundations

Transcription

1 Vibrations of Machine Foundations Richard P. Ray, Ph.D., P.E. Civil and Environmental Engineering University of South Carolina

2 Thanks To: Prof. Richard D. Woods, Notre Dame Univ. Prof. F.E. Richart, Jr.

3 Topics for Today Fundamentals Modeling Properties Performance

4 Foundation Movement Z Y θ φ ψ X

5 Design Questions (1/4) How Does It Fail? Static Settlement Dynamic Motion Too Large (0.02 mm is large) Settlements Caused By Dynamic Motion Liquefaction What Are Maximum Values of Failure? (Acceleration, Velocity, Displacement) Fundamentals-Modeling-Properties-Design-Performance

6 Velocity Requirements Massarch (2004) "Mitigation of Traffic-Induced Ground Vibrations"

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8 Design Questions (2/4) What Are Relations Between Loads And Failure Quantities Loading -Machine (Periodic), Impluse, Natural Relations Between Load, Structure, Foundation, Soil, Neighboring Structures Generate Model: Deterministic or Probabilistic

9 Design Questions (3/4) How Do We Measure What Is Necessary? Full Scale Tests Prototype Tests Small Scale Tests (Centrifuge) Laboratory Tests (Specific Parameters) Numerical Simulation

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11 Design Questions (4/4) What Factor of Safety Do We Use? Does FOS Have Meaning What Happens After There Is Failure Loss of Life Loss of Property Loss of Production Purpose of Project, Design Life, Value

12 r -2 r -2 r Rayleigh wave Vertical component Horizontal component + Shear - + wave + - Relative amplitude r -1 Shear window r -1 + Waves + Wave Type r Percentage of Total Energy Rayleigh 67 Shear 26 Compression 7

13 Modeling Foundations Lumped Parameter (m,c,k) Block System Parameters Constant, Layer, Special Impedance Functions Function of Frequency (ω), Layers Boundary Elements (BEM) Infinite Boundary, Interactions, Layers Finite Element/Hybrid (FEM, FEM-BEM) Complex Geometry, Non-linear Soil

14 Lumped Parameter P Po sin( ω t) r m m G ν ρ c k m&&+ z cz& + kz P 0 sin( ωt)

15 SDOF Mag A dynamic A static ω 1 ωn ω 2D ω n 2

16 Lumped Parameter System C z I ψ ψ Z m z &&+ z c z& + K z m z k z z K x P 0 sin( ω t) ω n X k m C x K ψ C ψ /2 C ψ /2 D c ccr ccr 2 km

17 Lumped Parameter Values Mode Vertical Horizontal Rocking Torsion Stiffness k Mass Ratio m mˆ Damping Ratio, D Fictitious Mass 4Gr 1 ν m(1 ν ) 3 4ρr / 2 mˆ 0.27m mˆ 8Gr 2 ν m(2 ν ) 3 8ρr / 2 mˆ 0.095m mˆ 3 8Gr 16Gr 3(1 ν ) 3 3Iψ (1 ν ) Iθ 5 8ρr 5 ρr / 2 (1+ mˆ ) mˆ 1+ 2mˆ 0.24I x 0.24 mˆ mˆ Dc/c cr GShear Modulus νpoisson's Ratio rradius ρmass Density I ψ,i θ Mass Moment of Inertia 3 I z

18 Mass Ratio

19 Design Example 1 VERTICAL COMPRESSOR Unbalanced Forces Vertical Primanry 7720 lb Vertical Secondary 1886 lb Horzontal Primary 104 lb Horizontal Secondary 0 lb Operating Speed 450 rpm Wt Machine + Motor lb DESIGN CRITERION: Smooth Operation At Speed Velocity <0.10 in/sec Displacement < in Soil Properties Shear Wave Velocity V s 680 ft/sec Shear Modulus, G psi Density, γ 110 lb/ft 3 Poisson's Ratio, ν 0.33 Jump to Chart

20 A r zs Q k z 72.8" (1 ) Q " 4Gr 0 ν 6.07' 0.667( ) r Try a 15 x 8 x 3 foundation block, Area 120 ft 2 and r 6.18 ft Weight 54,000 lb Total Weight (1 ν ) 3 4r mˆ A ˆ 3 m D A z dynamic W g g γ z static M z 0.002" ( 6.18) D 0.42 Jump to Figure

21 18' Design Example - Table Top W lb Q lb I ψ 2.88 x 10 6 ft-lb-sec 2 34' 18' 11' Soil Properties Shear Wave Velocity V s 770 ft/sec Shear Modulus, G psi Density, γ 110 lb/ft 3 Poisson's Ratio, ν 0.33 ψ DESIGN CRITERION 0.20 in/sec Horizontal Motion at Machine Centerline Ax in. from combined rocking and sliding Speed 160 rpm Slower speeds, Ax can be larger

22 Horizontal Translation Only in Gr Q k Q A Mag m D r m m ft cd r Equivanlent x static x x / ˆ ˆ ν ρ ν π π Rocking About Point "O" ˆ ) ˆ ( (12.04) (0.67) 8 ) 3(1 ˆ / / ) ( / lbs ft M Moment Static Mag m m D r I m sec rad I k ft lb Gr k sec rad rpm ft cd r Equivalent o n + ψ ψ ψ ψ ψ ψ ψ ψ ψ ρ ν ω ν ω π π

23 OmegaRatio Mag Damping 9% ) ( ) ( (0.67) in Resonance At in h A Motion Horizontal rad k M Deflection Angular Static s xs o s ψ ψ ψ

24 Impedance Methods Based on Elasto-Dynamic Solutions Compute 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

25 Impedance Functions P P e o iωt P o ( cos( ω t) + i sin( ωt) ) S z S z Rz 2K K + iωc STATIC SOIL Az ω ( K k( ω) ) + iω C+ D Radiation Damping Jump Wave Soil Damping

26 Impedance Functions a 0 ρ ω r G ωr V s ψ Luco and Westmann (1970)

27 Layer Effects

28 Impedance Functions ψ

29 Boundary Element Stehmeyer and Rizos, 2006

30 B-Spline Impulse Response Approach

31 [ M ]{&& z} + [ K]{ z} { p} i t e ω { z} { Z} e iωt then {[ ] 2 K ω [ M] }{ Z} { p} Finite/Hybrid Model ( ) β + 2β 1 G* G i β

32 Dynamic p-y Curves Tahghighi and Tonagi 2007

33 Soil Properties Shear Modulus, G and Damping Ratio, D Soil Type Confining Stress Void Ratio Strain Level Field: Cross-Hole, Down-Hole, Surface Analysis of Seismic Waves SASW Laboratory: Resonant Column, Torsional Simple Shear, Bender Elements

34 Oscilloscope Crosshole Testing ASTM D 4428 Pump t Test Depth Downhole Hammer (Source) packer Note: Verticality of casing must be established by slope inclinometers to correct distances x with depth. PVC-cased Borehole Shear Wave Velocity: V s x/ t Slope Inclinometer x PVC-cased Borehole Velocity Transducer (Geophone Receiver) Slope Inclinometer

35 Resonant Column Test G, D for Different γ

36 Torsional Shear Test Schematic Stress-Strain

37 Hollow Cylinder RC-TOSS

38 TOSS Test Results

39 Steam Turbine-Generator (Moreschi and Farzam, 2003)

40 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 Jump to Resonance Strength criteria: reinforced concrete design

41 STG Pedestal Structure

42 Vibration Properties Evaluation Identification of the foundation natural frequencies for the dominant modes Frequency 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

43 Finite Element Model Structure and Base Y Z X

44 Low Frequency Modes 1 st mode 6.5 Hz 95 % m.p.f. 2 nd mode 7.2 Hz 76 % m.p.f

45 High Frequency Modes 28 th mode 46.3 Hz 0.3% m.p.f Excitation frequency: Hz 42 nd mode 64.6 Hz 0.03% m.p.f

46 Local Vibration Modes Identification of natural frequencies for individual structural members Quantification of changes on vibration properties due to foundation modifications

47 ATST Telescope and FE Model

48 Assumptions in FE analyses Optics Lab mass/instrument weight 228 tons Wind mean force 75 N, RMS 89 N Ground base excitation PSD g 2 /hz Concrete Pier High Strength Concrete (E N/m 2, ν0.15) Soil Stiffness, k Four 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)

49 Frequency vs Soil Stiffness Stiffness units SI, frequency mode (hz) MODE Stiffness Kx Ky Kz Krx Kry Krz Soil property range: Shear modulus (30~75ksi), Poisson s ratio (0.35~0.45) Pier Footing: Diameter (23.3m) min m in+33.3% m in+66.6% max 1.19E E E E E E E E E E E E E E E E E E E E E E E E min for shear modulus of 30 ksi; max for 75 ksi

50 Summary and Conclusions (Cho, 2005) 1. High fidelity FE models were created 2. 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 behavior of the telescopes. 7. Fundamental Frequency (for a lowest soil stiffness): OSS20.5hz; OSS+base9.9hz; SS+base+Coude+soil6.3hz 8. A seismic analysis was made with a sample PSD 9. ATST structure assembly is adequately designed: 1. Capable of supporting the OSS 2. Dynamically stiff enough to hold the optics stable 3. Not significantly vulnerable to wind loadings

51 Free-Field Analytical Solutions R V z C r H a R L V i r u ω ρ β ω θ ) ( 2,0), ( R V r C r H a R M V i r u ω ρ β ω θ ) ( 2,0), ( u r u z

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57 Trench Isolation Karlstrom and Bostrom 2007

58 Chehab and Nagger 2003

59 Celibi et al (in press)

60 Thank-you Questions?

61 r -2 r -2 r Rayleigh wave Vertical component Horizontal component + Shear - + wave + - Relative amplitude r -1 Shear window r r Wave Type Percentage of Total Energy Rayleigh 67 Shear 26 Compression 7

62 Waves Rayleigh, R Surface Shear,S Secondary Compression, P Primary

63 Machine Performance Chart Performance Zones ANo Faults, New BMinor Faults, Good Condition C Faulty, Correct In 10 Days To Save $$ D Failure Is Near, Correct In 2 Days E Stop Now 450