Vibrations of Machine Foundations Richard P. Ray, Ph.D., P.E. Civil and Environmental Engineering University of South Carolina
Thanks To: Prof. Richard D. Woods, Notre Dame Univ. Prof. F.E. Richart, Jr.
Topics for Today Fundamentals Modeling Properties Performance
Foundation Movement Z Y θ φ ψ X
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
Velocity Requirements Massarch (2004) "Mitigation of Traffic-Induced Ground Vibrations"
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
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
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
r -2 r -2 r -0.5 + 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
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
Lumped Parameter P Po sin( ω t) r m m G ν ρ c k m&&+ z cz& + kz P 0 sin( ωt)
SDOF Mag A dynamic A static ω 1 ωn 2 1 2 + ω 2D ω n 2
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
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 0.425 1/ 2 mˆ 0.27m mˆ 8Gr 2 ν m(2 ν ) 3 8ρr 0.288 1/ 2 mˆ 0.095m mˆ 3 8Gr 16Gr 3(1 ν ) 3 3Iψ (1 ν ) Iθ 5 8ρr 5 ρr 0.15 0.50 1/ 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
Mass Ratio
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 10 900 lb DESIGN CRITERION: Smooth Operation At Speed Velocity <0.10 in/sec Displacement < 0.002 in Soil Properties Shear Wave Velocity V s 680 ft/sec Shear Modulus, G 11 000 psi Density, γ 110 lb/ft 3 Poisson's Ratio, ν 0.33 Jump to Chart
A r zs Q k z 72.8" (1 ) Q0 0.002" 4Gr 0 ν 6.07' 0.667(7720+ 1885) 4 11000 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 54 000 + 10 900 64 900 (1 ν ) 3 4r 0.425 mˆ A 0.66 0.67 64 900 ˆ 3 m D A z dynamic W g g γ z static 4 110 M z 0.002" ( 6.18) 1.0 1 2D 0.42 Jump to Figure
18' Design Example - Table Top W550 000 lb Q 0 400 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 14 000 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 0.0015 in. from combined rocking and sliding Speed 160 rpm Slower speeds, Ax can be larger
Horizontal Translation Only in Gr Q k Q A Mag m D r m m ft cd r Equivanlent x static x x 5 0 0 2 1/ 3 10 3.0 2 8 1.0 0.465 ˆ 0.288 0.38 8 2 ˆ 13.96 34 18 4 4 ν ρ ν π π Rocking About Point "O". 7200 18 400 5.6 0.09 ˆ ) ˆ (1 0.15 0.83 (12.04) 32.2 110 10 2.88 8 3(0.67) 8 ) 3(1 ˆ / 10 10 2.88 10 2.90 / 10 2.90 0.33 2 12.04 144) (14 000 8 2 8 / 12.5 120 12.0 3 9 17 16 3 16 5 6 5 6 8 8 4 3 4 3 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
0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 OmegaRatio Mag Damping 9%. 10 5.6 ) 10 5.6(1.0 10 1.0 12) (18 10 0.50 10 0.50 10 2.9 3(0.67) 7200 4 4 4 6 6 8 in Resonance At in h A Motion Horizontal rad k M Deflection Angular Static s xs o s ψ ψ ψ
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
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
Impedance Functions a 0 ρ ω r G ωr V s ψ Luco and Westmann (1970)
Layer Effects
Impedance Functions ψ
Boundary Element Stehmeyer and Rizos, 2006
B-Spline Impulse Response Approach
[ M ]{&& z} + [ K]{ z} { p} i t e ω { z} { Z} e iωt then {[ ] 2 K ω [ M] }{ Z} { p} Finite/Hybrid Model ( ) 2 2 1 2β + 2β 1 G* G i β
Dynamic p-y Curves Tahghighi and Tonagi 2007
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
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
Resonant Column Test G, D for Different γ
Torsional Shear Test Schematic Stress-Strain
Hollow Cylinder RC-TOSS
TOSS Test Results
Steam Turbine-Generator (Moreschi and Farzam, 2003)
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
STG Pedestal Structure
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
Finite Element Model Structure and Base Y Z X
Low Frequency Modes 1 st mode 6.5 Hz 95 % m.p.f. 2 nd mode 7.2 Hz 76 % m.p.f
High Frequency Modes 28 th mode 46.3 Hz 0.3% m.p.f Excitation frequency: 50-60 Hz 42 nd mode 64.6 Hz 0.03% m.p.f
Local Vibration Modes Identification of natural frequencies for individual structural members Quantification of changes on vibration properties due to foundation modifications
ATST Telescope and FE Model
Assumptions in FE analyses Optics Lab mass/instrument weight 228 tons Wind mean force 75 N, RMS 89 N Ground base excitation PSD 0.004 g 2 /hz Concrete Pier High Strength Concrete (E3.1 10 10 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)
Frequency vs Soil Stiffness Stiffness units SI, frequency mode (hz) MODE Stiffness Kx Ky Kz Krx Kry Krz 1 2 3 4 5 6 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+10 1.83E+10 2.48E+10 3.12E+10 1.19E+10 1.83E+10 2.48E+10 3.12E+10 1.48E+10 2.45E+10 3.41E+10 4.38E+10 1.34E+12 2.21E+12 3.09E+12 3.96E+12 1.34E+12 2.21E+12 3.09E+12 3.96E+12 1.74E+12 2.61E+12 3.49E+12 4.36E+12 6.3 7.0 7.4 7.5 6.4 7.1 7.5 7.7 9.4 9.7 9.9 10 9.4 10.3 11.1 11.8 10.4 11.9 12.6 13.3 11.2 13.0 13.6 13.7 min for shear modulus of 30 ksi; max for 75 ksi
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
Free-Field Analytical Solutions R V z C r H a R L V i r u ω ρ β ω θ 2 0 0 3 0 ) ( 2,0), ( R V r C r H a R M V i r u ω ρ β ω θ 2 1 0 3 0 ) ( 2,0), ( u r u z
Trench Isolation Karlstrom and Bostrom 2007
Chehab and Nagger 2003
Celibi et al (in press)
Thank-you Questions?
r -2 r -2 r -0.5 + Rayleigh wave Vertical component Horizontal component + Shear - + wave + - Relative amplitude r -1 Shear window r -1 + + r Wave Type Percentage of Total Energy Rayleigh 67 Shear 26 Compression 7
Waves Rayleigh, R Surface Shear,S Secondary Compression, P Primary
Machine Performance Chart Performance Zones ANo Faults, New BMinor Faults, Good Condition 0.002 C Faulty, Correct In 10 Days To Save $$ D Failure Is Near, Correct In 2 Days E Stop Now 450