FEA Theory, Method, Limitations and Examples

1

• FEA Presentation
• (FEA Theory, Method, Limitations and Examples)
• Robert Stone

2
FEA Theory

• Finite element method approximates the physics
  - numerical procedure for solving a continuum
  mechanics problem with acceptable accuracy
• Subdivide a large problem into small elements
  connected by nodes
• Use small strain theory and classical solutions
  (i.e. partial differential equation) to determine
  element strain as a function of displacement
• Assume a simple polynomial behavior in each
  element
• Write equilibrium at nodes ? matrix equations
  ?solvefor nodal values (displacements)
• B Matrix (DOF to Strain Transformation)
  expressed in terms of the nodal coordinates
• E Matrix (Stress to Strain Transformation)
  recall s Ee
• A Matrix (Stress to Force Transformation) A
  BT
• Develop elemental stiffness matrix Kn BT E B
• Assemble into Master Stiffness Matrix
• Solve F K?

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FEA Theory

• B Matrix (DOF to Strain Transformation)
• Each coefficient is a polynomial function that
  contains the nodal coordinates
• DOF Elemental Degrees of Freedom (typically 2
  6 depending on the element)
• 2D Bar / Spar / Truss 2 Node, 2 DOF
  (Translation)
• Spring 2 Node, 2 DOF (1 Translational 1
  Rotational)
• 3D Beam 3 Node, 6 DOF (3 Translational 3
  Rotational)
• Shell4 4 Node, 6 DOF (3 Translational 3
  Rotational)
• Solid (8 20 Node) 3 DOF (3 Translational)
• E Matrix (Stress to Strain Transformation)
• Material Properties
• Elastic Modulus (E)
• Poissons Ratio (?)

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FEA Method

• Subdivide the structure into simple elements
• More element more nodes more DOF improved
  accuracy
• As DOF increases the solution typically converges
  to a solution.
• Most element formulations are overly stiff
  flexibility increases as the DOF increases
• As you increase the mesh density the deformation
  increase
• Deformations converge rapidly (typically a course
  model will yield good deformation results)
• Stresses converge slowly (in most cases, a fine
  mesh model is required to capture accurate
  stresses

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FEA Method

• Linear Static (small displacement, linear
  materials)
• Nonlinear (large displacement, nonlinear material
  properties)
• Linear or nonlinear Dynamics
• Frequency analysis
• Random vibration analysis
• Buckling
• Thermal and heat transfer
• Fluid mechanics
• Electromagnetics
• Optimization

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FEA Method

• In general odd geometries are easily handled
• An accurate FEA models may have little or no
  resemblance to the CAD model
• Simple models have many advantages
• Easy to develop and easy to change
• Fast solution time
• Easy to optimize or perform What If studies
• Complex models are time consuming to develop
• Require precise modeling and care
• Difficult to change
• May require long run times
• Engineer / Analyst determines the complexity

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FEA Limitations

• Accurately modeling of load and boundary
  conditions is very difficult
• Complex joints are difficult
• Any form of member pre-tensioning requires a work
  around
• Symmetric problem with a mesh that lacks symmetry
  will produce asymmetric results

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FEA Model
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FEA Model

• COSMOS FEA Model Details
• High fidelity model
• Element Types
• 8 node solids, 3 node beams
• 4 node shells , 2 node springs
• Elements 20427
• Nodes 26105
• DOF 81035
• Materials ULE, Invar, steel
• Gravity simulates /- 8.5 from 35 zenith
• Z axis is the optical axis with origin at the
  vertex of M2

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FEA Model

• COSMOS FEA Model Details
• Constraint equations simulate whiffle tree rocker
  motion
• Springs simulate flexures
• Flexure Stiffness 165000 lbs/in
• Equivalent thickness for pretensioned spider arms
• Stiff massless beams and springs simulate
  torsional stiffness of pretensioned spider for
  frequency analysis
• Massless beams support vertex node for each
  mirror
• FEA Weight 1234 lbs

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FEA Model

• Spider Torsional Frequency
• Spider Vane Stiffness

• Vertex Node
• Rocker Arm Coordinate System
• Constraint Equations

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Frequency Response
Fundamental Frequency 32.5 Hz
2nd Frequency 33.3 Hz
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Frequency Response
4th Frequency 44.0 Hz
3rd Frequency 43.8 Hz
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Frequency Response
5th Frequency 44.3 Hz
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FEA Thermal Model

• COSMOS FEA Model Details
• Element Types
• 3D truss elements replace the 2 node spring
  elements
• Eliminated massless beams and springs from M4
  spider
• Materials ULE, Invar, steel, stainless steel
• Varied CTE of steel knuckles, then calculated an
  equivalent length of steel in M4 truss tubes
• Thermal simulates ?T 20C (36F)

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M4/M5 Athermalization
Invar Truss Tubes
Steel Knuckle

• Compensates for negative CTE of Invar truss
• Model simulates 6.5 of steel from midplane of
  strongback

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FEA Thermal Distortions

• Uniform distortion in Z direction
• Zero rotation about X axis and Y axis

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Optical System
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Optical Bench FEA

• Improved Fidelity
• All Optical Elements
• Bearing Stiffness
• Elements 9065
• Nodes 11692
• DOF 36910
• Element Types
• 8 Node Solids
• 3D Beams
• Thin Shells
• 2 Node Springs

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FEA Details

• Radial Bearing 1,000,000 lbs/in
• Duplex (Radial) 850,000 lbs/in
• Duplex (Axial) 690,000 lbs/in
• SM (Axial) 108,750 lbs/in
• SM (Tangent) 435,000 lbs/in
• FM (Axial) - 108,750 lbs/in
• FM (Tangent) 435,000 lbs/in

• Tangent Flexures 125,000 lbs/in
• Axial Flexures 165,000 lbs/in

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Optical Bench Frequency
Fundamental Frequency 34.58 Hz
Second Frequency 36.54 Hz
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Optical Bench Frequency
Third Frequency 39.13 Hz
Fourth Frequency 41.18 Hz
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Optical Bench/ Base FEA

• Improved Fidelity
• Elements 9559
• Nodes 12133
• DOF 39518
• Element Types
• 8 Node Solids
• 3D Beams
• Thin Shells
• 2 Node Springs
• Rigid Bars
• Translational Constraints (All Rotational DOF
  Released)

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FEA Details

• Bench Coupled to Base
• Bearing Stiffness Included

• Bench Coupled to Base
• Lead Screw Stiffness Included

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Optical Bench / Base FEA
Fundamental Frequency 25.00 Hz
Second Frequency 29.55 Hz
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Optical Bench / Base FEA
Third Frequency 31.46 Hz
Fourth Frequency 32.44 Hz
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Random Vibration Analysis

• 3 Axis Uniform Base Motion
• 1 100 Hz
• 15 Frequencies
• Modal Damping 0.01
• 40 Integration Steps Between Frequencies

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Random Vibration Performance
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LOS Jitter Performance
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Preliminary Thermal FE Model

• Simple Model to Get Close, Validate Method
• Main Structural Beams
• Radiatve Coupling to Surfaces at Top and Bottom
• Shell Elements
• Proper Radiation View Factor
• Elements 2310
• Nodes 2486
• DOF 2486

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Preliminary Thermal FE Model

• Model simulates a 2 K difference from top to
  bottom of chamber
• Shroud top (T531.0 R) radiating heat only to top
  surface of top I-beam
• Bottom surface of I-beam radiating heat only to
  bottom shroud (T527.4 R)
• e (shrouds) 0.90
• e (aluminum) 0.10
• a 0.13e-4/F
• k 0.22e-2 BTU/in-sec-F
• s .1714e-8 BTU/hr-ft2-R4
• s 3.306e-15 BTU/sec-in2-R4

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Preliminary Model -- Steady State Temperature
(for 2 K difference from top to bottom of
shroud)
Top Surface Average 69.39 ºF
Bottom Surface Average 69.03 ºF
dT 0.36 F
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Thermal Model for Hand Calculations

• Shroud Top
• T1 531.0 R (295K)
• A1 gt A2 (assumed as hemisphere)
• e1 0.90
• Upper I-beam Top Surface
• T2 Unknown
• A2 1248 in2
• e2 0.10
• Lower I-beam Btm Surface
• T3 Unknown
• A3 1248 in2
• e3 0.10
• Shroud Bottom
• T4 527.4 R (293K)
• A4 gt A3 (assumed as hemisphere)
• e4 0.90

For Steady State Equilibrium
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Preliminary Thermal Results Comparison
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Refined Thermal FE Model

• Shroud top (T531.0 R) radiating heat to top
  surface of both I-beams
• Bottom surface of both I-beam radiating heat
  to bottom shroud (T527.4 R)
• Blocking surfaces included
• e (shrouds) 0.90
• e (aluminum) 0.10
• a 0.13e-4/F
• k 0.22e-2 BTU/in-sec-F
• s 3.306e-15 BTU/sec-in2-R4

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Refined Thermal FE Model Steady State Temperature
Top Surface Temp 69.615 ºF
Bottom Surface Temp 69.473 ºF
dT 0.142 F
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Temperature 1ºK
Spec.
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Primary Mirror Support

• 3-pairs of axial rods on mirror back
• Minimize rigid body movement figure change
  during 7.5 rotation
• Rods thread into 1 dia pucks bonded to back
• Each pair connected by a seesaw flexure to
  provide 3-point mounting to cell
• Differential screws at each rod end
• Equalize pre-load
• Provide fine tilt adjustment

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Primary Mirror FEA

• FE model of mirror
• 3039 element
• 3629 nodes
• DOF 11,214
• Mirror modeled as 8-node solid elements
• Surface modeled as 4-node thin shell elements
• 1-g P-V RMS surface distortion predicted
• Zero degree position
• 7.5 position
• SigFit used to remove rigid body movement

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Primary Mirror FEA Results

• Surface distortion determined over 22.68 CA
• Zero degree position
• P-V figure 17.7 nm (0.028 ?)
• RMS figure 3.1 nm (0.005 ?)
• 7.5 rotated position
• P-V figure 17.5 nm (0.027 ?)
• RMS figure 3.1 nm (0.005 ?)
• Weight 205 (mirror cell)
• Resonance 52 Hz
• Flexure stress
• 1g V 0.5g H (ksi) 31 (Tang) 15 (Axial)
• 4g V (ksi) 73 (Tang ) 23 (Axial)

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PM Unit Forces Moments Analysis
Mz 1 in-lb
Fr 1 lb
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Tangent Rods

• 3 rods spaced 120 apart
• 0.5 square X-section
• 0.030 thick cross circular flexures at each end
• 1.5 dia pucks bonded to mirror with Milbond
• Thru hole for threaded rod end
• Fixed end clamped to mount using one screw and a
  pin to prevent slippage under shock

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Tangent Rod Analysis

• Normal Loads
• Axial stiffness 384K lb/in
• Bending stiffness (x-dir) 139 lb/in
• Bending stiffness (z-dir) 58 lb/in
• Torsional compliance 809 in-lb/rad
• Unit force moment analysis
• 0.010 displacement of rod end
• Shear force 1.39 lb 0.58 lb
• Moment 3.1 in-lb 1.29 lb
• 1 deg. displacement of rod end
• My 8.83 in- lb
• Mz 7.42 in-lb
• 0.01 radian torsion
• 8.1 in-lb moment

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Secondary Mirror Mount

• Open frame type mount
• 3 I/F points on orthogonal axes
• 4 safety stops
• 3-flat blade flexures support mirror
• 1 dia pucks

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SM FEA

• FE model of mirror
• 2682 element
• 3193 nodes
• DOF 10,461
• Mirror modeled as 8-node solid elements
• Surface modeled as 4-node thin shell elements
• 1-g P-V RMS surface distortion predicted
• Zero degree position
• 7.5 position
• SigFit used to remove rigid body movement

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SM FEA Results

• Surface distortion determined over 10.50 CA
• Zero degree position
• P-V figure 5.87 nm (0.0093 ?)
• RMS figure 0.72 nm (0.0011 ?)
• 7.5º rotated position
• P-V figure 6.73 nm (0.0106 ?)
• RMS figure 1.20 nm (0.0018 ?)
• Weight 35.5
• Resonance 142 Hz
• Flexure stress
• 1g Y 0.5g H 3,163 psi
• 4g V 5,444 psi

Zero degree position
7.5º rotated position
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Fold Mirror Mount
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FM FEA

• FE model of mirror
• 2676 element
• 3172 nodes
• DOF 10,893
• Mirror modeled as 8-node solid elements
• Surface modeled as 4-node thin shell elements
• 1-g P-V RMS surface distortion predicted
• Zero degree position
• 7.5 position
• SigFit used to remove rigid body movement

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FM FEA Results

• Surface distortion determined over 14.2 CA
• Zero degree position
• P-V figure 7.71 nm (0.012 ?)
• RMS figure 0.84 nm (0.001 ?)
• 7.5 rotated position
• P-V figure 11.83 nm (0.019 ?)
• RMS figure 2.33 nm (0.003 ?)
• Weight 64
• Resonance 277 Hz
• Flexure stress
• 1g Y 0.5g H 1890 psi
• 4g V 4477 psi

7.5º rotated position
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Counterweighted Design

• Counterweighted Design is required for
  optimum performance in multiple gravity
  orientations.

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Counterweighted Design Schematic
Horizon Pointing
Zenith Pointing

• Counterweighted Design Provides Support in Any
  Gravity Orientation

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Assembled Mirror and Cell(Outer Skin and Bottom
Skin Removed)
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Cell Design Details

• 24 Radial Counterweights
• Rods Aligned with CG
• 18 Axial Counterweights
• 6 _at_ Inner Radius 11.525 in.
• 12 _at_ Outer Radius 25.380 in.
• Material 6061-T6 Aluminum
• 3 Bi-Pod Supports
• Encapsulated Lead Weights
• Cell Material 6061 T6
• Bolted Construction
• Helicoil Threaded Inserts
• All Blind Holes Vented
• OAP WT Estimate 5215 lbs
• Flat WT Estimate 5500 lbs

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Connecting Rod Design
Optimum Counterweight Design Applies An Axial
Load At The Puck With Minimum Error Forces Due To
Moments, Shear and Torsion

• EMD Flexure
• Short Compact Design
• Well Defined Axial Stiffness
• Low Flexural Stiffness
• Rotary Joint
• Needle Thrust Bearing
• Limits Torsional Loads on Puck
• Short Design
• Timkin Tapered Roller Bearings (2)
• High Radial Stiffness
• Improves Frequency Response 2-3 Hz
• Static Friction 2.7 x Running Friction

Minimum Overall Length Required To Meet Envelop
Requirements
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OAP Finite Element Model

• ANSYS FEA Model Details
• High Fidelity Model
• EDM Flexure Stiffness Included
• Flex Pivot Stiffness Included
• Tapered Roller Bearing Stiffness Included
• Bipod Stiffness Included
• Breakaway Stiffness Included

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OAP Finite Element Model

• ANSYS FEA Model Details
• Element Types
• 8 Node Solids
• 2 Node Beams
• 4 Node Shells
• Elements 10234
• Nodes 8324
• DOF 39525
• Supports 6 Points Vertical 3 Points X Y or
  Tangential

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Modes 1 2
OAP Modal Performance
Mode 3
Mirror Translation (7.40 Hz)
Mirror Piston (9.46 Hz)
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Modes 4 5
OAP Modal Performance
Mode 6
Mirror Tilt (12.40 Hz)
Cell Deformation (17.90 Hz)
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Thermal PerformanceCTE Variations

• ANSYS FEA Model Details
• 8 Node Solid Elements
• Elements 3820
• Nodes 4565
• DOF 14211
• Supports 3 Tangential 3 Radial
• Model CTE 0.7 ?-?/C
• Temp. -0.5C to 0.5C
• PV Strain Difference 0.7 ?-?
• Assumed Spatial CTE Variations

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Thermal Analysis Results
Normalized Strain Distribution for Power
Distributed (2p2-1) Bending Gradient.
Surface Normal Displacement for the Power
Distributed Bending Gradient
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Thermal Analysis Results
Astigmatically Distributed Normalized Strain
Surface Normal Displacement for the
Astigmatically Distributed Bending Gradient
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Thermal Analysis Results(Power Removed)
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Thermal Analysis Results
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Thermal Analysis Results
Power
Irregularity
Power ? 50 nm rms /- 10?C
Irregularity ? 25 nm rms /- 10?C
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OptoMechanical Design Strength (Positive
Margins of Safety)

• FEA Analysis of Knife Edge Support
• 4 Node Shell Elements
• Cell Structure
• Knife Edge
• 3 Node Beam Elements
• Bolts
• 2g Gravity Load Applied as Pressure to Knife Edge

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OptoMechanical Design Strength (Positive
Margins of Safety)
Max Disp. 0.064 in.
Max Stress 8416 psi
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OptoMechanical Design Strength (Positive
Margins of Safety)